Exhibit TWENTY-ONE Hard Problem Registry v342   # Hard Problem Registry   **An Open Registry of Hard Problems in Science**   **This file holds every arrival whole in the fields' own words, which is the incoming face.** Resolving the Hard Problem Registry carries the same arrivals resolved in the living register. **One learning at two files.**   --- # The conditions prior Where there is science and a method for problems, five conditions hold before the method acts. The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters: what comes to the statement carries nothing of what it carried before. The first is required by replication, the second by measurement, the third by the inference, the fourth by replication. The fifth is the condition on what meets the statement, and the first four stand on it: a statement holds across readings where nothing arriving at it brings a prior reading. The five fix what enters and how it runs. A problem adds a sixth, its own, fixing what comes out: the outcome preserves a stated relation over a stated range. This condition is the problem's and not the method's, and it stands at Conserving relation and Conserving reach in every entry. Everything below follows from these and from nothing else. --- # 1 What stands open, and what stands fixed This is an open registry: a set of entries on a stated population, each carrying the same properties. **The problems stand open.** No entry carries a settled answer to the statement it holds, except where the record settles one and the entry says so. **The intake stands open.** Nothing is turned away. An arrival is entered, and the collection is added to as arrivals come. **The properties and the checks stand open.** Every property is stated at section 4 and every check at section 6. Any of them runs against the fields' literature, by anyone holding it. **Every line stands open at an address.** A filled property keys to the place its statement is verified. A line without an address makes a claim that cannot be met, and check C1 tests for it. Against these, one thing stands fixed: the properties themselves. Every entry carries the same properties, in the same order, whatever the field. Uniform elements are what make two entries comparable, and comparability is what a registry is for. The schema is closed once, at one place, so that nothing else has to be. --- # 2 Hard problems in science **Problems in science** are hypotheses. A hypothesis is an account with consequences following from it, tested together with the specifications it carries. Where a consequence fails, the conjunction of account and specifications is refuted, and no member of it is identified. Where a consequence holds, nothing follows about the account, a rival account entailing the same consequence. **Hard** is where testing continues and returns no refutation: either no consequence separating the accounts is derivable, or the accounts entail the same consequences. Both are properties of the hypothesis together with what it carries, and neither is a property of what the hypothesis is about. **Progress** is what stands: established, measured, proved, or ruled out. It accumulates whether or not a problem settles, every approach that ends returning a result about the approach. **Persistence** is the remainder between what stands and a settling result. Each entry states one problem as it is posed, and carries progress and persistence with the remainder between them visible. Every entry carries the same properties, filled from the record at which each is verified. --- # 3 Derivation ## 3.1 What a refutation identifies A refutation falls on the conjunction of account and specifications. No member of it is identified. A test therefore returns no indication of which specification is at issue where a settling result requires one not to hold. ## 3.2 Specification A hypothesis carries every specification a consequence derives from. A hypothesis carrying fewer entails no consequence. A specification is universal under a name where it holds across everything stated under that name, and local where it holds for one statement. No specification is identified as the one to withdraw, and specifications stand at the number brought in. ## 3.3 Repetition A result is replicated where the same procedure returns the same outcome independently of what precedes it. A test carrying its own history is not replicable, and no test reads what a prior test produced. Accumulation stands in the specifications. A result that stands becomes a specification of tests that follow. The inference reads the specifications at the start of a test and writes to them at no point. Where a problem is durable, the specifications stand unchanged across the tests run against them. The number of hypotheses tested measures the testing. ## 3.4 The two responses Where a test returns no refutation, refinement or revision follows. Refinement sharpens the statement, the accounts, or the instruments, within the specifications standing. Revision withdraws a specification, retains the accounts, and tests again. The inference licenses neither, having refuted a conjunction and identified no member. Both operate on what the specifications permit. A settling result the specifications do not permit is reached by neither. --- # 4 What an entry carries An entry carries two parts. The properties are of the problem, and stand whether or not work has been done on it. The report is of work done, and stands empty where none is recorded. Each states the record at its address, or stands empty. ## 4.1 The properties **Conditions prior.** The five, stated in every entry. Every unreachability result derives from these together with the specifications below. **Statement.** The form in which the problem is posed, in the words it is posed in. Variant formulations stand, unranked. **The field's account of its hardness.** Where a field states, in its own words, what is hard about its own problem — before any account is proposed. Stated from the field's review or methodological literature, or empty. No inference from the difficulty of the record. **Accounts.** What each account proposes to exist, on its own content, at the strength its record carries. The number is as posed. No relation between accounts stands here. A statement about the condition of the effort proposes no account. **Specifications.** Each specification the statement carries beyond the five, with the name it is universal under, or local. The number stands against the number of consequences derivable. **Specifications withdrawn.** Each specification the record carries as given up with the accounts retained. No reason stands, a refutation identifying no member. **Common requirement.** What the accounts require alike, where stated. **Conserving relation.** What the outcome is required to preserve, stated as the problem states it: an ordering that does not cycle, a procedure that returns, a predicate that divides cases, a value standing prior to measurement. **Conserving reach.** The range over which the relation is required, stated as the problem states it: all profiles, all programs, all statements of the language, all cases. **Reach.** What the statement quantifies over. What its instruments return values over. Where a relation between the two is stated — an equivalence, a proved limitation, an account of what a method does not deliver — that statement stands. **Unreachability.** A result establishing that the conserving relation is not preserved over the conserving reach. The result is about the conditions and stands whether or not an attempt was made. What was attempted stands at Approaches. **Persistence.** The settling result specified. **Addresses.** The location at which each property is verified, keyed property by property. ## 4.2 The report **Approaches.** Each direction taken, with its present reach in the measures used, and its standing: refuted; blocked by a proved limitation; equivalent by construction to a rival; answering an adjacent question by the method's own account; instruments disagreeing on the object; narrowing; open. **Narrowing recorded.** Where the record carries the reach narrowed, or the relation weakened, with results following. Each is stated as the record states it. **Progress.** What stands, with its measures. **Own record.** Where a statement about the problem's own standing is made, phrased as it is made. ## 4.3 Register Properties carry present standing: it is proved that; no algorithm is known; the bound stands at; the parameter space is excluded below; the estimates disagree by measured amounts. No person functioning as attribution, and no date. Eponyms that are technical vocabulary stand where that vocabulary puts them. Years stand at the addresses. ## 4.4 Statement rule A precise statement carries nothing the original statement together with its own specifications does not carry. Short verbatim phrases stand where an exact formulation is itself the object — a named thesis, a canonical question — marked as such. ## 4.5 What is not carried Priority and sequence. Contested wherever recorded, and no standing rests on it. Names and years stand at the addresses, locating a text. Importance, difficulty, promise. No statement carries these. An account not proposed. Any property of a problem read against another problem. --- # 5 The entries ## The form **Conditions prior.** **Statement.** **The field's account of its hardness.** **Accounts.** **Specifications.** **Specifications withdrawn.** **Common requirement.** **Conserving relation.** **Conserving reach.** **Reach.** **Unreachability.** **Persistence.** **Addresses.** *The report.* **Approaches.** **Narrowing recorded.** **Progress.** **Own record.** The five stand in every entry as they stand at the front, the problem being what was fixed in order for the method to act on it. **A property standing empty is a reading and not a gap.** Nothing recorded at that property, and the entry says so; the record carrying it and the entry not, and check C1 through C19 finds it. **An emptiness is recorded at the entry and never at the door.** **The entries are numbered from one, in the order they were entered.** **A number is where an arrival was entered and not what it is**, and the aggregate at section 7 counts over filled properties and not over numbers. ## 1 Sorites **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** One grain of sand is not a heap. Adding one grain to something that is not a heap does not make it a heap. Therefore no number of grains makes a heap. Which step fails? **The field's account of its hardness.** That no finite number of divisions seems adequate: the phenomenon driving the paradox suggests no more two sharp boundaries than one, and higher-order vagueness leaves no more ground for a sharp line between the true and the indeterminate than between the true and the false. Vague predicates are described in the field's literature as boundaryless. **Accounts.** A number exists at which one grain makes the difference, unknowable, vagueness being ignorance of a sharp boundary rather than absence of one. No such number exists, the predicate has no sharp extension, and classical logic does not apply to statements formed with it. **Specifications.** A predicate has an extension — universal under classical semantics. Classical logic applies to statements formed with the predicate — universal under classical logic. The induction step preserves truth — local. **Specifications withdrawn.** **Common requirement.** The predicate divides cases into those it applies to and those it does not, so that a boundary either exists or is absent. **Conserving relation.** The predicate divides cases into those it applies to and those it does not, the division standing without sway. **Conserving reach.** All cases the predicate admits, the borderline included. **Reach.** The statement quantifies over a boundary in the predicate's application. Uses of the predicate are what is measured, varying continuously between speakers and occasions. The two are not the same. **Unreachability.** A treatment assigning intermediate values introduces a boundary between the values assigned, and that boundary admits the same argument. The division is not preserved across all cases by any treatment that assigns values. **Persistence.** Exhibition of the boundary, or a demonstration that the predicate's use requires none. **Addresses.** Statement, hardness, accounts, approaches: Hyde and Raffman, "Sorites Paradox", Stanford Encyclopedia of Philosophy, substantive revision 26 August 2025, §§2–3.4. Higher-order vagueness recurrence: §3.3.1 and §3.3.3. Boundarylessness: §3.3.3, citing Sainsbury 1990. Response taxonomy: §3 opening. As read at v342: the entry's opening as a database carries an earlier version's abstract verbatim (the paradox the name of a class of little-by-little arguments arising from the indeterminacy surrounding the limits of application of the predicates; no one grain identifiable as making the difference; from apparently true premises by seemingly uncontroversial reasoning to an apparently false conclusion), the entry itself not fetched, its sections as keyed above read at an earlier session and not re-read at v342; the higher-order vagueness dispute as the encyclopedia's entry on vagueness carries it (several philosophers repudiating higher-order vagueness as an illusion; defenders replying that borderline behaves like a vague predicate and can be embedded in a sorites argument; vagueness characterized as boundarylessness), returned in the search, its author not carried. Persistence, no landing: the closing condition at the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the four types of response, denying that logic applies, denying a premise, denying validity, embracing the conclusion; not all sorites able to be sound unless one accepts a pandemic of contradictions; the supervaluationist denial that the conclusion expresses a sharp boundary of which we are ignorant, with the doubts about the adequacy of the logic that follow), the entry itself not fetched. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Ideal language approaches — logic held not to apply to soritical terms, vagueness treated as a defect of natural language to be eliminated; abandoned with the ideal language doctrines, and recorded in the field's own history of the problem. Epistemicism — a sharp boundary posited unknowable, its location fixed by an unknowable function of unknowable patterns of use; consistent, bivalence retained, and its proponents granting the account is intuitively implausible. Supervaluationism and its relatives — truth on all admissible precisifications, classical laws preserved without bivalence; higher-order vagueness recurs against it, and the subvaluationist dual treats the indeterminacy as overdetermination instead. Degree and many-valued theories — three-valued and infinite-valued semantics, truth-functional, the response depending on which values are designated; higher-order vagueness recurs, and the totally ordered truth-set is recorded as too simple for multi-dimensional predicates. Contextualism and its relatives — the extension shifting with context so that no boundary is ever encountered where one looks; criticised as reaching the forced-march form rather than the paradox proper. The multiple range theory — multiple equally permissible arbitrary stopping places relative to a single context, classical logic and bivalence retained, the major premise false relative to each range. Embracing the paradox — vague predicates held incoherent, or the argument held sound and the predicates applying to nothing; the cost recorded as prohibitive on both sides. The field states four broad response types available to any paradox of this form: denying that logic applies; rejecting a premise; denying validity; embracing the conclusion. Each approach above is placed by its own literature under one of the four. **Narrowing recorded.** Contextualism narrows the reach: the division stands within a context, and the reach is one context rather than all cases. Degree-theoretic and supervaluationist treatments weaken the relation: the division admits intermediate standing, and results follow under the weakening. **Progress.** The recurrence of higher-order vagueness against two families of treatment. Epistemicism's positing of an unknowable boundary, stated by the account itself. Whether every possible treatment faces one or the other is not established. **Own record.** The argument contains no material of any kind: sand is incidental, and the same argument runs for any predicate admitting small increments. ## 2 The concept of disease **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What makes a condition a disease: natural dysfunction, or harm judged against values. **The field's account of its hardness.** That the current taxonomy of illness could involve naturalist intuitions about some conditions and constructivist rationalizations about others, the folk concepts then possibly too confused to serve as constraints, the premise of an argument for revisionism; that constructivism seems correct about some putative diseases, societies having at times counted activities pathological because of values rather than scientific evidence, while showing it correct about the concept of disease is another task; and that the prospects for a general naturalism about disease depend on whether human biology can be understood as a set of structures whose functions can be discovered empirically. **Accounts.** Disease is statistical-biological dysfunction against a reference class, value-free in principle. Disease is essentially evaluative, harm belonging to the concept, no value-free account capturing the extension. Dysfunction and harm both belong, the hybrid standing recorded between the two. **Specifications.** Medicine's classificatory practice — universal under medicine. Function ascription as biology supplies it — universal under biology. The manuals' lists as the operational extension — local. **Specifications withdrawn.** **Common requirement.** A fact about the concept's content or its referent. **Conserving relation.** A criterion sorting conditions into disease and not-disease, the sorting standing without appeal to the case. **Conserving reach.** All conditions medicine classifies, the canonical cases and the borderline alike. **Reach.** The statement quantifies over the criterion of disease. Physiological function measured against reference classes is what is measured, the choice of reference class standing as the crux. The two are not the same. **Unreachability.** Measurement fixes function against a reference class and does not fix the class. The canonical cases sort differently under each account, and the sorting is not preserved across all conditions by any criterion that fixes function against a class. **Persistence.** A criterion sorting the canonical cases alike under conceptual analysis, or a stipulation adopted as the method of settlement. **Addresses.** Statement: Murphy, "Concepts of Disease and Health", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the current taxonomy of illness possibly involving naturalist intuitions about some conditions and constructivist rationalizations about others, the folk concepts possibly too confused to serve as constraints, the premise of the argument for revisionism; constructivism seeming correct about some putative diseases, its correctness about the concept of disease another task; the prospects for a general naturalism depending on whether human biology can be understood as structures with empirically discoverable functions), the entry itself not fetched, its revision date and section numbers not carried at the returns; an earlier archived edition carries objectivist where the later editions carry naturalist. Persistence, no landing: the closing condition at the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the two-stage picture parting whether the body malfunctions from whether that is bad for the person; the hybrid account holding both a causal and a normative basis; an eliminativist alternative replacing the notion with successor notions tied to the science; not a clear consensus among writers), the entry itself not fetched. Approaches: the four routes at the same entry's sections on naturalism, normativism, the hybrid and the revisionist and eliminativist alternatives, read at pass one and at the passages above. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The biostatistical formulation — stands, with its counterexample record beside it. Normativist formulations — stand, with their counterexample record. The hybrid — stands as the recorded middle, both records inherited. Revisionist proposals — elimination or stipulation-for-purposes, standing published. **Narrowing recorded.** Stipulation-for-purposes narrows the reach: the criterion stands for a stated purpose rather than for all conditions, and results follow under the stipulation. Elimination weakens the relation: no single criterion is required, and the classifying stands as several. **Progress.** The case taxonomy. The reference-class dependence. **Own record.** The manuals' shifting borderline lists, standing as the dispute's data. ## 3 The population concept **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What a population is — the unit over which selection, drift, and statistical inference are defined. **The field's account of its hardness.** That the population definitions commonly used under the ecological paradigm, which emphasizes demographic cohesion, and under the evolutionary paradigm, which emphasizes reproductive cohesion, are found on review to be none of them truly operational. **Accounts.** Populations are bounded by rates of causal interaction, one criterion serving ecology and evolution alike. Populations are relative to research questions, no single boundary criterion serving. **Specifications.** Selection and drift defined over populations — universal under evolutionary theory. The statistical concept of a population co-founded with the biological one — universal under biometry. Interaction rates as what bounds a unit — local. **Specifications withdrawn.** **Common requirement.** A fact about whether population boundaries are objective. **Conserving relation.** A boundary criterion dividing organisms into a population and its outside, the division standing independently of the question asked. **Conserving reach.** All groupings over which selection and drift are defined. **Reach.** The statement quantifies over the boundary of the unit. Interaction rates are what is measured, the thresholds that would bound a population from them standing as the dispute. The two are not the same. **Unreachability.** Rates are measured and the boundary threshold is not fixed by them. The division is not preserved across all groupings by any criterion drawn from interaction rates. **Persistence.** A threshold fixed from interaction rates, or the adoption of question-relative boundaries as the method of settlement. **Addresses.** Statement: Waples and Gaggiotti, "What is a population? An empirical evaluation of some genetic methods for identifying the number of gene pools and their degree of connectivity", Molecular Ecology 15:1419-1439, 2006, the invited review finding none of the definitions in use operational; Millstein, "Populations as Individuals", Biological Theory, 2009. Hardness: the invited review, at the opening sentence of its abstract as the publisher's page carries it verbatim (the population definitions commonly used under the ecological paradigm, emphasizing demographic cohesion, and under the evolutionary paradigm, emphasizing reproductive cohesion, reviewed and none found truly operational), issue 6, doi 10.1111/j.1365-294X.2006.02890.x, the paper itself not fetched and the rest of the abstract not carried at the return; the second source not read this pass. Persistence, no landing: the closing condition at the encyclopedia's entry on ecology, at its passage on the population concept as the encyclopedia's page and its archived editions carry it verbatim (the ambiguity in the term's use held problematic, leading to confusion and hampering research; the causal interactionist concept defining a population by survival or reproductive interactions over a generation, populations viewed as individuals; critiques on the basis that the ecological conception differs from the evolutionary), returned in the search, its author not carried; the demarcation of discrete boundaries between populations recorded as a problem at the encyclopedia's entry on race. Approaches: the unified proposal and the pluralist objections at that passage, and at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The unified proposal — stands defended against the published objections. Pluralist objections — stand published against it. The historical route — the co-founding of the biological and statistical concepts stands documented, the shared origin standing as the shared crux. **Narrowing recorded.** The pluralist position narrows the reach: a boundary stands relative to a research question rather than across all groupings. **Progress.** The measurement methods. The documented co-founding. **Own record.** That arbitrary populations make the core processes arbitrary, standing as the field's statement of what rides on it. ## 4 The rule-following problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What fact about a person makes it the case that they mean one function rather than another by a term. **The field's account of its hardness.** That the meaning-constituting facts the sceptic demands are held to the constraint that the meaningful use of words be revealed as intentional, and, on some readings of the sceptical argument, to an epistemological constraint on the justification of semantic judgments rather than on the rationality of the applications of expressions, the justifications required to be in some sense transparent. **Accounts.** A fact supplies it: dispositions, or normative practice, or irreducible meaning-facts. No such fact stands, meaning being attributed through communal practice rather than grounded in individual fact. **Specifications.** Past use and mental history as the candidate ground — universal under the skeptical formulation. Every candidate fact is finite — universal under the founding argument. Deviant continuations fit any finite record — universal under the founding argument. **Specifications withdrawn.** **Common requirement.** A fact, or a licensed practice in its absence. **Conserving relation.** A ground determining the term's application, the determination standing for applications not yet made. **Conserving reach.** All continuations, the deviant included. **Reach.** The statement quantifies over the ground of meaning. Use histories and dispositions are what is measured, finite by the argument's own point and fitting the deviant assignments. The two are not the same. **Unreachability.** Every candidate fact is finite and fits deviant continuations. The determination is not preserved across all continuations by any ground drawn from a finite record. **Persistence.** A fact of the required kind exhibited, or agreement on what a satisfying fact would be. **Addresses.** Statement: Miller and Sultanescu, "Rule-Following and Intentionality", Stanford Encyclopedia of Philosophy, 2022. The sceptical paradox as posed: Kripke, Wittgenstein on Rules and Private Language, Harvard, 1982. Hardness: the same entry, at passages of its text as the encyclopedia's archived editions carry them verbatim (the meaning-constituting facts required to accommodate the intentionality of meaningful use; some interpreters taking the sceptical argument to impose an epistemological constraint on the justification of semantic judgments, the justifications to be in some sense transparent), the entry itself not fetched, its section numbers and revision date not carried at the returns; the passages read also carry a non-reductionist reading of the triangulation argument, that no account of the constitution of the standards of correctness can be offered without presupposing a commitment to them already in place, a reading for Accounts at the next pass; the sceptical paradox as posed not read this pass. Persistence, no landing: the closing condition at the same entry, at its structure as the encyclopedia's page carries it (the sceptical challenge to facts about meaning and content; sceptical responses conceding the non-existence of the facts while preserving the propriety of the talk; straight responses denying an assumption of the argument), the entry itself not fetched; a 2025 paper at its abstract as the publisher's page carries it (despite numerous solutions a definitive resolution remaining elusive, suggesting some commonsense notions may need reconsidering; a version of radical conventionalism defended), Topoi, doi 10.1007/s11245-025-10210-7, its author not carried. Approaches: the dispositionalist, normativist and sceptical routes at the same entry's sections on the responses, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Dispositionalism — stands defended, the finitude and error objections recorded. Normativist accounts — stand defended, the regress objection recorded. The skeptical solution — stands as the canonical alternative, its coherence disputed. Kin structure — the projectibility kinship with the new riddle of induction, recorded. **Narrowing recorded.** The skeptical solution weakens the relation: no ground determines the application, and assertibility conditions within a community stand in its place, with results following. **Progress.** The argument's canonical form and its case structure. **Own record.** The canonical formulation and the standing replies, in the field's surveys. ## 5 Aggregation of preferences **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Can individual rankings of options be combined into a single group ranking that is fair by stated conditions? **The field's account of its hardness.** That the question which procedures there are for deriving, from what is known or can be found out about people's preferences, a social ordering of the alternatives from better to worse receives the answer that there are none at all meeting certain conditions concerning the autonomy of the people and the rationality of their preferences; and that relaxing the ordering condition by limiting the transitivity requirement to strict social preferences yields in every case an oligarchy, a group whose unanimous strict preference the society always follows and against any member of which it never goes, so that the relaxation does not seem a promising way of securing acceptable social welfare functions. **Accounts.** No such rule exists: with unrestricted domain, unanimity, independence of irrelevant alternatives, and no dictator, none is available, and the proof is complete. Groups combine preferences continually and the results are used. The independence condition excludes intensity information actual procedures use, and rules admitting cardinal comparability lie outside the stated conditions. **Specifications.** Individual preferences are orderings — universal under social choice theory. Aggregation is a function from profiles of orderings to a social ordering — universal under social choice theory. The function satisfies conditions stated in advance — universal under social choice theory. **Specifications withdrawn.** **Common requirement.** That aggregation is a function from profiles of individual orderings to a social ordering, and that the dispute is over which conditions it must satisfy. **Conserving relation.** A social ordering that does not cycle, produced by a rule meeting the stated conditions. **Conserving reach.** All profiles of individual orderings — unrestricted domain. **Reach.** The statement quantifies over all rules satisfying the stated conditions. Particular procedures and their outcomes are what is measured. The two are not the same. **Unreachability.** No rule meeting the four conditions produces a non-cycling social ordering over all profiles. The relation is not preserved over the reach. **Persistence.** Within the conditions, nothing: the results are proved. Outside them, which procedures satisfy which conditions, settled case by case. **Addresses.** Statement: Morreau, "Arrow's Theorem", Stanford Encyclopedia of Philosophy. The conditions as posed: Arrow, Social Choice and Individual Values, Wiley, 1951. Hardness: the same entry, first published 2014 and substantively revised at the Winter 2025 edition as the encyclopedia's citation page carries it, at its opening and at a passage of its text as the encyclopedia's page and its archived editions carry them verbatim (the question which procedures derive a social ordering from what is known of preferences, answered that there are none meeting the conditions on autonomy and rationality; relaxing the ordering condition to strict social preferences yielding in every case an oligarchy, and not seeming a promising way of securing acceptable social welfare functions), the entry itself not fetched, its section numbers not carried at the returns; the conditions as posed not read this pass. Persistence: the landing, the results proved, at the entry's statement read above that there are no procedures meeting the conditions, the theorem's own proof at the founding book named above, not fetched. Approaches: the routes and their endings at the same entry's sections on the ways around the theorem, weakening independence, restricting the domain, cardinal comparison and probabilistic rules, read at pass one and at the passages the hardness keys above. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Weakening independence — admits Borda and scoring rules, which become manipulable, and any non-dictatorial rule over three or more options is manipulable. Restricting the domain — single-peaked preferences admit majority rule, holding where the restriction holds, an empirical matter case by case. Cardinal utility — escapes the stated conditions, requiring interpersonal comparison for which no basis is established. Probabilistic rules — escape some conditions, constrained by the random dictatorship result. Judgment aggregation — the same structure recurring for propositions, with its own impossibility results. **Narrowing recorded.** Restricting the domain narrows the reach: single-peaked profiles admit majority rule, and results follow. Weakening independence weakens the relation: scoring rules produce an ordering, at manipulability, and results follow. Probabilistic rules weaken the relation to a distribution over outcomes. **Progress.** The four proved limitations on classes of rules, and the doctrinal paradox in judgment aggregation. Each is proved. **Own record.** That independence of irrelevant alternatives requires the social ranking of a pair to depend on individual rankings of that pair and nothing else, and that the proof proceeds by altering one individual's ranking with all others held fixed. ## 6 The halting problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is there a general procedure that, given any program and input, decides whether the program eventually stops? **The field's account of its hardness.** That the problem of enumerating the computable sequences is equivalent to the problem of finding out whether a given number is the description number of a circle-free machine, and there is no general process for doing this in a finite number of steps; the naive diagonal argument that the computable sequences are not enumerable fails at its assumption that the diagonal sequence is computable, and the diagonal process applied correctly shows that there cannot be any such general process, the most direct proof being that if the process existed there would be a machine computing the diagonal sequence. **Accounts.** No such procedure exists, and this is proved, no non-trivial property of a program's behaviour being decidable from the program. **Specifications.** A computation is a finite description operating on finite input — universal under computability theory. Properties of the computation are established from the description rather than by running it — universal under computability theory. **Specifications withdrawn.** **Common requirement.** That a property of an execution can be established without executing it. **Conserving relation.** A decision procedure returning a value on every input, the return standing without running the program. **Conserving reach.** All programs and inputs. **Reach.** The statement quantifies over all programs and inputs. Finite descriptions, examined finitely, are what is measured. The two are not the same. **Unreachability.** No procedure returns on all programs and inputs; verification succeeds on programs people write, termination checkers handle large classes with their incompleteness stated by their authors, and undecidable cases are constructed rather than encountered. The relation is not preserved over the reach. **Persistence.** Nothing within the conditions: the result is proved. The disputed content is scope, settled case by case. **Addresses.** Statement: Turing, "On computable numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society (2) 42:230-265, 1936/37, and the correction at 43:544-546, 1937. The problem's name and standard statement: Davis, Computability and Unsolvability, McGraw-Hill, 1958. Hardness: the founding paper, at passages of its section on the application of the diagonal process as three repository copies carry them verbatim (the enumeration of the computable sequences equivalent to finding out whether a given number is the description number of a circle-free machine, with no general process for doing this in a finite number of steps; the fallacy of the naive diagonal argument in its assumption that the diagonal sequence is computable; the diagonal process applied correctly showing there cannot be any such general process, the most direct proof that the process would yield a machine computing the diagonal sequence), received 28 May and read 12 November 1936 as the copies carry it, the paper itself not fetched; the correction dated 1938 at one return where the address carries 1937, recorded and not moved; the Davis text not read this pass. Persistence: the landing, the result proved, at the founding paper's diagonal-process section read above, its correction at the second citation. Approaches, in part: the termination-checker route at the field's review of termination proving, Cook, Podelski and Rybalchenko, "Proving program termination", Communications of the ACM 54(5):88-98, 2011, doi 10.1145/1941487.1941509, at its citation and at a citing passage as a repository carries it (ranking functions a sound and complete approach for non-probabilistic programs, transition invariants unifying termination metrics and invariants, the tool built on them verifying device drivers), the review's own abstract not returned; the language-restriction, bounded-checking and interactive-proof routes keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Restricting the language — total languages guarantee termination and lose expressiveness in exact proportion, a known and exact trade. Termination checkers — size-change and ranking-function methods succeed widely and are incomplete by construction, stated by their authors. Bounded checking — answers for a bounded run, which is not the question. Interactive proof — the burden shifted to a person supplying the invariant, not forbidden by the result and not scaling automatically. Probabilistic and average-case decidability — results for restricted distributions, the general statement unaffected. **Narrowing recorded.** Restricting the language narrows the reach: total languages guarantee termination, at an expressiveness cost stated exactly. Bounded checking narrows the reach to a bounded run. Interactive proof weakens the relation: the return requires an invariant supplied from outside the procedure. **Progress.** The theorem itself, its generalisation to non-trivial behavioural properties, and the exact expressiveness cost of total languages. Termination checking succeeds on large classes of programs written in practice, incomplete by construction. **Own record.** That the statement contains no subject matter, concerning procedures and their execution and nothing else. ## 7 The pseudogap **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What the pseudogap of the underdoped copper oxides is — the partial suppression of spectral weight above the superconducting transition. **The field's account of its hardness.** That there is no consensus on the physical origin of high-temperature superconductivity, in large part for a lack of understanding of the state of matter out of which it arises, that state in optimally and underdoped materials exhibiting a pseudogap at temperatures large compared to the transition temperature, and the physical origin of the pseudogap behaviour and whether it constitutes a distinct phase of matter described as still shrouded in mystery; spectral gaps arising from pairing precursors are qualitatively similar to those caused by competing states, which renders a standard approach to their analysis inconclusive. **Accounts.** Pairs form without coherence, the suppression being superconductivity's precursor. A distinct state — density-wave, nematic, or loop-current — owns the suppression. **Specifications.** The superconducting gap's framework as the reference — universal under superconductivity. The probes' onset determinations as the record — local. **Specifications withdrawn.** **Common requirement.** An identification of the state. **Conserving relation.** One state identified as owning the suppression, the identification standing across the probes. **Conserving reach.** All probes returning the suppression, and the doping and temperature range over which it is returned. **Reach.** The statement quantifies over the state's identity. The suppression is what is measured, by every probe, the probes returning different onsets. The two are not the same. **Unreachability.** The state stands measured by all probes and identified by none to the field's agreement, the onset disagreement standing. The identification is not preserved across the probes. **Persistence.** An identification of the state holding across the probes, or an account of the onset disagreement. **Addresses.** Statement: Norman, Pines and Kallin, "The pseudogap: friend or foe of high Tc?", Advances in Physics 54:715-733, 2005; the experimental survey: Timusk and Statt, "The pseudogap in high-temperature superconductors: an experimental survey", Reports on Progress in Physics 62:61, 1999; Keimer, Kivelson, Norman, Uchida and Zaanen, Nature 518:179-186, 2015. Hardness: the perspective at its abstract as the arXiv page, an abstract service and an index carry it verbatim (no consensus on the physical origin of high-temperature superconductivity, in large part for a lack of understanding of the state of matter out of which it arises; that state exhibiting a pseudogap in optimally and underdoped materials at temperatures large compared to the transition temperature; the origin of the pseudogap behaviour and whether it constitutes a distinct phase of matter still shrouded in mystery; a five-week gathering at a physics centre summarized), arXiv:cond-mat/0507031, the paper itself not fetched; the similarity clause: "Competition between the pseudogap and superconductivity in the high-Tc copper oxides", Nature, 2009, doi 10.1038/nature07644, at its abstract as the publisher's page carries it verbatim (spectral gaps from pairing precursors qualitatively similar to those from competing states, a standard approach to their analysis inconclusive), returned in the search, its authors not carried at the return; the survey and the review not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched, their authors carried at the copies where given: a photoemission study at its introduction as the arXiv copy carries it (precursor or competing state a matter of long-standing debate; the data and their interpretations still controversial), arXiv:1012.3205; a Josephson study at its abstract as the publisher's page carries it (a direct probe of the pairs the way to address the conundrum; the fluctuation view remaining to be tested), Nature Physics, 2008, doi 10.1038/nphys1017; a charge-order study at its introduction (the origin remaining unresolved; most probes indicating a crossover rather than a transition at the onset temperature), arXiv:2504.12077; a report at its closing (extremely controversial views, an absence of consensus), arXiv:1204.3804. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Pairing-precursor programs — supported by fluctuation measurements in ranges, the range's extent disputed. Competing-order programs — the orders detected, charge order established, their relation to the suppression standing as the question. The simulator route — the suppression realized in the cold-atom model, opening the model-side identification. **Narrowing recorded.** **Progress.** The suppression's universality in the family. The charge-order detections. The simulator realization. **Own record.** ## 8 The measurement problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How the deterministic, linear evolution of the quantum state yields the definite outcomes observed in measurement. The canonical statement is a trilemma: the wave-function's completeness as a physical description, the universality of the linear dynamics, and the definiteness of measurement outcomes are jointly inconsistent, and at least one must be given up. **The field's account of its hardness.** That several difficulties confront attempts to reconcile the fundamental quantum-mechanical dynamics with the Born rule, many of the proposed accounts of measurement failing at least one of them, only collapse theories and hidden-variables theories having a chance of succeeding and the modal interpretations among the latter failing, and any real solution demanding new physics. **Accounts.** The state description is complete and the unitary dynamics universal, definite outcomes to be recovered within the formalism. The description or the dynamics is amended, by additional variables carrying definiteness, or by a stochastic modification under which superpositions collapse. **Specifications.** Linear, unitary evolution of the state as the law of motion — universal under quantum mechanics. The Born rule linking states to outcome statistics — universal under quantum mechanics. The founding two-process structure, unitary evolution between measurements and reduction at them, as the structure the problem is posed against — local. The theory's writ running from the microscopic to the apparatus — universal under quantum mechanics. **Specifications withdrawn.** **Common requirement.** A fact about whether measurement outcomes are definite, and a description of measurement within the theory rather than alongside it. **Conserving relation.** Definite outcomes, recovered from the dynamics, the recovery standing without measurement entering as a primitive term. **Conserving reach.** All measurement processes, from the microscopic to the apparatus. **Reach.** The statement quantifies over all measurement processes. Matter-wave interference establishes superposition for objects of increasing mass, to beyond twenty-five kilodaltons; the parameter space of the spontaneous-collapse models is bounded and progressively excluded, with no collapse signature observed. The two are not the same. **Unreachability.** Interpretations reproducing the quantum statistics are empirically equivalent within the theory's domain, and no experiment definable within standard quantum mechanics discriminates them. The recovery is not preserved over all measurement processes by any interpretation the statistics admit. **Persistence.** Detection of a spontaneous-collapse signature, settling against universal unitarity. No null result at any sensitivity settles, the null bounding parameters and standing compatible with both accounts. **Addresses.** Statement, the trilemma as the field states it: Maudlin, "Three measurement problems", Topoi 14:7-15, 1995, p. 7. The survey: Myrvold, "Philosophical Issues in Quantum Theory", Stanford Encyclopedia of Philosophy, the measurement-problem section. Hardness: re-formed to the field's words at the trilemma paper, at its abstract as the publisher's page, an index and two repository copies carry it verbatim (several difficulties confronting attempts to reconcile the fundamental dynamics with the Born rule; many proposed accounts failing at least one; only collapse and hidden-variables theories with a chance, the modal interpretations failing; any real solution demanding new physics), issue 1, doi 10.1007/BF00763473, the paper itself not fetched; the line's former clauses, the standing objection that measurement should not appear as a primitive term in the formulation of a fundamental theory and the decoherence literature's own scope statement, not carried at the returns, keyed at the next pass at their own sources; the survey's measurement-problem section not returned this pass. Persistence, no landing: the closing condition at the field's review of the tests, Carlesso, Donadi, Ferialdi, Paternostro, Ulbricht and Bassi, "Present status and future challenges of non-interferometric tests of collapse models", Nature Physics 18(3):243-250, 2022, doi 10.1038/s41567-021-01489-5, arXiv:2203.04231, at its abstract as the publisher's page, the arXiv page, a group's page and an index carry it verbatim (whether the superposition principle applies to macroscopic systems or breaks down progressively a fundamental and still open question; collapse models predicting the latter and questioning the universality of the theory; non-interferometric experiments the most effective tests, cold atoms, optomechanics, X-rays, bulk heating and cosmological comparisons bounding the models; future dedicated experiments discussed), the review itself not fetched. Approaches: the collapse route at that review and the decoherence and hidden-variables routes at the trilemma paper's abstract read under check C18 (only collapse and hidden-variables theories with a chance), the decoherence program's own statements keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The decoherence program — environmental decoherence established, its rates measured and its suppression of interference confirmed, the program's own literature stating that it explains the appearance of definiteness and the preferred basis and does not resolve which outcome occurs. Spontaneous-collapse models — empirically distinguishable in principle, the excluded parameter region growing, no signature observed. Interference at scale — superposition established at growing mass and complexity, proposals extending toward mechanical and biological scales. No-go and reconstruction results — the constraint structure settled: impossibility results on local hidden variables and on non-contextual value assignments, the theorem on the reality of the quantum state under its assumptions, and the extended observer-chain results bounding interpretations treating agents as systems. Interpretation programs — each carrying its own recorded internal disputes, the probability and preferred-basis questions within the relative-state program, the relativistic and field-theoretic extension within the pilot-wave program. **Narrowing recorded.** **Progress.** Decoherence theory and its measured confirmations. Superposition at mesoscopic scale. The no-go constraint structure. The collapse-model exclusion bounds. **Own record.** The trilemma as the problem's canonical form. ## 9 P versus NP **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether every decision problem whose solutions can be verified in polynomial time can also be solved in polynomial time — whether the complexity classes P and NP coincide. Variant formulations stand alongside the discrete statement: an analogue over the complex numbers, and average-case formulations in which hardness is asked of random instances rather than worst cases. **The field's account of its hardness.** That proving lower bounds remains the most difficult of tasks in computational complexity theory, no nonlinear time lower bound being known on a general-purpose model for any problem in NP, satisfiability expected to take exponential time while no proof is known even that non-linear-time algorithms are required, and no non-linear lower bound proved for the non-monotone circuit complexity of any Boolean function in NP; and that the standard approach through Boolean circuits, on which a superpolynomial circuit lower bound for some NP problem would separate the classes, seems very hard for general circuits, exponential-size lower bounds having been shown only for restricted circuit models. **Accounts.** The classes are distinct, on the recorded grounds that decades of algorithm engineering and the parallel effort of cryptanalysis have produced no polynomial-time algorithm for any complete problem. The classes coincide, not excluded and defended as live: a proof of equality could be nonconstructive, the failure to prove lower bounds is pervasive across complexity theory rather than special to this question, a linear-time algorithm for satisfiability is consistent with present knowledge, and the known chain of class inclusions has provably distinct endpoints while no single adjacent inclusion in it has been proved proper. **Specifications.** The machine model of computation defines the computable functions — universal under computability theory, accepted since the model's introduction. Polynomial time defines feasible computation — universal under complexity theory, the class robust and coinciding across machine models. The feasibility identification's recorded caveats — problem families proposed as potential counterexamples, and the randomized and quantum models held as standing reservations against the thesis — local. **Specifications withdrawn.** **Common requirement.** A fact of the matter about the classes' relation, expressible in the framework. **Conserving relation.** A proof separating the classes or identifying them, the proof standing by the field's stated acceptance conditions. **Conserving reach.** All algorithms. **Reach.** The statement quantifies over all algorithms. The best proven algorithms for the standard complete problem run in exponential time; for unrestricted Boolean circuits computing explicit functions no super-linear lower bound stands, the strongest proven bounds being linear-factor. The two are not the same. **Unreachability.** Three obstructions stand as proved results about the methods themselves. Diagonalization's lower bounds relativize, and oracles exist relative to which the classes coincide and relative to which they differ, so the method alone cannot resolve the question by its own proved property. The natural-proofs obstruction stands over the lower-bound methods used so far: a property they share would, if extended to separate the classes, break cryptographic pseudorandomness widely believed secure. Algebrization extends the barrier over the interactive-proof-era techniques. The proof is not reached over all algorithms by any method the obstructions cover. **Persistence.** An accepted proof in either direction, by the stated settling discipline: publication in a qualifying outlet, a two-year period, and general acceptance in the mathematics community. **Addresses.** Statement: Cook, "The P versus NP problem", in The Millennium Prize Problems, Clay Mathematics Institute and American Mathematical Society, 2006, p. 87. The analogue over the complex numbers: Blum, Shub and Smale, "On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines", Bulletin of the American Mathematical Society 21:1-46, 1989. The average-case formulation: keyed at the next pass. Hardness: re-formed to the field's words at statements returned in the search, the official description not returned at its text: a Turing Award lecture's line that no nonlinear time lower bound is known on a general-purpose model for any problem in NP, and a later line that proving lower bounds remains the most difficult of tasks in the theory, satisfiability expected to take exponential time while non-linear-time algorithms cannot be proved required, both quoted verbatim at an arXiv paper on lower bounds in linear time and sublinear space, its authors not carried at the return; an institute's research vignette as its page carries it (the standard approach through Boolean circuits, a superpolynomial circuit lower bound for some NP problem sufficing to separate the classes, the question seeming very hard for general circuits, exponential-size bounds only in restricted models), its author not carried; a preprint's statement that no non-linear lower bound has been proved for the non-monotone circuit complexity of any Boolean function in NP, as the arXiv copy carries it; none fetched; the official description and the analogue paper not read this pass. Persistence, no landing: the closing condition at the Institute's own rules, adopted 26 September 2018, as the Institute's page carries them verbatim (before a proposed solution is considered, all three conditions must be satisfied: publication in a qualifying outlet, at least two years passed since publication, and general acceptance in the global mathematics community; direct submission not accepted), the rules document not fetched. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Algorithm construction — a standard toolkit stands, no polynomial-time algorithm for any complete problem stands. Diagonalization — closed as a sole route by a proved limitation. Circuit lower bounds — strong bounds in restricted models, monotone circuits and bounded depth; against unrestricted circuits the natural-proofs obstruction stands. Algebraic methods — the interactive-proof-era techniques evade relativization, with algebrization standing over them. The geometric program — a separation program through representation theory and algebraic geometry, with its architects' own stated assessment that its completion lies far off. **Narrowing recorded.** Restricted models narrow the reach: strong lower bounds stand for monotone circuits and for bounded depth, and results follow within the restriction. Average-case formulations narrow the reach: hardness is asked of random instances rather than of the worst case. **Progress.** The completeness theory: thousands of problems across the sciences are complete, and a polynomial algorithm for any one is a polynomial algorithm for all. Primality testing is in P. The hardness-of-approximation theory on the probabilistically checkable proofs theorem. The three obstructions as proved results about the methods. **Own record.** That the polynomial class is the robust choice for the question. That the feasibility identification carries stated reservations. ## 10 The problem of induction **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What justifies inference from observed instances to unobserved ones. The canonical statement: no demonstrative argument establishes that unobserved cases resemble observed ones, and any argument from experience for the reliability of experience presupposes the very inference in question. The canonical modern extension: which predicates are projectible, syntactic uniformity not selecting, since deviant predicates fit past observations equally. **The field's account of its hardness.** That the argument is a dilemma with both horns closed: a demonstrative argument establishes a conclusion whose negation is a contradiction, and the negation of the uniformity principle is not a contradiction; while any probable argument for that principle proceeds on the supposition it is meant to establish. The field's record carries a significant number who embrace the conclusion that the problem is insoluble. **Accounts.** Inductive inference is justified or justifiable, through de facto reliability, through pragmatic vindication, through probabilistic coherence and convergence, or through grounding in local matters of fact. It is not: the skeptical conclusion stands, and scientific method neither needs nor possesses inductive justification, corroboration being non-inductive. **Specifications.** Deductive validity as the standard of justification, against which the inductive gap is posed — universal under the framework. Matters of fact known through experience — universal under empiricism. The uniformity assumption, named in the founding statement as what the inference would require and cannot non-circularly obtain — local. **Specifications withdrawn.** **Common requirement.** A fact about whether inductive methods are truth-conducive, expressible in the framework's terms. **Conserving relation.** A justification of the inference, the justification standing without employing the inference it justifies. **Conserving reach.** All inductive inferences, those not yet made included. **Reach.** The statement quantifies over all inductive inferences. Arguments and observations return the past performance of the methods, and the founding statement records that employing past performance is the circle itself. The two are not the same. **Unreachability.** No argument definable within the framework discriminates the accounts without presupposing one. Averaged over all target functions, no learner outperforms any other, and uniform assumption-free justification is provably unavailable. The justification is not preserved over all inferences by any argument that does not employ the inference. **Persistence.** The record carries the split as found: one account holds that demonstrated reliability is justification and nothing further is owed; the other holds that any demonstration cites the method's outputs and the question survives it. **Addresses.** Statement, hardness, accounts, approaches: Henderson, "The Problem of Induction", Stanford Encyclopedia of Philosophy, substantive revision 22 November 2022, §§1–6. The dilemma's reconstruction as premises and conclusions: §2. The response structure by horn: §§3, 4, 5, 6. Learning-theoretic results: §3.4. As read at v342: the entry at passages of its text as the encyclopedia's page and seven archived editions carry them verbatim (the argument a dilemma appearing to rule out any reasoning from the premises to the conclusion of an inductive inference, the two possible types of argument, demonstrative and probable, neither serving; demonstrative reasoning establishing only conclusions that cannot be conceived to be false, the course of nature conceivably changing without contradiction; probable reasoning proceeding upon the supposition that the future will be conformable to the past, the uniformity principle), the entry itself not fetched. Persistence, no landing: the closing condition at the same entry, at the passages read above at v342 (the dilemma ruling out demonstrative and probable argument alike; the uniformity principle; the two responses the entry structures, rejecting a horn or accepting the conclusion), the record's split as the file states it standing at those passages. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The field structures its responses by which step of the dilemma is rejected. Against the first horn, that no demonstrative argument reaches the principle: synthetic a priori knowledge, the uniformity secured as a condition of experience; the nomological-explanatory route, inference to the best explanation taken as an a priori mode distinct from extrapolation, its independence from inductive inference disputed; the Bayesian route, the coherence and convergence results standing conditional on priors the framework does not select, with the recorded judgement that the choice of prior introduces empirical assumptions; partial solutions, Bayesian conditioning read as a consistency constraint given an empirical premise, with the learning-theoretic results establishing that no algorithm outperforms another averaged over all target functions and that model-relative guarantees stand where universal ones do not; the combinatorial route, the proportional syllogism applied to samples matching populations, its final step recorded as equivocating between two senses of high probability. Against the second horn, that any probable argument is circular: inductive justifications of induction, the circularity argued to be rule-circular rather than premise-circular and benign for a fundamental rule, with the counterinductive rule shown to support itself in the same fashion; and the no-rules route, each inductive inference resting on its own empirical presupposition so that the circularity becomes a regress, with the field divided on whether a regress is better. Against the conception of justification itself: postulates and hinges, principles held as entitlements requiring no supporting argument; the ordinary language dissolution, an inference counting as justified where it conforms to inductive standards, with the objection that conformity does not establish likely truth; the pragmatic vindication, the inductive rule justified as a necessary condition of success if success is possible, recorded as tied to the long run while inductive practice occurs in the short; formal learning theory, methods justified by optimality toward stated epistemic ends, relative to a given hypothesis space; and meta-induction, a weighted strategy over accessible prediction methods proved long-run optimal, with the transfer of that justification to object-level induction disputed. And accepting the skeptical conclusion: the deductivist account of science, holding that science makes no extrapolative inferences at all, met by the objection that unrefuted hypotheses are left on an equal footing and corroboration takes it beyond strict deductivism. **Narrowing recorded.** Assumption-relative learning narrows the reach: guarantees stand conditional on sampling assumptions rather than over all inferences, and results follow. Pragmatic vindication weakens the relation: the justification is asymptotic rather than for the inference at hand. The material theory narrows the reach: inductions are licensed locally rather than by a universal schema. **Progress.** The circle, as a meta-result of the founding statement. The failure of syntactic projectibility. The conditional convergence theorems. The no-free-lunch limitation and the assumption-relative learning guarantees. **Own record.** The founding fork between relations of ideas and matters of fact. The restatement that the classical problem is displaced into projectibility rather than dissolved. ## 11 The reference class problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A probability is assigned to a single case by placing it in a class of similar cases. Every case belongs to many classes with different frequencies. Which class? **The field's account of its hardness.** That the problem arises whenever a probability is to be assigned to a proposition that may be classified in various ways, its probability changing with the classification; that it is usually regarded as specific to the frequentist interpretation and often as fatal to it, while versions of the classical, logical, propensity and subjectivist interpretations fall prey to their own variants, and the versions that evade it are no-theory theories, leaving obscure why probability should function as a guide to life, the problem besetting those theories that are genuinely informative and plausibly constrain inductive reasoning and decision; and that of a metaphysical and an epistemological reference class problem distinguished, the epistemological is conceded to remain even where the metaphysical is dissolved. **Accounts.** The narrowest class for which reliable statistics exist, refined by a homogeneity requirement: narrow until no further partition changes the frequency, and stop there. The rule is applied daily in actuarial practice, medicine, and forecasting, where classes are chosen by this criterion and the resulting probabilities are calibrated against outcomes. No rule selects the class. Narrowing terminates in the single case, where the frequency is one or zero, and stopping earlier requires a judgment the rule does not supply. The problem recurs in every interpretation of probability, including subjectivist ones, where it appears as the choice of what to condition on. **Specifications.** A single case's probability derives from a frequency in a class — universal under frequentist probability. Classes are given prior to the assignment — local. **Specifications withdrawn.** **Common requirement.** That the single case has a probability and a class determines it. **Conserving relation.** A selection of the class, the selection standing without a judgment of relevance supplied from outside the rule. **Conserving reach.** All single cases to which a probability is assigned. **Reach.** The statement quantifies over which of many true descriptions of a case governs. Frequencies in classes, once a class is chosen, are what is measured. The two are not the same. **Unreachability.** Narrowing terminates at the single case, and every interpretation absorbs the problem at its own site, which each interpretation's own literature records. The selection is not preserved across all cases by any rule that does not supply a judgment of relevance. **Persistence.** A principled selection of the class, not itself resting on a judgment of relevance. **Addresses.** Statement: Hájek, "The reference class problem is your problem too", Synthese 156:563-585, 2007; the classical statement: Reichenbach, The Theory of Probability, 1949, p. 374. The survey: Hájek, "Interpretations of Probability", Stanford Encyclopedia of Philosophy. Hardness: the same paper, at its abstract as the publisher's page, an index, a university repository and an abstract service carry it verbatim (the problem arising when a probability is to be assigned to a proposition classifiable in various ways, the probability changing with the classification; usually regarded as specific to frequentism and often as fatal to it; versions of the classical, logical, propensity and subjectivist interpretations falling prey to their own variants, the versions that evade it no-theory theories leaving obscure why probability should guide life, the problem besetting the genuinely informative theories; a metaphysical and an epistemological problem distinguished, the epistemological conceded to remain), issue 3, doi 10.1007/s11229-006-9138-5, the paper itself not fetched; the classical statement and the survey not read this pass. Approaches: the routes and their endings at the paper the address names first, at its survey of the proposals, narrowest homogeneous class, propensity and subjective readings, read at pass one; the classical statement at the second source. Persistence, no landing: the closing condition at the paper the address names first, at its abstract read at pass two (a probability to be assigned to a proposition classifiable in various ways, the probability depending on the class chosen; the problem every interpretation's problem), a selection not resting on relevance standing there as what none of the interpretations supplies. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Narrowest homogeneous class — terminates at the single case unless stopped by a judgment about relevance. Propensity interpretations — probability assigned to the case's own physical setup, the class removed, facing the difficulty of identifying the setup, which is the same question restated. Subjective probability — the class removed in favour of degrees of belief, the choice of conditioning information being the same problem, as its own literature notes. Statistical model selection — chooses among candidate classes by fit and penalty, returning a class relative to a criterion. Legal and actuarial practice — classes fixed by convention and stipulation, which works, the stipulation being the thing in question. **Narrowing recorded.** Statistical model selection narrows the reach: a class stands relative to a stated criterion rather than across all cases, and results follow. Legal and actuarial practice narrows the reach by stipulation: classes are fixed by convention within a domain, and the practice is calibrated within it. **Progress.** That the problem recurs in every interpretation, documented across the literature, including by proponents of each. **Own record.** ## 12 The origin of life **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Living systems come from living systems. How did the first one arise from chemistry that was not living? **The field's account of its hardness.** That no criteria exist for evaluating the competing models: the field's own review states that what is needed is not a definition of life but a theory of life from which such criteria might emerge, and that currently no such criteria exist, with the historical particulars stated in the review literature as probably forever lost in the mists of time. **Accounts.** Genetic material first: replication with variation makes everything else possible, so a self-copying molecule came first and metabolism was recruited around it. Ribozymes are demonstrated, and an RNA world couples information and catalysis in one molecule. Metabolism first: a self-copying molecule of the required length is prohibitively improbable without a chemistry already running to build and concentrate its parts. Autocatalytic cycles at mineral interfaces supply that chemistry, and heredity arises within it. **Specifications.** Living systems and non-living chemistry are distinguishable by criteria stated in advance — local. The transition between them occurred as an event, with prebiotic conditions before it — local. **Specifications withdrawn.** **Common requirement.** That there is a threshold separating non-living chemistry from living, and that the question is which capacity crossed it first. **Conserving relation.** A continuous sequence from prebiotic conditions to a self-maintaining, heritably varying system, the sequence standing without a step supplied from outside the chemistry. **Conserving reach.** The transition as a single event, prebiotic chemistry before it and inheritance after. **Reach.** The statement quantifies over a transition, with prebiotic chemistry before it and inheritance after. Present chemistry, and preserved traces from later than the event, are what is measured. The two are not the same. **Unreachability.** The event is one occurrence, and what is measured is present chemistry and traces postdating it. The sequence is not reached over the transition by any measurement returning present chemistry or later traces. **Persistence.** A demonstrated route from plausible prebiotic conditions to a self-maintaining, heritably varying system, in one continuous sequence. **Addresses.** Statement and hardness: Walker, "Origins of Life: A Problem for Physics, A Key Issues Review", Reports on Progress in Physics 80(9):092601, 2017, §1. Historical loss: Pross and Pascal, "The origin of life: what we know, what we can know and what we will never know", Open Biology 3:120190, 2013, introduction. As read at v342: the review at its abstract as the publisher's page, a university index, a database, an index, the arXiv page and an archive copy carry it verbatim (the origins of life among the great open scientific questions of our time; proposals for starting points not achieving states of complexity anywhere near the simplest living systems; the key challenge identifying the properties that distinguish living from non-living matter; the hope of a theory akin to those of fundamental physics), published 14 August 2017, doi 10.1088/1361-6633/aa7804, arXiv:1705.08073, the review itself not fetched; the second source not searched this pass. Persistence, no landing: the closing condition at the review the address names, at its abstract read at v342 (proposals for starting points not achieving states of complexity anywhere near the simplest living systems; the key challenge identifying the properties distinguishing living from non-living matter). Approaches: the routes' endings at that abstract, in part, and at the sources the address names for each, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Prebiotic synthesis — amino acids, nucleobases, and activated nucleotides made under plausible conditions, some in one pot, ending short of a self-maintaining system, the gap stated by the field. In vitro evolution of ribozymes — polymerases copying substantial lengths, not yet themselves. Autocatalytic set theory — such sets form above a diversity threshold in model chemistries, the models not being the chemistry of the early Earth, which their authors state. Hydrothermal and mineral surface chemistry — gradients and organics produced, establishing plausibility of conditions rather than the transition. Search for traces — isotopic and morphological evidence contested at the relevant age, the earliest rocks metamorphosed. Panspermia — relocates the event without addressing it, as its proponents acknowledge. **Narrowing recorded.** Autocatalytic set theory narrows the reach: the result stands over model chemistries rather than over the transition, and results follow within the models. **Progress.** That the building blocks form abiotically under plausible conditions. That RNA can be both information and catalyst. That autocatalytic sets form above a threshold in model chemistries. **Own record.** ## 13 Underdetermination of theory by evidence **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether evidence can fail in principle to select among rival theories. **The field's account of its hardness.** That the evidence available at a given time may be insufficient to determine what beliefs to hold in response to it, which is the simple idea at the heart of the problem; that the response to the experimental or observational falsification of a theory is concluded to be always underdetermined in this way; and that the challenge posed by contrastive underdetermination is that there might be more than one account of the otherwise inaccessible workings of nature behind the phenomena well confirmed by the evidence, not merely that one may believe only a theory's observable consequences, a change of fundamental subject arising where a theory's Craigian reduction is proposed as an empirical equivalent, and those who do not share radical holism start from scratch in deciding whether contrastive underdetermination is worth taking seriously. **Accounts.** Yes, significantly: confirmation reaches theories only in blocks, auxiliary hypotheses absorb refutation, and empirically equivalent rivals stand constructible. No, not in any scientifically significant form: the equivalent rivals are artifacts or notational variants, and transient underdetermination resolves as inquiry proceeds. **Specifications.** Hypotheses meet experience in bodies, not singly — universal under the founding holism. **Specifications withdrawn.** **Common requirement.** A fact about the reach of evidential discrimination. **Conserving relation.** Evidence selecting among rivals, the selection standing on evidence rather than on a criterion supplied beside it. **Conserving reach.** All rival theories, the constructed equivalents included. **Reach.** The statement quantifies over the limits of evidence. The case record — historical rivalries and their resolutions — and the constructed-equivalence results are what is measured. The two are not the same. **Unreachability.** The construction results stand as existence results, and the case record is cited by both accounts, as evidence of transience and as evidence of theory-choice by extra-evidential virtue. The selection is not preserved over all rivals by evidence alone. **Persistence.** The record carries the split over what significance requires. **Addresses.** Statement: Stanford, "Underdetermination of Scientific Theory", Stanford Encyclopedia of Philosophy. Hardness: the same entry, substantively revised at the Summer 2023 edition as the encyclopedia's citation page carries it, at its opening, at passages of its text and at its notes as the encyclopedia's page and its archived editions carry them verbatim (the evidence available at a time possibly insufficient to determine what beliefs to hold, the simple idea at the heart of the problem; the response to an experimental or observational falsification of a theory concluded always underdetermined; the challenge of contrastive underdetermination that there might be more than one account of the inaccessible workings of nature behind the well-confirmed phenomena, a proposed empirical equivalent such as a theory's Craigian reduction changing the fundamental subject; those not sharing radical holism starting from scratch on whether contrastive underdetermination is worth taking seriously), the entry itself not fetched, its section numbers not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passages read at pass one and at the hardness above (the evidence available possibly insufficient to determine what to believe; the split over the scientific significance of constructed equivalents). Approaches: the constructed-equivalents, historical-case and virtues routes at the same entry's sections on the responses, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Constructed equivalents — stand as existence results, their scientific significance disputed. Historical cases — recorded as resolving, cited by both accounts. The virtues route — simplicity and unification stand as working criteria, their epistemic standing disputed. **Narrowing recorded.** The virtues route weakens the relation: selection stands on evidence together with simplicity and unification rather than on evidence alone, and the criteria are in working use. **Progress.** The holism as a logical point. The construction results. The case record. **Own record.** The founding statement of holism and the reply literature's transience statements. ## 14 The demarcation problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What distinguishes science from non-science. **The field's account of its hardness.** That the much-noted death certificate issued to the problem holds there is no hope of finding a necessary and sufficient criterion of something as heterogeneous as scientific methodology, after which work on the problem waned, its revival maintaining that the concept can be clarified by other means than necessary and sufficient criteria or that a definition is possible only when supplemented with discipline-specific criteria to become fully operative; that the canonical falsifiability criterion is criticized both for excluding legitimate science and for assigning scientific status to some pseudosciences, most proposers instead putting forward lists of criteria to be used in combination; and that the demarcation metaphor itself is argued misleading, giving the impression that whether a claim is pseudoscientific can be determined without specialized knowledge in the subject matter. **Accounts.** A criterion exists: falsifiability as the canonical proposal, with multicriterial refinements standing in the later literature. No single criterion exists: the problem stands dissolved or replaced, in the canonical dissolution statement and the family-resemblance accounts that followed. **Specifications.** Science's own boundary-keeping — journals, institutions, and courts demarcating operationally, whatever the theory of the boundary — local, and standing as the practice the problem is posed against. **Specifications withdrawn.** **Common requirement.** A fact about whether the operational practice tracks a principled boundary. **Conserving relation.** A criterion sorting cases into science and non-science, the sorting standing over the practice rather than reading off it. **Conserving reach.** All cases the practice sorts. **Reach.** The statement quantifies over the boundary of science itself. The case record is what is measured: institutional and legal demarcations, entered and functioning, and the criteria's counterexample record, standing in both directions. The two are not the same. **Unreachability.** The cases sort under practice, and the criterion dispute stands over the practice. The sorting is not preserved across all cases by any criterion standing over the practice, the counterexample record running in both directions. **Persistence.** The record carries the split over whether a principled criterion is owed or the practice suffices. **Addresses.** Statement: Hansson, "Science and Pseudo-Science", Stanford Encyclopedia of Philosophy. The problem's demise argued: Laudan, "The demise of the demarcation problem", in Cohen and Laudan, editors, Physics, Philosophy and Psychoanalysis, Reidel, 1983. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (work on the problem waned after the much-noted death certificate holding no hope of a necessary and sufficient criterion for something as heterogeneous as scientific methodology, then revitalized, the concept clarifiable by other means than such criteria or a definition possible only when supplemented with discipline-specific criteria; the falsifiability criterion criticized both for excluding legitimate science and for assigning scientific status to some pseudosciences, most proposers instead putting forward lists of criteria used in combination; the demarcation metaphor argued misleading, giving the impression that whether a claim is pseudoscientific can be determined without specialized knowledge in the subject matter), the entry itself not fetched, its section numbers and revision date not carried at the returns; the demise paper not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass one and at the hardness above (the death certificate holding no hope of a necessary and sufficient criterion for something as heterogeneous as scientific methodology; the revival on other means than such criteria). Approaches: the falsifiability, dissolution and multicriterial routes at the same entry's sections, read at pass one, and the legal record at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Falsifiability — stands as the canonical proposal, with its counterexample record standing in the literature. The dissolution — stands as the canonical counter-statement. The multicriterial revival — stands published, case-driven. The legal record — court demarcations standing as operational data, cited by the revival literature. **Narrowing recorded.** The multicriterial revival weakens the relation: several criteria stand together and case by case rather than one criterion over all cases, and results follow. **Progress.** The counterexample taxonomy. The operational record. **Own record.** The dissolution statement, the revival literature, and the operational record standing together. ## 15 The quantum interpretation debate **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Several interpretations reproduce every prediction of quantum mechanics and describe reality differently. Which is correct? **The field's account of its hardness.** That there is no consensus among physicists or philosophers of physics on what, if anything, the empirical success of the theory is telling about the physical world, the common core of interpretation consisting of recipes for calculating probabilities of outcomes of experiments on systems subjected to state preparation procedures and the interpretations differing on what, if anything, is added to that core; and that, if the Everettian approach is granted a solution to its evidential problem, none of the major lines of approach is favored in a straightforward way by the empirical evidence. **Accounts.** The question is not empirical and is set aside. Interpretations agreeing on every prediction differ only in what they add beyond the formalism, and adding unobservable structure is not what physics is for. The formalism is complete as an instrument, saying what will be measured with unmatched precision, a century of successful application having required no interpretive commitment. The interpretations differ in more than presentation — in whether the state is physical, whether there is one world, whether the theory is deterministic — and these are questions about what the theory says is the case. Setting them aside is not answering them, and the differences bear on extending the theory to new domains. **Specifications.** The formalism admits multiple descriptions of what is the case — universal under quantum mechanics. Theory choice is settled by prediction — universal under the framework. **Specifications withdrawn.** **Common requirement.** That the interpretations are genuinely empirically equivalent. **Conserving relation.** A selection among the interpretations, the selection standing on prediction. **Conserving reach.** All interpretations reproducing the formalism's predictions. **Reach.** The statement quantifies over what the world is like given the formalism. Predictions are what is measured, and they are identical across interpretations. The two are not the same. **Unreachability.** The interpretations are empirically equivalent by construction. The selection is not preserved over the interpretations by prediction, no discriminating prediction standing where the equivalence is built in. **Persistence.** A discriminating prediction, which empirical equivalence excludes by construction. **Addresses.** Statement: Myrvold, "Philosophical Issues in Quantum Theory", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at its opening and at a passage of its text as the encyclopedia's page and its archived editions carry them verbatim (no consensus among physicists or philosophers of physics on what, if anything, the theory's empirical success is telling about the physical world; a common core of interpretation consisting of recipes for calculating probabilities of outcomes, the interpretations differing on what, if anything, is added to it; if the Everettian approach is granted a solution to its evidential problem, none of the major lines of approach favored in a straightforward way by the empirical evidence), the entry itself not fetched, its section numbers and revision date not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (no consensus on what the theory's empirical success is telling about the physical world; a common core of recipes for calculating probabilities, the interpretations differing beyond it), the empirical equivalence of the interpretations standing there as the reason a discriminating prediction is excluded by construction. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Looking for a discriminating prediction — collapse models make one and are in the empirical arena; the interpretations proper, by construction, do not. Ontological models framework — the state-reality theorem constrains epistemic readings under stated assumptions, a genuine constraint, its assumptions contested. Extended observer-chain arguments — an inconsistency derived for certain single-world assumptions, contested in detail, the interpretations differing on which premise fails. Simplicity and coherence — each interpretation claiming economy on a different measure. Extension to gravity and cosmology — interpretations differing in fruitfulness, argued on both sides without resolution so far. **Narrowing recorded.** Simplicity and coherence weaken the relation: the selection stands on economy on a stated measure rather than on prediction, each interpretation claiming it on a different measure. **Progress.** The state-reality theorem under its assumptions. The observer-chain result and the mapping of which interpretation rejects which premise. That collapse models are distinguishable and interpretations are not. **Own record.** ## 16 Natural kinds **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether scientific classifications track mind-independent kinds, or serve purposes by convention. **The field's account of its hardness.** That the metaphysical commitment of a theory of natural kinds ranges from the out-and-out conventionalist claim that there is no natural sameness and difference in nature, or none that human classifications capture, a fortiori no classifications into natural kinds and no natural-kind essences, to the claim that there are natural distinctions between things, some of them distinctions of kinds, the kinds a certain species of irreducible entity with essences, intermediate views standing between; and that the homeostatic property cluster account leaves the mechanisms not clearly delineated, a fully realist account of the clusters being doubted where conventional elements enter into deciding when two mechanisms are of the same kind and where one mechanism ends and another begins. **Accounts.** Kinds carve nature at joints, essences or homeostatic property clusters grounding the classifications' inductive success. Classifications answer to interests: many cross-cutting groupings serve, and none is privileged by nature. **Specifications.** Induction's reliance on kinds — universal under inductive practice. Classification as the working instrument of every science — universal under scientific practice. **Specifications withdrawn.** **Common requirement.** A fact about what grounds projectibility. **Conserving relation.** A ground for the classifications, the ground standing independently of the interests the classifications serve. **Conserving reach.** All scientific classifications. **Reach.** The statement quantifies over the ground of kinds. The classificatory successes of the sciences are what is measured, recorded in the literature as compatible with both accounts. The two are not the same. **Unreachability.** The successes fit both accounts and the case record is shared, each case cited by both. The ground is not preserved across all classifications by any account the successes admit. **Persistence.** The record carries the split over what would count as settling. **Addresses.** Statement: Bird and Tobin, "Natural Kinds", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the Spring 2023 archived edition carries them verbatim (the metaphysical commitment of a theory of natural kinds ranging from the out-and-out conventionalist claim of no natural sameness and difference, or none that human classifications capture, to natural distinctions of kinds as irreducible entities with essences, intermediate views between; the homeostatic property cluster account leaving the mechanisms not clearly delineated, a fully realist account of the clusters doubted where conventional elements enter into deciding when two mechanisms are of the same kind and where one ends and another begins), the Winter 2025 edition a minor correction as the encyclopedia's citation page carries it, the entry itself not fetched, its section numbers not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the metaphysical commitment of a theory of natural kinds ranging from the conventionalist claim of no natural sameness and difference to the essentialist, with the positions between). Approaches: essentialism and the cluster accounts at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Essentialism — carries the chemical kinds as its exemplar, the record showing biology as resistant, with no universal essentialism standing. Cluster accounts — the homeostatic-property-cluster position standing as the recorded middle. Pluralist accounts — standing published, with the case studies as their evidence. Case studies — the species and element cases, each cited by both accounts. **Narrowing recorded.** Cluster accounts weaken the relation: the ground stands as a homeostatic cluster rather than an essence, and results follow. Pluralist accounts narrow the reach: a grouping stands relative to an interest rather than across all classifications. **Progress.** The case-study record. The failure of universal essentialism in biology, by the field's own record. **Own record.** The standing statements of the accounts, and this problem's own reflexive place: it is the ordering dispute about ordering disputes. ## 17 The definition of planet **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What a planet is — by which criterion bodies are classified as planets. **The field's account of its hardness.** That the adopted definition is held, in the field's proposal for its revision, to be problematic because it is vague and excludes exoplanets, inadequate for its vague, non-quantified terms and for not including exoplanets in a cohesive framework, with difficulties ingrained in the current recommendations that a mass-based framework is proposed to avoid. **Accounts.** Dynamically: the adopted definition requires orbit-clearing, and the classification follows the architecture of the system. Geophysically: planethood is intrinsic, hydrostatic equilibrium and geological character, under which the dwarf bodies and the large satellites classify as planets. **Specifications.** Taxonomy tracks scientifically useful kinds — universal under scientific classification. The union's chartered authority over nomenclature — local. **Specifications withdrawn.** **Common requirement.** A fact about which properties classify naturally or usefully. **Conserving relation.** A criterion classifying bodies as planets or not, the criterion standing on measurement rather than on adoption. **Conserving reach.** All bodies the classification applies to, within the solar system and beyond it. **Reach.** The statement quantifies over the criterion of classification. The bodies' orbits and intrinsic properties are what is measured; membership under each criterion follows from measurement, and the choice between criteria does not. The two are not the same. **Unreachability.** Measurement decides membership under each criterion and does not decide between criteria. The criterion is not reached over the bodies by measurement. **Persistence.** No measurement selects between the definitions on the record; adoption and usage stand as the recorded mechanisms of settlement. **Addresses.** Statement: International Astronomical Union, Resolution B5, "Definition of a Planet in the Solar System", XXVIth General Assembly, 2006. The standing dispute over the criterion: Margot, Gladman and Yang, "Quantitative Criteria for Defining Planets", Planetary Science Journal 5:159, 2024, doi 10.3847/PSJ/ad55f3, arXiv:2407.07590, the authorship given before this pass as Margot and Gladman, the two proposers of the resolution the paper reproduces, withdrawn. Hardness: the same paper, at its abstract as the publisher's page, the arXiv page, an index and a repository carry it verbatim (the adopted definition problematic because vague and excluding exoplanets; unsupervised clustering returning satellites as distinct from planets and dynamical dominance as a natural organizing principle; two quantitative unified frameworks proposed, the second mass-based and avoiding some of the difficulties ingrained in the current recommendations) and at the text of the proposed resolution as the publisher's page reproduces it (the definition inadequate for its vague, non-quantified terms and for not including exoplanets in a cohesive framework), the paper itself not fetched; the resolution not read this pass. Persistence, no landing: the closing condition at the paper the address names, at its abstract read at pass two (the adopted definition problematic because vague and excluding exoplanets; a classification by measured property proposed in its place), adoption and usage standing as the mechanisms the record carries. Approaches: the vote and the geophysical definition at that paper and at the sources the address names for the adoption and the dissent. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Adoption by vote — the definition adopted, with the dissent entered into the record at adoption. The geophysical definition — published and defended, used in the planetary-science literature's own practice, by that literature's account of itself. Exoplanet extension — the adopted definition's solar-system scope recorded as its limit, the dissenting literature recording the definition unusable beyond it. Usage studies — mixed usage across communities, recorded. **Narrowing recorded.** The adopted definition narrows the reach: the criterion stands within the solar system, its scope recorded as its limit. Adoption weakens the relation: the criterion stands by chartered decision rather than on measurement. **Progress.** The bodies' properties. Membership under each definition. **Own record.** ## 18 Biological individuality **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What counts as one organism — the criterion of biological individuality. **The field's account of its hardness.** That although biologists and philosophers of biology draw freely on the concept of an individual, there has been little explicit work devoted to the biological notion of an individual itself, the questions being how biological individuals should be thought of and what roles they play in natural selection, speciation and development; that the impressive variation amongst biological individuals generates many complexities in addressing the simple-sounding question what a biological individual is, knowing what to say about a striking case turning on what is packed into the conception of an organism; and that the sheer difficulty of identifying individuals in biological reality has generated a debate characterized in the field as a problem agenda of multiple interconnected questions of individuation, demarcation, parthood and unity. **Accounts.** Individuals are the units of selection and reproduction: bottlenecked lineages with germ-soma separation. Individuals are units of metabolic and immune integration, under which hosts with their symbionts count as one. **Specifications.** Biology counts individuals — fitness, development, and physiology attributed per individual — universal under biology, and standing as the science's presupposition. **Specifications withdrawn.** **Common requirement.** A criterion, the field's own case tables recording that the standing criteria cross-classify. **Conserving relation.** A criterion counting organisms, the count standing alike across the canonical cases. **Conserving reach.** All cases biology counts, the colonies, lichens, holobionts, and chimeras included. **Reach.** The statement quantifies over the unit of life's organization. The canonical cases are measured in full, and their sorting differs by criterion, by the field's own tables. The two are not the same. **Unreachability.** The hard cases sort differently under each criterion, and the measurements stand and do not select. The count is not preserved across the canonical cases by any criterion the measurements admit. **Persistence.** The record carries the split over whether one criterion is owed. **Addresses.** Statement: Wilson and Barker, "The Biological Notion of an Individual", Stanford Encyclopedia of Philosophy. Hardness: re-formed to the field's words at the same entry, at its introduction as an index and the Summer 2013 archived edition carry it verbatim (little explicit work devoted to the biological notion of an individual itself; how biological individuals should be thought of and what roles they play in selection, speciation and development; knowing what to say about a striking case turning on what is packed into the conception of an organism) and at the opening of its 2019 revision as the author's page carries it (the impressive variation amongst biological individuals generating many complexities in addressing the simple-sounding question what a biological individual is; a distinction between evolutionary and physiological individuals useful), the entry itself not fetched, its title carried without the article at the encyclopedia's pages and as Biological Individuals at the author's page and an index for the 2019 and later versions, recorded and not corrected, its section numbers not carried at the returns; a later paper's characterization of the debate as a problem agenda, at its text as the publisher's page carries it, Biology and Philosophy, doi 10.1007/s10539-021-09791-5, its authors not carried, returned in the search; the former line, the reviews' statements that individuality is plural in the science's own use, standing in the entry's section on pluralism in the face of biological heterogeneity, not returned at its text, keyed at the next pass. Persistence, no landing: the closing condition at the same entry, at its introduction and its later revision read under check C18 (little explicit work devoted to the notion itself; what to say about a striking case turning on what is packed into the conception of an organism; the debate a problem agenda of interconnected questions). Approaches: the evolutionary, physiological and pluralist routes at the same entry's sections on the accounts and on pluralism, read at pass one and keyed at the next pass at their text. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Evolutionary accounts — carry the paradigm cases, the symbiont cases standing against them. Physiological and immunological accounts — carry the symbiont cases, the colonial cases standing against them. Pluralism — standing published, the concept plural in use, by the reviews' own statements. **Narrowing recorded.** Pluralism narrows the reach: a criterion stands for a purpose rather than across all cases, and the reviews record the plurality as the science's own use. **Progress.** The case record. The criteria's extensions, tabulated in the field's literature. **Own record.** ## 19 The units of selection **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** At what level natural selection operates — genes, organisms, groups — and whether that question has a factual answer or an accounting one. **The field's account of its hardness.** That four distinct questions, what is the interactor, what is the replicator, what is the beneficiary, and what entity manifests the adaptations resulting from evolution by selection, appear under the one rubric and are often used in combination to represent the problem; that a serious ambiguity in the meaning of adaptation has had deep consequences for both the group selection and the species selection debates; that much of the confusion is held in the field to have arisen because the same terms have been used with different meanings by different authors, with the fear that the confusions will not easily be ended; and that the identification and status of the units under the four meanings is still a very active area of disagreement, the four questions approached differently in the two schools of multilevel selection theory. **Accounts.** Gene-level accounting suffices: higher-level selection is re-describable in gene-level terms, and the inclusive-fitness framework carries the cases. Multilevel selection is real and causally distinct: group-level selection operates under stated conditions, and the levels are not mere bookkeeping. **Specifications.** Natural selection as the framework — universal under evolutionary biology. Fitness attributed to entities at a level — universal under evolutionary biology. The covariance formalism as the general accounting instrument — universal under the field's formal practice. **Specifications withdrawn.** **Common requirement.** A fact about causal structure beyond the accounting, the field's own equivalence literature disputing whether there is one. **Conserving relation.** A level identified as where selection operates, the identification standing on trait dynamics rather than on choice of framework. **Conserving reach.** All cases the two frameworks describe. **Reach.** The statement quantifies over the causal level of selection. Trait dynamics that the inclusive-fitness and multilevel frameworks both fit are what is measured, with formal translations between them standing proved. The two are not the same, and the equivalence results state the gap. **Unreachability.** The frameworks are intertranslatable over the shared cases, by the field's own proved results, and fit the same data by construction. The identification is not preserved over those cases by trait dynamics. **Persistence.** The record carries the split over whether any empirical result could distinguish accounting schemes, the causal-versus-bookkeeping statements standing on both sides of the reviews. **Addresses.** Statement, the four questions the field parts: Lloyd, "Units and Levels of Selection", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (four distinct questions under the one rubric, often used in combination to represent the problem; a serious ambiguity in the meaning of adaptation with deep consequences for the group and species selection debates; the field's comment that much of the confusion arose from the same terms used with different meanings by different authors, with the fear the confusions will not easily be ended; the identification and status of the units under the four meanings still a very active area of disagreement, the four research questions approached differently in the two schools of multilevel selection theory), the entry itself not fetched, its section numbers and revision date not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (four distinct questions under the one rubric; a serious ambiguity in the meaning of the phrase; the causal-versus-bookkeeping question as the reviews state it), the split standing at those passages. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Kin-selection formalism — carries the social-evolution cases. Multilevel formalism — carries the same cases under the covariance partition. The equivalence results — the translations between frameworks standing proved, the route ending in accounting-equivalence. Empirical test cases — the canonical cases recorded under both descriptions. **Narrowing recorded.** **Progress.** The covariance framework. The formal translations. The case record. **Own record.** ## 20 Species boundaries **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Where does one species end and another begin? **The field's account of its hardness.** That the tree of life is segmented by different processes into different types of species lineages, lineages bound by interbreeding, lineages sharing a common and unique ancestry, and lineages exposed to common sets of stabilizing selection, the organic world being multifaceted on the well-substantiated theory, the plurality of species concepts held by its defenders to result from a fecundity of biological forces rather than a paucity of scientific information, and not everyone willing to accept the plurality. **Accounts.** At reproductive isolation: populations not exchanging genes under natural conditions are distinct species, and the criterion is testable. No single criterion works: ring species, asexual lineages, hybridising taxa, and pervasive introgression defeat it, and species concepts number in the dozens, chosen for the question at hand. **Specifications.** Organisms sort into species — universal under systematics. A criterion identifies the sorting — universal under systematics. The criterion applies across the relevant lineages — local. **Specifications withdrawn.** **Common requirement.** That populations divide into kinds, and that a criterion identifies the division. **Conserving relation.** A criterion dividing populations into species, the division standing across the lineages. **Conserving reach.** All lineages, sexual and asexual, extant and extinct. **Reach.** The statement quantifies over a boundary holding across all lineages. Populations varying continuously in every relevant respect are what is measured. The two are not the same. **Unreachability.** Concept choice changes species counts by large and measured factors, which is the instruments disagreeing on the same populations. The division is not preserved across all lineages by any criterion the measurements admit. **Persistence.** A criterion applying without exception across sexual and asexual, extant and extinct lineages. **Addresses.** Statement: Ereshefsky, "Species", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its section on species pluralism as the encyclopedia's page and its archived editions carry them verbatim (the tree of life segmented by different processes into different types of species lineages, those bound by interbreeding, those sharing a common and unique ancestry, and those exposed to common sets of stabilizing selection; the organic world multifaceted on the well-substantiated theory; the plurality of concepts held by its defenders to result from a fecundity of biological forces rather than a paucity of scientific information, and not everyone willing to accept it), the entry itself not fetched, its own posing of the species problem at its opening not carried at the returns, its section numbers and revision date not carried. Persistence, no landing: the closing condition at the same entry, at its section on species pluralism read at pass two (the tree of life segmented by different processes into different types of species lineages, no one criterion applying across them). Approaches: the biological, phylogenetic and pluralist concepts at the same entry's sections on the concepts, read at pass one and at that section. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Biological species concept — works for sexual, sympatric, extant populations, undefined for asexual and fossil lineages, which its author acknowledged. Phylogenetic concepts — apply universally and multiply species counts by large factors depending on the resolution chosen. Genomic thresholds — a percentage divergence stipulated, differing by taxon. Integrative taxonomy — combines criteria and returns agreement where they agree, which is where the question was not hard. Species as a pragmatic unit — abandons the boundary, conservation law and biodiversity accounting requiring one, which the field notes. **Narrowing recorded.** The biological species concept narrows the reach: the criterion stands for sexual, sympatric, extant populations, its author acknowledging it undefined elsewhere. Genomic thresholds weaken the relation: the division stands by a stipulated percentage, differing by taxon. The pragmatic unit weakens the relation: the boundary stands for a purpose rather than as a division. **Progress.** That introgression is documented between taxa standardly treated as distinct species, including in humans and hominins. That concept choice changes species counts by large and measured factors. **Own record.** ## 21 Group three of the periodic table **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which elements compose group 3 of the periodic table: scandium and yttrium with lanthanum and actinium, with lutetium and lawrencium, or scandium and yttrium alone with the third and fourth places left unassigned — the three forms the chartered project's report sets out. **The field's account of its hardness.** The chartered project's report states that there is no objective means to adjudicate between the two compositions; that the question is in the final analysis one of convention rather than one decidable on objective scientific grounds; that the criteria tried — chemical and physical properties and the atoms' electronic configurations — provide no clear-cut resolution; and that the question cannot be treated independently of the form of the periodic table as a whole. **Accounts.** Lanthanum and actinium, on chemical behaviour, and as the majority representation of the textbooks by the field's own survey. Lutetium and lawrencium, on electron-configuration regularity and the undivided d-block. **Specifications.** The periodic law, arrangement by recurring properties — universal under chemistry. The union's chartered role over the table's presentation — local. **Specifications withdrawn.** **Common requirement.** A criterion of arrangement that selects one composition. **Conserving relation.** A criterion of arrangement selecting one composition, the selection standing on the elements' measured properties. **Conserving reach.** All four candidate elements. **Reach.** The statement quantifies over the criterion of arrangement. The configurations and properties of all four elements are measured, and the chartered project's report states there is no objective means to adjudicate between the compositions. The two are not the same, and the field's own body states it. **Unreachability.** Both candidates carry a d-differentiating electron; measurement stands and does not select. The chartered body states that no objective criterion within the field decides the field's own arrangement. The selection is not reached over the four elements by measurement. **Persistence.** A criterion the field adopts; the project's report stands as the field's own statement that none is objectively forced. **Addresses.** Statement, hardness, reach, persistence, narrowing recorded, and the chartered project's approach: Scerri, "Provisional Report on Discussions on Group 3 of the Periodic Table", Chemistry International 43(1):31-34, 2021. As read at v342: the report at its text as the publisher's page carries it verbatim (no objective means to adjudicate between the two constitutions; a ruling the more important for that, the question in the final analysis one of convention rather than one decided on objective scientific grounds; quantum mechanics explaining the table without a full and exact reduction; some relativistic quantum chemists upholding fifteen elements in the f-block, findings not to be imposed on the majority of users), doi 10.1515/ci-2021-0115, the report itself not fetched; a follow-on examination critiquing the report, at its abstract as an index carries it, its author not carried, a reading for Narrowing at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Measuring the configurations — both candidates carry a d-differentiating electron, measurement standing and not selecting. The chartered project — convened to recommend, its provisional report stating the non-adjudicability and noting a compromise form, no recommendation standing adopted. Regularity arguments — published on both sides across the education and philosophy-of-chemistry literatures. **Narrowing recorded.** The compromise form noted in the project's report weakens the relation: a presentation stands without the selection being made. **Progress.** The configurations. The properties. The usage survey. **Own record.** ## 22 Rank-based and clade-based nomenclature **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How the names of taxa above the species are to be governed: by the rank-based codes, where a name's application is fixed by its rank and its type, or by the phylogenetic code, where a name's application is fixed by an explicit definition in terms of ancestry and descent. **The field's account of its hardness.** The phylogenetic code's preface states that rank assignment is subjective and of dubious biological significance, that the arbitrary nature of ranking is acknowledged by most taxonomists, and that a clade whose composition and diagnostic characters have not changed may be given a different name under the rank-based codes purely on considerations of rank; it states stability and continuity of names as the goal both systems share. **Accounts.** By ranks and types: the standing codes govern practice, stability of names being the stated purpose, secured through typification. By ancestry: names defined by phylogenetic relations, ranks being arbitrary and non-equivalent across groups, the phylogenetic code standing published and activated as the alternative instrument. **Specifications.** Names serve stable communication — universal under nomenclature. Taxa are evolutionary units — universal under systematics. The codes' institutional governance of naming — local. **Specifications withdrawn.** **Common requirement.** A fact about what the science's naming should track. **Conserving relation.** A naming instrument tracking what the science requires of names, the tracking standing on phylogeny. **Conserving reach.** All taxa named. **Reach.** The statement quantifies over the instrument of naming. Phylogenies are measured, and rank assignment is recorded as not fixed by them, ranks non-equivalent across groups. The two are not the same. **Unreachability.** Phylogeny is measured and the choice of naming instrument is not fixed by the measurement, by the field's own record of rank non-equivalence. The tracking is not reached over the taxa by phylogeny. **Persistence.** The record carries the split between stability and systematic correspondence as the criterion of good naming. **Addresses.** Statement, hardness, reach, and the phylogenetic code's approach: Cantino and de Queiroz, International Code of Phylogenetic Nomenclature (PhyloCode), version 6, CRC Press, 2020, preface. As read at v342: the code at its publisher's description as the publisher's page, a library record and booksellers carry it verbatim (a system for naming taxa by explicit reference to phylogeny, in contrast to the codes that define taxa by reference to ranks and types; governing the names of clades, species names under the traditional codes; designed for concurrent use, not to replace existing names), ratified on 20 January 2019 by the committee as the library record carries it, the preface the address keys not returned at its text, the code itself not fetched. Persistence, no landing: the closing condition at the code's own description read at v342 (a system naming taxa by explicit reference to phylogeny against the codes naming by rank and type; designed for concurrent use, not to replace existing names), the two criteria of good naming, stability and systematic correspondence, standing there as the two systems' own aims. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Revising the rank codes — successive editions, the rank structure standing. The phylogenetic code — drafted, published, activated, adopted by part of the field's practice. Dual usage — both systems in concurrent use, recorded. **Narrowing recorded.** Dual usage narrows the reach: each instrument stands over the part of practice adopting it rather than over all taxa named. **Progress.** The phylogenies. The non-equivalence of ranks across groups. **Own record.** The codes' own scope statements. ## 23 Psychiatric nosology **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How mental disorder is to be classified: as discrete diagnostic categories, or as dimensions of continuous variation. **The field's account of its hardness.** The dimensional model's founding paper states that the reliability and validity of the traditional taxonomies are limited by arbitrary boundaries between psychopathology and normality, unclear boundaries between disorders, frequent co-occurrence of disorders, heterogeneity within disorders, and diagnostic instability, and that those taxonomies went beyond the evidence available on the structure of psychopathology. The research-domain framework's founding statement records that the manuals have facilitated reliable clinical diagnosis and research, and that a framework for research on pathophysiology is needed to inform future classification. **Accounts.** Categorically: the diagnostic manuals' categories organize clinical practice, communication, records, and treatment, their validity carried by clinical utility and by iterative revision. Dimensionally: symptom measures distribute continuously, the measured comorbidity and within-category heterogeneity being the categorical structure misdescribing the variation, with the hierarchical-taxonomy and research-domain frameworks standing published as the alternative instruments. **Specifications.** Diagnosis tracks conditions of medical kind — universal under medicine. Inter-rater reliability as the founding standard of the modern manuals — local. The institutional authority of the manuals over practice — local. **Specifications withdrawn.** **Common requirement.** A fact about the structure of psychopathology, expressible by a classification. **Conserving relation.** A classification tracking the structure of psychopathology, the tracking standing on measured structure. **Conserving reach.** All conditions the manuals classify. **Reach.** The statement quantifies over the structure of disorder. Symptom covariation and reliability statistics are what is measured, with biomarkers not diagnostic at the category level, by the field's own record. The two are not the same. **Unreachability.** The instruments return covariation that both structures fit, and the institutional documents carry both accounts. The tracking is not preserved across the classified conditions by measured structure, no measurement recorded selecting between the criteria of a good classification. **Persistence.** The record carries the split: one account holds clinical utility the criterion of a good classification, the other holds measured structure the criterion; no measurement recorded selects between the criteria. **Addresses.** Statement and hardness: Kotov et al., "The Hierarchical Taxonomy of Psychopathology (HiTOP): A Dimensional Alternative to Traditional Nosologies", Journal of Abnormal Psychology 126(4):454-477, 2017, abstract and introduction; the research-domain framework's founding statement: Insel et al., "Research Domain Criteria (RDoC): Toward a New Classification Framework for Research on Mental Disorders", American Journal of Psychiatry 167(7):748-751, 2010. As read at v342: the taxonomy paper at its abstract as six university repositories and an index carry it verbatim (the reliability and validity of traditional taxonomies limited by arbitrary boundaries between psychopathology and normality, often unclear boundaries between disorders, frequent co-occurrence, heterogeneity within disorders and diagnostic instability; the taxonomies going beyond the evidence on the structure of psychopathology; the model constructing syndromes from the observed covariation of symptoms and combining co-occurring syndromes into spectra), doi 10.1037/abn0000258, the paper itself not fetched; the research-domain statement not searched this pass. Persistence, no landing: the closing condition at the taxonomy paper's abstract read at v342 (the reliability and validity of traditional taxonomies limited; the model constructing syndromes from the observed covariation of symptoms), measured structure standing there as its criterion and clinical utility as the criterion of the revisions it answers. Approaches: the revision route at the sources the address names for the manuals, the dimensional route at that abstract, the biomarker route keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Revising the categories — successive editions improving reliability, the validity disputes standing across revisions. Biomarker search — decades of search, no biomarker diagnostic at the category level, the research-domain framework's founding statement posing this as its reason. Dimensional frameworks — published, adopted in research practice, clinical adoption partial, the categorical instruments remaining the governing ones. Hybrids — the manuals carrying dimensional supplements, the two structures standing side by side in the field's own instruments. **Narrowing recorded.** Dimensional frameworks narrow the reach: the alternative instruments stand over research practice rather than over clinical classification. Hybrids weaken the relation: the categories stand with dimensional supplements beside them rather than tracking the structure alone. **Progress.** The reliability record. The measured comorbidity and heterogeneity. The continuous distribution of the major symptom measures. **Own record.** ## 24 Race as a biological category **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether human races are biological kinds. The field records a consensus against the older biological conception, on which races bear heritable essences shared by all and only their members, and a standing disagreement over whether a different conception — races as social constructions, or as biological populations without essences — refers to anything real. **The field's account of its hardness.** That the consensus against the old biological conception leaves the ontological status of any different conception of race disputed across three metaphysical camps, skepticism, constructivism and population naturalism, and two normative camps, eliminativism and conservationism; that the skeptic's argument proceeds by the scientific rejection, in turn, of essences, geography, phenotypes, post-Mendelian transmission genetics and genealogies as possible foundations for races, while a constructivist finds the term a perfectly clear referent in those socially ascribed to its categories; and that race talk overall is held by one deflationary view to be too ambiguous and contested to be salvaged in the search for a dominant understanding. **Accounts.** No: human genetic variation is predominantly within-group and clinal, no subspecies structure stands, and the category is social — the account carried in the field's institutional statements. A minimal biological realism: statistical clusters corresponding to ancestry groups stand in the measured structure, and a deflated kind-claim is defended on them in the published literature. **Specifications.** Population genetics' measures of variation — universal under population genetics. The population concept — universal under population genetics. The criterion of biological kindhood — local. **Specifications withdrawn.** **Common requirement.** A criterion of kindhood applied to the measured structure, the debate literature recording that the measurements themselves are shared between the accounts. **Conserving relation.** A criterion of kindhood applied to the measured structure, the application standing on the measurements. **Conserving reach.** The measured structure of human genetic variation. **Reach.** The statement quantifies over the category's biological standing. The apportionment of variation, the clinality, and the clustering behaviour under sampling are measured, and cited by both accounts. The two are not the same, the dispute being the criterion and not the data, by the field's own statement. **Unreachability.** The measurements stand and are shared, and the kindhood criterion is not fixed by them. The application is not reached over the measured structure by the measurements. **Persistence.** The record states the split is over the criterion of kindhood; the data are recorded as common ground. **Addresses.** Statement: the entry on race, Stanford Encyclopedia of Philosophy, at its account of the consensus against racial naturalism and the three-way disagreement over the ontological status of race, read as the encyclopedia's page carries it verbatim, its author keyed at the next pass. The institutional statement at Accounts and Approaches: Fuentes et al., "AAPA Statement on Race and Racism", American Journal of Physical Anthropology 169(3):400-402, 2019. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the consensus against racial naturalism leaving the ontological status of a different conception disputed across three metaphysical camps and two normative camps; the skeptic's argument summarizing the scientific rejection of essences, geography, phenotypes, post-Mendelian transmission genetics and genealogies as possible foundations; a constructivist finding a perfectly clear referent in those socially ascribed to the categories; a deflationary view holding race talk overall too ambiguous and contested to be salvaged in the search for a dominant understanding), the entry itself not fetched, its author, section numbers and revision date not carried at the returns; the earlier archived editions carry the consensus as that such races simply do not exist where the later carry the consensus against racial naturalism. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the consensus against racial naturalism leaving the ontological status of a different conception open; the criterion of kindhood the point at issue), the data standing there as common ground as the file states it. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Apportionment results — measured, cited by both accounts. Clustering analyses — measured, with their sampling dependence recorded, cited by both accounts. Philosophical formulations — the minimal-realist and anti-realist positions standing at strength in the published debate. Institutional statements — the professional associations' statements standing in the record as field documents. **Narrowing recorded.** The minimal-realist position weakens the relation: a deflated kind-claim stands on statistical clusters rather than a kindhood criterion applied in full. **Progress.** The variation structure. The clinality. The clustering behaviour and its sampling dependence. **Own record.** ## 25 The language and dialect boundary **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What distinguishes a language from a dialect, and whether anything linguistic does. In the field's own statement, the relation in which a language stands above a dialect may be either linguistic or social, and the ambiguity of the two terms is why the number of languages spoken in the world cannot be stated precisely. **The field's account of its hardness.** The field's founding paper on the distinction states that the impossibility of saying precisely how many languages or dialects are spoken in the world is due to the ambiguities of meaning in the terms themselves, and that the relation of language to dialect may be either linguistic or social, the social side falling to sociolinguistics. **Accounts.** Linguistic criteria can ground the boundary: sufficient structural distance makes a variety a language regardless of standardization, in the founding framework's distance concept, stated with the claim that a linguist would have to call such a variety a language even unwritten, and pursued in the intelligibility-measurement literature. No linguistic criterion suffices or is likely to be found: the boundary is sociopolitical, autonomy, standardization, and institutional roofing deciding, as the field's aphorism has long carried and its critics of the distance concept argue directly. **Specifications.** Varieties are counted and named — universal under the field's surveys, catalogues, and policy instruments, all requiring a unit. The dialect continuum as the measured background against which any boundary is drawn — universal under dialectology. **Specifications withdrawn.** **Common requirement.** A criterion of language-hood applied to the measured variation. **Conserving relation.** A criterion cutting the continuum into languages, the cut standing on the measured variation. **Conserving reach.** All varieties counted and named. **Reach.** The statement quantifies over the boundary of the field's own counting unit. Intelligibility is measured, and measured as gradient and asymmetric; structural distance is measured as continua. The record carries standardized varieties mutually intelligible across the boundary, and unstandardized varieties structurally distant within it, sorting oppositely under the two criteria sets. The two are not the same, the measurements returning continua the cut is not fixed by. **Unreachability.** The measurements return continua and asymmetries the cut is not fixed by, and the same varieties sort oppositely under the two criteria sets, which is the field's instruments disagreeing on its own unit. The cut is not preserved across the varieties by the measured variation. **Persistence.** The record carries the split over whether a linguistic criterion is owed or the sociopolitical account suffices. **Addresses.** Statement and hardness: Haugen, "Dialect, Language, Nation", American Anthropologist 68(4):922-935, 1966, at its abstract as the publisher's page carries it verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a paper on intelligibility as a demarcation criterion at its abstract as a repository carries it (dominant notions entirely sociopolitical, disregarding structural and communicative aspects, a stance argued no longer tenable), 2021, its author not carried; a chapter of a history of the conceptual pair at its conclusion as the publisher's page carries it (there might after all be an empirical measure, further exploration required; in-between situations a transition zone), Oxford University Press, 2020, its author not carried; a computational-linguistics paper at its introduction as the journal's page carries it (intelligibility often not symmetrical, exposure determined by historical and sociological factors that should not affect a linguistic classification), Computational Linguistics 45(4), 2019. Approaches: the two-criteria framework at a concept page's statement of it (linguistic distance against sociopolitical elaboration, introduced to handle standardized varieties of one continuum); the intelligibility measurement at the two papers above. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The two-criteria framework — distinguishes distance-languages from elaboration-languages, making the boundary double rather than deciding it, with the opposing classifications of the same varieties standing in its own literature. Intelligibility measurement — quantifies comprehension across varieties, returning gradients and asymmetries, with the threshold a stipulation. The autonomy-heteronomy analysis — systematizes the sociopolitical reading, accounting for the standard cases and giving up the linguistic criterion, which is the account. The no-criteria argument — argues directly that final, reliable, uniform linguistic criteria do not exist and are unlikely to be discovered. Catalogue practice — the language catalogues drawing the boundaries operationally and revising them, the revisions standing as the practice's own record of the criterion's unsettledness. **Narrowing recorded.** The two-criteria framework weakens the relation: two criteria stand where one cut was required, the boundary doubled rather than decided. Intelligibility measurement weakens the relation: the cut stands at a stipulated threshold on a gradient. Catalogue practice narrows the reach: boundaries stand operationally within a catalogue and are revised. **Progress.** The gradient and asymmetric structure of intelligibility. The continuum structure of variation. The opposing classifications under the two frameworks, documented in the field's own literature. **Own record.** ## 26 Dark matter **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What constitutes the nonluminous mass whose gravitational effects are measured at galactic, cluster, and cosmological scales — or whether the measured effects instead indicate a modification of the gravitational dynamics at low accelerations. **The field's account of its hardness.** The assessment that the hypothesis stands on a century of accumulating and consistent gravitational evidence while the constitution stands unidentified. The experiments' own statements of the neutrino floor as the instrument class's boundary. The galactic-regularity literature's statement that the regularities constrain any account, particle or modified. **Accounts.** A new particle species beyond the standard model constitutes the component, its abundance standing measured at roughly a quarter of the cosmic energy density within the standing cosmological framework. The galactic-scale phenomenology follows a modified acceleration law without new matter, the tight regularities of galaxy dynamics, including the radial acceleration relation, standing as its evidence. **Specifications.** General relativity, with its Newtonian limit, as the gravitational law across scales — universal under physics. The standard model as the established particle inventory, containing no cold dark component — universal under particle physics. The concordance cosmology as the framework within which the component's abundance and coldness are inferred — universal under cosmology. **Specifications withdrawn.** **Common requirement.** An account of the measured mass discrepancies expressible in the framework, as constitution or as law. **Conserving relation.** An account of the discrepancy holding at every scale it is measured at. **Conserving reach.** All scales the discrepancy is measured at — galactic, cluster, and cosmological. **Reach.** The statement quantifies over the constitution of the discrepancy. Direct-detection experiments exclude candidate couplings down to the neutrino fog, the leading experiment recording the fog entered and observing the coherent solar-neutrino signal; collider and indirect searches bound production and annihilation channels; the cosmological instruments measure the component's abundance and coldness. The two are not the same. **Unreachability.** The instruments return different verdicts at different scales, a standing scale-split recorded as the field records it: the cluster-collision separations of lensing mass from baryonic mass standing as evidence for constitution, the galactic regularities standing as evidence for modification-like phenomenology. The account is not preserved across the scales by the measurements. **Persistence.** A laboratory detection consistent with the astrophysical abundance, settling the constitution. In the other direction no terminal null stands recorded: candidates are excluded one by one, the parameter spaces closing without closing the question, the asymmetry stated in the field's own assessments. **Addresses.** Statement: Bertone and Hooper, "History of dark matter", Reviews of Modern Physics 90:045002, 2018. The modified-dynamics formulation: Milgrom, Astrophysical Journal 270:365-370, 1983. Hardness: the line stands as it stood, unkeyed with the reason told: the review the address names, at its abstract as the publisher's page, an abstract service and an index carry it verbatim (dark matter a central element of modern cosmology, the history of its acceptance often ignored or condensed into an anecdotal account; the aim a broader historical perspective on the observational discoveries and theoretical arguments that led the community to adopt it as an essential part of the standard cosmological model), issue 4, published 15 October 2018, doi 10.1103/RevModPhys.90.045002, arXiv:1605.04909, the review itself not fetched, carries the history's aim and not the line's three clauses; the clauses, the century of consistent gravitational evidence with the constitution unidentified, the neutrino floor as the direct-detection class's boundary, and the galactic regularities constraining any account, stand keyed at the next pass at their own sources; the modified-dynamics formulation not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the community's white paper on direct detection to the neutrino fog at its section on the fog as the arXiv copy carries it (the neutrino floor a boundary below which discovery becomes extremely challenging without further constraints on the backgrounds, its location evolving; the backgrounds not imposing a hard limit, the effect gradual, hence the fog), arXiv:2203.08084; a light-dark-sector paper at its section on the observation of solar neutrinos (no direct-detection experiment having yet confirmed a dark matter event; the discovery criterion an excess over the predicted neutrino events), arXiv:2508.05787; a collider paper's opening (the presence of dark matter strongly implied by astrophysical observations, its particle nature remaining elusive), arXiv:2512.12457; their authors not carried. Approaches: the direct-detection route and its ending at the white paper; the axion, indirect and collider routes keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Direct detection of weakly interacting candidates — the exclusion regions reaching the neutrino fog, the fog standing entered, the instrument class recording its own floor. Axion searches — the haloscope program scanning the motivated mass ranges. Collider and indirect bounds — production and annihilation channels bounded. Astrophysical discrimination — the lensing-baryon separations recorded as evidence for constitution, the galactic regularities recorded as evidence for modification-like phenomenology, the small-scale challenges standing with baryonic-feedback resolutions recorded. Cosmological inference and relativistic completion — the abundance and coldness measured, the multimessenger constraint on the speed of gravity excluding a class of relativistic modification theories, the surviving completions' cosmological fits recorded as the modification program's standing difficulty. **Narrowing recorded.** The modification account narrows the reach: the acceleration law stands at galactic scales, its relativistic completion recorded as constrained. **Progress.** The discrepancy itself, measured across scales and a century of instruments. The component's abundance and coldness within the standing framework. The lensing-baryon separations. The exclusion regions to the neutrino fog. The galactic regularities. **Own record.** ## 27 Heritability and the determination of traits **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What a heritability estimate measures — the fraction of a trait's variance in a population attributable to genetic variance, in that population and its range of environments — and what it licenses: whether the partition bears on the causes of the trait in an individual, on its malleability, or on the sources of difference between populations. Variant formulation, as commonly posed: for a trait that varies across people, is the variation produced by genetic differences or by environmental differences? **The field's account of its hardness.** That heritability, an estimable dimensionless population parameter comparing the relative importance of genes and environment to the variation of traits within and across populations, is among the oldest parameters in genetics and among the most misunderstood, its use and application attended by continuous misunderstandings and controversies; that it is not constant in a population, changes in the method of measurement, environmental change and the effects of migration, selection and inbreeding all altering it; and that it is defined in each calculation for a particular population and environment, predicting nothing about changes in a mean over time or about differences between groups, which are the two forms of the misconception that it is informative about the nature of between-group differences. **Accounts.** Genetic differences dominate: twin and adoption designs return substantial heritability for nearly every measured trait, replicated across cohorts, decades, and countries, and molecular data recover polygenic signal in the same direction. The statement as posed has no answer: heritability is a ratio specific to a population and its range of environments, not a property of a trait. Change the range and the ratio changes, and environmental interventions produce large trait changes no genetic account predicts. **Specifications.** A trait's variation in a population decomposes into components attributable to sources — universal under quantitative genetics. A design holding one source fixed estimates the other's contribution — universal under the design literature. **Specifications withdrawn.** **Common requirement.** That the outcome is to be attributed to a determining source, and that a design holding one source fixed while apportioning variance to the other bears on that attribution. **Conserving relation.** An attribution of the variation to a source, the attribution standing across environments and genetic compositions. **Conserving reach.** All populations and all ranges of environments. **Reach.** The statement quantifies over an attribution invariant across environments and genetic compositions. Conditional quantities specific to the range sampled are what is measured. The two are not the same. **Unreachability.** A ratio is specific to the population and range sampled, and larger samples return a more precise conditional rather than an invariant. The attribution is not preserved across environments and genetic compositions by any design apportioning variance within a sampled range. **Persistence.** An attribution holding across environments and genetic compositions, or the adoption of the conditional reading as what the designs deliver. **Addresses.** Statement: Visscher, Hill and Wray, "Heritability in the genomics era — concepts and misconceptions", Nature Reviews Genetics 9:255-266, 2008. The analysis-of-variance critique: Lewontin, American Journal of Human Genetics 26:400-411, 1974. Hardness: the review, at its abstract as the publisher's page, an index, a university repository, an abstract service and a repository copy carry it verbatim (heritability an estimable, dimensionless population parameter comparing the relative importance of genes and environment to the variation of traits within and across populations; continuous misunderstandings and controversies over its use and application, the parameter remaining key to the response to selection and to the prediction of disease risk) and at passages of its text as the publisher's page and a repository copy carry them (one of the oldest parameters in genetics and one of the most misunderstood; not constant in a population, altered by changes in the method of measurement, environmental change, migration, selection and inbreeding; defined in each calculation for a particular population and environment, predicting nothing about changes in a mean over time or about differences between groups, the two forms of the between-group misconception), issue 4, doi 10.1038/nrg2322, the review itself not fetched; the critique not read this pass. Persistence, no landing: the closing condition at the review the address names, at its abstract read at pass two (heritability an estimable population parameter, a ratio for the sampled range and not a measure of a trait's determination), the conditional reading standing there as what the designs deliver. Approaches: the twin-sample and association routes at that abstract in part, their endings at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Larger twin samples — estimates stabilise, measuring more of what was reached, a more precise ratio for the sampled range. Genome-wide association at scale — recovers increasing but still partial variance, the gap itself a standing problem of the field. Adoption and migration designs — vary the environmental range and the estimates move, which is what the second account predicts. Gene-environment interaction models — estimate interaction terms, design-dependent and typically underpowered, which the field states. Within-family designs — remove population stratification and reduce estimated effects, showing part of the earlier estimate was not what it was taken to be. **Narrowing recorded.** The conditional reading narrows the reach: a ratio stands for a population and a range of environments rather than across all of them, and the design literature states the restriction. **Progress.** The replicated estimates across cohorts, decades, and countries. The polygenic signal. The within-family reductions. The measured movement of estimates with environmental range. **Own record.** ## 28 Simultaneity and the present **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is there a single present moment shared by everything, or is "now" only relative to an observer's motion? **The field's account of its hardness.** That the ordinary structure of time, absolute simultaneity slicing spacetime into planes completely ordered by the relation of being earlier than, is the backdrop against which presentism and possibilism suppose one plane of simultaneity uniquely metaphysically important, as all that exists or as the locus of becoming; and that in special relativity each inertial frame picks out a distinct set of points as simultaneous with the origin, the relativity of simultaneity following from the constancy of the measured speed of light for every such observer and the complete equivalence of inertial frames for the formulation of the laws of physics. **Accounts.** There is a present: change is the most evident feature of experience, and an account on which all times are equally real must reconstruct change as a relation among tenseless facts, which its critics take to be no reconstruction of change. There is no shared present: the relativity of simultaneity is established and precisely confirmed, a universal present requires a privileged slicing, and no measurement identifies one. **Specifications.** The laws are the same in every inertial frame — universal under relativity. No measurement identifies a preferred frame — universal under relativity. Simultaneity is defined by a signalling convention within a frame — universal under relativity. **Specifications withdrawn.** **Common requirement.** That the question is which description is correct, and that one of them is. **Conserving relation.** A slicing identified as the real one, the identification standing on measurement. **Conserving reach.** All frames. **Reach.** The statement quantifies over which slicing, if any, is the real one. Measurements taken in a frame are what is measured. The two are not the same. **Unreachability.** Neo-Lorentzian reconstruction reproduces every prediction of relativity with a privileged frame added, so the accounts are empirically equivalent. The identification is not reached over the frames by measurement. **Persistence.** A measurement identifying a slicing, which the empirical equivalence excludes, or the adoption of one description as the field's adopted reading. **Addresses.** Statement: Savitt, "Being and Becoming in Modern Physics", Stanford Encyclopedia of Philosophy. The argument from relativity of simultaneity as the field states it: Putnam, "Time and Physical Geometry", Journal of Philosophy 64:240-247, 1967. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (absolute simultaneity an equivalence relation slicing spacetime into planes completely ordered by earlier-than, the ordinary structure of time and the backdrop for the three rival metaphysics; presentism and possibilism supposing one plane of simultaneity uniquely metaphysically important, as all that exists or as the locus of becoming; each inertial frame in special relativity picking out a distinct set of points as simultaneous with the origin, the relativity of simultaneity following from the constancy of the measured speed of light and the equivalence of inertial frames), the entry itself not fetched, its section numbers and revision date not carried at the returns; the argument from relativity of simultaneity not read this pass. Persistence: re-formed at v342 to release a landing word used in another sense, the resolving file reading a landing at that word; the line stood as: A measurement identifying a slicing, which the empirical equivalence excludes, or the adoption of one description as the field's settled reading. Approaches: the preferred-frame and neo-Lorentzian routes at the same entry's sections on the responses to the argument from relativity, read at pass one and at the passages the hardness keys. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Looking for a preferred frame — the microwave background rest frame exists and is measurable, and is not dynamically privileged, nothing in the laws referring to it. Neo-Lorentzian reconstruction — reproduces every prediction of relativity with a privileged frame added. Presentism with a preferred foliation — consistent, and requiring a foliation no measurement identifies. Tenseless reconstruction of passage — accounts for the temporal structure of experience within a block, its critics holding that it accounts for the appearance rather than the passage. Quantum gravity — some programmes reintroduce a preferred slicing, none established. **Narrowing recorded.** Presentism with a preferred foliation weakens the relation: a slicing stands by stipulation rather than by measurement. **Progress.** The relativity of simultaneity, established and precisely confirmed. The microwave background rest frame, measurable and not dynamically privileged. The empirical equivalence of the neo-Lorentzian reconstruction. **Own record.** ## 29 Free will and determinism **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** If every event is fixed by prior events and physical law, can a person act otherwise than they do? In the field's own framing, free will is a kind of control over one's choices and actions: an action is up to the agent in two senses, that she is able to do otherwise and that she is the source of the action, and whether each sense is required, and in what kind of ability or sourcehood it consists, is the standing controversy. The field's argument that the freedom to do otherwise is incompatible with determinism, in plain words: if determinism holds, present acts are consequences of the laws of nature and of events in the remote past; neither the remote past nor the laws are up to anyone; so neither are their consequences, the present acts among them. **The field's account of its hardness.** The field's survey states that the ability to do otherwise comes in different varieties, so that a claim that an agent has it is ambiguous until the sense of ability is spelled out; that a satisfactory account owes both an analysis of the kind of ability and an argument that this kind is the one constitutive of the freedom to do otherwise, a debt the survey says is sometimes left unpaid; that the categorical analysis, and with it incompatibilism, stays attractive because no compatibilist analysis has yet been furnished on which phobics clearly lack the ability while some merely determined agents have it; and that disputes about free will involve disputes about metaphysics and about ethics at once. **Accounts.** No: a person is a physical system, and physical systems evolve by law from prior states. Indeterminacy does not help, a randomly produced action being no more the person's own. Persons act otherwise in the sense that matters: deliberation and reasons are features of how a system's own states produce its actions, unaffected by whether those states were produced. The statement conflates being caused with being compelled. **Specifications.** Every physical event is fixed by prior physical events and law, or is not — universal under physics. A person is a physical system to which the same description applies — universal under the framework. **Specifications withdrawn.** **Common requirement.** That the action has a determining source, and that locating it settles the question. **Conserving relation.** A determination of what would have happened otherwise, standing on what did happen. **Conserving reach.** All actions. **Reach.** The statement quantifies over what would have happened under conditions that did not obtain. What did happen is what is measured. The two are not the same. **Unreachability.** Timing experiments address an adjacent question and their interpretation stands revised; indeterminism supplies randomness, which both accounts reject as agency. The determination is not reached over the actions by what did happen. **Persistence.** The record carries the split over whether the ability in dispute is the one the compatibilist analysis defines. **Addresses.** Statement, hardness, persistence, and at Approaches the compatibilist analysis, the alternative-possibilities cases, the luck arguments and the timing experiments' contested interpretation: O'Connor and Franklin, "Free Will", Stanford Encyclopedia of Philosophy, first published 7 January 2002, substantive revision 3 November 2022, §§2.1, 2.2, 2.3, 3.1, read whole; the incompatibility argument's informal statement as the entry carries it: van Inwagen, An Essay on Free Will, 1983, p. 16; the review of the timing evidence as the entry names it: Schurger et al., 2021. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Neuroscience of decision — preparatory activity found preceding reported intention, the interpretation contested within the field, and later work showing the signal partly an artefact of averaging. Quantum indeterminacy — supplies randomness, which both accounts agree is not agency. Compatibilist analysis — redefines the ability in dispute, its critics holding that the redefinition changes the subject. Frankfurt cases — argue that alternative possibilities are not required for responsibility, widely discussed. Experimental philosophy — reports what people judge, which is not the statement. **Narrowing recorded.** The compatibilist analysis weakens the relation: the ability stands as a feature of how a system's own states produce actions rather than as a determination of what would have happened otherwise. **Progress.** The timing record and its revision. The mapping of which account rejects which premise in the Frankfurt cases. **Own record.** ## 30 Goodhart's law **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A statistical regularity used as a target for control ceases to hold. Why does it break exactly when it is relied on? The law's original formulation stands verbatim in part: an observed statistical regularity "will tend to collapse once pressure is placed upon it for control purposes"; its popular formulation: "when a measure becomes a target, it ceases to be a good measure". **The field's account of its hardness.** That the law's statement was, in the field's own review of it, a jocular aside rather than the main point of the paper in which it arose, and since it was an aside its interpretation is open to some questions; that it identifies the problem for policymakers as the unreliability of any statistical regularity that is subsequently used for control purposes and raises the question whether it dooms all statistical regularities to fail, a theorem which, if generally valid, would face central bank policy with a virtually insurmountable difficulty, and which its author is recorded as definitely not claiming to be generally valid; that its essential feature is the one it shares with the critique it is compared to and with the general problem of researchers' interventions in social science, the lack of invariance of the system to the observer's intervention, a relationship having the appearance of a regularity by its stability over a period and tending to break down when it ceases to be an ex post observation of related variables and becomes an ex ante rule for control purposes; and that a fair test has never been made, no one having compared the stability of money-demand functions in targeting and non-targeting countries, the argument being always circular because countries with the most stable relationships are more likely to choose targeting, and the stable functions found since arguably being stable only because no authority now puts pressure on them. **Accounts.** It was never a mechanism, only a correlation obtained where nobody optimised against it. Applying pressure changes those conditions, and the lesson is not to use correlations as targets. It was a mechanism, and pressure alters the mechanism. Those measured respond to being measured, and the response is part of the system. No care in selecting the measure avoids this, because the response is to being measured at all. **Specifications.** A statistical relation estimated over observed instances holds over instances not yet occurring — universal under statistical practice. Acting on the relation does not alter it — local. **Specifications withdrawn.** **Common requirement.** That a relation estimated over past instances can be asserted over instances not yet occurring, and acted on there. **Conserving relation.** A regularity holding under the pressure applied to it. **Conserving reach.** All instances occurring after the assertion. **Reach.** The statement quantifies over instances occurring after the assertion. Instances occurring before it are what is measured. The two are not the same. **Unreachability.** Each replacement measure is gamed in turn, the record across public administration, education, and health standing consistent, and the general mechanism-design case is bounded by the manipulability result. The regularity is not preserved over instances after the assertion by any measure selected before it. **Persistence.** A measure holding under pressure, or a mechanism-design scheme applying where the setting supplies the required knowledge of preferences. **Addresses.** Statement, the original formulation: Goodhart, "Problems of Monetary Management: The U.K. Experience", Papers in Monetary Economics, Volume 1, Reserve Bank of Australia, 1975, reprinted in Courakis, editor, Inflation, Depression, and Economic Policy in the West, Barnes and Noble, 1981, p. 116, as the field's citations carry the sentence verbatim and the venue and reprint, the paper's opening carried at an abstract service (a new approach to monetary management adopted in 1971, the subsequent experience of operating it troublesome and at times unhappy, the purpose to illustrate more general analytical themes), the paper itself not fetched; the popular formulation: Strathern, 1997, as the field attributes it. Hardness: filled at the review's own text, Chrystal and Mizen, "Goodhart's Law: Its Origins, Meaning and Implications for Monetary Policy", prepared for the Festschrift at the Bank of England and dated 12 November 2001, published in Central Banking, Monetary Theory and Practice: Essays in Honour of Charles Goodhart, volume 1, Edward Elgar, 2003, pp. 221-243, doi 10.4337/9781781950777.00022 as the repository carries it, fetched whole at an academic repository's copy, the one fetch of the v342 session, at its sections on what the law is and how it arose, on the law and the critique, and on the wider implications (the statement a jocular aside, the interpretation open to questions; the law a statement that the collapse will always happen when policymakers use statistically estimated relationships as the basis for policy rules; the problem identified as the unreliability of any regularity used for control, and the question whether all are doomed, a central banker's reflection quoted that a generally valid theorem would be a virtually insurmountable difficulty, its author not claiming general validity; the lack of invariance of the system to the observer's intervention central, shared with the critique and with the researcher's-intervention problem in social science; the ex post observation becoming an ex ante rule; no test known comparing targeting and non-targeting countries, the argument circular; stable functions found since perhaps stable only because no pressure is put on them). The original statement, the paper's own note records, stood in the second of two papers delivered in July 1975 to a conference at the Reserve Bank of Australia, reproduced in Papers in Monetary Economics, volume 1, under the titles Monetary Relationships: A View from Threadneedle Street and Problems of Monetary Management: The UK Experience. Two earlier searches at the review literature returned only a repository's summary and the field's near-statements, recorded at the ledger. Persistence, no landing: the closing condition at the review's own text fetched at v342 (the question whether the law dooms all statistical regularities used for control; a regularity tending to break down when it becomes an ex ante rule; no fair test ever made), a measure holding under pressure standing there as what the review finds untested; the mechanism-design alternative keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Picking a better measure — each replacement gamed in turn, the record consistent across sectors. Using multiple measures — distributes the response without removing it, composite indices gamed at the weights. Keeping the measure secret — works while secret, secrecy recorded as unstable in institutions and foreclosing accountability, which was the purpose. Measuring outcomes rather than proxies — outcomes also responded to, more slowly, delaying rather than removing. Mechanism design — incentive-compatible schemes existing for restricted settings and requiring knowledge of preferences the setting does not supply, the general case bounded. **Narrowing recorded.** Mechanism design narrows the reach: incentive-compatible schemes stand for restricted settings where preferences are known, and results follow within them. Secrecy narrows the reach: the regularity stands while the measure is unpublished. **Progress.** The consistency of the record across sectors. The bound on the general mechanism-design case. The measured gaming of composite indices at the weights. **Own record.** ## 31 The Lucas critique **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Relations estimated from economic data change when policy acts on them. Can any estimated relation be used for policy? The critique's own summary: the structure of an econometric model consists of the optimal decision rules of economic agents; those rules vary systematically with changes in the structure of the series relevant to the decision maker; so any change in policy systematically alters the structure of the model. **The field's account of its hardness.** That the procedures of econometric policy evaluation are faulted for failing to recognize that the optimal decision rules of economic agents vary systematically with changes in policy, the coefficients of relations estimated from past data being not invariant but changing along with the decision rules in response to a new policy; and that the issue in essence is whether an econometric model isolates invariants of the economic process, the parameters of policy rules entering the agents' optimization problems through their expectations of policy behaviour. **Accounts.** Not as estimated: the relations aggregate decisions made under the regime in force during estimation, and a new policy changes the decisions. Policy evaluation requires parameters invariant to policy, and estimated relations are not those. In practice, with care: some relations are stable over relevant horizons, and the programmes built to supply invariant parameters have not supplied them. A critique no available method satisfies is a counsel of perfection. **Specifications.** An economy is described by parameters estimable from data — universal under econometrics. Some of those parameters are invariant to the policy regime — local, and what policy evaluation requires. **Specifications withdrawn.** **Common requirement.** That there exist, or could exist, parameters describing the system that are invariant to the policies acting on it. **Conserving relation.** Parameters describing the system, standing invariant under the policies acting on it. **Conserving reach.** All policy regimes, those not yet enacted included. **Reach.** The statement quantifies over behaviour under policies not yet enacted. Behaviour under policies enacted is what is measured. The two are not the same. **Unreachability.** Natural experiments return the estimation problem one step out, and the microfoundations programme assumes the invariance it was to supply. The invariance is not reached over the regimes by estimation from enacted regimes. **Persistence.** Parameters demonstrated invariant across regimes rather than assumed so. **Addresses.** Statement: Lucas, "Econometric Policy Evaluation: A Critique", in The Phillips Curve and Labor Markets, Carnegie-Rochester Conference Series on Public Policy 1:19-46, 1976, p. 41, as the field's citations carry the sentence verbatim, the paper itself not fetched. Hardness: the field's dictionary entry on the critique, at its abstract as the publisher's page carries it verbatim (a criticism of policy evaluation procedures failing to recognize that optimal decision rules vary systematically with changes in policy; estimated relationships from past data faulted for forecasting a new policy's effects, the estimated coefficients not invariant but changing with the decision rules), and a chapter abstract as the publisher's page carries it (the issue in essence whether an econometric model isolates invariants of the economic process, the parameters of policy rules entering the agents' optimization through their expectations of policy behaviour), both returned in the search, neither fetched, their authors not carried at the returns; the sentence at p. 41 verified at both, and the volume's editors, Brunner and Meltzer, carried at the returns. Persistence, no landing: the closing condition at the dictionary entry read at pass two (optimal decision rules varying systematically with policy, any change in policy altering the structure of econometric models), invariance demonstrated rather than assumed standing there as what the critique demands. Approaches: the deep-parameter and natural-experiment routes at the sources the address names, read at pass one, and at that entry in part. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Deep parameters — microfoundations in preferences and technology, held policy-invariant, the invariance assumed rather than tested, estimates shifting across samples, which the field states. Natural experiments — identify effects of enacted changes, bearing on regimes that occurred. Structural vector autoregression — requires identifying restrictions themselves not estimated. Reduced-form forecasting — abandons policy evaluation and forecasts, performing adequately until a regime change. Robustness to model uncertainty — designs policies performing acceptably across a set of models, requiring the set to be specified. **Narrowing recorded.** Reduced-form forecasting narrows the reach: the relation stands within a regime rather than across regimes. Robustness to model uncertainty weakens the relation: acceptable performance across a specified set stands in place of invariance. **Progress.** The measured shifts of deep-parameter estimates across samples. The identified effects of enacted changes. **Own record.** ## 32 The replication crisis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A large fraction of published findings fail to reproduce when the study is repeated. Are the original findings wrong? In the field's own posing, reproducibility is a defining feature of science, and the extent to which it characterizes current research is unknown. **The field's account of its hardness.** The field's largest replication project reports its result under several criteria at once — whether the replication reached significance, whether the original effect lay within the replication's confidence interval, whether the effect was rated as replicated, and what the original and replication results give combined — and the criteria return different rates for the same hundred studies, the combined rate depending on whether bias in the originals is assumed. **Accounts.** Many are: publication incentives, analytic flexibility, and small samples enrich the literature in false positives, and direct replication is the corrective. Failure rates are measured and mechanisms identified. Failure to replicate does not establish error: effects are conditional on populations, settings, and procedures that differ between original and replication, and a null replication is one study with its own error rate. **Specifications.** A finding obtained in one sample estimates a population effect — universal under statistical inference. An independent sample from the same population estimates the same effect — universal under statistical inference. **Specifications withdrawn.** **Common requirement.** That a finding obtained in one study is assertable at studies not yet run, and that the second bears on the first. **Conserving relation.** A finding holding at studies not yet run, the holding standing independently of the setting sampled. **Conserving reach.** All studies, those not yet run included. **Reach.** The statement quantifies over studies not run. Each study's own sample is what is measured. The two are not the same. **Unreachability.** Replication returns more studies, each with its own error rate, and the interpretation of a failed replication is where the field's instruments disagree on the same finding. The holding is not reached over studies not run by any study run. **Persistence.** Agreement on what a failed replication establishes about the original finding. **Addresses.** Statement, hardness, and the measured failure rates at Progress: Open Science Collaboration, "Estimating the reproducibility of psychological science", Science 349(6251):aac4716, 2015, at its abstract as the journal's page carries it verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the field's review at its abstract as a tracking service carries it verbatim (replication an important, uncommon and misunderstood practice; systematic multi-site projects observing surprising failures; replication framed as a personal attack rather than a healthy practice), Annual Review of Psychology 73:719-748, 2022, doi 10.1146/annurev-psych-020821-114157, its authors not carried; a commentary at its opening as an archive copy carries it (a failure to replicate simply meaning similar results were not found, one reason being that the original is correct and the replication flawed), eNeuro, 2016, doi 10.1523/ENEURO.0072-16.2016, its author carried; the reproducibility project's conclusion as an archive copy carries it (weaker evidence despite the original authors' materials and advance review; the strength of the initial evidence more predictive of success than the replicating team), Science, 2015; the project leader's statement at a news report (failing to replicate not meaning the original was wrong or flawed). Approaches: the multi-site route at the project and the review; preregistration at the review's practices section as the abstract summarizes them. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Large multi-site replication — reproducibility projects returning rates well below expectation, the interpretation of a failed replication remaining the disputed point. Preregistration — removes analytic flexibility, preregistered studies reporting lower effect sizes, which is evidence about the mechanism, and does not settle the status of the earlier literature. Raising thresholds — reduces false positives and raises false negatives by a computable amount, a known trade. Meta-analysis — pools studies and inherits the publication bias it corrects for by models that are themselves assumptions. Many-analysts studies — the same data to independent teams returning divergent results, quantifying analytic flexibility without adjudicating any single finding. **Narrowing recorded.** The conditional reading narrows the reach: an effect stands for the population, setting, and procedure sampled rather than across studies not yet run. Raising thresholds weakens the relation: the holding stands at a stipulated error rate, at a computable cost in false negatives. **Progress.** The measured failure rates. The identified mechanisms. The lower effect sizes under preregistration. The quantified analytic flexibility. **Own record.** ## 33 Levinthal's paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** The protein-folding problem, as the field's own review poses it, is three questions: by what physical code a sequence dictates its native structure; how a polypeptide chain reaches that structure in seconds, when the conformations available to it exceed what could be sampled in the age of the universe — Levinthal's paradox; and whether an algorithm can predict the structure from the sequence. **The field's account of its hardness.** That the problem is three questions rather than one, by the field's fiftieth-year review: the physical code by which a sequence dictates its native structure, how proteins fold so fast, and whether an algorithm can predict structure from sequence, with the review's own journal stating that it is no longer useful to talk about solving the protein-folding problem, the problem having grown into a field of research. **Accounts.** Folding is not a search: random or exhaustive sampling would require times exceeding the age of the universe by many orders of magnitude, and the arithmetic is not disputed. Folding is a search on a structured landscape: funnelled energy landscapes, local structure formation, and nucleation reduce the accessible volume enormously, and the question becomes an empirical one about landscape shape. **Specifications.** A protein's conformations form a space defined by its degrees of freedom — universal under structural biology. The native state is the global minimum of a free energy defined over that space — universal under the thermodynamic hypothesis. **Specifications withdrawn.** **Common requirement.** That there is a conformational space, enumerable in principle, over which folding is defined, the dispute concerning the measure on it. **Conserving relation.** The native state determined from a sequence, the determination standing in advance of folding it. **Conserving reach.** All sequences, orphan sequences and novel folds included. **Reach.** The statement quantifies over the native state of an arbitrary sequence, in advance of folding it. Solved structures and simulated trajectories are what is measured. The two are not the same. **Unreachability.** Simulation reaches further timescales without reaching general sequences; landscape theory accounts for how fast folding is possible and does not supply the native state for a given sequence, by its authors' statement; learned prediction is refinement within the observed domain, disputed as such. The determination is not reached over all sequences by any method drawing on solved structures. **Persistence.** The native state determined for an arbitrary sequence, orphan sequences and novel folds included. **Addresses.** Statement, hardness and the three questions: Dill and MacCallum, "The Protein-Folding Problem, 50 Years On", Science 338(6110):1042-1046, 2012, abstract and the issue's editorial summary. Deep learning from the structure database and the dispute over what it reaches: Jumper et al., "Highly accurate protein structure prediction with AlphaFold", Nature 596:583-589, 2021; the 2024 Nobel Prize in Chemistry citation. Co-translational folding, the chain folding as it is made: reviewed at Cassaignau, Cabrita and Christodoulou, Annual Review of Biochemistry 89:389-415, 2020. As read at v342: the review at its abstract as the publisher's pages, an abstract service, two indexes and a research-project page carry it verbatim (the problem first posed about half a century before, the term referring to three broad questions, the physical code by which a sequence dictates the native structure, how proteins fold so fast, and whether an algorithm can predict structures from sequences; simulations achieving the accurate folding of small proteins in a few cases; folding rapid because random thermal motions lead energetically downhill, captured in funnel-shaped landscapes; prediction far more successful than thought possible) and at the issue's editorial summary as the publisher's page and an index carry it (no longer useful to talk about solving the protein-folding problem, the field grown into one where the next fifty years promise to be as exciting as the last), Science 338(6110):1042-1046, 23 November 2012, doi 10.1126/science.1219021, the review itself not fetched; the other sources not searched this pass. Approaches: the simulation, landscape and prediction routes and their endings at the review's abstract read at v342 (simulations achieving accurate folding of small proteins in a few cases; folding rapid on funnel-shaped landscapes; prediction far more successful than thought possible). Remaining properties: keyed at the next pass. **Which of the three the properties key to.** Reach, unreachability and persistence key to the third question — whether an algorithm returns the native state for an arbitrary sequence. The second question, how folding is so fast, is answered at the resolving (2.3) and is not what the properties quantify over. The first, the physical code by which a sequence dictates its structure, is addressed at no property here and stands unworked in this entry. *The report.* **Approaches.** Molecular dynamics from first principles — reaches microseconds to milliseconds for small proteins with special-purpose hardware, general sequences remaining out of range. Energy landscape theory — accounts for how fast folding is possible, its authors stating it does not supply the native state for a given sequence. Homology modelling — predicts structures from solved relatives, refinement within the observed domain by construction. Deep learning from the structure database — predicts many structures at experimental accuracy, obtained by learning correlations across solved structures, whether it constitutes a solution or a very good interpolation disputed within the field, its failures on orphan sequences and novel folds standing as the evidence in that dispute. De novo design — designs sequences that fold as intended, the inverse problem, which succeeds, the field distinguishing the two directions. Deep learning past the structure database — diffusion architectures extending prediction to complexes with ligands, nucleic acids and other proteins, and the hardest ligand cases at low similarity to training data remaining the reported edge; the failures on orphan sequences and novel folds standing where they stood. **What the approaches leave untouched.** Every one of them returns a single structure. The measurement is an ensemble: many models fit the same data equally, and the disordered proteins fold to no single native state at all. The single-native-state assumption the statement inherits from the thermodynamic hypothesis is carried by every approach listed and by the statement itself. **Narrowing recorded.** Homology modelling narrows the reach: the determination stands for sequences with solved relatives. Molecular dynamics narrows the reach: the determination stands for small proteins at reachable timescales. **Progress.** The arithmetic, undisputed. Structures predicted at experimental accuracy across the observed domain. The measured timescales reached by simulation. The success of the inverse direction. **Own record.** ## 34 Aging **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Do organisms age because damage accumulates, or because aging is produced by the organism's own regulation? In the field's own posing: whether ageing is programmed in the strict sense in which development is, a specific sequence of events under direct genetic control and so a product of positive selection for a controlled termination of life, or arises non-adaptively and has to be explained indirectly; against the review that poses it so, a reply holding ageing to be a particular case of programmed organismal death. **The field's account of its hardness.** The field's review states that compelling arguments eliminate the idea that death is generally programmed by genes for ageing, while a widespread tendency remains to interpret data in terms of loosely defined age regulation that usually makes neither evolutionary nor mechanistic sense; that the non-adaptive theories therefore have to explain the evolution of ageing indirectly; and that in exceptional circumstances the possibility exists for selection to favour limiting survival. **Accounts.** Accumulation: damage occurs continuously, repair is imperfect, and the failure to repair is what aging is. Selection weakens with age, so no programme is needed and none would be maintained. Regulation: aging shows species-typical timing, regulated pathways alter lifespan reproducibly across taxa, and single-gene interventions produce large coordinated changes. **Specifications.** Aging is a change in an organism's state over time with a determinable cause — local. Damage and regulation are separable sources of that change — local. **Specifications withdrawn.** **Common requirement.** That aging has a determining source, and that identifying it as damage or as regulation settles the question. **Conserving relation.** One of two coupled processes identified as driving the other, the identification standing on measurement. **Conserving reach.** All organisms measured. **Reach.** The statement quantifies over which of two coupled processes drives the other. The two together, in every organism measured, are what is measured. The two are not the same. **Unreachability.** Interventions establish modulation of rate rather than the driver, and the epigenetic clocks restate the dispute in a new instrument, whether the clock reads damage or a programme being the dispute itself. The identification is not reached over the organisms by measurements returning the two together. **Persistence.** A measurement separating the two coupled processes, or agreement that modulation of rate is what the question asks. **Addresses.** Statement and hardness: Kirkwood and Melov, "On the programmed/non-programmed nature of ageing within the life history", Current Biology 21(18):R701-R707, 2011, at its abstract as the journal's page carries it verbatim, the paper itself not fetched; the reply holding the other side: Skulachev, "Aging as a particular case of phenoptosis, the programmed death of an organism", Aging 3(11):1120-1123, 2011, at its opening as the journal's page carries it. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the critique of the hallmarks paradigm at passages as two university copies carry them (advances in the genetics of lifespan not matched by advances in understanding the causes and mechanisms; a field in transition; the hallmarks account insinuating rather than stating damage as primary cause), Ageing Research Reviews 70:101407, 2021, its authors carried; a review at its text as the publisher's page carries it (the framework largely descriptive, not answering why aging occurs; error-based and programmatic models the two seemingly opposing models, not necessarily exclusive if the rate is modulated by regulatory networks), Ageing Research Reviews, 2026; a review at its opening as the journal's page carries it (no definitive consensus on which of the two points of view is right), Frontiers in Aging, 2024; their authors not carried except as given. Approaches: the lifespan-mutant and caloric-restriction routes at the critique's passage on the longevity pathways and drugs, in part, their endings at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Single-gene lifespan mutants — extend lifespan severalfold in nematodes, less in mice, showing regulation modulates rate, which both accounts accept and read differently. Caloric restriction — extends lifespan across taxa, primate results mixed and diet-dependent. Damage-clearance interventions — senolytics and repair-directed approaches producing measurable improvements in some tissues, bearing on rate rather than the driver. Comparative biology — negligible-senescence species existing, showing aging is not compulsory without parting the two accounts. Epigenetic clocks — predict age accurately from methylation, whether the clock reads damage or a programme being the dispute restated in a new instrument. **Narrowing recorded.** **Progress.** The lifespan extensions and their taxon dependence. The measured tissue improvements. The negligible-senescence species. The predictive accuracy of the clocks. **Own record.** ## 35 The glass transition **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A cooling liquid becomes rigid over a narrow range with no change in its structure. Is there a phase transition there? **The field's account of its hardness.** That supercooled liquids fall out of equilibrium well before the putative transition is reached, whether that transition is dynamic or thermodynamic, so the regime the question concerns is not equilibrated in any experiment. The field's reviews record that the leading thermodynamic account reproduces the salient experimental facts while direct and indisputable experimental validation is missing. **Accounts.** Yes, an underlying one: relaxation times grow by fourteen orders of magnitude over a modest temperature range, and theories positing an ideal transition below the accessible range fit that growth. No: nothing identifying a phase transition is observed, no diverging static correlation length and no thermodynamic discontinuity, and the glass temperature depends on cooling rate, which a genuine transition would not. **Specifications.** Phases are distinguished by order parameters and thermodynamic discontinuities — universal under statistical mechanics. A transition is identified by those signatures — universal under statistical mechanics. Measurements are taken on equilibrated systems — universal under thermodynamics. **Specifications withdrawn.** **Common requirement.** That rigidity is either a phase transition or it is not, and that the structural signatures determine which. **Conserving relation.** A transition identified by its signatures, the identification standing on equilibrated measurement. **Conserving reach.** All temperatures the question concerns, those where equilibration exceeds any experiment's duration included. **Reach.** The statement quantifies over behaviour at temperatures where equilibration exceeds any experiment's duration. Equilibrated states above that range are what is measured. The two are not the same. **Unreachability.** Equilibration time grows faster than any experiment can be extended, so slower cooling is bounded by construction, and two theories fit the accessible range and diverge only outside it. The identification is not reached over the temperatures concerned by equilibrated measurement. **Persistence.** A signature observed at a temperature the theories diverge at, which equilibration bounds by construction. **Addresses.** Statement, its first sentence in substance: a later perspective on the glass transition at its abstract as an index carries it (the dynamics of glass-forming liquids dramatically slowing down upon cooling toward the glass transition temperature without any obvious significant change in their static structure), its authors not confirmed at the return; the field's review, Berthier and Biroli, "Theoretical perspective on the glass transition and amorphous materials", Reviews of Modern Physics 83:587-645, 2011, doi 10.1103/RevModPhys.83.587, arXiv:1011.2578, at its abstract as the publisher's page, the arXiv page and two indexes carry it verbatim (the glass transition in molecular liquids at thermal equilibrium; the main theoretical approaches presented, their successes and failures critically assessed), carrying the field and not the posing's question, which stands keyed at the next pass at the review's introduction; neither fetched; the Statement standing as it stood. Hardness, and the out-of-equilibrium statement: the colloidal random-pinning literature, arXiv:1406.6478, introduction. Thermodynamic account's standing and its dynamical extension: Biroli and Bouchaud, "The RFOT Theory of Glasses: Recent Progress and Open Issues", arXiv:2208.05866, abstract and §1. Facilitation as a purely dynamical account: the same, and the dynamic facilitation reviews, arXiv:2003.03700. Mode-coupling range and its relation to facilitation: the colloid review, arXiv:1608.08924, §3. Structural approaches and point-to-set: the hard-sphere review, arXiv:2305.02452, §A. Persistence, no landing: the closing condition in part at the review's abstract read at v342 (the main theoretical approaches presented and their successes and failures critically assessed, the transition studied in liquids at thermal equilibrium), the equilibration bound on any observed signature keyed at the next pass at the review's text. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Cooling more slowly — equilibration time growing faster than any experiment can be extended. Mode-coupling theory — predicts a dynamical transition at nonzero temperature, recovered as a crossover beyond mean field, its range of validity recorded as extending from the onset of glassy dynamics to that crossover. Random first-order transition theory — an ideal transition below the accessible range, its static aspects broadly supported by recent investigation, its standard dynamical extension recorded by its own reviewers as struggling in relation to facilitation. Dynamical facilitation and kinetically constrained models — the slowdown reproduced with no transition at any temperature and thermodynamics assigned no role, the approach rooted in finite dimensions, and the colloid record reporting facilitation growing with density and with the fraction of pinned particles. Structural and geometric approaches — locally favoured structures inhibiting crystallisation, a distinct mechanism for arrest. Point-to-set correlations — a static length measured by the effect of a frozen disordered boundary, one of the instruments by which a growing static length is sought. Search for a growing correlation length — dynamic heterogeneity measured and growing modestly, no diverging static length found. Random pinning — freezing a subset of particles as an alternative route to the transition, with simulation supporting the thermodynamic account's predictions and the dynamical account recorded as less explored. Vapour-deposited ultrastable glasses — states equivalent to millennia of cooling, above the proposed ideal temperature. Computational equilibration at long effective timescales — modern methods reaching deeper into the supercooled regime than direct simulation. **Narrowing recorded.** Vapour deposition narrows the reach: states equivalent to millennia of cooling are reached, above the proposed ideal temperature, and results follow within that range. **Progress.** The fourteen orders of magnitude of relaxation growth. The dynamic heterogeneity, measured. The ultrastable states reached. The absence of a diverging static length in the accessible range. **Own record.** ## 36 Efficient markets **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Do asset prices reflect all available information? The hypothesis in the field's own statement, verbatim: "security prices fully reflect all available information"; its weaker form, that prices reflect information to the point where the marginal benefit of acting on information does not exceed the marginal cost. **The field's account of its hardness.** The field's own review states that market efficiency is not testable on its own: it must be tested jointly with a model of equilibrium expected returns, so anomalous evidence on returns cannot be split unambiguously between market inefficiency and a bad model of equilibrium, and precise inferences about the degree of efficiency are likely to remain impossible. **Accounts.** Substantially yes: prices respond to news within moments, active managers do not outperform indexes net of costs over sustained periods, and anomalies shrink after publication. No: persistent deviations — momentum, value, volatility clustering, bubbles with identifiable dynamics — exceed what risk adjustment explains, and the behavioural mechanisms producing them are demonstrated experimentally. **Specifications.** An asset's price reflects expected future payoffs discounted by a model of required return — universal under asset pricing. Tests of price behaviour are conducted against such a model — universal under asset pricing. **Specifications withdrawn.** **Common requirement.** That prices either do or do not incorporate information, and that returns data can distinguish these. **Conserving relation.** Efficiency established alone, separated from the model of expected returns it is tested with. **Conserving reach.** All tests of price behaviour. **Reach.** The statement quantifies over efficiency alone. The conjunction of efficiency and a model of expected returns is what is measured. The two are not the same. **Unreachability.** Any rejection is a rejection of the joint hypothesis, which the field names and does not dispute, and each anomaly can be absorbed by a factor, the factor zoo documented in the field's own literature as the consequence. Efficiency is not separated from the return model by any test of returns data. **Persistence.** A test bearing on efficiency alone, which the joint hypothesis excludes. **Addresses.** Statement, hardness, reach and persistence: Fama, "Efficient Capital Markets: II", Journal of Finance 46(5):1575-1617, 1991, §I, read at its opening as the publisher's and repositories' pages carry it verbatim, the paper itself not fetched; the 1970 definition: Fama, "Efficient Capital Markets: A Review of Theory and Empirical Work", Journal of Finance 25(2), 1970, as the 1991 review cites it. Approaches: the anomaly-testing route at the review's opening read at pass one (any test of efficiency a test of the joint hypothesis with a model of expected returns); the risk-factor route and the factor zoo keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Testing an anomaly — any rejection being a rejection of the joint hypothesis, which the field names and does not dispute. Adding risk factors — each anomaly absorbable by a factor, the factor zoo documented as the consequence. Event studies — measure response speed to news, which both accounts accept and read differently. Experimental markets — produce bubbles reliably under controlled conditions, disputed as to transfer. Post-publication decay of anomalies — consistent with efficiency arriving and with data mining, both readings fitting. **Narrowing recorded.** Adding risk factors weakens the relation: efficiency stands relative to a stated factor model rather than alone, and results follow within the model. **Progress.** The measured response speed to news. The net-of-costs performance record. The documented factor proliferation. The reliably produced laboratory bubbles. **Own record.** ## 37 The tragedy of the commons **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** When a resource is open to all and each user's gain is private while the cost is shared, is depletion inevitable? The essay's own statement, verbatim: "Freedom in a commons brings ruin to all" — each user, pursuing his own best interest in a society that believes in the freedom of the commons, is locked into a system that compels him to increase his use without limit in a world that is limited. **The field's account of its hardness.** The essay places the problem in the class it names problems with no technical solution: the ruin follows from each user's rational choice under the logic of the commons, and the remedy the essay offers is coercion agreed on by the affected majority. The journal's own fiftieth-anniversary record states that later research has documented contexts, cases and principles by which groups govern common resources collectively, challenging the essay's emphasis on property rights and regulation. **Accounts.** Yes, absent assigned rights or an external authority: each user's individually rational choice is to take more, the aggregate exceeds renewal, and depletion follows from the payoff structure. No: long-lived commons are documented, governed neither by private property nor external authority, using locally devised rules with monitoring and graduated sanctions. The outcome depends on institutional arrangements the payoff model omits. **Specifications.** Each user acts to maximise its own return — universal under the payoff model. The resource has a renewal rate — local. The outcome follows from the payoff structure facing individual users — local. **Specifications withdrawn.** **Common requirement.** That the outcome is determined by the payoffs facing individual users, so that arrangements either alter the payoffs or are irrelevant. **Conserving relation.** An outcome determined by the payoff structure, the determination standing across arrangements. **Conserving reach.** All arrangements. **Reach.** The statement quantifies over outcomes across all arrangements. Cases, each with one arrangement, are what is measured. The two are not the same. **Unreachability.** The folk theorems establish that cooperation is sustainable in equilibrium and equally that defection is, the multiplicity being the result; and design principles extracted from surviving commons face a selection objection. The determination is not preserved across arrangements by cases each carrying one. **Persistence.** A determination of the outcome from the payoff structure holding across arrangements, or agreement that arrangements are what determines it. **Addresses.** Statement and hardness: Hardin, "The Tragedy of the Commons", Science 162(3859):1243-1248, 1968, at p. 1244, as the field's pages carry the passage verbatim, the paper itself not fetched; the field's record of the later work, at hardness and the field studies at Approaches: "Tragedy revisited", Science, 2018, doi:10.1126/science.aaw0911. Persistence, no landing: the closing condition at the founding paper's passage read at pass one (the outcome following from the payoff structure under freedom in a commons) and at the field's revisiting collection the address names at Approaches (the arrangements the field studies record as determining outcomes), the two standing as the two readings the line states. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Formal modelling — yields depletion under the stated payoffs, by construction. Repeated-game folk theorems — establish that cooperation is sustainable in equilibrium, and equally that defection is, the multiplicity being the result. Field studies — durable self-governance established, with design principles extracted, whether the principles are predictive or descriptive of survivors disputed as a selection question. Laboratory commons games — communication and sanctioning raising cooperation substantially, transfer to field settings contested. Privatisation and state control — both working in some cases and failing in others, which is the field's own summary. **Narrowing recorded.** Design principles narrow the reach: the account stands over the documented long-lived cases rather than across all arrangements, and the selection objection stands against the extension. **Progress.** The documented long-lived commons. The design principles, extracted. The measured effect of communication and sanctioning. The record of privatisation and state control working and failing. **Own record.** ## 38 The arrow of time **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** The laws of physics are symmetric in time. The world is not. Why? **The field's account of its hardness.** That without some new physics that eliminates or explains the Past Hypothesis, the posit of a special low-entropy initial condition, or some satisfactory third way, what is left is a bald posit of special initial conditions, the probability of the requisite state under the standard measure on phase space estimated as astronomically small, while one reading assimilates the posit to the conspiratorial initial conditions every special-science law requires and whether the posit is itself needy of explanation is disputed; that a critic finds the posit not coherent enough to be false, its main problem that it cannot be stated in the language of general relativity, with the related question of stating it when self-gravitation is included; and that a persistent subsystem worry asks how the global entropy increase of the universe implies local entropy increase among the subsystems, which are what caused the second law to be posited in the first place. **Accounts.** The past was low in entropy: the asymmetry is in the initial condition, not the laws, and everything asymmetric follows from it plus statistics. Granting the initial condition is the problem: a state of that specificity is overwhelmingly improbable under any standard measure, and naming it an initial condition relocates the puzzle. **Specifications.** The fundamental laws are time-symmetric — universal under physics. Thermodynamic behaviour follows from those laws together with a specification of the boundary conditions — universal under statistical mechanics. **Specifications withdrawn.** **Common requirement.** That the asymmetry lies either in the laws or in the boundary condition, and that assigning it settles the matter. **Conserving relation.** The asymmetry accounted for, the account standing without stipulating the condition it accounts for. **Conserving reach.** The initial condition of the universe. **Reach.** The statement quantifies over the initial condition of the universe. Its consequences are what is measured. The two are not the same. **Unreachability.** Statistical mechanics derives the second law given the past hypothesis, requiring what is in question, and the cosmological proposals each contain a stipulation at the same place. The account is not reached over the initial condition by measurements of its consequences. **Persistence.** An account of the initial condition not itself stipulating the specificity in question. **Addresses.** Statement: Callender, "Thermodynamic Asymmetry in Time", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (without new physics eliminating or explaining the Past Hypothesis, or a satisfactory third way, a bald posit of special initial conditions left; the probability of the requisite initial state under the standard phase-space measure estimated astronomically small, against a reading assimilating the posit to the conspiratorial initial conditions every special-science law requires, and the dispute whether the posit is itself needy of explanation; a critic finding the posit not coherent enough to be false, unstatable in the language of general relativity, with the related question of stating it under self-gravitation; the subsystem worry, how global entropy increase implies local increase among the subsystems), the entry itself not fetched, its section numbers and revision date not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (without new physics eliminating or explaining the past hypothesis, a bald posit; whether the posit is needy of explanation disputed; the hypothesis not statable in the relativistic theory). Approaches: the statistical-mechanical, inflationary and other routes at the same entry's sections on the past hypothesis and its explanation, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Statistical mechanics — derives the second law given the past hypothesis. Inflation — accounts for homogeneity, whether it accounts for low entropy disputed within cosmology, the inflationary state itself being special. Measured symmetry violation in weak decays — real and measured, too small and too specific to produce the thermodynamic arrow, which the field states. Cosmological accounts of the initial state — proposals standing, none established, each containing a stipulation at the same place. Two-time boundary conditions — consistent constructions existing, not favoured by evidence. **Narrowing recorded.** **Progress.** The measured symmetry violation and its magnitude. The derivation of the second law given the past hypothesis. The homogeneity accounted for by inflation. **Own record.** ## 39 The frame problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** When something changes, most other things do not. How can a system determine what is relevant to update without checking everything? In the field's own two statements: for logic-based artificial intelligence, how to represent the effects of an action in logic without explicitly representing the large number of obvious non-effects; for philosophy, whether it is possible in principle to limit the scope of the reasoning required to derive the consequences of an action. **The field's account of its hardness.** The field's own survey states that in a domain of M actions and N properties nearly M × N frame axioms would be needed, an unwelcome burden whether stored in a computer's memory or merely part of the designer's specification; that a full solution must work in the presence of concurrent actions, actions with non-deterministic effects, continuous change, and actions with indirect ramifications; and that solutions adequate for logic-based artificial intelligence now exist for the technical problem, the wider epistemological question standing apart from them. **Accounts.** By explicit machinery: default assumptions, persistence axioms, and circumscription formalise what stays the same unless stated otherwise, and working systems use them. The technical answers address the narrow logical version and leave the general one: relevance depends on the whole context including what was not anticipated, and no enumeration of defaults reaches this. **Specifications.** A system's knowledge is a set of represented propositions — universal under the representational framework. Action changes some and leaves others — universal under the framework. What changes must be derivable from a specification made in advance — local. **Specifications withdrawn.** **Common requirement.** That relevance can be settled by a procedure specifiable in advance of the situation it applies to. **Conserving relation.** Relevance settled by a procedure, the settling standing for situations not anticipated when the procedure was specified. **Conserving reach.** All situations, those not anticipated included. **Reach.** The statement quantifies over situations not anticipated. Specifications made in advance are what is measured. The two are not the same. **Unreachability.** The formalisms solve the logical and representational versions, distinguished by their authors from the general one, and learned models generalise within the distribution sampled, out-of-distribution behaviour standing as the open case the field measures. The settling is not reached over unanticipated situations by any procedure specified in advance. **Persistence.** Relevance determined for situations not anticipated at specification, or agreement that the specified versions are what the question asks. **Addresses.** Statement, hardness, and the working systems at Accounts: "The Frame Problem", Stanford Encyclopedia of Philosophy, first published 23 February 2004, at its opening and its account of the technical problem, read as the encyclopedia's pages carry them verbatim; its author and revision date keyed at the next pass. Persistence: re-formed at v342 to release a landing word used in another sense, the resolving file reading a landing at that word; the line stood as: Relevance settled for situations not anticipated at specification, or agreement that the specified versions are what the question asks. Approaches: the circumscription, default-logic and successor-state routes at the same entry's account of the technical problem read at pass one, the entry distinguishing the logical formulation from the general problem. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Circumscription and default logics — solve the logical formulation, their authors distinguishing this from the general problem. Successor state axioms — solve the representational version efficiently, for a specified set of fluents. Probabilistic state estimation — handles uncertainty over a specified state space, the space being specified. Learned world models — acquire relevance from data and generalise within the distribution sampled, out-of-distribution behaviour standing as the open case, which the field measures and does not solve. Embodied approaches — argue relevance is fixed by the task and the body rather than represented, reframing rather than settling, as its proponents accept. **Narrowing recorded.** Successor state axioms narrow the reach: the settling stands for a specified set of fluents. Probabilistic state estimation narrows the reach: the settling stands over a specified state space. Learned world models narrow the reach: the settling stands within the distribution sampled. **Progress.** The logical and representational versions, solved. The measured out-of-distribution behaviour of learned models. **Own record.** ## 40 Simpson's paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** An association between two variables in a population can emerge, disappear or reverse when the population is divided into subpopulations. Which association is the one to act on, the whole population's or the subpopulations', and what decides it: the probabilities alone, or the causal structure that produced them? The field's review poses the problem as three questions: in what sense it is a paradox, what its proper analysis is, and how to proceed when confronted with a case of it. **The field's account of its hardness.** The field's review states that cases of the paradox are unproblematic from the standpoint of mathematics and probability theory yet strike many as surprising; that whether to partition a population cannot be decided from the probabilities alone, since the same data arise under different causal structures; and that whether the causal analysis is the correct explanation of the paradox remains a topic of continued debate. **Accounts.** The subgroup reading: the aggregate mixes populations differing in ways relevant to the outcome, and conditioning on the relevant variable gives the effect. Neither, in general: which reading answers a question depends on the causal structure, and identical numbers support opposite conclusions under different structures. **Specifications.** An association between variables is estimated from data — universal under statistical practice. A correct estimate requires conditioning on the variables relevant to the outcome — universal under statistical practice. **Specifications withdrawn.** **Common requirement.** That there is a correct reading to be identified. **Conserving relation.** A reading identified as correct, the identification standing on the data. **Conserving reach.** All causal structures consistent with the data. **Reach.** The statement quantifies over the causal structure. The data, identical under both structures, are what is measured. The two are not the same. **Unreachability.** Conditioning on everything available can introduce collider bias, which is proved; the back-door criterion determines which sets to condition on given the graph, and the graph is an input; causal discovery recovers structures only up to an equivalence class, which is proved. The identification is not reached over the structures by the data. **Persistence.** A structure identified from the data alone, which the equivalence-class limit excludes, or a graph supplied from outside the data. **Addresses.** Statement, hardness and reach: Sprenger and Weinberger, "Simpson's Paradox", Stanford Encyclopedia of Philosophy, first published 24 March 2021, substantive revision 6 June 2026, §§1, 3.4, 3.5 and 4. As read at v342: the entry at passages of its text as the encyclopedia's page and seven archived editions carry them verbatim (mathematics not ruling out associations being reversed at the level of subpopulations; not a paradox in the sense of an inconsistent set of plausible propositions of which one must be rejected; a cause able to raise the probability of its effect in one population and lower it in another, different populations having different sets of interactive background factors; the challenges to accounts of causation based on probability-raising, what background factors determine when a relationship is causal, what reversals imply for inference, whether the objectivity of causal relationships is threatened) and at the earlier edition's opening (the principle called Simpson's Reversal of Inequalities; the back-door criterion used to help resolve whether to base a decision on the aggregate or the partitioned statistics), the entry itself not fetched. Persistence, no landing: the closing condition at the same entry, at the passages read at v342 (the back-door criterion used to resolve whether to base a decision on the aggregate or the partitioned statistics; what background factors determine when a relationship is causal). Approaches: the conditioning and graphical routes at those passages and at the causal-inference sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Conditioning on everything available — can introduce collider bias, which is proved, more conditioning not being monotonically better. Graphical criteria — the back-door criterion determining which sets to condition on given the graph, the graph being an input. Randomisation — settles it where feasible, by removing the structure in question, not available for many questions. Sensitivity analysis — bounds the conclusion over a range of assumed structures, returning a range. Causal discovery algorithms — recover structures up to an equivalence class from data, the equivalence class being the limit, and proved. **Narrowing recorded.** Randomisation narrows the reach: the identification stands where randomisation is feasible, by removing the structure in question. Sensitivity analysis weakens the relation: a bounded range stands over a set of assumed structures rather than one reading identified. **Progress.** The proof of collider bias. The back-door criterion. The equivalence-class limit of causal discovery. **Own record.** ## 41 Externalities **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Production imposes costs on people outside the transaction. Can those costs be brought into the price? In the field's own posing: actions of firms that have harmful effects on others, analysed as a divergence between the private and the social product of the activity, with liability for the damage, a tax equal in money terms to the damage, or exclusion of the activity as the standard remedies — and, against that analysis, the reciprocal nature of the harm, since to avoid the harm to one party is to inflict harm on the other. **The field's account of its hardness.** The founding paper of the modern treatment states that the problem is reciprocal in nature, which the standard analysis in terms of a divergence between private and social product does not recognise, and that the courts have often been aware of this where economists have not. **Accounts.** Yes, by pricing them: assign the right or levy the charge and the cost enters the calculation, magnitudes being estimable and the mechanisms working where tried. Not in general: which effects count as external is set by where the transaction's boundary is drawn, and the boundary is chosen rather than found. Internalising some prices those and leaves the rest. **Specifications.** A transaction has parties, and effects falling on parties to it and on others — universal under the framework. The boundary between the two is determinable — local. **Specifications withdrawn.** **Common requirement.** That there is a boundary distinguishing effects internal to a transaction from effects external to it. **Conserving relation.** All effects of a production brought into the price, the bringing standing without a boundary chosen from outside. **Conserving reach.** All effects of a production. **Reach.** The statement quantifies over all effects of a production. Effects inside a drawn boundary are what is measured. The two are not the same. **Unreachability.** Each boundary extension reveals further effects at the new edge, and the efficiency result holds under zero transaction costs, a condition that does not obtain, which its author states. The pricing is not reached over all effects by any boundary drawn. **Persistence.** A boundary determined rather than chosen, or agreement that pricing within a chosen boundary is what the question asks. **Addresses.** Statement and hardness: Coase, "The Problem of Social Cost", Journal of Law and Economics 3:1-44, 1960, §I, as the field's pages carry the opening verbatim, the paper itself not fetched; the standard treatment it names: Pigou, The Economics of Welfare, as cited there. Persistence, no landing: the closing condition at the founding paper's opening read at pass one (the reciprocal nature of the problem; the standard treatment's pricing of the harmful effect and the question whether the harm is greater than the harm of restraining it), a boundary determined rather than chosen standing there as what neither treatment supplies. Approaches: the Pigouvian and Coasean routes at that opening and at the standard treatment it names; the later routes keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Pigouvian taxation — prices the identified effect, requiring the magnitude, estimated under assumptions about the boundary. Coasean bargaining — establishes that allocation is efficient under zero transaction costs regardless of assignment, transaction costs not being zero, which its author states. Tradable permits — work well for a bounded pollutant with measurable emissions, the boundedness being the condition. Full-cost accounting — extends the boundary, each extension revealing further effects at the new edge. Contingent valuation — prices non-market effects by survey, the estimates sensitive to elicitation format by amounts the field measures. **Narrowing recorded.** Tradable permits narrow the reach: the pricing stands for a bounded pollutant with measurable emissions. Pigouvian taxation narrows the reach: the pricing stands for the identified effect under assumptions about the boundary. **Progress.** The efficiency result under its stated condition. The measured sensitivity of contingent valuation to elicitation format. The performance record of permit markets for bounded pollutants. **Own record.** ## 42 The Collatz conjecture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Take any positive integer. Halve it if even; triple it and add one if odd. Repeat. Does every starting number reach 1? In the field's own statement: the Collatz map sends an odd number N to 3N + 1 and an even number N to N/2, and the conjecture asserts that the orbit of every positive integer under the map reaches 1. **The field's account of its hardness.** The field's own paper states that the full resolution of the conjecture remains well beyond the reach of current methods, and that even sharpening its result — almost all orbits attain almost bounded values — to an absolute bound holding for almost all starting numbers is likely to be almost as hard to settle as the conjecture itself. **Accounts.** Every starting number reaches 1: every number tested reaches 1, past 10²⁰, and heuristics from the average multiplicative effect predict decrease over long runs. **Specifications.** The map is defined on all positive integers — universal under arithmetic. A question about the behaviour of all trajectories is well posed — universal under arithmetic. The iterate composes into a single reaching — local. **Specifications withdrawn.** **Common requirement.** That the iterate composes into a single reaching, so a question can be posed about all starting values at once. **Conserving relation.** A proof that every trajectory reaches 1, the proof standing over all starting values at once. **Conserving reach.** All positive integers. **Reach.** The statement quantifies over all positive integers. Finitely many, and the structure of the map, are what is measured. The two are not the same. **Unreachability.** No quantity is known to decrease monotonically along trajectories, so no induction runs; and a class of generalised maps is proved undecidable, which the field records as indicating the difficulty is structural. The proof is not reached over all positive integers by computation or by the known monotone quantities. **Persistence.** A proof that every trajectory reaches 1, or a counterexample. **Addresses.** Statement, hardness, the verified range at Accounts and Progress, and the density result at Approaches: Tao, "Almost all orbits of the Collatz map attain almost bounded values", Forum of Mathematics, Pi, 2022, arXiv:1909.03562, §1, as the journal's and the archive's pages carry it verbatim, the paper itself not fetched; the surveys as it cites them: Lagarias, The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010. Persistence, no landing: the closing condition at the paper's introduction read at pass one (the conjecture that every trajectory reaches one; the density result stated explicitly as almost all and not all), a proof or a counterexample standing there as what the field poses. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Computation — verified past 10²⁰, returning more of the same domain. Density results — almost all orbits attain almost bounded values, a genuine advance and explicitly not all. Cycle exclusion — non-trivial cycles excluded below very large lengths, bounds rather than exclusion. Generalised Collatz maps — a class proved undecidable, indicating the difficulty is structural. Reformulation as a stopping-time problem — restates it, the same obstacle appearing at the new form. **Narrowing recorded.** Density results narrow the reach: the statement stands for almost all orbits attaining almost bounded values rather than for all starting values reaching 1. Cycle exclusion narrows the reach: the exclusion stands below very large lengths. **Progress.** The verification past 10²⁰. The density results. The cycle-exclusion bounds. The undecidability of the generalised class. **Own record.** ## 43 The neural code **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Neurons signal with discrete spikes. Is the information carried by how many spikes occur, or by when they occur? In the field's own analysis, the information in a population's spike times separates into the information in the spike counts and additional terms contributed by timing, and the question is whether, over the observation window at issue, the timing terms carry information beyond the counts. **The field's account of its hardness.** The field's own analysis states that the answer turns on the observation window: for windows shorter than the timescale of the stimulus-induced response fluctuations the purely temporal information is of third order in time and spike counts dominate, and for windows much longer than that timescale timing information of first order can appear, so the two regimes are parted by a timescale the stimulus sets and the analyst's window selects. A theoretical treatment of the rate side states that, because spike generation is stochastic, transmitting even the simplest signals reliably by rate would require observation periods too long for the speed of sensory processing or numbers of redundant neurones too large for the anatomy of sensory pathways. **Accounts.** Rate: rates correlate reliably with stimuli and behaviour across systems, decoding from rates succeeds, and the system is noisy enough that fine timing would not survive transmission. Timing: millisecond-precision timing carries information rates discard, is reproducible across trials in several systems, and is read by coincidence-sensitive mechanisms downstream. **Specifications.** Neurons carry information about stimuli in their spiking — universal under systems neuroscience. The code is identified by which feature of the spiking carries the information — local. A window over which to define a rate is available — local. **Specifications withdrawn.** **Common requirement.** That there is a code, and that identifying its variable settles the question. **Conserving relation.** The variable the downstream system uses, identified from the spiking. **Conserving reach.** All systems the code is asked of. **Reach.** The statement quantifies over what the downstream system uses. Spike trains, analysed over a window the analyst chooses, are what is measured. The two are not the same. **Unreachability.** Decoding returns amounts of information in rate and in timing, both non-zero, rather than the used variable; and the rate window is set by the analyst, which is where the instruments differ. The identification is not reached over the systems by decoding from spike trains. **Persistence.** The used variable identified downstream rather than the information available upstream, generalised beyond the systems where timing manipulation has behavioural effects. **Addresses.** Statement, hardness, reach, and the analyst's window at Unreachability: Panzeri and Schultz, "A unified approach to the study of temporal, correlational and rate coding", Neural Computation 13(6):1311-1349, 2001, arXiv:physics/9908027, at its abstract and opening as the archive's pages carry them verbatim, the paper itself not fetched; the rate side's cost at hardness: "Rate coding versus temporal order coding: a theoretical approach", 1998, at its abstract as the publisher's page carries it verbatim, its authors and journal keyed at the next pass. Persistence, no landing: the closing condition in part at the unified-approach paper's abstract and opening read at pass one (information quantified in rate, timing and correlation, the amounts returned rather than the variable used), the downstream identification keyed at the next pass at the manipulation studies the address names at Approaches. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Information-theoretic decoding — quantifies information in rate and in timing, both non-zero, returning amounts rather than the used variable. Optogenetic timing manipulation — alters timing with rate controlled, with behavioural effects in specific systems, system-specific so far. Downstream biophysics — coincidence detection demonstrated in auditory brainstem, establishing timing is usable there, generalisation standing open. Population decoding — shifts the question to population states, where the same choice recurs at the population level. Metabolic argument — precise timing being expensive, so used where it pays, a constraint rather than a determination. **Narrowing recorded.** Optogenetic timing manipulation narrows the reach: the identification stands for the specific systems manipulated. Downstream biophysics narrows the reach: usability stands for the auditory brainstem. **Progress.** The measured information in rate and in timing. The behavioural effects of timing manipulation in specific systems. The demonstrated coincidence detection. **Own record.** ## 44 Missing heritability **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Twin studies attribute a large share of variance in a trait to genetics. Genome studies find variants accounting for much less. Where is the rest? **The field's account of its hardness.** That the two estimates are of a ratio whose numerator is measured directly from observed effects and whose denominator is inferred indirectly from population data, so the gap can lie in either term. The field records that the indirect estimate assumes no interaction among loci, that models with interaction are also consistent with the observable data, and that under such models the total may be much smaller — a portion of the gap arising from the denominator rather than the numerator. Several mechanisms have been proposed and none accounts for the gap in full. **Accounts.** In variants not yet found: effects are small and numerous, rare variants are undersampled, and larger studies recover more of the gap steadily. In the estimate: twin-based heritability assumes equal environments and no gene-environment interaction, and where these fail the estimate is inflated and there is less to find. **Specifications.** There is a single quantity, the genetic contribution to variance, that both twin designs and molecular designs estimate — local. The designs' assumptions hold in the populations sampled — local. **Specifications withdrawn.** **Common requirement.** That a single quantity — the genetic share of variance — exists and is estimated by both methods. **Conserving relation.** One quantity estimated alike by both designs, the agreement standing across the designs' assumptions. **Conserving reach.** All traits the two designs are applied to. **Reach.** The statement quantifies over a variance component defined for a population and environment range. Two designs with different assumptions, each returning its own estimate, are what is measured. The two are not the same. **Unreachability.** Both designs return estimates and they do not agree, and the intermediate estimates are consistent with both accounts. The agreement is not preserved across the traits by designs carrying different assumptions. **Persistence.** The gap closed by variants recovered, or the twin estimate shown inflated by a measured amount sufficient to account for it. **Addresses.** Statement, in substance: the founding review of the question, Manolio et al., "Finding the missing heritability of complex diseases", Nature 461(7265):747-753, 2009, doi 10.1038/nature08494, published 8 October 2009, twenty-seven authors, at its abstract as the publisher's page, an abstract service, a university repository and three indexes carry it verbatim (genome-wide association studies identifying hundreds of variants; most conferring relatively small increments in risk and explaining only a small proportion of familial clustering; many led to question how the remaining, missing heritability can be explained) and at the journal's summary (a large component of heritability somehow evading detection; the possible explanations, many small-effect variants yet to be found, rare structural or epigenetic variation undetected, gene-gene and gene-environment interactions hard to detect), the review itself not fetched; the Statement standing as it stood. Hardness, the numerator-denominator structure, and the phantom-heritability argument: Zuk, Hechter, Sunyaev, and Lander, "The mystery of missing heritability: Genetic interactions create phantom heritability", PNAS 109(4), 2012, abstract and introduction. Rare-variant contribution in height and gene expression: the eLife commentary on the yeast diallel panel, eLife 2019;8:e53018, and the studies it reviews. Stratification confounding rare-variant estimates: the stratification analysis, bioRxiv 048181. Non-additive partition in controlled crosses: the diallel panel of 3025 hybrids, reported at the same eLife address. Proposed mechanisms not accounting for the gap in full: the microbiome review, PMC5468393. Persistence, no landing: the closing condition at the founding review's abstract and the journal's summary read at v342 (how the remaining, missing heritability can be explained; the possible explanations, many small-effect variants yet to be found, rare structural or epigenetic variation undetected, interactions hard to detect), the gap closed or the estimate shown inflated standing there as the two directions. Approaches: the larger-study, whole-genome and twin-reappraisal routes at that summary in part, their endings at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Larger association studies — recover increasing variance as power rises, the newly found variants having smaller average effect so their contribution is small relative to those of moderate effect. Whole-genome sequencing for rare variants — rare variants with effect sizes an order larger than common ones, recorded as accounting for most of the gap in height and a substantial share of gene-expression heritability, with subtle population stratification recorded as confounding rare-variant estimates. Estimation from all markers jointly — returns values between the two, consistent with both accounts. Within-family designs — reduce estimates substantially, showing part of the earlier figure was population structure rather than direct genetic effect. Interaction modelling — underpowered at available sample sizes, which the field states. The phantom-heritability argument — the indirect estimate shown to be inflated where loci interact, relocating part of the gap to the denominator, with the interaction models recorded as consistent with the same observable data. Model-organism crosses — panels of controlled hybrids partitioning additive from non-additive contribution directly, with a substantial non-additive share measured and dominance recorded as having a key role. Other proposed contributions — epigenetic transmission, sequencing depth, and the host microbiome, each proposed and each recorded as not accounting for the gap in full. **Narrowing recorded.** Within-family designs narrow the reach: the estimate stands with population structure removed rather than across the sampled population. **Progress.** The variance recovered by association studies. The measured reduction under within-family designs. The intermediate SNP-based values. **Own record.** ## 45 Is and ought **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Can a statement about what ought to be done follow from statements about what is the case? The canonical passage, verbatim in part: in every system of morality the author proceeds in the ordinary way of reasoning until, of a sudden, "instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not"; and a reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others entirely different from it. **The field's account of its hardness.** The founding passage states that the change from is to ought is imperceptible but of the last consequence, and that a reason is owed for what it calls altogether inconceivable: how the new relation can be a deduction from others entirely different from it. **Accounts.** No: a conclusion containing an evaluative term cannot be validly derived from premises containing none, since validity preserves only what the premises contain. The point is logical rather than substantive, holds whatever the facts turn out to be, and is what makes moral criticism of any factual arrangement possible at all — if ought followed from is, whatever is the case would be what ought to be. The gap is bridged by the meanings of terms: institutional and functional facts carry evaluative content, and inferences from them to obligations are ordinary. The prohibition assumes a separation of vocabularies language does not observe. **Specifications.** Statements sort into descriptive and evaluative — local. Validity is a relation between premises and conclusion preserving content in the premises — universal under logic. **Specifications withdrawn.** **Common requirement.** That descriptive and evaluative statements are two kinds, and that the question is whether one entails the other. **Conserving relation.** A determination of whether the entailment holds, standing over all possible derivations. **Conserving reach.** All possible derivations. **Reach.** The statement quantifies over all possible derivations. Exhibited derivations, one at a time, are what is measured. The two are not the same. **Unreachability.** The prohibition is proved for restricted logical languages, its scope being exactly the restriction; and each exhibited derivation is met by locating an evaluative premise in it. The determination is not reached over all derivations by exhibited derivations. **Persistence.** A derivation whose premises are agreed to contain no evaluative term, or a proof of the prohibition for an unrestricted language. **Addresses.** Statement and hardness: Hume, A Treatise of Human Nature, Book III, Part I, Section I, final paragraph, 1739-40, at the text as davidhume.org carries it. Persistence, no landing: the closing condition at the founding passage read at pass one (the transition from is to ought that seems altogether inconceivable without a reason given), a derivation with no evaluative premise or a proof of the prohibition standing there as the two directions the passage leaves. Approaches: the derivation route and the naturalising route keyed at the next pass at the promising argument and its critics the address does not yet name. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Exhibiting a derivation — the promising argument, its critics locating an evaluative premise in the institutional description. Naturalising the evaluative terms — defines goodness in natural terms, the open question argument pressed against each definition, the dispute old and live. Denying the evaluative side has truth values — expressivism, facing the embedding problem, which its own proponents work on and have not closed. Thick concepts — terms carrying description and evaluation together, claimed by both accounts, whether they can be factored being the dispute restated. Formalising the prohibition — proofs existing for restricted logical languages, their scope being exactly the restriction. **Narrowing recorded.** Formalisation narrows the reach: the prohibition stands proved for restricted logical languages. Expressivism weakens the relation: the evaluative side carries no truth value, and the entailment question is replaced by an account of what evaluative utterance does. **Progress.** The proofs for restricted languages. The counterexample record against each naturalising definition. The embedding problem, stated and open. **Own record.** ## 46 Placebo effects **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Patients given an inert substance sometimes improve. Is the improvement produced by the administration? In the field's own posing: placebo treatments have been reported to help patients with many diseases, and whether the evidence supports their having powerful clinical effects is tested by the trials in which patients were assigned at random to placebo or to no treatment. **The field's account of its hardness.** The field's systematic review of placebo against no treatment states that the quality of the evidence for placebo effects had not been rigorously evaluated; that it found little evidence in general of powerful clinical effects, none significant on objective or binary outcomes and possible small benefits on continuous subjective outcomes and for pain; and that the inverse relation it found between trial size and effect size qualifies its pooled estimate for subjective continuous outcomes. The correspondence on the review states that such a meta-analysis is restricted by the studies chosen and the sensitivity of their measures, binary outcomes having less power to detect effects and the populations varying widely. **Accounts.** Yes, and measurably: open-label trials show improvement without deception, ritual dose-response is documented, and imaging shows endogenous opioid release in placebo analgesia. Much of what is called a placebo effect is not an effect: regression to the mean, natural history, and reporting bias account for a large share, and trials with no-treatment arms find the difference small outside pain and subjective outcomes. **Specifications.** A trial's arms differ only in the substance administered — local. The difference between arms estimates the substance's effect — universal under trial methodology. **Specifications withdrawn.** **Common requirement.** That the inert substance's administration either does or does not produce the improvement, and that a comparison isolates which. **Conserving relation.** The administration's contribution isolated, the isolation standing with context removed. **Conserving reach.** All outcomes the effect is asked of. **Reach.** The statement quantifies over what the administration contributes. Arms of a trial, each including a context that cannot be removed while the trial runs, are what is measured. The two are not the same. **Unreachability.** No-treatment is itself a context, so the comparison does not remove what it is to isolate. The contribution is not isolated over the outcomes by any arm of a running trial. **Persistence.** The contribution isolated with context removed, or agreement that context is part of what the question asks about. **Addresses.** Statement, hardness, the no-treatment comparison at Accounts and Approaches, and the effect sizes by outcome type at Progress: Hróbjartsson and Gøtzsche, "Is the Placebo Powerless? An Analysis of Clinical Trials Comparing Placebo with No Treatment", New England Journal of Medicine 344(21):1594-1602, 2001, at its abstract as the journal's page carries it verbatim and at p. 1599 as the journal's pages carry it, the paper itself not fetched; its update: Hróbjartsson and Gøtzsche, "Is the placebo powerless? Update of a systematic review with 52 new randomized trials comparing placebo with no treatment", Journal of Internal Medicine 256(2):91-100, 2004; the correspondence on the review's method: New England Journal of Medicine, 25 October 2001, doi:10.1056/NEJM200110253451712. Persistence, no landing: the closing condition at the trial analysis read at pass one (placebo compared with no treatment across clinical trials, small effects except for pain and subjective outcomes, no treatment itself a context), the contribution isolated with context removed standing there as what the design cannot deliver. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Three-arm trials with no-treatment control — small effects except for pain and subjective outcomes, the strongest route, its own limitation being that no-treatment is itself a context. Open-label placebo — improvement without deception, replicated, narrowing the mechanism while the control problem persists. Neuroimaging and antagonist blockade — blockade of placebo analgesia establishing a mechanism for that outcome, specific to it. Conditioning paradigms — measurable immune and endocrine responses in animals, where reporting bias is absent, the strongest evidence for a real effect, in a different setting. Nocebo studies — symmetric findings, same methodological position. **Narrowing recorded.** Antagonist blockade narrows the reach: the mechanism stands established for placebo analgesia. Conditioning paradigms narrow the reach: the effect stands established in animals, where reporting bias is absent. **Progress.** The measured effect sizes by outcome type. The replicated open-label improvement. The blockade result. The animal conditioning responses. **Own record.** ## 47 Turbulence and the Navier–Stokes equations **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Do smooth solutions of the equations of fluid motion in three dimensions remain smooth for all time, or can they develop a singularity? And the turbulence problem the title also names, posed by the field on its own: whether a predictive theory of turbulent flow follows from the equations, the averaged equations for the mean flow carrying more unknowns than equations at every order — the closure problem. **The field's account of its hardness.** Supercriticality, in the field's own formalisation: the a priori bounds available are not scale-invariant in three dimensions, classical energy methods do not prevent concentration at small scales, and any proposed positive solution to the regularity problem which does not use the finer structure of the nonlinearity cannot possibly be successful, established through finite-time blow-up of an averaged equation sharing the known bounds. **Accounts.** They remain smooth: no singularity has been observed in any physical fluid or reliable simulation, and the equations describe fluids accurately across the range where they are tested. Not established, and blow-up is plausible: the scaling of the nonlinear term against dissipation is supercritical in three dimensions, which is exactly the regime where the available methods fail, two dimensions being settled precisely because it is not supercritical. **Specifications.** The equations describe fluid motion at all scales of the flow — universal under fluid dynamics. A solution means a function satisfying them for all time from given initial data — universal under the framework. **Specifications withdrawn.** **Common requirement.** That the equations continue to describe the fluid at every scale, so that a question about all time and all initial data is well posed. **Conserving relation.** Smoothness holding for all time from given initial data, the holding established by proof. **Conserving reach.** All initial data, for all time. **Reach.** The statement quantifies over all initial data, for all time. Finite simulations at finite resolution, and specific classes of initial data, are what is measured. The two are not the same. **Unreachability.** An averaged version of the equations admits blow-up, showing that any proof of regularity must use structure the averaged version discards, so a class of arguments is proved unable to reach it. The holding is not reached over all initial data by methods that do not use that structure. **Persistence.** A proof of global regularity, or a constructed solution developing a singularity. For the closure problem, a closure derived from the equations rather than supplied by a model. **Addresses.** Statement, regularity: Fefferman, "Existence and smoothness of the Navier–Stokes equation", in The Millennium Prize Problems, Clay Mathematics Institute and American Mathematical Society, 2006, p. 57. Statement, turbulence and closure: Pope, Turbulent Flows, Cambridge, 2000, the mean-flow equations chapter; Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge, 1995. Hardness and the barrier: Tao, "Finite time blowup for an averaged three-dimensional Navier-Stokes equation", Journal of the American Mathematical Society 29:601-674, 2016, §1.1. Scaling exposition: the partial-regularity and quantitative-regularity literature's own state-of-the-art statements. As read at v342: the official statement at its text as the Institute's copy and a university copy carry it verbatim (the equations to be solved for an unknown velocity vector and pressure; fluids important and hard to understand; since we do not even know whether these solutions exist our understanding at a very primitive level; standard methods from partial differential equations appearing inadequate to settle the problem), the volume's pages 57-67 as a citing paper gives them, and at the Institute's page on the problem (no proof for the most basic questions one can ask, whether solutions exist and are unique; our understanding of the equations remaining minimal), the statement itself not fetched; the turbulence and closure sources and the blowup paper not searched this pass. Persistence, no landing: the closing condition at the official statement read at v342 (whether solutions exist and are unique the most basic questions, our understanding at a very primitive level, standard methods appearing inadequate) and at the Institute's rules for a proposed solution, read at v342 (publication in a qualifying outlet, two years, general acceptance in the global mathematics community), the closure problem's condition at the sources the address names for it. Approaches: the energy and partial-regularity routes at the official statement's account of what is known, in part, read at v342; the closure routes at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Energy methods — give global weak solutions, uniqueness and smoothness of weak solutions standing open, which is the same question. Partial regularity — the singular set bounded to parabolic Hausdorff dimension zero, a strong constraint that does not exclude. Numerical search for blow-up — candidate scenarios proposed and repeatedly resolved as resolution improved, recent computer-assisted work establishing blow-up for related equations and boundary cases, not for the free three-dimensional problem. Modified and averaged equations — an averaged construction with blow-up, showing that any proof of regularity must use structure the averaged version discards. Two-dimensional case — settled, its methods not carrying to three. **Narrowing recorded.** Partial regularity narrows the reach: the singular set stands bounded rather than excluded. The two-dimensional case narrows the reach: the holding stands proved in two dimensions, its methods not carrying to three. Weak solutions weaken the relation: global existence stands without uniqueness or smoothness. **Progress.** The global weak solutions. The parabolic Hausdorff bound on the singular set. The two-dimensional result. The blow-up results for related equations. The averaged construction and what it proves about admissible arguments. **Own record.** ## 48 Black hole information **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Matter falls into a black hole and the hole evaporates into radiation. Is the information about what fell in preserved? **The field's account of its hardness.** That the breakdown of the classical concepts of space and time, and of all the known laws formulated on a classical background, at the singularities any reasonable theory of gravitation admits is claimed to be not merely a result of ignorance of the correct theory but a fundamental limitation on the ability to predict the future, analogous but additional to the limitation the uncertainty principle imposes, arising because general relativity allows the causal structure of spacetime to be very different from that of flat spacetime. **Accounts.** Yes: quantum evolution is unitary, and abandoning unitarity would break the framework everywhere it is tested. Calculations of the entropy curve reproduce the expected turnover. The semiclassical calculation says no: the derivation gives thermal radiation independent of what formed the hole, and it uses only physics well established at the horizon, where curvature is low. **Specifications.** Quantum evolution is unitary — universal under quantum mechanics. Semiclassical gravity applies where curvature is low, including at the horizon of a large black hole — universal under the framework. Both hold together in one consistent theory — local. **Specifications withdrawn.** **Common requirement.** That the question has one answer determinable from a consistent theory covering the horizon and the interior together. **Conserving relation.** One answer determinable from a theory covering horizon and interior together. **Conserving reach.** The interior, and the correlations between interior and exterior. **Reach.** The statement quantifies over the interior, and the correlations between interior and exterior. No evaporation has been observed, and all routes are theoretical. The two are not the same. **Unreachability.** Unitarity, effective field theory at the horizon, and no drama at the horizon are proved not all able to hold, with no consensus on which to give up; and evaporation timescales for astrophysical holes exceed the age of the universe by enormous factors. The one answer is not reached over the interior by any route standing. **Persistence.** A theory covering horizon and interior together from which the answer follows, the premises shown incompatible being resolved by giving one up. **Addresses.** Statement: Hawking, "Breakdown of predictability in gravitational collapse", Physical Review D 14:2460, 1976. Review: Calmet and Hsu, "A brief history of Hawking's information paradox", Europhysics Letters 139:49001, 2022. Hardness: the founding paper, at its abstract as the publisher's page carries it verbatim (gravity always attractive by the principle of equivalence and the positivity of energy, leading to singularities in any reasonable theory of gravitation; a singularity a place where the classical concepts of space and time and all known laws formulated on a classical background break down; the breakdown claimed not merely a result of ignorance of the correct theory but a fundamental limitation on the ability to predict the future, analogous but additional to the uncertainty principle's, arising because general relativity allows the causal structure of spacetime to be very different from that of Minkowski space), pages 2460–2473, issue 10, published 15 November 1976, doi 10.1103/PhysRevD.14.2460, the paper itself not fetched; the review not read this pass. Persistence: re-formed at v342 to release a landing word used in another sense, the resolving file reading a landing at that word; the line stood as: A theory covering horizon and interior together from which the answer follows, the premises proved incompatible being resolved by giving one up. Approaches: the semiclassical route at the founding paper's abstract read at pass two; the holographic and later routes at the brief history the address names, read at pass one, their endings keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Semiclassical calculation — gives information loss, its own domain of validity at late times disputed. Holographic duality — evolution unitary in the dual description where the duality is established, whether the setting applies to astrophysical holes standing open. Island formulas and the entropy curve — reproduce the expected turnover using gravitational path integral techniques, what they imply about the interior actively disputed. Firewalls — the proved incompatibility of unitarity, effective field theory at the horizon, and no drama, with no consensus on which to give up. Remnants and baby universes — consistent constructions with acknowledged costs. Observation — evaporation timescales for astrophysical holes exceeding the age of the universe by enormous factors. **Narrowing recorded.** Holographic duality narrows the reach: unitarity stands in settings where the duality is established rather than for astrophysical holes. **Progress.** The entropy curve turnover, reproduced. The proved incompatibility of the three premises. The unitarity of the dual description where the duality is established. **Own record.** ## 49 Bell correlations **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Two particles prepared together are measured far apart. The results are correlated more strongly than any account giving each particle its own properties allows. What produces the correlation? **The field's account of its hardness.** That the essential difficulty is created by the requirement of locality, more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, the additional variables the paradox called for having been meant to restore causality and locality to the theory, and the idea, formulated mathematically, shown to be incompatible with the statistical predictions of quantum mechanics. **Accounts.** Nothing local does, and the theorem settles it: an inequality is satisfied by any theory in which each particle carries its own state and no influence travels faster than light, experiments violate it, and loophole-free experiments close detection and locality simultaneously. The theorem constrains a class of theories, and which assumption to give up is not settled: nonlocality, denial of measurement independence, retrocausality, and superdeterminism each preserve the correlations at a different cost, and the field has no agreed criterion for choosing among them. **Specifications.** Each system carries its own state — universal under the local framework. Measurement outcomes are fixed by that state and the local setting — universal under the local framework. Settings are chosen independently of the state — universal under experimental practice. No influence propagates faster than light — universal under relativity. **Specifications withdrawn.** The conjunction of the four is refuted by the theorem together with loophole-free experiment, and the record carries no agreement on which is given up. **Common requirement.** That the correlation is produced by something specifiable at one side or transmitted between the sides. **Conserving relation.** An account of what produces the correlation, the account standing on the coincidence counts. **Conserving reach.** All accounts preserving the observed correlations. **Reach.** The statement quantifies over what produces the correlation. Coincidence counts at two stations are what is measured. The two are not the same. **Unreachability.** The surviving accounts are empirically equivalent to each other, each preserving the correlations at a different cost, and the field carries no agreed criterion for choosing among them. The account is not reached over the surviving class by the coincidence counts. **Persistence.** A criterion for choosing among the surviving accounts, or a distinguishing prediction between them. **Addresses.** Statement: Bell, "On the Einstein Podolsky Rosen paradox", Physics 1:195-200, 1964. Review: Brunner, Cavalcanti, Pironio, Scarani and Wehner, "Bell nonlocality", Reviews of Modern Physics 86:419-478, 2014. Hardness: the founding paper, at its introduction as the publisher's copy and an abstract service carry it verbatim (the paradox advanced as an argument that quantum mechanics could not be complete but should be supplemented by additional variables restoring causality and locality; the idea formulated mathematically and shown incompatible with the statistical predictions; the requirement of locality, more precisely that the result of a measurement on one system be unaffected by operations on a distant system it has interacted with in the past, creating the essential difficulty), the journal carried at the publisher's page as Physics Physique Fizika, volume 1, issue 3, published 1 November 1964, the paper itself not fetched; the review not read this pass. Persistence, no landing: the closing condition at the founding paper's introduction read at pass two (the requirement of locality creating the essential difficulty; the argument advanced that the theory could not be complete), a distinguishing prediction among the surviving accounts standing there as what the theorem's own logic leaves. Approaches: the loophole-closing route at the experimental sources the address names, read at pass one; the interpretive routes at the founding paper. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Closing the loopholes — detection, locality, and freedom-of-choice loopholes closed, the last using cosmic photons and human random choices, the correlation standing and the interpretation not following. Local hidden variables — excluded by the theorem plus experiment, the route closed outright. Nonlocal hidden variables — reproduce everything, explicitly nonlocal, empirically equivalent. Superdeterminism — preserves locality by denying setting independence, not excluded by any experiment, its critics holding it undermines the practice of experiment. Retrocausal accounts — consistent constructions existing, no distinguishing prediction so far. Information-theoretic reconstructions — derive the correlations from principles such as information causality and reach the observed bound, accounting for the magnitude without settling what produces it. **Narrowing recorded.** Information-theoretic reconstructions weaken the relation: the observed bound is derived from principles rather than the producing account identified. **Progress.** The theorem. The loophole-free violations. The exclusion of local hidden variables. The derived bound from information-theoretic principles. **Own record.** ## 50 Wave and particle **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether a quantum entity is a wave, a particle, both, or neither, given that it shows interference in one experimental arrangement and localized detection in another — and whether the question is well-posed, the field's standing answer, complementarity, holding the two descriptions mutually exclusive and jointly required, and the delayed-choice arrangements leaving what that answer establishes in dispute. **The field's account of its hardness.** That two kinds of complementary description stand, those ascribing either kinematic or dynamic properties and those ascribing either wave or particle properties to a single object, and how the two kinds are related was never indicated, with no agreement even among the interpretation's own spokesmen; that the description of light as either particles or waves was already a classical dilemma, which the definition of the photon did not solve, the momentum of the photon as a particle depending on the frequency of the light as a wave; and that the phenomena are complementary in that their manifestations depend on mutually exclusive measurements while the information gained through those experiments exhausts all possible objective knowledge of the object, the truth conditions of an ascription then including reference to the experimental setup as well as the outcome. **Accounts.** Neither, and the statement is malformed: the formalism is complete, wave and particle are classical pictures, and the two descriptions apply to mutually exclusive experimental arrangements. Complementarity names the situation rather than accounting for it: delayed-choice and quantum-eraser experiments show the arrangement can be settled after the entity has passed, and accounts differ on what this establishes. **Specifications.** An entity has a determinate nature specifiable independently of the measurement made on it — local. The two classical pictures exhaust the available descriptions — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about what the entity is, or a demonstration that there is not. **Conserving relation.** A fact about what the entity is between preparation and detection, standing independently of the arrangement. **Conserving reach.** All arrangements. **Reach.** The statement quantifies over what the entity is between preparation and detection. Preparation settings and detection events are what is measured. The two are not the same. **Unreachability.** Obtaining path information destroys interference, quantitatively, by measured duality relations, which establishes the trade exactly without settling the nature; and the interpretations are empirically equivalent by construction. The fact is not reached over the arrangements by preparation settings and detection events. **Persistence.** A determination of the entity's nature between preparation and detection, which the empirical equivalence of the interpretations excludes. **Addresses.** Statement: Faye, "Copenhagen Interpretation of Quantum Mechanics", Stanford Encyclopedia of Philosophy. The delayed-choice arrangements: Ma, Kofler and Zeilinger, "Delayed-choice gedanken experiments and their realizations", Reviews of Modern Physics 88:015005, 2016. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (two kinds of complementary description, kinematic against dynamic and wave against particle, how they are related never indicated and no agreement among the interpretation's own spokesmen; the description of light as either particles or waves already a classical dilemma, the definition of the photon not solving it since the photon's momentum as a particle depends on the frequency of the light as a wave; the phenomena complementary in that their manifestations depend on mutually exclusive measurements while the information gained exhausts all possible objective knowledge of the object; the truth conditions of an ascription including reference to the experimental setup, the indefinability thesis), the entry itself not fetched, its section numbers and revision date not carried at the returns; the delayed-choice review not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (two kinds of complementary description; the entity's nature between preparation and detection not determinable by the theory's own account). Approaches: the which-path and delayed-choice routes at the same entry's sections on the experiments, read at pass one and at those passages. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Which-path measurement — obtaining path information destroys interference, quantitatively, by measured duality relations, establishing the trade exactly. Delayed choice — the experiment performed, the outcome matching quantum prediction under every arrangement, the interpretation disputed. Quantum eraser — interference recoverable by correlation with the marker, widely misreported as retrocausal, the field stating the correlations require the joint measurement. Weak measurement of trajectories — reconstructs average trajectories in a two-slit apparatus, what an average trajectory establishes about individual entities being disputed. Interpretation — each giving its own answer, and they are empirically equivalent. **Narrowing recorded.** The duality relations weaken the relation: the trade between path information and interference visibility stands quantified in place of the nature determined. **Progress.** The measured duality relations. The delayed-choice outcomes. The reconstructed average trajectories. **Own record.** ## 51 Incompleteness and Hilbert's programme **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Can a formal system strong enough for arithmetic prove every arithmetical truth, and prove its own consistency? **The field's account of its hardness.** That the two theorems concern the limits of provability in formal axiomatic theories: in any consistent formal system within which a certain amount of arithmetic can be carried out there are statements of its language that can be neither proved nor disproved in it, and such a system cannot prove its own consistency, assuming it is consistent; that the programme called for a formalization of all of mathematics together with a consistency proof carried out by finitary methods only, and the attempt to prove the consistency of analysis with the resources of arithmetic required the notion of truth and met the conclusion that arithmetical truth cannot be defined in arithmetic; and that with the first theorem the truth of the unprovable statement follows given the consistency of the system. **Accounts.** No, proved: a true sentence is unprovable in the system, and consistency is unprovable within it. The programme as stated cannot be carried out. The programme survives in modified form: the consistency of arithmetic is proved by transfinite induction to a specified ordinal, and reverse mathematics has established exactly which axioms are needed for which theorems. What was lost is the finitary requirement, not the enterprise. **Specifications.** Mathematical truth is captured by derivability in a formal system — local. Consistency can be established by means weaker than the system itself — local. **Specifications withdrawn.** The finitary requirement stands given up in the modified programme, by that programme's own account, the enterprise retained. **Common requirement.** That there is a determinate set of arithmetical truths, and a system either captures it or does not. **Conserving relation.** A system capturing the arithmetical truths and proving its own consistency. **Conserving reach.** All arithmetical truths. **Reach.** The statement quantifies over all arithmetical truths. Derivations in a specified system are what is measured. The two are not the same. **Unreachability.** Each strengthening carries its own unprovable sentence, by construction at every stage, and consistency is unprovable within the system. The capture is not reached over the arithmetical truths by derivation in any system strong enough for arithmetic. **Persistence.** Within the conditions, nothing: the results are proved. Outside them, which axioms suffice for which theorems, settled case by case. **Addresses.** Statement: Raatikainen, "Gödel's Incompleteness Theorems", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at its opening as an index and the encyclopedia's archived editions carry it verbatim (the theorems concerning the limits of provability in formal axiomatic theories; in any consistent formal system within which a certain amount of arithmetic can be carried out, statements of its language neither provable nor disprovable in it; such a system unable to prove its own consistency, assuming it consistent) and at passages of its text as the encyclopedia's page and its archived editions carry them (the attempt to prove the consistency of analysis with the resources of arithmetic requiring the notion of truth and meeting the conclusion that arithmetical truth cannot be defined in arithmetic; the truth of the unprovable statement following from the system's consistency; some incompleteness not totally unexpected), the entry itself not fetched, its section numbers and revision date not carried at the returns; the programme's own statement, a formalization of all of mathematics with a consistency proof by finitary methods only, at the encyclopedia's entry on the programme as its page carries it, returned in the search, its author not carried at the return. Persistence: the landing, the theorems proved, at the entry's opening read above, the theorems' own papers keyed at the next pass. Approaches: the strengthening and transfinite-consistency routes at the same entry's sections on the second theorem and on the consistency proofs, read at pass two and at the opening the hardness keys. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Strengthening the system — each strengthening carrying its own unprovable sentence, by construction at every stage. The transfinite consistency proof — succeeds, using induction to an ordinal not available within arithmetic, requiring at a higher level what it was meant to secure at a lower. Reverse mathematics — maps exactly which axioms suffice for which theorems, with striking regularity across five subsystems, a substantial positive result addressing a different question. Ordinal analysis — extends the transfinite route upward, each system's proof-theoretic ordinal established, each needing a stronger metatheory. Set-theoretic realism — holds truth outruns provability and the unprovable sentences are true, consistent, and supplying no method. **Narrowing recorded.** The transfinite consistency proof weakens the relation: consistency stands proved by means stronger than the system rather than weaker. Reverse mathematics narrows the reach: which axioms suffice stands theorem by theorem rather than for the truths as a whole. **Progress.** The two theorems. The transfinite consistency proof. The reverse-mathematics map across five subsystems. The established proof-theoretic ordinals. **Own record.** ## 52 The continuum hypothesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is there a set larger than the integers and smaller than the real numbers? **The field's account of its hardness.** That very little progress was made on the hypothesis in the early era of set theory, the lack of progress explained by the independence results, which showed that the hypothesis cannot be resolved on the basis of the axioms mathematicians were employing; that the pluralist maintains the independence results effectively settle the undecided questions by showing that they have no answer; and that the strongest known local case for axioms that settle the hypothesis is a case for its failure, an equally good case for the hypothesis being put beside it, the hope of settling the issue resting on optimistic global scenarios. **Accounts.** The statement has no answer: the hypothesis is consistent with the standard axioms and so is its negation. It is independent, and independence is the answer. Independence from one axiom set is not the end: the axioms are incomplete, and the question is which further axioms are true. Programmes on inner models and on forcing axioms aim at settling it, and some favour a definite value. **Specifications.** The sets form a determinate universe about which questions have answers — local. The standard axioms describe that universe — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about the universe of sets, or a demonstration that there is not. **Conserving relation.** A determination of the hypothesis, standing on the axioms. **Conserving reach.** All models of the standard axioms. **Reach.** The statement quantifies over whether a further axiom is true of the sets, beyond being consistent with them. Consequences of chosen axiom systems are what is measured. The two are not the same. **Unreachability.** Independence is proved twice over, and large-cardinal axioms provably do not settle it, forcing preserving them. The determination is not reached over the models by the standard axioms or by the large-cardinal extensions. **Persistence.** A criterion for the truth of a further axiom, beyond its consistency. **Addresses.** Statement: Koellner, "The Continuum Hypothesis", Stanford Encyclopedia of Philosophy. Consistency: Gödel, Proceedings of the National Academy of Sciences 24:556-557, 1938. Independence: Cohen, Set Theory and the Continuum Hypothesis, Benjamin, 1966, chapter IV. Hardness: the same entry, at passages of its text as the encyclopedia's page, its archived editions and a mirror carry them verbatim (very little progress made on the hypothesis in the early era of set theory, the lack of progress explained by the independence results showing it cannot be resolved on the basis of the axioms mathematicians were employing; the pluralist maintaining that the independence results effectively settle the undecided questions by showing they have no answer; the strongest known local case for axioms that settle the hypothesis a case for its failure, an equally good case for the hypothesis considered beside it, and optimistic global scenarios investigated as the hope of settling the issue), the entry itself not fetched, its section numbers and revision date not carried at the returns; the consistency and independence sources not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (very little progress in the early era; the independence results closing proof from the axioms; the search for new axioms and the question of their justification beyond consistency). Approaches: the axiom-proof, large-cardinal and forcing-axiom routes at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Proving it from the axioms — excluded, closed by proof. Large cardinal axioms — settle many questions, provably not this one, since forcing preserves them. Forcing axioms — imply a definite value for the continuum, the choice of forcing axiom itself unsettled. Inner model programme — aims at a canonical model for large cardinals, ongoing, technically formidable, its architects describing it as incomplete. Further criterion programmes — more than one criterion proposed and revised, including reversals of direction, the field not regarding the matter as decided. **Narrowing recorded.** Forcing axioms narrow the reach: a definite value stands under a chosen forcing axiom rather than over the models. **Progress.** The two independence results. The proof that large-cardinal axioms do not settle it. The values implied under specific forcing axioms. **Own record.** ## 53 Gettier **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Knowledge was defined as justified true belief. Cases exist in which a belief is justified, true, and by accident. What is missing? **The field's account of its hardness.** That a lesson of the cases is that even true beliefs that are justified can nevertheless be epistemically lucky in a way inconsistent with knowledge, the hope being that a fourth condition articulated beyond justification, truth and belief would give an analysis that did not succumb to counterexamples; that those who take the approach to be on the right track must choose between strengthening the justification condition to rule the cases out as justified and amending the analysis with a fourth condition preventing a justified true belief from being gettiered; and that the natural proposal, that the belief not be inferred from any falsehood, is not widely accepted, apparent cases needing no inference, and any non-conclusive justification is observed to permit such cases. **Accounts.** A fourth condition: candidate additions — no false lemmas, defeasibility, causal connection, reliability, safety, sensitivity — each handle the original cases, and the project is to find the right one. No fourth condition works: each proposal has generated counterexamples within a few years, for fifty years, and the analysis project itself is argued to be misconceived, knowledge being unanalysable. **Specifications.** Knowledge decomposes into individually necessary and jointly sufficient conditions — local. Intuitions about cases adjudicate proposals — local. **Specifications withdrawn.** **Common requirement.** That cases sort into knowledge and non-knowledge, and that a correct analysis reproduces the sorting. **Conserving relation.** An analysis reproducing the sorting, the reproduction standing over cases not yet constructed. **Conserving reach.** The concept's extension over all possible cases. **Reach.** The statement quantifies over the concept's extension over all possible cases. Cases constructed and judged, one at a time, are what is measured. The two are not the same. **Unreachability.** Each added condition has been met by a counterexample, the pattern documented and consistent over decades. The reproduction is not reached over all possible cases by analyses adjudicated against cases constructed one at a time. **Persistence.** An analysis surviving the construction of further cases, or agreement that the concept admits none. **Addresses.** Statement: Ichikawa and Steup, "The Analysis of Knowledge", Stanford Encyclopedia of Philosophy. The cases as posed: Gettier, "Is Justified True Belief Knowledge?", Analysis 23, 1963. Hardness: the same entry, at passages of its text and notes as the encyclopedia's page and its archived editions carry them verbatim (a lesson of the problem that even true beliefs that are justified can be epistemically lucky in a way inconsistent with knowledge; much of the literature's focus on a fourth condition, the hope that an analysis so amended would not succumb to counterexamples; two strategies for those who take the approach to be on the right track, strengthening the justification condition or amending the analysis with a degettiering condition; the no-false-lemmas proposal not widely accepted, apparent cases needing no inference; a note's observation that any non-conclusive justification will permit such cases), the entry itself not fetched, its section numbers and revision date not carried at the returns, the notes page carrying a 2026 copyright; the cases as posed not read this pass, the paper's issue and pages carried at a citing paper as 23(6):121–123. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (justified true beliefs epistemically lucky in a way incompatible with knowledge; the pattern of proposed conditions met by further cases), an analysis surviving further cases standing there as what the record has not yet supplied. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Adding a condition — each met by a counterexample, the pattern documented and consistent over decades. Knowledge first — knowledge taken as primitive and the others analysed from it, reversing the direction and contested, supplying no analysis. Contextualism — standards varying with context, accounting for variation, the original cases persisting within a fixed context. Experimental philosophy — reports variation in case judgments across groups, bearing on the adjudication method rather than the analysis. Denying the phenomenon — few take it, the cases widely found compelling. **Narrowing recorded.** Contextualism narrows the reach: the sorting stands within a context rather than over all cases. Knowledge first weakens the relation: knowledge stands primitive and the other terms are analysed from it, in place of an analysis of knowledge. **Progress.** The counterexample record across five decades of proposals. The measured variation in case judgments across groups. **Own record.** ## 54 The Münchhausen trilemma **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Any claim can be asked for its grounds. The grounds can be asked in turn. Where does this end? In the field's own statement the demand for a foundation must end at one of three horns, verbatim: "infinite regress, circularity, or that kind of dogmatism that is seen as a necessary evil", the third a solution one resigns oneself to accepting in view of the obvious unserviceability of the other two. **The field's account of its hardness.** The treatise that names the trilemma states that it must necessarily arise wherever a foundation is sought, for workable inferential procedures and for the ultimate linguistic or extralinguistic authorities used as a basis for inference alike; that the third horn is accepted only as a necessary evil, the other two being obviously unserviceable; and that a principle of the self-confidence of reason in the truth of its immediate knowledge, which claims freedom from error for certain assertions, is merely a psychologistic form of dogmatism. **Accounts.** It ends at basic beliefs: some are self-justifying or justified non-inferentially, and the regress stops there. It does not end: every proposed foundation either takes its warrant from something further, which continues the regress, or from itself, which is circular, or from stipulation, which is arbitrary. Circularity is accepted in one form, regress in another. **Specifications.** Justification is a relation between claims — local. A claim's warrant derives from its relation to other claims or from itself — local. **Specifications withdrawn.** **Common requirement.** That justification terminates, circles, or continues, and that these three exhaust the options. **Conserving relation.** A termination of the regress, standing without warrant taken from further, from itself, or from stipulation. **Conserving reach.** The structure of justification in general. **Reach.** The statement quantifies over the structure of justification in general. Particular justifications, each already inside the structure, are what is measured. The two are not the same. **Unreachability.** Each candidate class of basic belief has been argued to require justification of its own, and each horn is developed and carries a standing objection. The termination is not reached over the structure by particular justifications taken from inside it. **Persistence.** A class of basic beliefs surviving the demand for its own justification, or agreement on which horn is to be accepted. **Addresses.** Statement and hardness: Albert, Treatise on Critical Reason, Princeton University Press, 1985, chapter I, at its section on the problem of foundation, as a repository page carries the passage verbatim, the book itself not fetched; the section as the field's citations carry it: chapter I, section 2. Persistence, no landing: the closing condition at the founding section read at pass two (the three horns, regress, circle and the breaking off of the procedure at a point taken as self-evident; the demand for justification not stopping of itself). Approaches: the foundationalist, coherentist and infinitist routes at the same section's statement of the three horns, their endings at the sources the address names for the later literature. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Foundationalism — identifies basic beliefs, each candidate class argued to require justification of its own. Coherentism — accepts mutual support without foundations, facing the isolation objection, that a coherent set may be unconnected to anything outside it. Infinitism — accepts infinite chains, facing finite-mind objections, its proponents arguing these confuse having a chain with traversing one. Contextualism — grounds stopping where the context does not question them, accounting for practice, the trilemma recurring when the context is questioned. Rejecting the demand — holds the regress question malformed, its critics replying that the demand arises from ordinary practice. **Narrowing recorded.** Contextualism narrows the reach: the termination stands within a context rather than over the structure in general. **Progress.** The taxonomy of the three horns. The standing objection against each. The documented failure of each candidate class of basic belief. **Own record.** ## 55 The liar **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** "This sentence is false." If it is true it is false, and if it is false it is true. What is wrong? **The field's account of its hardness.** That the hierarchy of languages and metalanguages faces the challenge of explaining why truth cannot be defined for the whole hierarchy by quantifying over levels, a predicate such as true at some level returning the paradox, a problem for all hierarchical views, many concluding that the hierarchy buys a solution at the cost of implausible restrictiveness; that approaches leaving classical logic fixed share the hallmark of restricting the range of application of capture and release to block the paradoxical reasoning; and that revenge paradoxes are rejected by some approaches, met with additional apparatus by others, and seen by contextualist views not as a separate problem but as the core phenomenon. **Accounts.** The sentence is not well formed for truth evaluation: a consistent truth predicate for a language cannot be defined within that language, and the hierarchy of metalanguages removes the sentence from the domain. Natural language has one level and speakers use "true" of their own sentences constantly: hierarchies are stipulated rather than found, and revenge sentences reconstruct the paradox at every level of every proposed solution. **Specifications.** Sentences bear truth values — universal under semantics. A truth predicate applies to sentences of the language it occurs in — local. **Specifications withdrawn.** **Common requirement.** That the sentence either has a truth value or is shown to fall outside the domain. **Conserving relation.** A treatment removing the paradox, the removal standing against sentences constructed within the treatment. **Conserving reach.** The truth predicate as used, at every level of every treatment. **Reach.** The statement quantifies over the truth predicate as used. Formal languages, each with a stipulated structure, are what is measured. The two are not the same. **Unreachability.** A consistent truth predicate for a language is proved undefinable within that language, and every family of solution has been met by a revenge sentence, which its own authors record. The removal is not preserved over the treatments by any treatment of a stipulated language. **Persistence.** A treatment against which no revenge sentence is constructible, or agreement that the hierarchy's cost is to be accepted. **Addresses.** Statement: Beall, Glanzberg and Ripley, "Liar Paradox", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the hierarchy of languages and metalanguages challenged to explain why truth cannot be defined for the whole hierarchy by quantifying over levels, a problem for all hierarchical views, many concluding the hierarchy buys a solution at the cost of implausible restrictiveness; approaches leaving classical logic fixed sharing a willingness to restrict the range of application of capture and release; revenge rejected by some approaches, met by additional apparatus by others, and seen by contextualist views as the core phenomenon rather than a separate problem), the entry itself not fetched, its section numbers and revision date not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the hierarchy challenged to explain why truth cannot be defined for the whole; the revenge problem recurring for each treatment), a treatment against which no revenge sentence is constructible standing there as the entry's own standard. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The hierarchy — proved and consistent, at the cost of denying that a language contains its own truth predicate. Fixed points — assign the liar no truth value, consistently, a revenge sentence reconstructing the problem, which its author states. Paraconsistent treatments — accept the sentence as both true and false and restrict inference to prevent explosion, consistent as a system, their critics holding the cost is the inference rules everyone uses. Contextual treatments — the truth predicate shifting with context of evaluation, accounting for the reasoning and facing their own revenge cases. Denying truth-bearing — each form met by a strengthened liar. **Narrowing recorded.** The hierarchy narrows the reach: a consistent truth predicate stands for an object language, defined in a metalanguage, rather than for the predicate as used. Paraconsistent treatments weaken the relation: inference is restricted so that both truth values stand without explosion. **Progress.** The undefinability theorem. The consistent fixed-point constructions. The graph of which revenge sentence meets which treatment. **Own record.** ## 56 Yablo's paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** An infinite list of sentences, each saying only that all the ones after it are untrue. No sentence refers to itself. The paradox arises anyway. What produces it? **The field's account of its hardness.** That the paradox demonstrates logical paradoxes without self-reference, only a certain kind of non-wellfoundedness being needed to obtain a contradiction, the ordinary paradoxes of self-reference involving a cyclic structure of reference where this one involves an acyclic but non-wellfounded structure; that building explicit hierarchies suffices to avoid circularity and to block the standard paradoxes, while blocking this one requires the hierarchy further to be well-founded, with a lowest level, the paradox being formalisable in a descending hierarchy of languages otherwise; and that whether liar-type paradox really requires some sort of circularity has been the subject of extensive debate. **Accounts.** Self-reference is not required: the construction is explicitly free of it, so whatever produces liar-type paradox is more general than circularity, quantification over an unbounded tail being enough. Circularity is present in a hidden form: the construction requires a fixed point and is therefore circular at the level of the property defined. **Specifications.** Paradox arises from a sentence's reference structure — local. Self-reference is identifiable in a construction — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about whether the construction is circular. **Conserving relation.** A determination of what produces paradox, standing over constructions generally. **Conserving reach.** All constructions producing paradox. **Reach.** The statement quantifies over what produces paradox in general. Particular constructions, analysed one at a time, are what is measured. The two are not the same. **Unreachability.** Whether the construction is circular is where the field's analyses disagree on one object, the graph-theoretic criteria classifying both liar and this construction while whether the graph property is circularity is the same dispute in new terms. The determination is not reached over constructions by analyses of particular ones. **Persistence.** A criterion of circularity the analyses agree on, applied to the construction. **Addresses.** Statement: Bolander, "Self-Reference and Paradox", Stanford Encyclopedia of Philosophy. The dispute over its circularity: Beall, "Is Yablo's paradox non-circular?", Analysis 61(3):176-187, 2001. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the paradox demonstrating logical paradoxes without self-reference, only a certain kind of non-wellfoundedness needed to obtain a contradiction; the ordinary paradoxes involving a cyclic structure of reference where this one involves an acyclic but non-wellfounded structure; explicit hierarchies sufficient to avoid circularity and block the standard paradoxes, this one blocked only where the hierarchy is further required to be well-founded, the paradox formalisable in a descending hierarchy of languages otherwise), the title carried as Self-Reference at the Spring 2010 and Spring 2015 editions and as Self-Reference and Paradox at the Fall 2024 and later editions, the entry itself not fetched, its section numbers and revision date not carried at the returns; the extensive debate whether liar-type paradox really requires some sort of circularity, at the encyclopedia's entry on the liar as its page carries it, returned in the search; the circularity paper not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the paradox demonstrating logical paradox without self-reference, a certain non-wellfoundedness sufficing; the dispute whether circularity is present in the defining property). Approaches: the inspection, fixed-point and graph-theoretic routes at those passages and at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Inspecting for self-reference — no sentence refers to itself, by construction, which is agreed. Fixed-point analysis — locates circularity in the defining property, contested and not resolved. Graph-theoretic analysis — characterises paradoxicality by properties of the reference graph, giving a criterion that classifies both liar and this construction, whether the graph property is circularity being the same dispute in new terms. Arithmetical versions — constructed, showing the phenomenon in a formal setting. **Narrowing recorded.** Graph-theoretic analysis weakens the relation: paradoxicality stands characterised by a graph property in place of circularity determined. **Progress.** The construction. Its arithmetisation. The graph-theoretic criteria. The circularity question, documented as open. **Own record.** ## 57 Newcomb's problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A reliable predictor has already filled a box based on a prediction of your choice. Take one box or two? In the field's own posing the case sets two principles of choice against each other, the expected-utility principle recommending one box and the dominance principle recommending two; the paper that posed it records, verbatim: "To almost everyone it is perfectly clear and obvious what should be done", and that those who consider it divide almost evenly. **The field's account of its hardness.** The paper that posed the problem states the difficulty as the division of those who consider it: nearly everyone finds the right choice obvious, yet they split almost evenly, each half taking the other to be silly. It states that in the example the action is referred to in the explanation of which state obtains, that the state's obtaining and the performance of the action are both partly explained by a third thing, an earlier state of the agent, and that a fuller investigation would have to pursue yet more complicated examples. The field's later record states that a more rigorous survey finds a similar divide. **Accounts.** Take one: those who take one box receive far more, reliably, and a decision theory whose recommendation is refuted by the outcome distribution is not a good decision theory. Take two: the contents are already fixed and the choice does not change them, and taking both dominates — whatever is in the opaque box, more is got by taking both. **Specifications.** The predictor is reliable — local. The contents are fixed before the choice — local. Both principles — maximise expected utility given evidence, and dominance — are correct — local. **Specifications withdrawn.** **Common requirement.** That the two principles are both valid and that one applies here. **Conserving relation.** A decision principle governing where the two conflict, the governing standing across the cases. **Conserving reach.** All cases the principles are applied to. **Reach.** The statement quantifies over which decision principle governs when they conflict. Cases, judged by the principles in dispute, are what is measured. The two are not the same. **Unreachability.** Each decision theory handles the cases the other fails, which is two instruments returning opposite verdicts on the same cases. The governing is not preserved across the cases by either principle. **Persistence.** A principle handling the cases both theories are pressed with, or a criterion for which principle applies where. **Addresses.** Statement, hardness, and the recorded divide at Approaches: Nozick, "Newcomb's Problem and Two Principles of Choice", in Rescher (ed.), Essays in Honor of Carl G. Hempel, Reidel, Dordrecht, 1969, 114-146, at p. 117 as the field's reviews carry the passage verbatim and at its closing paragraph as the publisher's page carries it, the paper itself not fetched; the later survey as the field records it: Notre Dame Philosophical Reviews, review of Newcomb's Problem, 14 August 2019. Persistence, no landing: the closing condition at the founding paper's passage read at pass two (the two principles of choice each giving a determinate recommendation and the two recommendations conflicting, the divide the field records), a principle handling the cases both theories are pressed with standing there as what the divide asks. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Evidential decision theory — recommends one box, facing the smoking lesion case, where it gives an answer widely judged wrong. Causal decision theory — recommends two, facing this case, and facing cases where it recommends against outcomes it can foresee. Functional and updateless variants — developed to handle both, each with its own cases, none settled. Denying the setup — argues perfect prediction is incompatible with free choice, the problem restated with a merely reliable predictor. Empirical study — reports what people choose, split, which is not the statement. **Narrowing recorded.** Functional and updateless variants weaken the relation: a principle stands for a stated class of cases rather than governing where the two conflict. **Progress.** The case map of which theory fails which case. The constructed variants and the cases each meets. **Own record.** ## 58 Zeno's dichotomy **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** To cross a distance one must first cross half, and half of the remainder, without end. How is the crossing completed? **The field's account of its hardness.** That the resolution by convergence assumes the complete run can be broken down into an infinite series of half runs which can be summed, and the question the infinity-machine strand presses is whether any infinite series of actions, a supertask, can really be completed, its point being that the paradoxes offer no problem to mathematics but show mathematics inapplicable to space, time and motion, and if such a series cannot be completed the completed runs cannot correctly be described as modern mathematics describes them; that the two senses of completing a task, carrying out a final step and carrying out every step, coincide for finite tasks, where the intuitions about tasks are formed, and come apart for supertasks, the dichotomy completable in the second sense and not the first; and that the ancient answer by potential rather than actual infinity is hard to see as completely satisfactory, assuming a distinction never fully achieved. **Accounts.** The series converges: the infinite sum of intervals is finite, and the mathematics of limits dissolves the puzzle entirely. Convergence shows the sum is finite, not how infinitely many tasks are completed: the supertask question is distinct from the arithmetic and has its own literature, including constructions where a supertask yields a determinate but paradoxical final state. **Specifications.** A distance is composed of parts — local. Traversal is composed of traversals of parts — local. **Specifications withdrawn.** **Common requirement.** That the traversal decomposes into the stated sequence. **Conserving relation.** The final state of an infinite sequence of tasks, determined by the sequence. **Conserving reach.** All supertasks the decomposition admits. **Reach.** The statement quantifies over the completion of an infinite sequence of tasks. The arithmetic of the sequence, and physical measurements at finite resolution, are what is measured. The two are not the same. **Unreachability.** A supertask can leave the final state undetermined by the sequence, which the lamp constructions establish. The determination is not preserved over the supertasks by the arithmetic of the sequence. **Persistence.** A determination of the final state from the sequence for supertasks generally, or agreement that the arithmetic is what the question asks. **Addresses.** Statement: Huggett, "Zeno's Paradoxes", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the resolution assuming the complete run can be broken down into an infinite series of half runs which can be summed; the infinity-machine strand asking whether any infinite series of actions, a supertask, can really be completed, meant to show the paradoxes no problem to mathematics but mathematics inapplicable to space, time and motion; the ancient answer by potential infinity hard to see as completely satisfactory, assuming a distinction never fully achieved), the entry itself not fetched, its section numbers and revision date not carried at the returns; the two senses of completing a task, a final step carried out and every step carried out, coinciding for finite tasks and parting for supertasks, at the encyclopedia's entry on supertasks as its page carries it, returned in the search, its authors carried at a citing page as Manchak and Roberts. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the resolution assuming the run can be broken into an infinite series of half runs summed by arithmetic; the supertask literature holding the final state a distinct question). Approaches: the convergence and lamp-construction routes at the same entry's sections on the dichotomy and on supertasks, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Convergence — settles the arithmetic completely, the supertask literature holding this to be a distinct question. Lamp constructions and successors — show that a supertask can leave the final state undetermined by the sequence, establishing that the arithmetic does not always fix the end state. Discrete space or time — removes the decomposition, not established, and facing its own difficulties in relativistic settings. Denying the decomposition — holds the motion is not composed of sub-motions, reframing, the decomposition being what the argument constructs. **Narrowing recorded.** Convergence narrows the reach: the determination stands for the arithmetic of the sequence rather than for the completion of the tasks. **Progress.** Convergence. The supertask results, showing final states can be underdetermined. **Own record.** ## 59 The three-body problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether the motion of three bodies under their mutual gravitation admits a general solution. In the classical sense — an expression in known functions giving the positions at any later time — the field holds the question settled in the negative: no algebraic integrals exist beyond the classical ten, and the convergent series that does exist converges too slowly for use. In the qualitative sense the field now poses: what the long-term behaviour is — which orbits are stable, which periodic families exist, and how sensitivity to initial conditions bounds prediction. **The field's account of its hardness.** That the system is not integrable in general and its orbits are often chaotic, sensitive to initial conditions, periodic orbits existing in special cases, only three families of periodic solutions having been found in three centuries before a numerical search returned thirteen more; and that a reliable arbitrary-precision check finds at least seven of the fifteen reported orbits departing greatly from periodicity after a long enough interval, their reported initial conditions not accurate enough to predict periodic orbits and the orbits unstable, possibly the computational periodicity the check names. **Accounts.** No: there are no sufficient integrals for the general problem, and the dynamics is chaotic, nearby initial conditions diverging exponentially, so no closed form of the sought kind exists. **Specifications.** A solution means a closed-form expression in known functions — local. The initial conditions are exactly specified — local. **Specifications withdrawn.** **Common requirement.** That the system's future is determined by its present state, and that the question is whether that determination is expressible. **Conserving relation.** An expression giving positions at arbitrary later times, usable for arbitrary initial data. **Conserving reach.** All initial data, at arbitrary later times. **Reach.** The statement quantifies over positions at arbitrary later times, for arbitrary initial data. Numerical integration at finite precision, over finite times, is what is measured. The two are not the same. **Unreachability.** No additional algebraic integrals exist, proved; and the convergent series converges so slowly as to be useless for computation, which the literature states, answering the letter and not the use. The expression is not reached over the initial data by integrals or by the series. **Persistence.** An expression usable at arbitrary later times, or agreement on which sense of solution the question asks. **Addresses.** Statement, the integrals: Bruns, "Über die Integrale des Vielkörper-Problems", Acta Mathematica 11:25-96, 1887. The series: Sundman, "Mémoire sur le problème des trois corps", Acta Mathematica 36:105-179, 1912/13. The qualitative programme: Poincaré, Acta Mathematica 13:1-270, 1890. The periodic families: Šuvakov and Dmitrašinović, "Three Classes of Newtonian Three-Body Planar Periodic Orbits", Physical Review Letters 110(11):114301, 2013, doi 10.1103/PhysRevLett.110.114301, arXiv:1303.0181, at its abstract as the publisher's page, the arXiv page and an index carry it verbatim (a numerical search for periodic orbits of three equal masses in a plane with zero angular momentum; a topological classification into families falling into four classes, the three previously known families in one; fifteen initial conditions, thirteen distinct orbits), the paper itself not fetched. Hardness: the introduction of a later search paper as the arXiv copy carries it (the system not integrable in general and its orbits often chaotic, sensitive to initial conditions, periodic orbits existing in special cases; only three families of periodic solutions found in three centuries before the thirteen), Li and Liao, arXiv:1705.00527, 2017, and the stability check of the thirteen at its abstract as the publisher's page and the arXiv page carry it (at least seven of the fifteen reported orbits departing greatly from periodicity after a long enough interval under an arbitrary-precision check, their initial conditions not accurate enough to predict periodic orbits, the seven unstable, possibly the computational periodicity named there), Li and Liao, Science China Physics, Mechanics and Astronomy, 2014, doi 10.1007/s11433-014-5563-5, arXiv:1312.6796, both returned in the search, neither fetched; the integrals, series and qualitative-programme sources not read this pass. Persistence, no landing: the closing condition at the search paper's introduction read at pass two (the system not integrable in general, its orbits often chaotic and sensitive to initial conditions, periodic orbits existing in special cases), an expression usable at arbitrary later times standing there as what non-integrability and sensitivity withhold. Approaches: the integral search, the convergent series and the numerical routes at that introduction in part and at the sources the address names for the series and the integrals. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Search for integrals — no additional algebraic integrals, closed by proof. The convergent series — converges, and so slowly as to be useless for computation, which is stated in the literature. Numerical integration — accurate over intervals set by the Lyapunov time, after which precision requirements grow exponentially. Perturbation and stability theory — establish stability for most initial conditions in near-integrable regimes, a major positive result covering a restricted region. Restricted and special cases — solved exactly, each a measure-zero set. **Narrowing recorded.** Stability theory narrows the reach: the result stands for most initial conditions in near-integrable regimes. Special cases narrow the reach: exact solutions stand on measure-zero sets. Numerical integration narrows the reach: accuracy stands over intervals set by the Lyapunov time. **Progress.** The non-existence proof for additional integrals. The convergent series. The stability results in near-integrable regimes. The exact special solutions. **Own record.** ## 60 The Fermi paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** The galaxy is old and large. If technological life is common, some should be evident. None is. Why? **The field's account of its hardness.** That the paradox's premise is a search completeness the field has quantified as extremely low: the fraction of the multidimensional search volume examined to date is stated as similar to the ratio of the volume of a large hot tub or small swimming pool to that of the Earth's oceans, with the earlier estimate placing the sampled fraction between 5 × 10⁻²⁶ and 1.5 × 10⁻¹⁸ of the nine-dimensional haystack. **Accounts.** Life or technology is rare: a step in the sequence from chemistry to detectable technology is improbable enough to make us close to alone, and the absence of evidence is what rarity looks like. The absence is not evidence yet: the volume searched, in frequency, sky coverage, sensitivity, and duration, is a negligible fraction of the space, and quantitative accounting of the searched fraction shows it close to zero. **Specifications.** Technological civilisations would produce detectable signatures — local. Expansion or emission over galactic timescales is the expected behaviour — local. **Specifications withdrawn.** **Common requirement.** That a civilisation would be detectable by the means used, so that absence bears on abundance. **Conserving relation.** An inference from absence to abundance, standing on the fraction searched. **Conserving reach.** The search space in frequency, sky coverage, sensitivity, and duration. **Reach.** The statement quantifies over abundance. A small and quantified fraction of the search space is what is measured. The two are not the same. **Unreachability.** The searched fraction is quantified and small, which both accounts accept, and every route narrows without closing. The inference is not reached over the search space by the fraction searched. **Persistence.** A detection, or a searched fraction large enough that absence bears on abundance. **Addresses.** Statement, in substance: the field's review of the paradox, Ćirković, "Fermi's Paradox - The Last Challenge for Copernicanism?", Serbian Astronomical Journal 178:1-20, 2009, arXiv:0907.3432, at its abstract and introduction as a directory, an index and an arXiv rendering carry them verbatim (the paradox, or the Great Silence problem, arguably the oldest and crucial problem for the search for extraterrestrial intelligence; the key argument following the remark where is everybody; a formidable challenge for any framework assuming a naturalistic origin of life and intelligence; equivalent to the absence of cosmic miracles), the review itself not fetched; the Statement standing as it stood. Hardness and the searched fraction: Wright, Kanodia and Lubar, "How Much SETI Has Been Done? Finding Needles in the n-Dimensional Cosmic Haystack", Astronomical Journal 156:260, 2018, abstract and conclusions. Prior estimate: Tarter et al. 2010, as restated in the decadal search literature. Persistence, no landing: the closing condition at the field's review read at v342 (the paradox a formidable challenge for any naturalistic framework; the absence equivalent to the absence of cosmic miracles), a detection or a searched fraction bearing on abundance standing there as what the review's own posing asks. Approaches: the radio and technosignature routes at the sources the address names, read at pass one, their searched fractions there. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Radio search — sixty years, no confirmed detection, the searched fraction quantified as a small volume of a large space, which both accounts accept. Technosignature searches — waste heat, megastructures, atmospheric markers, null so far, with sensitivity improving. Great Filter reasoning — locates the improbable step, where it lies unconstrained by any measurement, which its proponents state. Rare Earth arguments — assemble contingencies for our own case, sample size one, which the field notes. Anthropic and selection-effect arguments — show what our existence does and does not license inferring, constraining inference and supplying no abundance. **Narrowing recorded.** **Progress.** The quantified searched fraction. The sixty-year null record with its sensitivity bounds. The technosignature limits. **Own record.** ## 61 Poverty of the stimulus **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Children acquire grammar rapidly from limited and imperfect input. Is grammatical knowledge substantially innate? In the field's own posing of the argument's strongest form: speakers attain knowledge of the structure of their language for which the evidence, though an adult linguist searching intensively could find it, does not occur in conversational data often enough to guarantee that any particular child encounters it; the conclusion the argument is meant to reach is that infants are equipped with innate mental mechanisms specific to language acquisition. **The field's account of its hardness.** The field's assessment of the argument states that the one clear thing about it is its conclusion, that the facts of acquisition support the nativist rather than the empiricist view, and that what is not clear at all is the structure of the reasoning meant to support that conclusion; that a comprehensive critical discussion would take a full-length book, so the assessment selects one argument type; and that its finding is that linguists have additional work to do to sustain the claim of support for nativism, with reasons to think the further work will undermine the position rather than support it. **Accounts.** Yes: the input underdetermines the grammar acquired, children converge on structure-dependent rules without evidence distinguishing them from structure-independent alternatives, and formal learnability results show unconstrained learners cannot converge from positive evidence alone. No, or much less than claimed: corpus studies find the allegedly absent evidence present at low but non-zero frequency, statistical learners trained on realistic corpora acquire the structures in question, and the learnability results assume a class of grammars and a learning criterion that are choices. **Specifications.** There is a determinate grammar acquired — local. The input is characterisable as evidence for or against it — local. Learning is convergence on that grammar — local. **Specifications withdrawn.** **Common requirement.** That the acquired grammar and the available input can each be specified well enough to compare. **Conserving relation.** A separation of what the learner brings from what the input supplies, the separation standing on the corpora and the models. **Conserving reach.** All the structures acquired. **Reach.** The statement quantifies over what the learner brings, as against what the input supplies. Corpora, acquisition timelines, and trained models are what is measured. The two are not the same. **Unreachability.** The corpus frequency and modelling results are read oppositely by the two accounts, which is the instruments disagreeing on the same evidence, and the learnability results hold relative to a class and a criterion, which the literature states. The separation is not reached over the structures by corpora, timelines, or trained models. **Persistence.** A measurement separating what the learner brings from what the input supplies, or agreement on what frequency of evidence suffices. **Addresses.** Statement and hardness: Pullum and Scholz, "Empirical assessment of stimulus poverty arguments", The Linguistic Review 19(1-2):9-50, 2002, at its opening as the field's pages carry it verbatim and at its abstract as the publisher's and repositories' pages carry it, the paper itself not fetched; the argument's explicit form as the paper cites it: Hornstein and Lightfoot, 1981, p. 9. Persistence, no landing: the closing condition at the assessment paper's opening and abstract read at pass one and at the hardness (the empirical premises of the argument unassessed; the frequency of relevant constructions in the input the point at issue). Approaches: the corpus-frequency and formal-learnability routes at that paper and the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Corpus frequency studies — find relevant constructions at low frequency, whether low frequency suffices being the dispute restated quantitatively. Formal learnability — a negative result, and positive results under different criteria and priors, each relative to its class and criterion, which the literature states. Computational modelling — statistical learners acquiring several target structures from realistic input, whether the architecture encodes the constraint in question being the recurring objection. Large-scale language models — acquire structure-dependence from text at large scale, the data-volume objection met with a matched-scale programme whose targeted results show generalization on the canonical phenomena at child-scale input without direct positive evidence, while remaining less data-efficient than children, with weaker generalizations; each account reads its half. Cross-linguistic and atypical acquisition studies — constrain timelines, both accounts accommodating the findings. **Narrowing recorded.** Formal learnability narrows the reach: the negative result stands for a stated class of grammars and a stated learning criterion. **Progress.** The measured frequencies of the relevant constructions. The learnability results under stated criteria. The matched-scale modelling results and their measured data-efficiency gap. **Own record.** ## 62 Morphogenesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A fertilised egg becomes a body with parts in the right places. What determines where each part goes? In the field's own posing: how the structures and shapes of organisms arise, the two most influential ideas being reaction-diffusion and positional information, with confusion remaining about the relationship between them. **The field's account of its hardness.** The field's review states that, much as has been written about the two concepts, some confusion still remains, in particular about the relationship between them, and it addresses that relationship by proposing a scheme of three distinct ways in which the two ideas work together to shape biological form rather than by parting them. **Accounts.** Positional information: cells read their position from concentration gradients of signalling molecules and interpret it against thresholds, gradients being measured, thresholds identified, and manipulating a gradient relocating the structure. Self-organisation: reaction-diffusion systems generate pattern with no pre-existing coordinates, produce the observed spacing and its scaling, and account for regeneration and for pattern in tissues where no gradient is found. **Specifications.** Pattern is specified by information distinguishable from the tissue that carries it — local. Position is a variable cells can read — local. **Specifications withdrawn.** **Common requirement.** That there is a determinant of pattern locatable either in a signal or in the dynamics. **Conserving relation.** A determinant of placement located in one of the two, the location standing across the tissues. **Conserving reach.** All tissues the question is asked of. **Reach.** The statement quantifies over what determines placement. Concentrations, expression domains, and the effects of perturbation are what is measured, all in tissue where both processes run together. The two are not the same. **Unreachability.** Both frameworks fit most datasets, which is the recurring finding, and hybrid models fit better than either. The location is not preserved across the tissues by measurements taken where both processes run together. **Persistence.** A measurement in tissue separating the two processes, or agreement that both operate and the question is their proportion. **Addresses.** Statement, hardness and persistence: Green and Sharpe, "Positional information and reaction-diffusion: two big ideas in developmental biology combine", Development 142(7):1203-1211, 2015, at its abstract as the journal's page carries it verbatim, the paper itself not fetched; the two founding papers as the field cites them: Turing, "The chemical basis of morphogenesis", Philosophical Transactions of the Royal Society B 237:37-72, 1952; Wolpert, "Positional information and the spatial pattern of cellular differentiation", Journal of Theoretical Biology 25:1-47, 1969. Approaches: the gradient-manipulation and reaction-diffusion routes at the review the address names, at its abstract read at pass one (the two big ideas combining), their endings there in part. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Gradient manipulation — relocates structures, establishing gradients are instructive where tested, without excluding self-organising dynamics operating on them. Reaction-diffusion identification — candidate Turing systems identified in digit spacing, hair follicle spacing, and palatal ridges, with predicted perturbation outcomes confirmed, establishing the mechanism in those cases. Precision measurement of gradients — readout precision near physical limits in the fly embryo, constraining how much information a gradient can carry. Synthetic reconstitution — organoids generating axes and structures without maternal gradients, establishing self-organisation is sufficient in those settings. Modelling — both frameworks fitting most datasets, hybrid models fitting better than either. **Narrowing recorded.** Reaction-diffusion identification narrows the reach: the mechanism stands established in the specific tissues identified. Synthetic reconstitution narrows the reach: sufficiency stands in the organoid settings. **Progress.** The measured gradients and their readout precision. The identified Turing systems with confirmed perturbation outcomes. The organoid axes formed without maternal gradients. **Own record.** ## 63 Homochirality **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Biological amino acids are almost all left-handed and sugars right-handed. Ordinary chemistry produces both equally. Why one hand? In the field's own posing: how biological homochirality, a signature of life, arose from a prebiotic world that presumably contained equal amounts of both mirror-image forms, chemical reactions, physical processes, or a combination of both supplying prebiotically relevant mechanisms for the enrichment of one form over the other. **The field's account of its hardness.** The field's review states that the fact has been known for more than a century while experimental studies began to probe its origin only in the late twentieth century, and that the mechanisms it surveys are mechanisms for the enrichment of one form over the other from a presumably racemic prebiotic world; the field's later record states that the origins of homochirality remain an unsolved puzzle. **Accounts.** Amplification of a small initial bias: autocatalytic reactions amplify tiny enantiomeric excesses to near-purity, demonstrated in a known reaction, and small biases are available from circularly polarised light or crystal surfaces. The initial bias may be accidental. A systematic cause: the weak interaction is parity-violating and makes one enantiomer very slightly more stable, and this is not accidental, a universal bias accounting for why the same hand appears everywhere rather than by chance. **Specifications.** The asymmetry originated once and was inherited — local. A mechanism either selects a hand or amplifies whatever bias is present — local. **Specifications withdrawn.** **Common requirement.** That the observed excess traces to a determinable origin. **Conserving relation.** The origin of the excess determined, the determination standing on measurable chemistry. **Conserving reach.** The initial excess, and the amplification acting on it. **Reach.** The statement quantifies over the initial excess, and whether a bias that small is amplifiable. Present chemistry, laboratory amplification, and meteoritic samples are what is measured. The two are not the same. **Unreachability.** Amplification is demonstrated and does not supply the initial bias; the parity-violation energy difference is calculated at order 10⁻¹⁷ in relative terms, and whether amplification can act on a bias that small over available timescales is disputed and is the crux. The determination is not reached over the initial excess by present chemistry or laboratory amplification. **Persistence.** A demonstration that amplification acts on a bias of the calculated magnitude over available timescales, or a measurement of the initial excess. **Addresses.** Statement and hardness: Blackmond, "The Origin of Biological Homochirality", Cold Spring Harbor Perspectives in Biology 11(3):a032540, 2019, at its abstract as the journal's and repositories' pages carry it verbatim, the paper itself not fetched; its earlier form: Blackmond, "The Origin of Biological Homochirality", Cold Spring Harbor Perspectives in Biology 2(5):a002147, 2010, at its abstract; the field's later record of the standing: Ozturk et al., "Origin of biological homochirality by crystallization of an RNA precursor on a magnetic surface", Science Advances, 2023, doi:10.1126/sciadv.adg8274, at its opening as the journal's page carries it. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (the origin of the bias and its amplification the two questions the field parts; amplification demonstrated, the initial bias not supplied). Approaches: the autocatalytic-amplification and parity-violation routes at that abstract and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Autocatalytic amplification — demonstrated conclusively, establishing amplification without supplying the initial bias. Parity-violation energy difference — calculated, of order 10⁻¹⁷ in relative terms, its sufficiency being the crux. Circularly polarised light — produces measurable excesses, astronomical sources identified in star-forming regions. Meteoritic amino acids — left-handed excesses in some samples, supporting an extraterrestrial bias and relocating rather than settling the origin. Crystallisation and surface effects — produce excesses readily, without a preferred direction. **Narrowing recorded.** **Progress.** The demonstrated amplification. The calculated energy difference. The measured excesses from polarised light and the identified astronomical sources. The meteoritic excesses. **Own record.** ## 64 Matter and antimatter asymmetry **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** The laws produce matter and antimatter nearly symmetrically. The universe is matter. Where did the antimatter go? In the field's own posing: the origin of the matter-antimatter asymmetry remains unknown, and the question is which scenario for baryogenesis produced it, advances in theory and improved experimental limits having ruled some scenarios out and put others in sharper focus. **The field's account of its hardness.** The field's review states that the origin of the asymmetry remains unknown; that continuing advances in theory and improved experimental limits have ruled out some scenarios for baryogenesis, among them sphaleron baryogenesis at the electroweak phase transition in the Standard Model; and that the success of cosmological inflation and the prospects for discovering supersymmetry at the collider have put other models in sharper focus, the review confining itself to the scenarios it considers most plausible. **Accounts.** Known physics is insufficient and new physics is required: the three conditions are agreed, and the measured CP violation in the Standard Model is smaller than the observed asymmetry by roughly ten orders of magnitude. **Specifications.** The initial state was symmetric — local. The asymmetry was generated dynamically — local. The three conditions are necessary and sufficient — universal under the framework. **Specifications withdrawn.** **Common requirement.** That the asymmetry was produced by a mechanism operating in the early universe. **Conserving relation.** A mechanism identified as having produced the asymmetry, the identification standing on measurement. **Conserving reach.** Conditions at the energies and epochs the mechanism operated at. **Reach.** The statement quantifies over conditions at energies and epochs not accessible. Collider energies, precision CP measurements, and cosmological observations of the resulting abundance are what is measured. The two are not the same. **Unreachability.** The measured Standard Model CP violation is insufficient by orders, and every proposed mechanism narrows without closing, several sitting at scales far above collider reach, which their proponents state. The identification is not reached over the epoch by measurements at accessible energies. **Persistence.** A mechanism distinguished from the others by measurement, or a detection at a scale one of them predicts. **Addresses.** Statement, hardness, and the three conditions at Accounts: Dine and Kusenko, "Origin of the matter-antimatter asymmetry", Reviews of Modern Physics 76(1):1-30, 2004, arXiv:hep-ph/0303065, at its abstract as the field's pages carry it verbatim and at its opening as a repository copy carries it, the paper itself not fetched. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (the three conditions; the Standard Model's violation quantitatively insufficient; the mechanisms each requiring physics beyond it), a mechanism distinguished by measurement standing there as what the review's survey does not yet find. Approaches: the Standard-Model, electroweak and leptogenesis routes at that abstract and the review's sections the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Standard Model CP violation — measured precisely in kaon, B, and D systems, established and quantitatively insufficient by many orders. Electroweak baryogenesis — requires a first-order transition, excluded in the minimal model by the measured Higgs mass, extensions remaining viable and constrained by collider and electric dipole moment searches. Leptogenesis — consistent and attractive, its scale typically far above collider reach, which its proponents state, with neutrinoless double beta decay supporting the required nature and searches null so far. Electric dipole moment searches — improving by orders of magnitude and null, excluding parameter space for several mechanisms. Direct search for baryon number violation — proton decay searches null, with lifetime limits now extremely long. **Narrowing recorded.** Electroweak baryogenesis narrows the reach: the mechanism stands for extended Higgs sectors, the minimal model excluded by the measured mass. **Progress.** The precisely measured CP violation and its magnitude relative to the asymmetry. The exclusion of the minimal electroweak mechanism. The dipole moment and proton decay limits. **Own record.** ## 65 The strong CP problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A parameter in the strong interaction could take any value between zero and 2π. Measurement bounds it below 10⁻¹⁰. Why is it so small? **The field's account of its hardness.** That the bound from the neutron's electric dipole moment puts the parameter many orders of magnitude below the natural expectation of order one in the absence of a protective mechanism, this hierarchy constituting the problem, sharpened by the existence of CP violation in the weak interactions, and therefore within the Standard Model there is no symmetry reason enforcing an approximately vanishing value; that the term is a total divergence one might imagine irrelevant to physics, which is not the case; and that anthropic ideas very clearly fail to explain it, any value in the allowed range having little influence on life, the problem appearing in multiverse theories more serious than the other fine-tuning problems and requiring a dynamical solution. **Accounts.** A mechanism sets it: a symmetry makes the parameter dynamical and relaxes it to zero, predicting a new particle, the mechanism being economical and having other uses. It may require no explanation: the parameter is a free constant and small values are permitted, the demand for explanation resting on a measure over parameter space that is stipulated, and decades of searches having returned nothing. **Specifications.** A dimensionless parameter's value calls for explanation when it is far from a typical value under an assumed measure — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about whether a value requires explanation. **Conserving relation.** A determination of whether the value requires explanation, standing on measurement. **Conserving reach.** The space of possible parameter values. **Reach.** The statement quantifies over a measure over possible parameter values. One measured value, bounded, is what is measured. The two are not the same. **Unreachability.** One value is measured and a measure over the space is not, the demand for explanation resting on a measure that is stipulated. The determination is not reached over the space by the one bounded value. **Persistence.** Detection of the predicted particle, or agreement on a measure over the parameter space. **Addresses.** Statement, the problem as posed with its proposed resolution: Peccei and Quinn, "Constraints imposed by CP conservation in the presence of pseudoparticles", Physical Review D 16:1791-1797, 1977, doi 10.1103/PhysRevD.16.1791, published 15 September 1977, the title carried at an abstract service and an index, and the companion letter, "CP conservation in the presence of pseudoparticles", Physical Review Letters 38:1440-1443, 1977, as the field's citations carry it. Hardness: filled at the review literature returned in the search, none fetched: a status review at its opening as the arXiv copy carries it (the bound many orders of magnitude below the natural expectation of order one absent a protective mechanism; the hierarchy constituting the problem; sharpened by weak CP violation; therefore no symmetry reason within the Standard Model), arXiv:2609.00620, its authors not carried; lectures on the problem at their opening (the term a total divergence one might imagine irrelevant, which is not the case), arXiv:hep-ph/0011376, its author not carried; a review of fine-tuning and anthropic mechanisms at its section on the lack of anthropic constraints (anthropic ideas very clearly failing for this parameter; the problem more serious than the other fine-tuning problems in multiverse theories, requiring a dynamical solution), arXiv:0710.4080, its authors not carried; a 2025 paper at its opening (a priori order one; no anthropic explanation; two classes of solutions), arXiv:2510.18951. The founding papers stand as read before: their abstracts as an abstract service and an index carry it verbatim and the letter's opening as an abstract service carries it stating the proposed mechanism's condition (strong CP conservation remaining a natural symmetry if the full Lagrangian possesses a chiral U(1) invariance; a natural result for any theory where at least one flavour of fermion acquires its mass through a Yukawa coupling) and no statement of what is hard about the parameter's smallness before the mechanism, neither fetched; the review literature's own posing of the smallness keyed at the next pass. Persistence, no landing: the closing condition at the review literature read at v342 (the problem requiring a dynamical solution in the fine-tuning review's judgment; two classes of solutions at the 2025 paper), detection of the predicted particle standing there as the test of the leading class. Approaches: the particle searches, the massless-quark route and the spontaneous-violation route at the status review and the 2025 paper read at v342, in part, their endings at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Searches for the predicted particle — haloscopes, helioscopes, and astrophysical bounds excluding substantial mass ranges and continuing. Massless up quark — would remove the problem, excluded by lattice calculation, closed. Spontaneously broken CP mechanisms — viable, with model-building costs, and no distinguishing observation. Anthropic reasoning — argues no explanation is needed, its critics noting the parameter has little anthropic relevance, which weakens the argument here specifically. Neutron dipole moment measurement — the bound that defines the problem, improving steadily and null. **Narrowing recorded.** **Progress.** The bound below 10⁻¹⁰. The excluded mass ranges. The lattice exclusion of the massless up quark. **Own record.** ## 66 The problem of time in quantum gravity **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Quantum mechanics treats time as an external parameter. General relativity treats it as part of the dynamical geometry. What is time in a theory of both? In the field's own statement, the problem originates in the fundamental conflict between the way the concept of time is used in quantum theory and the role it plays in a diffeomorphism-invariant theory like general relativity; the schemes for resolving it fall into three categories, time identified before quantising, time identified after quantising, and time playing no fundamental role at all. **The field's account of its hardness.** The field's survey states that the problem originates in the conflict between time's use in quantum theory and its role in a diffeomorphism-invariant theory, and that ten distinct schemes stand across three categories of resolution. The field's own proposal states the problem as the difficulty of defining a Hilbert space structure on states, and local observables on it, for a theory in which the spacetime metric is treated as a quantum field, so that no classical metrical or causal structure is present on spacetime. **Accounts.** Time is emergent and the fundamental equation is timeless: the constraint equation contains no time parameter, and apparent time arises from correlations between subsystems, one variable serving as a clock for the rest. A timeless fundamental theory has not been shown to recover ordinary dynamics, and the conditional-probability route has known technical problems: some programmes reintroduce a preferred time and reproduce general relativity's predictions in the tested regime. **Specifications.** A quantum theory requires a time parameter for evolution — universal under quantum mechanics. General covariance forbids a preferred one — universal under general relativity. **Specifications withdrawn.** **Common requirement.** That the two treatments must be reconciled in one theory covering both regimes. **Conserving relation.** One treatment of time covering both regimes, the covering standing where each theory is confirmed. **Conserving reach.** Both regimes, at the scales the reconciliation concerns. **Reach.** The statement quantifies over the structure of a theory at scales not accessible. The two theories separately, each confirmed in its own regime, are what is measured. The two are not the same. **Unreachability.** Semiclassical emergence recovers a time parameter in the limit where both theories already work, and no experiment probes the regime, which the field states. The covering is not reached over the regimes by measurements confirming each theory separately. **Persistence.** An observation reaching the regime, or a derivation of ordinary dynamics from a timeless theory agreed to succeed. **Addresses.** Statement and hardness, with the timeless category at Accounts: Isham, "Canonical Quantum Gravity and the Problem of Time", in Integrable Systems, Quantum Groups and Quantum Field Theories, NATO ASI Series C 409, Kluwer, 1993, 157, arXiv:gr-qc/9210011, at its abstract as the publisher's and the archive's pages carry it verbatim, the paper itself not fetched; the problem as the field's own proposal states it: Wald, "A Proposal for Solving the 'Problem of Time' in Canonical Quantum Gravity", Physical Review D 48:2377-2381, 1993, arXiv:gr-qc/9305024, at its abstract as the archive's page carries it verbatim. Persistence, no landing: the closing condition at the founding review's abstract and sections read at pass one and at the hardness (the disappearance of time from the constraint equation; the three categories of response, time before, after, or timeless). Approaches: the internal-clock, conditional-probability and timeless routes at those three categories as the review states them. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The constraint equation with an internal clock — faces operator ordering, the measure, and the question of which variable is the clock. Conditional probabilities — recovers evolution relative to a clock subsystem, early objections addressed and the interpretation remaining disputed. Shape dynamics and preferred-foliation approaches — reproduce general relativity where tested, at the cost of the symmetry the other account takes as fundamental. Semiclassical emergence — recovers a time parameter in the appropriate limit, with corrections, the limit being where both theories already work. Observation — no experiment probes the regime, which the field states. **Narrowing recorded.** Shape dynamics narrows the reach: the reproduction stands in the tested regime, at the cost of the symmetry. Semiclassical emergence narrows the reach: the recovery stands in the limit where both theories already work. **Progress.** The constraint equation and its structure. The conditional-probability recovery relative to a clock subsystem. The reproduction of general relativity in the tested regime by preferred-foliation approaches. **Own record.** ## 67 Chronic pain **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Pain persists in many patients after tissue damage has healed, or with no damage found. What produces it? **The field's account of its hardness.** That the central mechanism cannot be assumed in a patient: what can be assessed clinically are its features — spreading pain, hypersensitivity, sensitivity to touch, movement, pressure, or temperature — and not the mechanism itself. The field's third mechanistic descriptor is defined by absence of evidence for the other two, patients commonly present with overlapping features of more than one, and the descriptor is recorded as applied residually to those not fully meeting the other criteria. The field states that the aetiology of that category is not yet settled. **Accounts.** Central sensitisation: changes in the nervous system's own processing amplify and sustain signalling independently of peripheral input, the changes being measurable, and the pain real and neurally produced. Peripheral drivers persist and are missed: small-fibre pathology, occult inflammation, and nerve injury are found in substantial fractions of patients previously classed as centrally driven, and improved detection keeps moving cases from one category to the other. **Specifications.** Pain has a locatable driver, peripheral or central — local. Finding it determines the treatment — local. **Specifications withdrawn.** **Common requirement.** That the pain is produced by an identifiable process at one location or the other. **Conserving relation.** The driver located at one site or the other, the location standing on measurement in the individual patient. **Conserving reach.** All patients the question is asked of. **Reach.** The statement quantifies over which process drives the persistence in a given patient. Imaging, quantitative sensory testing, and biopsy are what is measured, each returning findings compatible with both accounts in most patients. The two are not the same. **Unreachability.** Imaging establishes central change without separating cause from consequence, which the field states, and biopsy moves a measured fraction between categories without closing. The location is not reached over the patients by imaging, sensory testing, or biopsy. **Persistence.** A measurement separating cause from consequence in the individual patient, or agreement that both drivers operate and the question is their proportion. **Addresses.** Statement, in substance: the field's review, Cohen, Vase and Hooten, "Chronic pain: an update on burden, best practices, and new advances", The Lancet 397(10289):2082-2097, 2021, doi 10.1016/S0140-6736(21)00393-7, at its abstract as the publisher's page, two university repositories, a database and an index carry it verbatim (chronic pain affecting more than thirty per cent of people worldwide by some studies; unlike acute pain, which carries survival value, best considered a disease; nociceptive, neuropathic and nociplastic, the last from a sensitised nervous system, with considerable overlap so that many experts consider the classification a continuum) and at a passage of its text as the publisher's page carries it (once the acute danger period has passed the pain a burden, a disease unto itself; no clear threshold of when acute becomes chronic; pain persisting beyond the expected healing period generally accepted as chronic), the review itself not fetched; the Statement standing as it stood. Hardness, and the statement that the central mechanism cannot be assumed while its features can be assessed: the terminology editorial, Brazilian Journal of Physical Therapy, 2023, misconceptions table. The descriptor's definition by absence and its residual application: the same, and the non-neuronal review, PMC12886048, introduction. Overlapping presentation and unsettled aetiology: PMC12886048. The multifactorial model and the osteoarthritis case: the central sensitisation review, Lancet Rheumatology 2021. Prognostic association with outcomes after peripherally focused intervention: the critical reflection, Frontiers in Medicine 2026. Persistence after joint replacement: the arthroplasty study record, NCT05575310. Approaches: the blockade and biopsy routes at the review read at v342 in part (nociceptive, neuropathic and nociplastic pain, the last from a sensitised nervous system, the classification a continuum), their endings at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Peripheral blockade — relieves some patients and not others, read by both accounts as supporting theirs for those patients. Skin biopsy for small-fibre density — moves a measured fraction of patients into the peripheral category. Functional imaging — identifies central changes reliably, whether cause or consequence not settled by the imaging, which the field states. Clinical assessment of sensitization features — quantitative sensory testing and symptom-based instruments, assessing the features rather than the mechanism, which the field distinguishes explicitly. The third mechanistic descriptor — introduced as a pain phenotype alongside the nociceptive and neuropathic, defined by absence of clear evidence for either, and applied residually where the other criteria are not fully met. The multifactorial model — peripheral drivers and central mechanisms held to combine in the same condition, with osteoarthritis recorded as the worked case, and the field stating that the presence of central features should not stop the search for peripheral dysfunction. Prognostic use — sensitization features associated with poorer outcomes and greater risk of persistent pain after peripherally focused interventions, used to direct referral rather than to identify the driver. Persistence after structural removal — pain continuing after joint replacement where the implicated structures are no longer present, recorded as evidence for a central contribution. Non-neuronal mechanisms — glial and immune contributions to sensitization at both sites, a distinct mechanistic route. Psychological and multidisciplinary treatment — improves outcomes substantially without identifying a driver. Channel-targeted and growth-factor drugs — efficacy in specific conditions, defining subgroups rather than settling the general question. **Narrowing recorded.** Channel-targeted drugs narrow the reach: efficacy stands for specific conditions, defining subgroups. Biopsy narrows the reach: the peripheral location stands established for the measured fraction. **Progress.** The measurable central changes. The measured fraction reclassified by biopsy. The outcome improvements from multidisciplinary treatment. The subgroup efficacy results. **Own record.** ## 68 The axiom of choice **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether the axiom of choice is to be accepted: that for any collection of non-empty sets a set exists containing one element from each. The axiom is independent of the remaining axioms of set theory, so its standing is settled by no proof, and the field's question is whether it is true, or which of its consequences — every set well-orderable, non-measurable sets, a ball decomposed into finitely many pieces reassembling into two — are to be accepted at the cost of the others. **The field's account of its hardness.** That the axiom, hailed as probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to the axiom of parallels, appears humdrum and even self-evident as usually stated, amounting to the claim that a transversal can be assembled from any collection of mutually disjoint nonempty sets; and that although its usefulness quickly became clear, doubts about its soundness remained, reinforced by strikingly counterintuitive consequences, most spectacularly the paradoxical decompositions of the sphere into finitely many pieces reassembling into two spheres of the same size or into a sphere of arbitrary size, the question of soundness put to rest only by the proof of its consistency relative to the other axioms. **Accounts.** Yes, and the axiom is indispensable: without it standard results fail — every vector space having a basis, every field an algebraic closure, arbitrary products of compact spaces compact, countable unions of countable sets countable — and mathematics as practised uses it constantly. Its consequences include results widely regarded as unacceptable: a solid ball decomposes into finitely many pieces reassembling into two balls of the same size. The pieces are non-measurable, so no contradiction follows, and constructive mathematics proceeds without the axiom at known cost. **Specifications.** A selection may be asserted without a rule producing it — local. Sets exist independently of any construction of them — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about which axioms hold of the sets. **Conserving relation.** A determination of whether the axiom holds of the sets, standing on the other axioms. **Conserving reach.** All models of the other axioms. **Reach.** The statement quantifies over whether a selection may be asserted where no rule produces it. Consequences of chosen axiom systems are what is measured. The two are not the same. **Unreachability.** Independence is proved from both directions; and the dispute reduces to whether non-measurable sets are acceptable, which is the original dispute. The determination is not reached over the models by consequences of chosen systems. **Persistence.** A criterion for the truth of the axiom beyond its consistency, or agreement on whether non-measurable sets are acceptable. **Addresses.** Statement: Bell, "The Axiom of Choice", Stanford Encyclopedia of Philosophy. Consistency: Gödel, Proceedings of the National Academy of Sciences 24:556-557, 1938. Independence: Cohen, Set Theory and the Continuum Hypothesis, Benjamin, 1966, chapter IV. Hardness: the same entry, at its opening as an index carries it verbatim and at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the axiom hailed as probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to the axiom of parallels, yet humdrum, even self-evident as usually stated, amounting to the claim that a transversal can be assembled from any collection of mutually disjoint nonempty sets; its usefulness quickly clear while doubts about its soundness remained, reinforced by strikingly counterintuitive consequences, most spectacularly the paradoxical decompositions of the sphere; the question of soundness put to rest with the proof of consistency relative to the other axioms), the entry itself not fetched, its section numbers and revision date not carried at the returns; the consistency and independence sources not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the axiom probably the most interesting and most discussed since the parallel postulate; its consistency and independence proved; the non-measurable sets among its consequences), a criterion for truth beyond consistency standing there as what the independence leaves. Approaches: the proof-or-refutation, weaker-principle and determinacy routes at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Proving or refuting from the other axioms — consistency and independence both proved, closed by proof. Weaker choice principles — countable choice, dependent choice, the ultrafilter lemma, each sufficing for part of the practice and not all, the hierarchy mapped precisely. Determinacy axioms — contradict full choice, imply all sets of reals are measurable, and remove the decomposition, consistent relative to large cardinals, the conflict with choice being the trade. Constructive mathematics — proceeds without the axiom, the obtainable results precisely characterised, a known and exact cost. Locating the fault in measurability — the decomposition using non-measurable sets, whose existence follows from the axiom, the dispute becoming whether such sets are acceptable. **Narrowing recorded.** Weaker choice principles narrow the reach: each stands for the part of the practice it suffices for, the hierarchy mapped precisely. Constructive mathematics narrows the reach: the obtainable results stand characterised without the axiom. **Progress.** The two independence results. The mapped hierarchy of weaker principles. The characterised constructive results. The determinacy alternative and its stated trade. **Own record.** ## 69 Skolem's paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Set theory proves uncountable sets exist, and has a countable model. How can a countable model contain uncountable sets? **The field's account of its hardness.** That on the surface there is a straightforward contradiction, an element of a countable model looking uncountable from one perspective and clearly countable from another; and that as long as the basic set-theoretic notions are characterized simply by the model theory of first-order axiomatizations of set theory, many of those notions, countability and uncountability in particular, turn out to be unavoidably relative, which is the content of the claim that axiomatizing set theory leads to a relativity of set-theoretic notions, a claim to be distinguished from the more trivial relativity by which the elements that count as sets in one model may not count as sets in another. **Accounts.** There is no paradox: uncountability is a statement about the absence of a bijection, and the required bijection exists outside the model but not inside it, countability being relative to a model. The relativity is the difficulty, not its removal: if "uncountable" holds relative to a model, then either there is an absolute notion the models approximate, or set-theoretic language does not determine its own subject matter. **Specifications.** First-order axioms characterise the structures they are about — local. Cardinality is a property of a set — local. **Specifications withdrawn.** **Common requirement.** That the axioms are about something, and the question is whether they determine it. **Conserving relation.** A determination of the subject matter by the axioms, standing across the models. **Conserving reach.** All models satisfying the axioms. **Reach.** The statement quantifies over whether the axioms fix their subject matter. Models satisfying the axioms, of which there are many non-isomorphic ones, are what is measured. The two are not the same. **Unreachability.** Relativisation removes the contradiction and both accounts accept it, the disagreement being about what follows; and second-order categoricity presupposes the set theory in question. The determination is not reached over the models by the models themselves. **Persistence.** A characterisation of the subject matter not presupposing it, or agreement on what follows from the relativity. **Addresses.** Statement: Bays, "Skolem's Paradox", Stanford Encyclopedia of Philosophy; the paradox's source, Skolem 1922, as that entry carries it. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the simple formulation, an element of a countable model looking uncountable from one perspective and clearly countable from another, on the surface a straightforward contradiction; as long as the basic set-theoretic notions are characterized simply by the model theory of first-order axiomatizations of set theory, countability and uncountability among the notions turning out unavoidably relative, the content of the claim that axiomatizing set theory leads to a relativity of set-theoretic notions, distinguished from the more trivial relativity by which what counts as a set in one model may not in another), the entry itself not fetched, its section numbers and revision date not carried at the returns; the paradox's source not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (an element of a countable model looking uncountable from one perspective and countable from another; the relativity of set-theoretic notions; the dispute over what follows). Approaches: the relativisation and second-order routes at the same entry's sections on the resolutions, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Relativisation — removes the contradiction completely, accepted by both accounts, the disagreement being about what follows. Second-order logic — categorical characterisations becoming available, its semantics presupposing the set theory in question, which is the standing objection. Categoricity results — full second-order set theory quasi-categorical, given a background notion of set, the same objection at one remove. Structuralism — holds that only structure is determined and nothing further is missing, whether it answers the determinacy question disputed. The relativity conclusion — that set-theoretic notions are relative, held by a minority and not refuted. **Narrowing recorded.** Second-order categoricity narrows the reach: the characterisation stands given a background notion of set. **Progress.** The relativisation. The categoricity results and their stated presupposition. The map of non-isomorphic models. **Own record.** ## 70 Grue **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Evidence that all observed emeralds are green equally supports the prediction that they are grue — green if observed before a time, blue if after. Why does one projection succeed and not the other? **The field's account of its hardness.** That the paradox makes clear that something is wrong with instance confirmation and enumerative induction as initially characterized, no one taking the grue hypothesis to be well supported, neither the grue evidence statements nor the hypothesis entailing that any emerald changes colour, and the paradox not resolvable by looking to background knowledge, its unlikelihood merely restating the point; that the time parameter in the definition is sometimes advanced as a criticism, while similar examples are plentiful and it is commonly agreed that a purely syntactical definition of confirmation will not do; and that the proposed distinction of projectible predicates by their entrenchment comes very close to saying that the projectible predicates are the ones that are in fact projected. **Accounts.** Predicates differ in projectibility, and entrenchment settles which: predicates that have figured in successful past inductions are entrenched, and entrenchment is a matter of record. Entrenchment describes practice without justifying it: the grue predicate is definable from green and a time, and green is equally definable from grue and bleen, so the asymmetry is not in the logic, and every proposed non-circular criterion of naturalness has been contested. **Specifications.** Evidence supports a hypothesis by a relation statable in logical terms — universal under confirmation theory. The vocabulary is given — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about which predicates support projection. **Conserving relation.** A criterion of projectibility, standing without presupposing the vocabulary it selects. **Conserving reach.** All predicates the projection is asked of. **Reach.** The statement quantifies over which vocabulary the world is described in. Observations, describable in either vocabulary, are what is measured. The two are not the same. **Unreachability.** The interdefinability makes the asymmetry symmetric in the logic; entrenchment describes practice; and naturalness rankings and priors encode the asymmetry rather than deriving it, which their proponents accept. The criterion is not reached over the predicates by observations describable in either vocabulary. **Persistence.** A criterion of projectibility not presupposing the vocabulary it selects. **Addresses.** Statement: Goodman, Fact, Fiction, and Forecast, Harvard, 1955, chapter 3. The survey: Crupi, "Confirmation", Stanford Encyclopedia of Philosophy. Hardness: the encyclopedia's entry on the problem of induction, at passages of its text on the grue paradox as its archived editions carry them verbatim (the paradox making clear that something is wrong with instance confirmation and enumerative induction as initially characterized; no one taking the grue hypothesis well supported; neither the grue evidence statements nor the hypothesis entailing that any emerald changes colour; the paradox not resolvable by looking to background knowledge, its unlikelihood merely restating the point; the time parameter sometimes advanced as a criticism of the definition; the account of projectibility, roughly, that projectible hypotheses are made up of predicates with a history of use in projections), its author not carried at the returns, and the Internet Encyclopedia of Philosophy's entry on confirmation and induction as its page carries it (similar examples plentiful and a purely syntactical definition of confirmation commonly agreed not to do; projectibility by entrenchment coming very close to saying the projectible predicates are the ones in fact projected), its author not carried at the return, both returned in the search, neither fetched; the survey the address names not returned at its passages on the riddle, and the founding chapter, not read this pass. Persistence, no landing: the closing condition at the induction entry's passages on the paradox read at pass two (something wrong with instance confirmation and enumerative induction as stated; the projectible predicates to be parted from the others on some ground). Approaches: the entrenchment and naturalness routes at those passages and at the founding source the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Entrenchment — describes practice, its status as justification being the dispute. Natural kinds and natural properties — a naturalness ranking supplying the asymmetry, at the cost of a primitive the account does not derive. Simplicity of definition — the grue predicate requiring a time reference, the interdefinability showing this is symmetric, which the construction is built to show. Causal or nomic accounts — project only predicates in laws, laws being stated in projectible predicates, which is the circle. Bayesian priors — assign lower prior to grue-like hypotheses, the prior encoding the asymmetry rather than deriving it, which its proponents accept. **Narrowing recorded.** Naturalness rankings weaken the relation: the asymmetry stands supplied by a primitive rather than derived. Bayesian priors weaken the relation: the asymmetry stands encoded in a prior rather than derived. **Progress.** The interdefinability result. The entrenchment record. The mapped costs of each proposed criterion. **Own record.** ## 71 The ravens **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** "All ravens are black" is logically equivalent to "all non-black things are non-ravens." Does observing a white shoe confirm that all ravens are black? **The field's account of its hardness.** That one would want to say the hypothesis is not confirmed alike by a black raven and by a non-black non-raven, and the satisfaction theory of confirmation is unable to draw the distinction; that the hypothetico-deductive view blocks the derivation, the hypothesis not entailing that a generally sampled object will be a black raven, and reads the instance-confirmation insight differently, the evidence confirming only relative to the auxiliary assumption that the object was taken among ravens; and that the view fares well enough with the paradox at the troubling cost that the observation of a white swan, or of a million, does not confirm on it the prediction that a further swan will be white. **Accounts.** Yes, by a negligible amount: the confirmation is real and tiny, proportional to the relative sizes of the classes, and the appearance of absurdity is a failure of intuition about small quantities. The calculation depends on assumptions about the sampling and the class sizes that are supplied rather than derived, and reverses under different but equally reasonable assumptions: the relation between a generalisation and its instances is not settled by the calculation. **Specifications.** Confirmation is a relation between statements, invariant under logical equivalence — universal under confirmation theory. Instances confirm generalisations — universal under confirmation theory. **Specifications withdrawn.** **Common requirement.** That there is a confirmation relation with determinate values. **Conserving relation.** A confirmation relation with determinate values, standing independently of assumptions supplied. **Conserving reach.** The confirmation relation in general. **Reach.** The statement quantifies over the confirmation relation in general. Particular calculations under particular assumptions are what is measured. The two are not the same. **Unreachability.** The calculation reverses under equally reasonable assumptions, and whether the shoe was drawn from non-black things or from shoes changes the result, which establishes that the relation depends on how evidence is gathered, not stated in the original statement. The determinacy is not reached over the relation by calculations under supplied assumptions. **Persistence.** A confirmation relation whose values do not depend on assumptions supplied from outside the statement. **Addresses.** Statement: Hempel, "Studies in the Logic of Confirmation", Mind 54:1-26 and 97-121, 1945. The survey: Crupi, "Confirmation", Stanford Encyclopedia of Philosophy; the Bayesian treatments: Fitelson and Hawthorne, "How Bayesian confirmation theory handles the paradox of the ravens", in The Place of Probability in Science, Springer, 2010, 247-275. Hardness: the survey the address names, at passages of its text as the encyclopedia's page, its archived editions and a mirror carry them verbatim (the paradox stated at a black raven and a non-black non-raven, one wanting to say the hypothesis is not confirmed by both alike and the satisfaction theory unable to draw the distinction; the hypothetico-deductive view blocking the derivation, the hypothesis not entailing that a generally sampled object will be a black raven, and reading the instance-confirmation insight differently, the evidence confirming relative to the auxiliary assumption that the object was taken among ravens; the view faring well enough with the paradox at the troubling conclusion that a white swan, or a million, does not confirm on it that a further swan will be white), the entry itself not fetched, its section numbers and revision date not carried at the returns; the founding paper and the Bayesian treatments not read this pass. Persistence, no landing: the closing condition at the survey read at pass two (the paradox stated at a black raven and a non-black non-raven; the Bayesian resolution's dependence on assumptions about the relative sizes of the classes), a confirmation relation not depending on assumptions supplied from outside standing there as what the survey records each resolution as lacking. Approaches: the Bayesian and equivalence-denying routes at the same survey's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Bayesian quantification — yields tiny confirmation under standard assumptions, the assumptions being the disputed point, reversals exhibited in the literature. Denying equivalence-invariance — blocks the paradox and gives up a condition most accounts regard as basic. Sampling considerations — establish that the relation depends on how evidence is gathered, which is not in the original statement. Selective confirmation — restricts confirmation to instances of the subject class, ad hoc by its critics' account, and blocks the paradox. Rejecting the confirmation-relation framework — taken by some, the alternative accounts carrying their own open problems. **Narrowing recorded.** Selective confirmation narrows the reach: confirmation stands for instances of the subject class. Denying equivalence-invariance weakens the relation: a condition most accounts regard as basic is given up. **Progress.** The quantified confirmation under standard assumptions. The exhibited reversals. The demonstrated dependence on sampling. **Own record.** ## 72 Personal identity over time **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What makes a person at one time the same person as a person at another time — the persistence question — and what it is for that relation to hold across gradual replacement of parts, loss and recovery of memory, and the imagined cases of division. Variant formulation, as posed of any thing: a thing has all its parts replaced gradually; is it the same thing? — the question the field carries separately as material constitution. **The field's account of its hardness.** That reactions to the brain-transplant case may support only the claim that psychological continuity is what matters practically, compatible with other accounts of persistence, leaving it open whether there is any reason to accept the psychological approach; that fission is sometimes said to afflict all answers to the persistence question equally, apart perhaps from anticriterialism, whether it does being a hard question, and it looks especially worrying for the psychological approach, threatening that view's support without affecting the arguments for rivals, the non-branching view suggesting that the practical concerns one ordinarily has for oneself can apply to someone other than oneself; and that psychological-continuity views appear to rule out our being biological organisms, no sort of psychological continuity appearing either necessary or sufficient for a human organism to persist. **Accounts.** Psychological continuity: overlapping chains of memory, intention, and character constitute identity, the account matching judgments about ordinary cases and about imagined transfers. Biological continuity: a person is an organism, and the organism persists through psychological discontinuity — dreamless sleep, coma, early infancy — where the psychological account gives the wrong result or none. **Specifications.** There is a determinate fact about whether a later thing is the earlier one — local. A criterion states it — local. **Specifications withdrawn.** **Common requirement.** That identity over time is determinate and criterion-governed. **Conserving relation.** A criterion of identity over time, the criterion standing across the cases. **Conserving reach.** All cases, the fission cases included. **Reach.** The statement quantifies over a relation across time. States at times, and the relations between them, are what is measured, all compatible with both criteria. The two are not the same. **Unreachability.** A person divided into two psychologically continuous successors defeats simple versions of both criteria, which both accounts accept as decisive. The criterion is not preserved across the cases by either account in its simple form. **Persistence.** A criterion surviving the fission cases, or agreement that identity is not what the question asks about. **Addresses.** Statement: Olson, "Personal Identity", Stanford Encyclopedia of Philosophy. The division cases: Parfit, Reasons and Persons, Oxford, 1984, Part Three. Material constitution as the field parts it: Wasserman, "Material Constitution", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (reactions to the brain-transplant case possibly supporting only that psychological continuity is what matters practically, compatible with other accounts, leaving it open whether there is any reason to accept the psychological approach; fission sometimes said to afflict all answers to the persistence question equally, apart perhaps from anticriterialism, whether it does a hard question, and especially worrying for the psychological approach as threatening its support without affecting rivals; the non-branching view suggesting the practical concerns one has for oneself can apply to someone else; psychological-continuity views appearing to rule out our being biological organisms, no sort of psychological continuity necessary or sufficient for a human organism to persist), the entry itself not fetched, its section numbers and revision date not carried at the returns, the earlier archived editions carrying the Simple View where the later carry anticriterialism; the division cases and the constitution entry not read this pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Fission cases — both criteria giving unacceptable results, the case agreed decisive against simple versions. The what-matters response — identity not being what matters, continuity admitting of degree, widely influential and widely rejected as changing the question. Four-dimensionalism — persons extended in time and fission being two overlapping persons, consistent, at the cost of a metaphysics the other account rejects. Animalism — solves fission by the organism, facing transplant intuitions, which its proponents accept and bite. Denying determinacy — some cases having no answer, its critics holding that first-person survival does not admit of indeterminacy. **Narrowing recorded.** The what-matters response weakens the relation: continuity admitting of degree stands in place of a determinate criterion of identity. Denying determinacy narrows the reach: the criterion stands for the cases admitting an answer. **Progress.** The fission result against simple versions of both criteria. The mapped costs of each response. **Own record.** ## 73 Path dependence and lock-in **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Some technologies and institutions persist though better alternatives exist. Is the persistence inefficiency, or evidence that the alternative is not better? In the field's own posing: where two or more technologies with increasing returns to adoption compete for a market of potential adopters, insignificant events may by chance give one an early lead, which improves it further and widens its appeal, so that it may corner the market and lock the others out; static analysis locates the multiple equilibria but cannot say which will be selected, and the question is how random historical events during adoption cumulate to select one outcome. **The field's account of its hardness.** The founding paper states that allocation problems with increasing returns are known to exhibit multiple equilibria, so multiple outcomes are to be expected; that static analysis can typically locate the equilibria but usually cannot tell which one will be selected; and that only a dynamic approach admitting random events during adoption can examine how some sets of historical events cumulate to drive the process toward one outcome and others toward another. **Accounts.** Lock-in is real and inefficient: increasing returns, network effects, and switching costs make an early advantage self-reinforcing, the outcome needing to bear no relation to merit, and formal models producing it reliably. Claimed cases dissolve on inspection: re-examination of the standard examples found the alleged superior alternatives not superior on the relevant margins, and no case of remediable inefficiency demonstrated. **Specifications.** The alternatives can be compared on a standard independent of what was adopted — local. Adoption reflects or fails to reflect that standard — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about which alternative is better, statable independently of which was adopted. **Conserving relation.** A comparison of adopted and unadopted alternatives, standing at the scale the adopted one reached. **Conserving reach.** All cases the comparison is asked of. **Reach.** The statement quantifies over a counterfactual: how the unadopted alternative would have performed at scale. The adopted path, and small-scale or laboratory comparisons, are what is measured. The two are not the same. **Unreachability.** The counterfactual performance at scale is unobservable by construction, and the historical cases were re-examined and the re-examinations contested in turn, which is the field's instruments disagreeing on the same episodes. The comparison is not reached over the cases by the adopted path or by small-scale comparison. **Persistence.** A case of remediable inefficiency demonstrated at scale, or agreement on a standard of comparison independent of adoption. **Addresses.** Statement, hardness, and the increasing-returns mechanism at Accounts: Arthur, "Competing Technologies, Increasing Returns, and Lock-In by Historical Events", The Economic Journal 99(394):116-131, 1989, at its opening as the field's pages carry it verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the founding paper's opening read at pass one and at the hardness (competing technologies under increasing returns locked in by historical events, the outcome possibly inefficient and not predictable from the technologies' merits), a remediable inefficiency demonstrated at scale standing there as the empirical question the paper leaves. Approaches: the formal-modelling route at that paper; the historical-case route at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Formal modelling — produces lock-in under stated conditions, whether the conditions obtain being the empirical question. Historical case studies — the standard cases re-examined and contested, the re-examinations contested in turn, no case agreed. Laboratory experiments — produce lock-in reliably under controlled network effects, transfer to markets with entry and side payments disputed. Looking for remediable inefficiency — requires the counterfactual performance at scale, unobservable by construction. Ex post comparison after a switch — comparison available where switches occur, selection into switching being the confounding, which the field notes. **Narrowing recorded.** Laboratory experiments narrow the reach: lock-in stands established under controlled network effects. Ex post comparison narrows the reach: the comparison stands where switches occurred. **Progress.** The formal models and their stated conditions. The laboratory results under controlled network effects. The contested re-examination record. **Own record.** ## 74 The hermeneutic circle **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Understanding a part of a text requires understanding the whole; understanding the whole requires understanding the parts. How does interpretation begin? In the field's own statement, the circle's classical formulation is that interpretive experience involves a circular relation of whole and parts: a text must be understood as a whole to understand any of its parts, and understood in each of its parts to be understood as a whole; the foundational law of understanding, in the formulation the field records as the first, is to find the spirit of the whole through the individual and to grasp the individual through the whole. **The field's account of its hardness.** The field's survey states that the circle has been a prominent and recurring theme of the discussion since the circularity of interpretation was first drawn attention to, that in contemporary hermeneutics the circle is rarely restricted to text interpretation and its circularity is not necessarily cast in terms of whole and parts, and that even within hermeneutics it has been challenged whether interpretive experience is circular at all. **Accounts.** The circle is productive rather than vicious: interpretation proceeds by provisional understanding of the whole, revised as parts are read, and revised again — a spiral rather than a circle, and it is how understanding works. Naming it productive does not remove the problem of validation: if every reading is revisable in the light of the whole and the whole in the light of readings, no reading is ever adjudicated, and the criterion distinguishing better interpretation from worse is what the circle withholds. **Specifications.** A text has a determinate meaning — local. Interpretation approaches it — local. Parts and whole are distinguishable — local. **Specifications withdrawn.** **Common requirement.** That there is something the interpretation is of, against which readings can be better or worse. **Conserving relation.** A criterion for the correctness of a reading, standing outside the readings it adjudicates. **Conserving reach.** All readings the criterion is asked of. **Reach.** The statement quantifies over a criterion for the correctness of a reading. Readings, each already inside the circle, are what is measured. The two are not the same. **Unreachability.** Intention is reconstructed by interpretation and context is a text requiring interpretation, so each proposed criterion is itself inside the circle; and community-of-interpreters accounts give up the independent criterion, which their proponents accept. The criterion is not reached over the readings by readings taken from inside. **Persistence.** A criterion standing outside the readings it adjudicates, or agreement that no independent criterion is owed. **Addresses.** Statement and hardness: the entry "Hermeneutics", Stanford Encyclopedia of Philosophy, at its account of the hermeneutical circle as the encyclopedia's current page carries it verbatim and at the passage on the circle's history as the encyclopedia's archived editions carry it verbatim, the entry itself not fetched, its author and revision date keyed at the next pass; the first formulation as the entry cites it: Ast, 1808, p. 178; the principle as the entry cites it: Schleiermacher, lecture of 1829. Persistence, no landing: the closing condition at the same entry, at its account of the circle read at pass one and at the hardness (the understanding of the whole depending on the parts and of the parts on the whole; the circle's history from a method to a condition of understanding), a criterion standing outside the readings standing there as what the circle denies. Approaches: the intention and context criteria at the same entry's sections on the debate, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Authorial intention as the criterion — faces the difficulty that intention is itself reconstructed by interpretation. Historical context as the criterion — context being a text requiring interpretation, at one remove. Community of interpreters — validity being what the community accepts, accounting for practice and giving up the independent criterion, which its proponents accept. Philological method — establishes what the words meant at the time, with real results, underdetermining the readings in dispute, which are usually about significance rather than lexis. Rejecting the whole-part framing — reframes, the dependence reappearing at whatever units are chosen. **Narrowing recorded.** Philological method narrows the reach: the criterion stands for questions of lexis rather than for the readings in dispute. Community-of-interpreters accounts weaken the relation: validity stands as community acceptance rather than as an independent criterion. **Progress.** The philological results. The mapped position of each proposed criterion relative to the circle. **Own record.** ## 75 Hard cases in law **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Some legal cases are not settled by the applicable rules. Does the law contain an answer, or does the judge make one? In the field's own posing, a hard case is one that cannot be resolved by an unequivocal legal rule set out by the appropriate body before the event, where on the positivist theory the judge then has a discretion to decide the case either way; the thesis posed against that theory holds that the judge in such a case does not make law but is bound by standards the law already contains. **The field's account of its hardness.** The field's own guide to the debate states that even the core issue is unsettled: whether the debate is about the law containing principles as well as rules, about whether judges have discretion in hard cases, about the proper way to interpret legal texts, or about the very possibility of conceptual jurisprudence, so that identifying what the two sides are arguing about is itself the guide's task. **Accounts.** The law contains an answer: beyond rules there are principles with weight, and the correct answer is the one fitting and justifying the legal materials as a whole, judges seeking that fit rather than exercising discretion. Where rules run out the judge exercises discretion and makes new law: legal materials are open-textured, they underdetermine outcomes in exactly the hard cases, and disagreement among competent judges over the same materials is the evidence. **Specifications.** The legal materials either determine an outcome or do not — local. A judge's task is to apply the law — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about whether the materials determine the case. **Conserving relation.** A determination of whether the materials fix the outcome, standing separately from the outcome reached. **Conserving reach.** All hard cases. **Reach.** The statement quantifies over whether the materials determine an outcome. Decided cases, in which an outcome was reached either way, are what is measured. The two are not the same. **Unreachability.** Every hard case gets an outcome, which both accounts predict and read oppositely, and judges write as though finding rather than making, which both accounts accept and read differently. The determination is not separated from the outcome by decided cases. **Persistence.** A measurement separating whether the materials fixed the outcome from the fact that an outcome was reached. **Addresses.** Statement: Dworkin, "Hard Cases", Harvard Law Review 88:1057, 1975, at its definition of a hard case as the field's pages carry the sentence verbatim, the paper itself not fetched. Hardness: the working paper on the Hart-Dworkin debate, Public Law and Legal Theory Working Paper Series no. 77, at its opening as the repository copy carries it verbatim, its author keyed at the next pass. Persistence, no landing: the closing condition at the working paper's opening read at pass one and at the hardness (the debate whether the law runs out in hard cases and judges make it, or the materials fix the answer), a measurement separating the fixing from the reaching standing there as what neither side has produced. Approaches: the outcome-pointing and reasoning-analysis routes at that paper's account of the two positions. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Pointing to the decided outcome — every hard case getting an outcome, which both accounts predict and read oppositely. Analysing judicial reasoning — judges writing as though finding rather than making, both accounts accepting the practice and reading it differently. Empirical study of judicial behaviour — ideology predicting outcomes in contested cases with measurable effect, constraining the finding account without settling whether an answer existed. Inclusive positivism — admits moral criteria into the rule of recognition where a legal system so provides, narrowing the dispute and leaving the determinacy question. Critical legal studies — holds indeterminacy pervasive, its critics noting most cases are not litigated because the answer is clear, which is measurable and true. **Narrowing recorded.** Inclusive positivism narrows the reach: the account stands for systems whose rule of recognition admits moral criteria. **Progress.** The measured effect of ideology on contested outcomes. The litigation rates bearing on how many cases are clear. **Own record.** ## 76 Natural law and legal positivism **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is an unjust rule, properly enacted, law? In the field's own posing, legal positivism is the thesis that the existence and content of law depend on social facts and not on law's merits, formulated in the field's canonical sentence, verbatim: "The existence of law is one thing; its merit and demerit another"; the thesis does not hold law's merits unimportant to the philosophy of law but holds that they do not determine whether laws or legal systems exist, a society's having a legal system depending on the presence of certain structures of governance and not on the extent to which it satisfies ideals of justice. **The field's account of its hardness.** The field's survey states that the thesis is a thesis about the relation among laws, facts and merits and not otherwise a thesis about the individual relata; that the authority of principles of logic or of morality is not something legal philosophy has to explain, whereas the authority of enactments is, and that accounting for the difference is a central task of the philosophy of law; and that although the thesis is not an evaluation of its subject, it does not follow that legal philosophy offers a value-free description of it, there being no such thing, every description being value-laden. **Accounts.** Yes: law and morality are separate questions, what the law is being settled by social facts about enactment and acceptance, whether it is just being a further question, and keeping them separate being what makes criticism of law possible. Not entirely: a rule of sufficient injustice fails as law, and legal systems have an internal morality without which they do not function as legal systems at all, the post-war tribunals and the treatment of enacted atrocity standing as the cases. **Specifications.** There is a fact about what the law is, determined by criteria — local. The criteria are either exhausted by social facts or include moral ones — local. **Specifications withdrawn.** **Common requirement.** That "law" has a determinate extension. **Conserving relation.** Criteria of legal validity sorting rules into law and not-law, the sorting standing across the cases. **Conserving reach.** All rules the question is asked of, the enacted atrocities included. **Reach.** The statement quantifies over the criteria of legal validity. Legal practice, in which officials apply criteria they may state or not, is what is measured. The two are not the same. **Unreachability.** The grudge informer exchange produced two coherent treatments of one set of events reaching different characterisations, and comparative study documents both patterns extensively, supporting both readings. The sorting is not preserved across the cases by legal practice. **Persistence.** A case sorting differently under the two accounts and agreed to have been sorted correctly by one. **Addresses.** Statement and hardness: the entry "Legal Positivism", Stanford Encyclopedia of Philosophy, at its opening and at its statement of the positivist thesis as the encyclopedia's current page and archived editions carry them verbatim, the entry itself not fetched, its author and revision date keyed at the next pass; the canonical sentence, which the entry attributes to Austin, as the entry cites it: 1832, p. 157; the natural-law side's own posing not read this sitting and keyed at the next pass. Persistence, no landing: the closing condition at the same entry, at its opening and its statement of the thesis read at pass one and at the hardness (the existence and content of law depending on social facts and not on merits; the natural-law denial), a case sorting differently under the two accounts standing there as what the thesis's test would be. Approaches: the grudge-informer exchange and the rule-of-recognition route at the same entry's sections and the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The grudge informer exchange — each account producing a coherent treatment and reaching a different characterisation of the same events, documented and unresolved. Rule of recognition — locates validity in official practice, whether the practice can incorporate moral criteria being the inclusive-positivism debate, ongoing. Internal morality of law — the eight desiderata, their necessity widely accepted, whether they constitute a moral rather than an efficacy constraint disputed. Constitutional practice — many systems incorporating moral criteria explicitly, which one account accommodates as a contingent fact and the other reads as recognition. Comparative and historical study — documents both patterns extensively, supporting both readings. **Narrowing recorded.** Inclusive positivism narrows the reach: moral criteria stand incorporated where a system's rule of recognition so provides. **Progress.** The documented exchange and its two treatments. The eight desiderata and their accepted necessity. The comparative record. **Own record.** ## 77 Information as fundamental **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is physical reality made of information, or is information a description we apply to physical things? In the field's own posing, the question is how existence comes about, and the working hypothesis is stated in three words verbatim: "it from bit", that every physical quantity derives its ultimate significance from binary yes-or-no indications, no element in the description of physics showing itself as closer to primordial than the elementary quantum phenomenon, the device-intermediated act of posing a yes-no question and eliciting an answer. **The field's account of its hardness.** The founding report states that no escape is evident from four conclusions: that the world cannot be a giant machine ruled by any pre-established continuum physical law; that at the microscopic level there is no such thing as a space, time or spacetime continuum; that the probability function and wave equation of standard quantum theory provide mere continuum idealizations and by that circumstance conceal the information-theoretic source from which they derive; and that no element in the description of physics shows itself as closer to primordial than the elementary quantum phenomenon; and it states that, in default of a working hypothesis, the questions and clues do not move the search ahead, the hypothesis taken being the most effective one to have survived formulation after formulation of the central issues. **Accounts.** Information is fundamental: erasure carries a thermodynamic cost, measured experimentally at the predicted bound; black hole entropy scales with area in units of information; and quantum theory has been reconstructed from information-theoretic axioms. These are results about physical systems described informationally, not evidence that information is the substrate: information requires a physical carrier and an encoding, and the reconstruction results show quantum theory follows from principles about what can be done with systems, which presupposes systems. **Specifications.** There is a fact about what is fundamental — local. The relation between information and its carrier is one of dependence in one direction — local. **Specifications withdrawn.** **Common requirement.** That "fundamental" has a determinate sense in which one of these is it. **Conserving relation.** A dependence relation established in one direction between two descriptions. **Conserving reach.** All physical systems the question is asked of. **Reach.** The statement quantifies over a dependence relation between two descriptions, both of which apply. Physical systems, measured, and describable both ways, are what is measured. The two are not the same. **Unreachability.** The erasure bound and the area scaling are established and read both ways, and the reconstruction axioms quantify over systems, which is the standing objection. The dependence is not established over the systems by measurements describable in either description. **Persistence.** A distinguishing prediction between the two readings, which no informational ontology has produced. **Addresses.** Statement and hardness: Wheeler, "Information, Physics, Quantum: The Search for Links", in Proceedings of the 3rd International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, Physical Society of Japan, Tokyo, 1990, 354-368, reprinted in Zurek (ed.), Complexity, Entropy and the Physics of Information, Addison-Wesley, 1990, at its abstract as the repository's page carries it verbatim and at its section 19.2 as the field's pages carry the passage verbatim, the paper itself not fetched; the repository records the proceedings paper at pages 354-358 and 1989, the field's citations at pages 354-368 and 1990, each keyed as it carries it. Persistence, no landing: the closing condition at the founding essay read at pass one and at the hardness (the search for links between information, physics and quantum theory; every physical quantity deriving its ultimate significance from yes-no questions), a distinguishing prediction standing there as what the essay proposes to seek and no informational ontology has produced. Approaches: the erasure-principle and black-hole-thermodynamics routes at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The erasure principle — verified experimentally at the predicted bound, establishing a link and read both ways. Black hole thermodynamics — entropy scaling with area, a major result, its interpretation being the dispute at one more site. Information-theoretic reconstructions of quantum theory — derive the formalism from axioms about information processing, the axioms quantifying over systems, which is the standing objection. Informational ontology programmes — proposed, no distinguishing prediction produced. Digital physics and cellular automaton models — constrained by Lorentz invariance and by Bell correlations, specific proposals excluded. **Narrowing recorded.** Information-theoretic reconstructions narrow the reach: the derivation stands from axioms about what can be done with systems rather than establishing the dependence. **Progress.** The verified erasure bound. The area scaling of black hole entropy. The reconstruction results and their stated presupposition. The exclusion of specific digital proposals. **Own record.** ## 78 Meno's paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** If you know what you are looking for, you need not look. If you do not, you would not recognise it. How is inquiry possible? In the field's own posing, the paradox targets knowledge grounded in explanatory definitions, on which strict requirements are placed, and the challenge uses those constraints to argue that one can neither propose definitions nor recognise them; the field records the paradox in two versions and asks why there are two, and records the reply that inquiry is possible in the absence of knowledge where one relies on true belief, the theory of recollection making the distinct point of how inquiry can be successful. **The field's account of its hardness.** The field's studies state that the challenge is built from the constraints placed on knowledge and definition, knowledge requiring explanatory definitions that meet strict requirements, so that on those constraints definitions can neither be proposed nor recognised; that the response to the challenge has to be read as part of a larger disagreement about the value of inquiry; that the paradox stands in two versions whose relation is itself asked after; and the field's review records that a reply resting on true belief as sufficient for inquiry can seem too tidy for a puzzle that exercised the thinkers of antiquity. **Accounts.** By partial knowledge: inquiry begins from knowing enough to specify the question and not enough to answer it, the two horns falsely assuming knowledge is all or nothing. A detective knows the crime and not the culprit, a chemist knows the reaction and not the mechanism, and in each case the object is specified under one description and sought under another, recognition being by the specifying description, which the inquirer has. Specifying the question already presupposes the concepts the inquiry is meant to acquire, and the difficulty recurs at the level of concepts: acquiring a genuinely new concept cannot be described as recognising an answer to a question posed in prior terms. **Specifications.** Inquiry is the search for something specifiable in advance — local. Recognising a result requires a prior criterion — local. **Specifications withdrawn.** **Common requirement.** That inquiry has an object identifiable by the inquirer. **Conserving relation.** An account of acquisition, standing for what was not already available to the inquirer. **Conserving reach.** All acquisitions, the acquisition of primitives included. **Reach.** The statement quantifies over acquisition of what was not already available. Learners already equipped with concepts, studied acquiring further ones, are what is measured. The two are not the same. **Unreachability.** Partial knowledge dissolves the classical version and does not reach the acquisition version; statistical learning accounts for acquisition from data with the architecture supplying the primitives, which is the objection; and the nativist argument stands unrefuted rather than accepted. The account is not reached over the acquisitions by learners already equipped. **Persistence.** An account of primitive acquisition not presupposing the primitives, or a refutation of the argument that none is possible. **Addresses.** Statement and hardness: Ebrey, "Meno's Paradox in Context", British Journal for the History of Philosophy 22(1):4-24, 2014, at its abstract as the journal's and the repositories' pages carry it verbatim, the paper itself not fetched; the reply on true belief and recollection: Fine, The Possibility of Inquiry: Meno's Paradox from Socrates to Sextus, Oxford University Press, 2014, as the field's reviews carry its reading (Bryn Mawr Classical Review 2015.10.03; Notre Dame Philosophical Reviews, review of the book, its date not read), the book itself not fetched; the two versions as the field poses them: Shepardson, "Why Are There Two Versions of Meno's Paradox?", The Southern Journal of Philosophy 60(3):465-486, 2022, at its opening as the publisher's page carries it. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the paradox as an argument about inquiry into what one does not know; the classical dissolution by partial knowledge leaving the acquisition version), an account of primitive acquisition not presupposing the primitives standing there as what the acquisition version demands. Approaches: the partial-knowledge and recollection routes at that paper and the learning routes at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Partial knowledge — dissolves the classical version, not reaching the concept-acquisition version. Recollection — not taken up as literal by later accounts. Statistical and associative learning — accounts for acquisition of many categories from data, whether the primitives are learned or presupposed being the dispute, the architecture supplying them, which is the objection. Radical concept nativism — concludes primitives are innate, widely resisted, with no generally accepted refutation of the argument itself. Developmental studies — document the sequence of acquisition, establishing when rather than how the recognition problem is solved. **Narrowing recorded.** Partial knowledge narrows the reach: the account stands for acquisitions specifiable in prior terms. Statistical learning narrows the reach: acquisition stands accounted for where the architecture supplies the primitives. **Progress.** The dissolution of the classical version. The categories acquired by statistical learners from data. The documented acquisition sequences. **Own record.** ## 79 The equity premium puzzle **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Stocks have returned far more than safe bonds over long periods. Standard models say the gap should be small. Why is it large? **The field's account of its hardness.** That no reasonable values of the risk-aversion coefficient or the time discount factor reproduce the observed premium when the model's parameters are fitted to the historical consumption-growth data, so the discrepancy is between a prediction and a measurement rather than between two measurements. The problem's originators, reviewing the literature two decades on, reviewed the main classes of proposed explanation and recorded that none had produced a fully satisfactory resolution; the field's own survey states that the proposed resolutions fail along crucial dimensions. **Accounts.** The models are wrong about preferences: matching the observed premium with standard utility requires implausibly high risk aversion, and alternative preference specifications — habit formation, recursive utility, loss aversion — reproduce the premium with plausible parameters. The premium may be smaller than measured: survivorship bias in the sample of surviving markets, a possible peso problem in which rare disasters did not occur in the sample, and the ex post rather than ex ante nature of the measurement all inflate the estimate, and rare disaster models reproduce the premium with standard preferences. **Specifications.** Asset prices are set by a representative agent maximising expected utility over consumption — universal under asset pricing. Historical returns estimate expected returns — local. **Specifications withdrawn.** **Common requirement.** That there is a true expected premium the models should reproduce. **Conserving relation.** An expected premium reproduced by a model, the reproduction standing on parameters independently measured. **Conserving reach.** Expected returns, and preferences over states that did not occur. **Reach.** The statement quantifies over expected returns, and preferences over states that did not occur. Realised returns in samples that did occur are what is measured. The two are not the same. **Unreachability.** Alternative preference specifications fit by introducing parameters not independently measured, which the field states, and disaster calibration requires rates estimated from other histories and is not testable within the sample. The reproduction is not reached over the expected premium by realised returns. **Persistence.** A model reproducing the premium on parameters independently measured, or a measurement of expected rather than realised returns. **Addresses.** Statement, in substance: the founding paper, Mehra and Prescott, "The equity premium: A puzzle", Journal of Monetary Economics 15(2):145-161, 1985, at its abstract and introduction as the author's page, a university index and a copy carry them verbatim (restrictions that a class of general equilibrium models place upon the average returns of equity and Treasury bills strongly violated by the data over ninety years, the result robust to specification and measurement; for such economies the average real annual yield on equity at most four-tenths of a per cent above short-term debt, in sharp contrast to the six per cent premium observed; the largest premium obtainable in the model thirty-five hundredths of a per cent; an equilibrium model that is not an Arrow-Debreu economy most likely the one to rationalize both), the paper itself not fetched; the Statement standing as it stood. Hardness, and the failure of proposed resolutions along crucial dimensions: Mehra, "The Equity Premium: Why is it a Puzzle?", NBER Working Paper 9512, 2003, abstract. The twenty-year review of explanation classes and the originators' later leaning: Mehra and Prescott, "The equity premium in retrospect", Handbook of the Economics of Finance, 2003. The parametric-restriction counter-argument: Kocherlakota 1996, as reported at arXiv:1909.13019 §1. Disaster calibration to thirty-five countries: Barro 2006, as reported at the same address. Recent European decrease and the standard model's fit: the European-markets analysis reported at ResearchGate 228291610. Survivorship bias: Jorion and Goetzmann 1999, as cited at arXiv:cs/0512027 §D. Persistence, no landing: the closing condition at the founding paper's abstract and introduction read at v342 (the models' restrictions strongly violated by the data; an equilibrium model that is not an Arrow-Debreu economy most likely the one to rationalize both returns), a model reproducing the premium on independently measured parameters standing there as the paper's own closing. Approaches: the risk-aversion and non-standard-model routes at that introduction and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Raising risk aversion — reproduces the premium and produces the risk-free rate puzzle, the same problem at the other price, with the recorded counter-argument that both puzzles follow from the parametric restrictions the original model imposed on the discount factor and the aversion coefficient. Habit formation — a fitted specification, recorded in later work as predicting a value much higher than observed in recent European data. Recursive preferences — separate risk aversion from intertemporal substitution, each introducing parameters not independently measured. Rare disasters — the historical consumption sample recorded as unrepresentative, the originating country having experienced no peacetime peak-to-trough decline of the relevant size while many others did, with calibration to an international dataset of thirty-five countries reproducing the premium under standard preferences and not testable within the sample. Heterogeneous agents with borrowing constraints — the representative-agent assumption relaxed and a fraction of the population constrained from holding the optimal portfolio, generating larger premia and requiring additional structural assumptions about who is constrained and why; recorded as the originators' own later leaning. Behavioural explanations — loss aversion and myopic evaluation, surveyed as a class. Survivorship bias — the surviving-market sample proposed as inflating the estimate. Broadening the sample — cross-country and long-horizon datasets reducing the estimate somewhat, with a substantial decrease in European markets recorded after the recent recession and a standard model then reproducing the premium without extreme aversion. **Narrowing recorded.** Rare disaster models narrow the reach: the premium stands reproduced under disaster rates estimated from other histories. Heterogeneous-agent models weaken the relation: the required aversion falls at the cost of fitting parameters not independently measured. **Progress.** The measured premium across samples. The reduction under broadened samples. The fits obtained under each specification, with their parameter costs stated. **Own record.** ## 80 The Easterlin paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Within a country, richer people report greater wellbeing. Over time, as a country grows richer, average reported wellbeing does not rise correspondingly. Which is the relation? In the field's own posing, the paradox is the contradiction between observations on the relation of happiness to income at a point in time, across persons within a country, and the evidence on happiness and income over time: over the long term, usually a period of ten years or more, happiness does not increase as a country's income rises, while in the short term happiness and income go together, happiness tending to fall in economic contractions and to rise in expansions. **The field's account of its hardness.** The field's own statement of the paradox holds that the evidence for the long-term nil relation had been limited to developed countries and is presented also for a number of developing countries, for the countries transitioning from socialism to capitalism, and for a wider sample of developed countries; and that the critiques claiming a positive time-series relation between happiness and income are the result either of a statistical artefact or of a confusion of the short-term relation with the long-term one. **Accounts.** Relative income governs: wellbeing depends on position within a reference group, so raising everyone's income leaves positions unchanged and average reports flat, the cross-sectional and time-series findings then being consistent. The time-series flatness is a measurement artefact: larger international datasets under a log specification find no satiation point and a consistent income-wellbeing relation within and across countries and over time, the paradox arising from short series and bounded scales. **Specifications.** Reported wellbeing measures a quantity comparable across persons and times — local. A bounded scale can register unbounded change — local. **Specifications withdrawn.** **Common requirement.** That the reports measure the same thing in both comparisons. **Conserving relation.** One quantity measured alike in both comparisons, the sameness standing across populations and times. **Conserving reach.** Wellbeing, compared across times and populations. **Reach.** The statement quantifies over wellbeing, compared across times and populations. Responses on bounded scales, whose interpretation may shift with the population, are what is measured. The two are not the same. **Unreachability.** Both accounts read the extended series differently, which is the instruments disagreeing, and whether the log form is correct is not settled by data that fits both. The sameness is not preserved across populations and times by responses on bounded scales. **Persistence.** A measure whose interpretation is demonstrably stable across populations and times. **Addresses.** Statement and hardness: Easterlin, Angelescu McVey, Switek, Sawangfa and Zweig, "The happiness-income paradox revisited", Proceedings of the National Academy of Sciences 107(52):22463-22468, 2010, at its abstract as the journal's page carries it verbatim, the paper itself not fetched; the paradox stated as the cross-section against the time series: "The Easterlin Paradox", IZA Discussion Paper no. 13923, at its opening as the institute's page carries it, its authors not read. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the paradox holding over the long term and not the short in the authors' reading; the dispute over the series and the measure continuing), a measure demonstrably stable across populations and times standing there as what both sides read the series as lacking. Approaches: the series-extension and functional-form routes at that paper and the sources the address names for the other side. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Extending the time series — both accounts doing so and reading the extended series differently, the dispute continuing through successive datasets. Changing the functional form — log income removing the apparent satiation, whether the log form is correct not settled by data that fits both. Distinguishing evaluative from experienced wellbeing — the two measures behaving differently, evaluative tracking income further, refining the question without settling the original comparison. Anchoring vignettes — correct for scale interpretation across cultures and change some cross-country rankings, a genuine advance applied to a subset of datasets. Objective correlates — health, longevity, and time-use measures rising with income, accepted by both accounts and disputed as to whether they are the subject. **Narrowing recorded.** Anchoring vignettes narrow the reach: the correction stands for the subset of datasets to which it has been applied. Distinguishing evaluative from experienced wellbeing narrows the reach: each measure stands for its own construct rather than for the comparison as posed. **Progress.** The extended series. The measured behaviour of the two wellbeing constructs. The corrected cross-country rankings. The objective correlates. **Own record.** ## 81 Allais and Ellsberg **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** People make choices that violate expected utility axioms, consistently and on reflection. Are the axioms wrong, or the choices? In the field's own posing, the ambiguity side asks the question its founding paper sets as its first section heading, verbatim: "Are there uncertainties that are not risks?", against the sets of constraints on choice under uncertainty, all more or less equivalent in spirit, which carry the implication that for a rational man every uncertainty can be reduced to a risk; the feeling that people behave as though they assigned numerical probabilities even where the frequencies are unknown is what the constraints formalise. **The field's account of its hardness.** The founding paper of the ambiguity side states that there is no obvious basis for asserting that a decision-maker who violates the axioms will in the long run reach worse outcomes than he could expect if he reversed some of his preferences to conform to them; and the field's history records the reply that the violations are due to mistakes or misinterpretations and would disappear if these were avoided or corrected, and that the reply was countered. **Accounts.** The axioms describe rationality and the choices are errors: independence follows from a sure-thing principle most endorse when stated abstractly, a person violating it can be presented with a sequence of trades leaving them worse off by their own lights, and many shown the inconsistency revise. A theory of rational choice answers to what people would endorse on reflection, not to what they do. Many do not revise on reflection, and the ambiguity pattern reflects a real distinction between risk with known probabilities and ambiguity without them, which the axioms cannot represent. A normative theory that competent reflective agents reject is not obviously the normative one. **Specifications.** Preferences over outcomes are represented by a utility function and probabilities — universal under decision theory. The independence axiom holds — local. **Specifications withdrawn.** **Common requirement.** That there is a standard of correct choice against which the pattern is right or wrong. **Conserving relation.** A standard of correct choice, standing against the pattern rather than fitted to it. **Conserving reach.** All the choice patterns the axioms are asked of. **Reach.** The statement quantifies over whether the axioms state the standard. Observed choices, and reports after being shown the inconsistency, are what is measured. The two are not the same. **Unreachability.** Some revise and some do not, and the split is stable across studies; and the accommodating theories introduce functions fitted to the data they explain, which their authors state. The standard is not reached over the patterns by observed choices or by post-reflection reports. **Persistence.** A standard the competent reflective agents who reject the axioms would endorse, or agreement on what settles a normative question empirically. **Addresses.** Statement, the ambiguity side, and hardness: Ellsberg, "Risk, Ambiguity, and the Savage Axioms", The Quarterly Journal of Economics 75(4):643-669, 1961, at its section headings as the publisher's page carries them verbatim and at its opening and closing passages as a reader's page carries them verbatim, the paper itself not fetched; the reply and its counter as the field's history records them: Moscati, "Ellsberg 1961: text, context, influence", at its opening as the repository page carries it, its venue and year not read; the risk side: Allais, "Le comportement de l'homme rationnel devant le risque: critique des postulats et axiomes de l'ecole americaine", Econometrica 21(4):503-546, 1953, as the society's page carries the citation, the paper's own posing not returned this sitting and keyed at the next pass. Persistence, no landing: the closing condition at the founding paper's opening read at pass one and at the hardness (the deliberate violation of the axioms by reflective agents, the question whether the axioms or the choices should yield), a standard such agents would endorse standing there as the paper's own question. Approaches: the re-asking and independence-weakening routes at that paper and the sources the address names for the later theories. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Showing the inconsistency and re-asking — some revising, some not, the split stable across studies. Weakening independence — rank-dependent and prospect theories accommodating the first pattern, each introducing functions fitted to the data they explain, which their authors state. Non-additive beliefs and multiple priors — accommodate the ambiguity pattern by representing ambiguity, establishing representability and not which representation is correct. Raising the stakes — violations persisting at substantial stakes in several studies, the highest-stakes evidence limited. Neuroeconomic measurement — distinct activity for ambiguity and risk, establishing the distinction is registered without adjudicating the normative question. **Narrowing recorded.** Weakening independence weakens the relation: the standard stands as a fitted representation of the pattern rather than a criterion against it. Multiple-prior representations weaken the relation: ambiguity stands represented rather than the correct representation identified. **Progress.** The stable revision split. The measured persistence at substantial stakes. The distinct neural registration of ambiguity and risk. The accommodating representations and their stated fitting costs. **Own record.** ## 82 Time preference and preference reversal **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** People discount future rewards at rates that fall as the delay grows, and reverse earlier choices as a date approaches. Is discounting a stable rate? In the field's own posing, the discounted-utility model carries anomalies, the empirical regularities inconsistent with its theoretical predictions, and the review of three decades of empirical research on intertemporal choice finds a spectacular variation in implicit discount rates across studies, stressing throughout the importance of distinguishing time preference as such from the many other considerations that also influence intertemporal choices. **The field's account of its hardness.** The field's critical review states that the implicit discount rates estimated across studies vary spectacularly, and discusses the reasons; that time preference as such has to be distinguished from many other considerations that also influence intertemporal choices; and that if people discount utility from different sources at different rates, the notion of a unitary time preference is meaningless, time preference then having to be labelled according to the object being delayed. **Accounts.** Not a single rate: hyperbolic discounting fits the reversals directly and is documented across humans and animals, the reversals following from the shape of the curve rather than from any change in preference. Measured hyperbolic patterns can arise from aggregation over heterogeneous exponential discounters, from uncertainty about receipt, and from procedural features of the elicitation: individual-level data are less clearly hyperbolic than aggregate data, and the reversals may reflect changing information rather than changing preference. **Specifications.** A person's valuation of a delayed reward is a function of delay alone — local. Elicited choices measure that function — local. **Specifications withdrawn.** **Common requirement.** That there is a discount function to be measured. **Conserving relation.** A discount function measured for an individual, the measurement standing across elicitation formats and domains. **Conserving reach.** All individuals and domains the function is asked of. **Reach.** The statement quantifies over an individual's valuation function. Elicited choices, aggregated, and sensitive to format, are what is measured. The two are not the same. **Unreachability.** Estimated rates vary by elicitation format and by domain, and field estimates vary by orders of magnitude across studies, which is the instruments disagreeing on the same quantity. The measurement is not preserved across formats and domains by elicited choices. **Persistence.** A measurement of the function stable across elicitation formats and domains. **Addresses.** Statement and hardness: Frederick, Loewenstein and O'Donoghue, "Time Discounting and Time Preference: A Critical Review", Journal of Economic Literature 40(2):351-401, 2002, at its abstract as the journal's page carries it verbatim and at its passage on unitary time preference as a copy the field's pages carry verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the critical review's abstract read at pass one and at the hardness (the discount rate varying by orders of magnitude across studies and by elicitation, the construct's validity the question), a measurement stable across formats and domains standing there as what the review finds absent. Approaches: the individual-level and format-varying routes at that review's sections, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Eliciting at individual level — reduces the aggregation artefact, heterogeneity within individuals across domains remaining. Varying elicitation format — changes the estimated rates substantially, documented, and the standing methodological problem. Controlling for uncertainty of receipt — reduces apparent hyperbolicity in some designs, not eliminated. Field measurement from real decisions — estimates from borrowing and saving behaviour varying by orders of magnitude across studies and domains. Commitment device studies — show demand for commitment, supporting the reversal account behaviourally, the interpretation of demand contested. **Narrowing recorded.** Individual-level elicitation narrows the reach: the measurement stands for an individual within a domain rather than across domains. Controlling for uncertainty narrows the reach: the reduction stands in the designs where it is applied. **Progress.** The documented hyperbolic fit across humans and animals. The measured format sensitivity. The measured demand for commitment devices. **Own record.** ## 83 The productivity paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Large investments in information technology were followed by measured productivity growth that did not rise correspondingly. Where did the gains go? In the field's own posing, the relationship between information technology and productivity is widely discussed but little understood: delivered computing power in the economy rose by more than two orders of magnitude over the preceding two decades while productivity, especially in the service sector, stagnated, the observation carried in the field's canonical sentence, verbatim: "You can see the computer age everywhere but in the productivity statistics". **The field's account of its hardness.** The field's review states that the relationship is widely discussed but little understood; that the increased interest in the paradox has engendered a significant amount of research which has thus far only deepened the mystery; that measurement problems are even more acute in services than in manufacturing, many service transactions being idiosyncratic and not subject to statistical aggregation, and classifications sometimes seeming arbitrary even where abundant data exist; and that the benefits of investment in infrastructure, while they can be large, are indirect and often not immediate, lags of two to three years having been found before the strongest organisational impacts are felt. **Accounts.** Measurement misses them: output quality, variety, and free digital services are poorly captured in national accounts, the gains being real but unmeasured, with successive studies finding substantial unmeasured consumer surplus. Mismeasurement is insufficient in magnitude: the slowdown is broad across countries and sectors, including sectors with little digital output, and plausible mismeasurement adjustments do not close the gap. The gains may be smaller than expected, or delayed by complementary reorganisation. **Specifications.** Productivity is output per input, both measurable — universal under national accounting. Technology raises output for given inputs — local. **Specifications withdrawn.** **Common requirement.** That there is a true productivity change the statistics approximate. **Conserving relation.** A true productivity change approximated by the statistics, the approximation standing for output the accounts do not price. **Conserving reach.** Output including what the accounts do not price. **Reach.** The statement quantifies over output including what the accounts do not price. Priced transactions are what is measured. The two are not the same. **Unreachability.** Estimates of unmeasured surplus are large and, by the accounting against them, still short of the gap; and the slowdown appears where digital output shares differ widely, which constrains the mismeasurement account. The approximation is not reached over unpriced output by priced transactions. **Persistence.** A measurement of unpriced output closing the gap, or the predicted acceleration observed. **Addresses.** Statement and hardness: Brynjolfsson, "The Productivity Paradox of Information Technology", Communications of the ACM 36(12):66-77, 1993, at its opening as the field's pages carry it verbatim, at its passage on measurement in services as the publisher's page carries it, and at its passage on lags as the working-paper version carries it ("The Productivity Paradox of Information Technology: Review and Assessment", MIT Center for Coordination Science Working Paper 130), the paper itself not fetched; the canonical sentence as a general reference page carries it verbatim, attributed there to Solow, its original place of publication keyed at the next pass. Persistence, no landing: the closing condition at the founding paper's opening and its measurement passage read at pass one and at the hardness (the delivered computing power against the productivity statistics; mismeasurement in services among the explanations), a measurement of unpriced output closing the gap or the acceleration observed standing there as the two directions the paper's explanations leave. Approaches: the quality-adjustment and cross-country routes at that paper and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Adjusting for quality and free goods — estimates of unmeasured surplus large and, by the accounting against them, still short of the gap. Cross-country comparison — the slowdown appearing where digital output shares differ widely, constraining the mismeasurement account. Firm-level studies — large returns to IT combined with organisational change, establishing complementarity, the aggregate gap remaining. Lag hypotheses — general purpose technologies diffusing over decades, the record for electricity supporting a lag, predicting eventual acceleration, not observed and remaining available. Intangible capital measurement — raises measured output somewhat and shifts timing. **Narrowing recorded.** Firm-level studies narrow the reach: the returns stand at firm level where organisational change accompanies the investment. Intangible capital measurement narrows the reach: the correction stands for the intangibles it prices. **Progress.** The estimated unmeasured surplus and its magnitude against the gap. The cross-country and cross-sector distribution of the slowdown. The firm-level returns under complementary reorganisation. **Own record.** ## 84 The transformation problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** In a theory where value derives from labour time, prices are observed to deviate systematically from labour values. How do the two relate? In the field's own posing, the problem originates in the difference between two central abstractions of classical political economy, the labour theory of value and the equalisation of the rate of profit through competition among capitals; the theory proposed to reconcile the two by distinguishing the production of surplus value through the exploitation of labour in production from the realisation of surplus value through the price system, the equalisation of the rate of profit being viewed as a redistribution of a given surplus value among sectors. **The field's account of its hardness.** The field's survey states that the problem, read from an axiomatic perspective, is an impossibility result for a specific interpretation of value theory resting on specific assumptions and definitions; and the field's review states that modern commentary based on a dual system of parallel prices and embodied-labour coefficients loses sight of the theory's roots and of its function as a theory of money, the single-system approaches that identify the monetary expression of labour time with the ratio of money value added to living productive labour being proposed to address these problems. **Accounts.** The transformation can be completed: successive formulations, including single-system interpretations that value inputs and outputs at prices rather than values, preserve the aggregate equalities and remove the inconsistency the original procedure carried. The transformation is redundant: prices and the profit rate are determined by physical input-output data and the real wage alone, with no role for labour values in the determination, and the value magnitudes do no work. **Specifications.** There is a magnitude, labour value, standing behind price — local. Aggregate equalities between the two systems should hold — local. **Specifications withdrawn.** **Common requirement.** That labour values are determinate magnitudes and that their relation to prices is a question with an answer. **Conserving relation.** A determination of prices by value magnitudes, the determination standing alongside the physical determination. **Conserving reach.** All prices the theory is asked of. **Reach.** The statement quantifies over whether one determination underlies the other. Observed prices and physical input-output tables are what is measured. The two are not the same. **Unreachability.** The aggregate equalities cannot all be preserved simultaneously in the standard framework, which is proved; and prices and profit are determined without value magnitudes, which is not disputed as mathematics. The determination by value magnitudes is not reached over the prices by observed prices and physical tables. **Persistence.** A role for value magnitudes in the determination not already discharged by the physical data, or agreement that the equalities are definitional. **Addresses.** Statement: Foley, "The Marxian Transformation Problem", at its opening as a repository page carries it verbatim, its venue and year keyed at the next pass; hardness, and the impossibility result at Approaches: Mohun (second author as the journal's page lists), "Value, Price, and Exploitation: The Logic of the Transformation Problem", Journal of Economic Surveys, 2017, doi:10.1111/joes.12223, at its abstract as the publisher's page carries it verbatim; Foley, "Recent Developments in the Labor Theory of Value", Review of Radical Political Economics 32(1):1-39, 2000, at its abstract as the journal's page carries it verbatim; none of the three fetched. Persistence, no landing: the closing condition at the opening and the impossibility result the address names, read at pass one and at the hardness (the aggregate equalities not all simultaneously preservable in the standard framework; the single-system readings preserving them by redefinition), a role for value magnitudes not already discharged by the physical data standing there as what the impossibility result puts in question. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The corrected procedure — the aggregate equalities proved not all simultaneously preservable in the standard framework. Single-system interpretations — preserve the equalities by redefining the value of inputs, their critics holding the redefinition makes the equalities definitional. The physical-determination critique — establishes that prices and profit are determined without value magnitudes, not disputed as mathematics. Empirical correlation studies — high correlation between labour values and prices across sectors, contested as an artefact of sector size, examined and not settled. Rejecting the framework — alternatives determining prices differently, the dispute then being between frameworks rather than within one. **Narrowing recorded.** Single-system interpretations weaken the relation: the equalities stand under a redefinition of input value, which their critics hold makes them definitional. **Progress.** The proof that the aggregate equalities cannot all hold. The physical determination result. The measured correlations and the sector-size objection to them. **Own record.** ## 85 Money illusion **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Do people respond to nominal amounts rather than to real purchasing power? In the field's own posing, money illusion is a tendency to think in terms of nominal rather than real monetary values; it has significant implications for economic theory, yet it implies a lack of rationality that is alien to economists; the founding survey proposes that people often think about economic transactions in both nominal and real terms, the illusion arising from an interaction between the two representations that results in a bias toward a nominal evaluation. **The field's account of its hardness.** The founding survey states that money illusion has significant implications for economic theory yet implies a lack of rationality that is alien to economists, and that the survey questions reviewed, on people's reactions to variations in inflation and prices, are designed to shed light on the psychology that underlies the illusion, the proposal being that nominal and real representations interact and that the bias falls toward the nominal evaluation. **Accounts.** Yes: experimental and survey evidence shows systematic responses to nominal framing, downward nominal wage rigidity is documented and hard to account for otherwise, and it is a load-bearing assumption in models that fit inflation-output dynamics. Apparent illusion is often rational under uncertainty about relative prices, or an artefact of framing in surveys where real magnitudes are not salient: nominal rigidity is accountable by contracting and fairness considerations without any error about real values. **Specifications.** Agents have preferences over real quantities — local. Nominal magnitudes are transformations of them — universal under the framework. **Specifications withdrawn.** **Common requirement.** That there is a fact about which magnitude the agent responds to. **Conserving relation.** The magnitude the agent responds to, identified from choices. **Conserving reach.** All decisions the question is asked of. **Reach.** The statement quantifies over what the agent is responding to. Choices and reports under nominal presentation, with the real magnitude inferable but not presented, are what is measured. The two are not the same. **Unreachability.** Wage rigidity is consistent with fairness norms about nominal terms as well as with error, nominal anchoring in transaction data admits explanation by liquidity constraints, and the diminishing of illusion under salient inflation is predicted by both accounts. The identification is not reached over the decisions by choices under nominal presentation. **Persistence.** A decision setting in which the two accounts predict differently, or a measurement of the decision weight rather than the registration. **Addresses.** Statement and hardness: Shafir, Diamond and Tversky, "Money Illusion", The Quarterly Journal of Economics 112(2):341-374, 1997, at its abstract as the journal's page carries it verbatim, the paper itself not fetched; the framing results at Approaches: "Revisiting 'money illusion': Replication and extension of Shafir, Diamond, and Tversky (1997)", at its abstract as the publisher's page carries it, its journal and authors keyed at the next pass. Persistence, no landing: the closing condition at the founding paper's abstract read at pass one and at the hardness (the nominal representation registered alongside the real, with a decision weight the experiments elicit), a setting where the two accounts predict differently standing there as what the framing evidence has not supplied. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Framing experiments — robust nominal effects found, whether laboratory framing transfers to consequential decisions being the standing question. Wage rigidity evidence — nominal cuts rare and real cuts via inflation accepted, consistent with illusion and with fairness norms about nominal terms. Housing and asset markets — nominal anchoring effects documented in transaction data, alternative explanations by liquidity constraints available. High-inflation environments — illusion diminishing where inflation is salient, which both accounts predict. Neuroimaging — distinct response to nominal gains at equal real value, establishing registration and not the decision weight. **Narrowing recorded.** Framing experiments narrow the reach: the effects stand established in the laboratory, transfer to consequential decisions being the standing question. **Progress.** The measured framing effects. The documented wage rigidity. The nominal anchoring in transaction data. The distinct neural registration. **Own record.** ## 86 The two-fold cost of sex **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** An asexual lineage produces twice as many reproducing offspring per parent. Sexual reproduction is nonetheless nearly universal in multicellular life. What pays the cost? In the field's own posing, sexual reproduction entails a number of costs and yet the majority of eukaryotes engage in sex at least occasionally; the demographic cost is that populations of sexual organisms investing half their resources in producing male offspring that do not contribute to reproduction should grow at only half the rate of their asexual counterparts, and whether that cost is truly twofold for sexual females is recorded as unanswered. **The field's account of its hardness.** The field's review states that the early models to explain the evolution of sex failed to do so; that more recent efforts attempt to account for the complexities of evolution in the real world, with selection that varies over time and space, with differences among individuals in the tendency to reproduce sexually, and with populations limited in size; and that these efforts have clarified the conditions most likely to explain why sex is so common, the field heading its own standing as recent resolutions and remaining riddles; the cost itself is recorded as depending on the number of males produced, a female-biased sex ratio reducing it below twofold, so that sex can evolve with far fewer benefits than previously assumed. **Accounts.** Recombination clears deleterious mutations and combines beneficial ones: a ratchet operates in asexual lineages, and the mutational-deterministic account requires only that mutations interact in a specified way. Parasites adapt to common host genotypes and recombination generates rare combinations: field systems where sexual and asexual forms coexist show sexual forms favoured under high parasite pressure, which the mutational account does not predict. **Specifications.** The cost is correctly stated at two-fold — local. An advantage must be located that repays it in each generation — local. **Specifications withdrawn.** **Common requirement.** That the maintenance of sex is accounted for by a benefit exceeding the cost. **Conserving relation.** A benefit measured against the two-fold cost, the measurement standing across the taxa. **Conserving reach.** All multicellular taxa where sex is maintained. **Reach.** The statement quantifies over the advantage, measured against a stated two-fold cost. Particular systems where the comparison can be made are what is measured. The two are not the same. **Unreachability.** Each mechanism is demonstrated in its own system, and pluralist accounts combine them with weights not independently measured; the magnitude of the recombination benefit relative to the two-fold cost is contested. The measurement is not preserved across the taxa by particular systems. **Persistence.** A benefit measured against the two-fold cost across taxa, or independently measured weights for the combined mechanisms. **Addresses.** Statement and hardness: Otto, "The Evolutionary Enigma of Sex", The American Naturalist 174(S1):S1-S14, 2009, at its abstract as the journal's page carries it verbatim, the paper itself not fetched; the cost as stated and measured, and the systems at Reach: Kobayashi and Hasegawa, "A female-biased sex ratio reduces the twofold cost of sex", Scientific Reports 6:23982, 2016, at its abstract and opening as the archive's page carries them verbatim, the paper itself not fetched; the experimental measurement in a natural system as the field cites it: Gibson, Delph and Lively, "The two-fold cost of sex: experimental evidence from a natural system", Evolution Letters 1(1):6-15, 2017, its abstract not read. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (the two-fold cost stated and measured; the candidate benefits each supported in particular systems, their combination unweighted). Approaches: the mixed-reproduction and experimental-evolution routes at that review and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Systems with mixed reproduction — support the parasite account in those systems, with measured parasite pressure, system-specific. Experimental evolution with and without recombination — recombination speeding adaptation under stress in several organisms, establishing a benefit, its magnitude relative to the cost contested. Mutation accumulation experiments — confirm the ratchet in asexual lines, establishing the mechanism, the rates required for the deterministic account being at the edge of measured values. Ancient asexual lineages — the standing counterexample weakened by subsequent findings of horizontal transfer and other genetic exchange, which complicates the categories. Theoretical pluralism — most current accounts combining mechanisms, fitting, the combination's weights not independently measured. **Narrowing recorded.** The parasite account narrows the reach: the advantage stands measured in the systems where parasite pressure is measured. Experimental evolution narrows the reach: the benefit stands under stress in the organisms tested. **Progress.** The measured parasite pressure and its association with sexual forms. The confirmed ratchet in asexual lines. The measured speed of adaptation under recombination. **Own record.** ## 87 Peto's paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Cancer arises from mutations in dividing cells. Large long-lived animals have vastly more cell divisions and do not have proportionally more cancer. Why? In the field's own posing, the evolution of multicellularity required the suppression of cancer; if every cell has some chance of becoming cancerous, large long-lived organisms should have an increased risk of developing cancer compared with small short-lived ones, and the lack of correlation between body size and cancer risk is the paradox, animals with a thousand times more cells than humans not exhibiting an increased risk, which suggests that natural mechanisms can suppress cancer a thousand times more effectively than is done in human cells. **The field's account of its hardness.** The field's review states that the expectation of increased risk follows from the premise that every cell has some chance of becoming cancerous, that the lack of correlation between body size and cancer risk suggests natural mechanisms of suppression a thousand times more effective than in human cells, and that the review seeks, as pharmaceutical companies mine natural products, to understand how evolution has suppressed cancer; the field's modelling records that the assertion that body mass, a surrogate for cell number, should predict incidence has some support in humans and does not hold over the range of natural animal species where cancer incidence is known. **Accounts.** Evolved additional suppression: larger species carry additional tumour-suppressor mechanisms — many copies of a key suppressor gene in one lineage, alterations in DNA repair pathways in another — and the suppression scales with the risk. The framing may overstate the paradox: multistage models with species-specific parameters can produce the observed pattern without additional mechanisms, and cross-species incidence data are sparse, biased toward captive and veterinary populations, and not standardised. **Specifications.** Cancer risk is a function of cell number and division rate — local. Species are comparable on a standardised incidence measure — local. **Specifications withdrawn.** **Common requirement.** That a cross-species incidence comparison is meaningful. **Conserving relation.** Lifetime risk compared across species, the comparison standing on standardised incidence. **Conserving reach.** All species the comparison is asked of. **Reach.** The statement quantifies over lifetime risk across species. Necropsy and veterinary records, unevenly sampled across species and conditions, are what is measured. The two are not the same. **Unreachability.** Comparative genomics establishes mechanisms without quantifying their sufficiency, and multistage models fit with parameters estimated from the same data; the sampling bias is acknowledged in the studies themselves. The comparison is not reached over the species by unevenly sampled records. **Persistence.** A standardised cross-species incidence measure, or an independent quantification of the suppressor mechanisms' sufficiency. **Addresses.** Statement and hardness: Caulin and Maley, "Peto's Paradox: evolution's prescription for cancer prevention", Trends in Ecology & Evolution 26(4):175-182, 2011, at its abstract as the journal's page carries it verbatim, the paper itself not fetched; the paradox's original statement as the review cites it: Peto and others, "Cancer and ageing in mice and men", British Journal of Cancer 32:411-426, 1975; the standing across the species where incidence is known: "Peto's paradox revisited: theoretical evolutionary dynamics of cancer in wild populations", at its abstract as the index's page carries it, PubMed identifier 23396800, its journal, authors and year keyed at the next pass. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (cancer incidence not scaling with cell number and lifespan across species; the suppressor mechanisms proposed, their sufficiency unmeasured). Approaches: the comparative-genomics and database routes at that review and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Comparative genomics — identifies suppressor duplications and repair alterations in large species, establishing mechanisms exist, their quantitative sufficiency unmeasured. Zoo and necropsy databases — assemble incidence across many species and confirm weak scaling with body size, sampling bias acknowledged in the studies themselves. Multistage modelling — reproduces patterns under fitted parameters, the parameters not independently measured. Cell-level experiments — cells from large species showing different sensitivity to damage, establishing differences, the link to whole-organism incidence unmeasured. Within-species scaling — larger individuals within a species showing higher risk, consistent with the framing and not resolving the cross-species case. **Narrowing recorded.** Within-species scaling narrows the reach: the relation stands within a species rather than across species. Cell-level experiments narrow the reach: the differences stand at cell level, the link to organism incidence unmeasured. **Progress.** The identified suppressor duplications and repair alterations. The confirmed weak scaling with body size. The measured within-species scaling. The cell-level sensitivity differences. **Own record.** ## 88 The C-value paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Genome size varies more than a thousandfold among organisms of similar complexity. What determines it? In the field's own posing, the genome size diversity among organisms, more than two hundred thousandfold among eukaryotes, bears no relationship to organismal complexity, and both the origins and the reasons for the clearly non-random distribution of this variation remain unclear; the field now names the problem an enigma rather than a paradox, and sorts the theories proposed for it into mutation-pressure theories, which read the large non-coding portion of eukaryotic genomes as junk or selfish DNA, and optimal-DNA theories, which emphasise the strong link between DNA content and cell and nuclear volumes. **The field's account of its hardness.** The field's review states that variation in DNA content has been largely ignored as a factor in evolution, particularly following the advent of sequence-based approaches to genomic analysis; that both the origins and the reasons for the non-random distribution of the variation remain unclear; that each of the approaches to the enigma is problematic for a variety of reasons, the preponderance of the available evidence instead favouring a causal link between bulk DNA amount and cell volume, under which the variation is under direct selection through its effects on cellular and organismal parameters; and that no satisfactory mechanism had until then been presented for it. **Accounts.** Non-coding content is largely neutral: genome size tracks the accumulation of repetitive elements, and effective population size governs how efficiently selection removes them, no functional role being required. Much non-coding DNA is functional: regulatory elements, structural roles, and biochemical activity across most of the genome indicate function, and the assumption of neutrality was a placeholder rather than a finding. **Specifications.** Genome content is either functional or not — local. Function is identifiable by an operational criterion — local. **Specifications withdrawn.** **Common requirement.** That "functional" has a criterion both accounts accept. **Conserving relation.** A criterion of function sorting sequence into functional and not, the sorting standing across the instruments. **Conserving reach.** All sequence the question is asked of. **Reach.** The statement quantifies over whether sequence has a fitness-relevant role. Biochemical activity, conservation, and effects of deletion are what is measured — three criteria giving different answers on the same sequence. The two are not the same. **Unreachability.** Three criteria of function return different fractions on the same sequence, and the field debated exactly whether activity is function in print. The sorting is not preserved across the instruments by any of the three criteria. **Persistence.** A criterion of function the three instruments agree on, applied to the same sequence. **Addresses.** Statement and hardness: Gregory, "Coincidence, coevolution, or causation? DNA content, cell size, and the C-value enigma", Biological Reviews 76(1):65-101, 2001, at its abstract as the journal's and the index's pages carry it verbatim, the paper itself not fetched; the parallel review as the field cites it: Gregory, "The C-value enigma in plants and animals: a review of parallels and an appeal for partnership", Annals of Botany 95:133-146, 2005, its abstract not read. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (genome size uncorrelated with organismal complexity; the enigma reposed as the coevolution of DNA content and cell size), a criterion of function the instruments agree on standing there as what the later dispute over activity and function turns on. Approaches: the activity-assay and conservation routes at the sources the address names for the dispute in print. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Biochemical activity assays — report activity over most of the genome, whether activity is function being the dispute, debated in print. Comparative conservation — constrains functional fraction to a much smaller figure, a different criterion giving a different number. Deletion experiments — large deletions of conserved non-coding regions producing viable animals in some cases, establishing dispensability under laboratory conditions. The onion comparison and relatives — species with similar biology and vastly different genome sizes, a standing challenge to functional accounts, not answered. Population-genetic modelling — reproduces size variation from effective population size, fitting, and depending on parameters estimated from the same data. **Narrowing recorded.** Comparative conservation narrows the reach: the functional fraction stands under a conservation criterion. Deletion experiments narrow the reach: dispensability stands under laboratory conditions. **Progress.** The measured activity fraction. The conserved fraction. The viable deletions. The measured genome-size variation across similar biology. **Own record.** ## 89 The paradox of the plankton **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Competition theory says the number of coexisting species cannot exceed the number of limiting resources. Plankton communities show dozens of species on a handful of resources. How? **The field's account of its hardness.** That the fluctuation-dependent mechanisms remove the theoretical limit rather than supplying a count: one of them can support a number of species growing quadratically with the number of discrete resources, the other an arbitrary number. The field states the consequence itself — the mechanisms resolved the paradox and imposed additional problems, since with no theoretical limit to diversity the question becomes why the observed number is the number observed, and answering it requires a framework capable of measuring the relative importance of the competing explanations. The field also records that the mechanism originally proposed fails as stated, temporal variation alone being insufficient, while continuing to be cited as the explanation. **Accounts.** Equilibrium never arrives: environmental fluctuation, mixing, and predation reset competition before exclusion completes, and the exclusion principle applies to an equilibrium the system does not reach. The niches are not being counted correctly: spatial structure, temporal partitioning, and species-specific predation supply more limiting factors than the resource count records, and coexistence is niche-based after all. **Specifications.** Coexistence requires niche differences — local. Limiting factors are enumerable — local. **Specifications withdrawn.** **Common requirement.** That the number of limiting factors is countable in principle. **Conserving relation.** A count of limiting factors sufficient for the observed coexistence, the count standing on measurement. **Conserving reach.** All limiting factors operating. **Reach.** The statement quantifies over all limiting factors operating. Measured resource concentrations and identified predators are what is measured. The two are not the same. **Unreachability.** Each added measured factor raises the ceiling and whether the count converges is open; and the coexistence framework accommodates both accounts without measuring their weights in natural communities. The count is not reached over the operating factors by measured concentrations and identified predators. **Persistence.** A count of limiting factors demonstrated sufficient, or measured weights for the fluctuation-dependent and fluctuation-independent mechanisms in natural communities. **Addresses.** Statement, in substance: the founding paper, Hutchinson, "The Paradox of the Plankton", American Naturalist 95(882):137-145, 1961, doi 10.1086/282171, at its opening as two indexes and citing reviews carry it verbatim (the problem presented by the phytoplankton essentially how it is possible for a number of species to coexist in a relatively isotropic or unstructured environment all competing for the same sorts of materials; the very paradoxical situation of the plankton of relatively large bodies of water), the paper itself not fetched; a later review at its opening as the publisher's page carries it (the paradox resting on two implicit, perhaps naive postulates, that all plankton species have similar requirements and that the marine environment is relatively homogeneous in space and time), Biology Letters, 2022, doi 10.1098/rsbl.2022.0207, its authors not carried; the Statement standing as it stood. Hardness, the theoretical species counts, and the statement that the mechanisms resolved the paradox and imposed additional problems: the conceptual-issues paper on modern coexistence theory, arXiv:2201.07926, introduction. The failure of the originally proposed mechanism, and its continued citation: Li and Chesson, American Naturalist 188(2), 2016, and the storage-effect analysis at arXiv:1904.10845 §1. Storage effect and buffered growth: arXiv:2201.06691. Time-series selection estimates across 148 series: Fox, Ecology 2010, abstract. Spatial, turbulence, and toxin routes: the plankton-paradox review, ResearchGate 263267841. Trait-structured ocean models and slowed exclusion: ICES Journal of Marine Science 71(2), 2014. Persistence, no landing: the closing condition at the founding paper's opening read at v342 (how a number of species can coexist in a relatively unstructured environment all competing for the same materials; the mechanism proposed, permanent failure to reach equilibrium, the argument recorded as incomplete) and at the later review's two implicit postulates. Approaches: the originally proposed mechanism at that opening and the later routes at the review the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The originally proposed mechanism — coexistence by permanent failure to reach equilibrium as external factors change; the argument recorded as incomplete, with no model validating it, and the specific mechanism shown by later modelling to fail, while continuing to be cited. The coexistence framework — partitions mechanisms into stabilising and equalising, and into fluctuation-dependent and fluctuation-independent, each measurable in principle. The storage effect — species specialising on different aspects of a variable environment, with buffered growth when not actively taking up resources, supporting an arbitrary number of species in theory and tested in the field in specific systems. Relative nonlinearity — species specialising on resource variance rather than mean level, the variation generated externally by seasonal turnover or internally by consumer-resource dynamics, supporting a number growing quadratically with the resources. Chaotic dynamics — competition among few resources generating chaos permitting many species, established in models. Spatial and self-generated heterogeneity — segregation, mesoscale turbulence and coherent vortices, and toxin production, each proposed as a distinct route. Neutral and equalising accounts — diversity patterns reproduced with no niche differences, and trait-structured ocean models in which exclusion remains an intrinsic property but its rate is slowed. Time-series estimation of selection — net and fluctuating selection estimated across ninety-nine annual zooplankton series and forty-nine algal series, with within-year net selection found statistically significant and ecologically weak. Adding measured factors — each addition raising the ceiling, whether the count converges being the open part. Long-term microcosm experiments — sustained coexistence observed beyond the resource count. **Narrowing recorded.** Chaotic dynamics narrows the reach: sufficiency stands established in models. Microcosm experiments narrow the reach: the coexistence stands observed under laboratory conditions. **Progress.** The partitioned coexistence framework. The chaotic sufficiency result in models. The observed microcosm coexistence beyond the resource count. **Own record.** ## 90 The binding problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Different features of an object are processed in separate brain regions. Experience presents one object. What binds them? In the field's own posing, perceptual representations depend on distributed neural codes relaying the parts and properties of objects, and some mechanism is needed to bind the information relating to each object and to distinguish it from others; the standard example, as the field's survey gives it, is that one sort of visual feature, such as an object's shape, must be correctly associated with another, such as its location, to provide a unified representation of that object. **The field's account of its hardness.** The field's review states that despite the problem's prominence across psychology, neuroscience, computational modelling and philosophy, what binding means is rarely made explicit, the notions of binding being many; and the field's survey lists the candidate mechanisms, cells tuned to conjunctions of features, spatial attention, and synchronised firing across separate but interconnected areas of the brain, with deficits in neurological patients suggesting a role for the parietal cortex, and several current models combining these ideas. **Accounts.** Temporal synchrony: neurons responding to features of one object fire in synchrony while those responding to different objects do not, supplying a tag available without dedicated conjunction cells and scaling combinatorially where convergence does not. Gamma-band coherence is the documented signature, appearing across species and tasks, and tracking perceptual grouping in the predicted direction. Synchrony is a correlate, and no mechanism reading it has been identified: hierarchical convergence onto neurons responsive to conjunctions, plus attention, accounts for the phenomena without a separate binding process, and the problem may be posed by an assumption about how experience must be constructed. **Specifications.** Features are represented separately — universal under systems neuroscience. Unified experience requires a process combining them — local. **Specifications withdrawn.** **Common requirement.** That the unity of the object in experience corresponds to a combining operation. **Conserving relation.** A combining operation identified, the identification standing as cause rather than correlate. **Conserving reach.** All the cases unity is asked of. **Reach.** The statement quantifies over what makes the features one object in experience. Neural activity and behavioural report are what is measured. The two are not the same. **Unreachability.** Synchrony is documented and whether it is causal or a consequence of shared drive is where the field's analyses disagree, with the disputes in print on both sides; and no mechanism reading the synchrony has been identified. The identification is not reached over the cases by neural activity and behavioural report. **Persistence.** A mechanism reading the synchrony identified, or a manipulation separating cause from consequence across the cases. **Addresses.** Statement, hardness and the candidate mechanisms at Approaches: Treisman, "The binding problem", Current Opinion in Neurobiology 6(2):171-178, 1996, at its abstract as the index's page carries it verbatim; Roskies, "The Binding Problem", Neuron 24(1):7-9, 1999, at its opening as the journal's page carries it verbatim; neither fetched; the problem's original formulation as the latter cites it: von der Malsburg, 1981. Persistence, no landing: the closing condition at the two sources the address names, read at pass one and at the hardness (the problem of how separately represented features are combined; the candidate mechanisms, synchrony among them, their causal standing disputed), a mechanism reading the synchrony or a manipulation parting cause from consequence standing there as what the disputes in print ask. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Gamma synchrony studies — documented across species and tasks, whether causal or a consequence of shared drive disputed, the disputes in print on both sides. Illusory conjunction studies — binding failures under attentional load, establishing attention's role, not a mechanism. Convergence and conjunction cells — found in higher visual areas, handling some conjunctions, combinatorial coverage being the objection. Optogenetic manipulation of synchrony — behavioural effects in specific paradigms, the strongest causal route, scope so far narrow. Dissolving the problem — unity argued to be a feature of report rather than a construction requiring a process, reframing, the phenomena remaining to be accounted. **Narrowing recorded.** Optogenetic manipulation narrows the reach: the causal result stands for the specific paradigms manipulated. Conjunction cells narrow the reach: the account stands for the conjunctions convergence covers. **Progress.** The documented gamma coherence across species and tasks. The measured binding failures under attentional load. The identified conjunction cells. The behavioural effects of synchrony manipulation. **Own record.** ## 91 Sleep **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Sleep is universal among animals with nervous systems, costly, and dangerous. What is it for? In the field's own posing, sleep is universal, tightly regulated, and cannot be deprived without deleterious consequences, which suggests that searching for a core function of sleep, particularly at the cellular level, remains a worthwhile exercise; the null hypothesis set against that search holds sleep to be merely a way of imposing a quiet, immobile state, not important by itself in mammals and birds. **The field's account of its hardness.** The field's essay states that the purpose of sleep is not understood although it must be important because all animals do it, no clear evidence standing of an animal species that does not sleep; that the search for a core function can seem as elusive as the search for a mythical bird; that the null hypothesis, sleep as the imposition of rest with no importance of its own, has to be rejected before a core function is sought; and that even the presence of sleep in a species with one-sided sleep, a specialisation strongly suggesting an essential function that cannot be eliminated, has been questioned on several lines of evidence. **Accounts.** Restoration and clearance: metabolic waste clearance rises during sleep, specific molecular and cellular repair processes are sleep-dependent, and deprivation produces measurable physiological damage and, in some species, death. Synaptic and informational function: sleep supports memory consolidation and synaptic downscaling, with replay of waking activity documented, and the informational account addresses why brain activity is patterned rather than merely reduced. **Specifications.** Sleep has a function — local. Identifying its primary function is the question — local. **Specifications withdrawn.** **Common requirement.** That there is a primary function to be identified. **Conserving relation.** A primary function identified, the identification standing separately from the state that carries it. **Conserving reach.** All animals with nervous systems. **Reach.** The statement quantifies over the contribution of sleep, separable from the state that carries it. Correlates and effects of deprivation are what is measured, both of which the two accounts predict. The two are not the same. **Unreachability.** Deprivation produces cognitive and physiological deficits together, which both accounts predict, and the clearance results have been contested in technique, which is the instruments in dispute. The identification is not reached over the animals by correlates and deprivation effects. **Persistence.** A measurement separating the two functions, or agreement that both operate and the question is their proportion. **Addresses.** Statement, hardness and the one-sided sleep case at Approaches: Cirelli and Tononi, "Is Sleep Essential?", PLoS Biology 6(8):e216, 2008, at its abstract as the journal's page carries it verbatim, at its passage on one-sided sleep as the repository copy carries it, and at the journal's press summary as the field's news pages carry it, the essay itself not fetched; the null hypothesis as the essay cites it: Rial and others, "The trivial function of sleep", Sleep Medicine Reviews 11:311-325, 2007; the survey of perspectives as the essay cites it: Rechtschaffen, "Current perspectives on the function of sleep", Perspectives in Biology and Medicine 41:359-390, 1998. Persistence, no landing: the closing condition at the review's abstract and its one-sided-sleep passage read at pass one and at the hardness (the question whether sleep is essential, the functions proposed and the tests each would need), a measurement parting the two functions standing there as the review's own missing test. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Deprivation studies — produce cognitive and physiological deficits together, predicted by both accounts. Clearance measurement — clearance rising in sleep in rodents, the magnitude and the technique disputed in subsequent work. Replay and consolidation studies — documented in hippocampus and cortex, with targeted reactivation improving retention, establishing an informational role, whether primary not settled. Comparative studies — sleep duration and architecture varying widely with ecology and phylogeny, constraining single-function accounts. Unihemispheric and reduced sleep species — showing the requirement is modifiable, the residual requirement not zero. **Narrowing recorded.** Replay studies narrow the reach: the informational role stands established in the structures where replay is documented. Clearance measurement narrows the reach: the result stands in the species and technique measured. **Progress.** The measured deprivation effects. The documented replay and the retention gains from targeted reactivation. The comparative variation in duration and architecture. The modifiable requirement in reduced-sleep species. **Own record.** ## 92 Other minds **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** I know my own experience directly. I infer everyone else's from behaviour. What justifies the inference? **The field's account of its hardness.** That any knowledge of other minds, or any justification for belief in them, is held by the defenders of the argument from analogy and of the argument from best explanation alike to be indirect; that a thin sceptical problem, grappled with in everyday life, is parted from a thick one, distinctively philosophical, which leads to the possibility of solipsism; that thinking of mental states as lying behind and causing behaviour may break what some see as a deep connection between mentality and behaviour, opening up that possibility; and that a conceptual problem, how a concept of mental states other than one's own is possible at all, is raised from the same materials as the epistemological problem and is held by some to be the difficult one. **Accounts.** Inference to the best explanation: other bodies resemble mine, behave as mine does, and share my physiology, and experience in them is by far the best explanation of the behaviour. The inference is from a single case, which is the weakest form of induction there is, and no independent check on it is available: alternatives — that experience is a way of describing behaviour, or that the demand for justification is malformed — have been developed precisely because the argument from analogy is regarded as insufficient. **Specifications.** Experience is directly known in one's own case and not in others' — local. The asymmetry is a fact about access rather than about the concepts — local. **Specifications withdrawn.** **Common requirement.** That there is something in others whose presence the inference is about. **Conserving relation.** A justification of the inference, standing on a check independent of the single case. **Conserving reach.** All others the inference is made about. **Reach.** The statement quantifies over experience in another. Behaviour and physiology are what is measured. The two are not the same. **Unreachability.** The inference is from one case with no check, formally weak by a result both accounts accept; inference to the best explanation depends on the framework already admitting experience as an explanans, which is what is in question; and criterial accounts remove the inference and are contested as changing the question. The justification is not reached over the others by behaviour and physiology. **Persistence.** A check on the inference independent of the single case, or agreement that no justification is owed. **Addresses.** Statement, the epistemological and the conceptual problem parted: Avramides, "Other Minds", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and the Fall 2025 edition carry them verbatim (any knowledge of other minds, or any justification for belief in them, held by the defenders of the argument from analogy and of best explanation alike to be indirect; a thin sceptical problem of everyday life parted from a thick, distinctively philosophical one leading to the possibility of solipsism; thinking of mental states as lying behind and causing behaviour possibly breaking a deep connection between mentality and behaviour), the entry itself not fetched, its section numbers and revision date not carried at the returns; the archived editions before 2019 carry a different text, whose passages on the conceptual problem (how a concept of mental states other than one's own is possible; the same materials producing both problems; some holding the conceptual problem the difficult one) are read here verbatim, their author not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (any knowledge of other minds or justification for belief in them held by the defenders of the argument from analogy to rest on the single case; the criterial alternative removing the inference), a check independent of the single case standing there as what the argument's defenders acknowledge it lacks. Approaches: the analogy and criterial routes at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Argument from analogy — one case, no check, its weakness acknowledged by its defenders. Criterial accounts — behaviour in context as a criterion rather than evidence, removing the inference and contested as changing the question. Inference to the best explanation — depends on the explanatory framework already admitting experience as an explanans, which is what is in question. Neurophysiological similarity — establishes similarity, which is the analogy premise measured. Denying the asymmetry — self-knowledge held also to be inferential, levelling the two cases without supplying the missing check. **Narrowing recorded.** Criterial accounts weaken the relation: behaviour in context stands as a criterion in place of an inference to be justified. **Progress.** The measured neurophysiological similarity. The formal weakness of one-case induction, accepted by both accounts. **Own record.** ## 93 Symbol grounding **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A system manipulates symbols by rules. How do its symbols come to be about anything? In the field's own posing, the problem is how the semantic interpretation of a formal symbol system can be made intrinsic to the system rather than parasitic on the meanings in the heads of those who interpret it, and how the meanings of meaningless symbol tokens, manipulated solely on the basis of their arbitrary shapes, can be grounded in anything but other meaningless symbols, the problem being likened to learning a language from a dictionary written wholly in that language; the candidate solution sketched grounds symbolic representations bottom-up in non-symbolic representations of two kinds, iconic and categorical. **The field's account of its hardness.** The founding paper states that the meaningfulness of a symbol system's interpretation is parasitic on the meanings its interpreter projects onto it from a grounded head, an effect the paper names a hall of mirrors and the reverse side of the problem; that there is an irresistible tendency to over-interpret toy task performance, the parasitism being confirmed by a further problem lurking around the corner; and that even a system whose symbol meanings are intrinsic offers no guarantee of having captured subjective meaning, closeness being claimed only where behavioural capacities are lifesize. **Accounts.** By causal connection to the world: grounding is achieved through sensorimotor interaction, symbols acquired through perception and action being connected to what they denote. Causal connection underdetermines reference: any causal chain admits multiple assignments of what the symbol is about, the indeterminacy arguments have not been answered, and adding sensors relocates the question to what the sensor states are about. **Specifications.** Symbols have determinate content — local. Content is fixed by a relation between the system and the world — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about what a symbol is about. **Conserving relation.** Content fixed by a relation between system and world, the fixing standing against alternative assignments. **Conserving reach.** All symbols the content relation is asked of. **Reach.** The statement quantifies over the content relation. The system's behaviour and its causal history are what is measured. The two are not the same. **Unreachability.** Indicator theories face the disjunction problem and misrepresentation cases, documented and unresolved; and content fixed by selection history inherits indeterminacy of the selected function. The fixing is not reached over the symbols by behaviour and causal history. **Persistence.** A relation fixing content against the alternative assignments, or agreement that content is not fixed by such a relation. **Addresses.** Statement, hardness, and the room argument and the candidate solution at Approaches: Harnad, "The Symbol Grounding Problem", Physica D 42(1-3):335-346, 1990, at its abstract as the publisher's page and the repository copy carry it verbatim and at its closing passages as the repository copy carries them, the paper itself not fetched; the earlier form of the problem as the paper cites it: Searle, 1980. Persistence, no landing: the closing condition at the founding paper's abstract read at pass one and at the hardness (how the meanings of the symbols in a system can be made intrinsic rather than parasitic on interpretation; the candidate solution grounding elementary symbols in categorical representations), a relation fixing content against the alternative assignments standing there as the paper's own question. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Robotic grounding — builds systems whose categories arise from interaction, establishing categories are acquirable, whether they are about anything being the original question. Causal and informational semantics — asymmetric dependence and indicator theories, each facing the disjunction problem and misrepresentation cases, documented and unresolved. Teleosemantics — fixes content by selection history, facing indeterminacy of the selected function, contested. The room argument — holds syntax insufficient for semantics, the systems reply and the robot reply standing as responses, the exchange not converged. Large language models — acquire behaviour previously thought to require grounding, what this shows about content actively disputed. **Narrowing recorded.** Teleosemantics narrows the reach: content stands fixed by selection history where a function was selected. Robotic grounding narrows the reach: categories stand acquired through interaction in the systems built. **Progress.** The acquirable categories in built systems. The documented disjunction and misrepresentation cases. The behaviour acquired by models without sensorimotor interaction. **Own record.** ## 94 Interpretability of built systems **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A trained network produces outputs. Can its computation be explained in terms a person can check? In the field's own posing, models should be not only good but also interpretable, yet the task of interpretation appears underspecified: the literature provides diverse and sometimes non-overlapping motivations for interpretability and offers myriad notions of what attributes render a model interpretable, transparency to humans and post-hoc explanation standing as competing notions. **The field's account of its hardness.** The field's review states that the task of interpretation appears underspecified; that the motivations underlying interest in interpretability are diverse and occasionally discordant; that despite the ambiguity many papers proclaim interpretability axiomatically, absent further explanation; that it is not clear what common properties unite the techniques offered; and that transparency to humans and post-hoc explanations are competing notions, some interpretations explaining predictions without elucidating the mechanism. **Accounts.** Yes, increasingly: circuits have been identified, features localised, and specific behaviours traced to specific components, sparse decomposition methods extract features more interpretable than raw units, and the methods improve steadily. Explanations obtained so far cover a small fraction of computation, and the standard for success is not agreed: explanations that predict behaviour under intervention are rare, many are post hoc narratives fitting observed activity, and the field's own evaluations find explanations that appear plausible and fail causal tests. **Specifications.** The computation decomposes into components with stateable roles — local. A person-checkable explanation exists — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about what a component does, independent of the explanation offered. **Conserving relation.** An explanation of the computation, standing under intervention rather than fitting observed activity. **Conserving reach.** The whole of the system's computation. **Reach.** The statement quantifies over what the system's computation is. Activations, ablations, and behaviour under intervention are what is measured. The two are not the same. **Unreachability.** Probing was shown by the field to recover information the system does not use, a proved limitation on its own instrument; and causal results are component-specific and do not compose into whole-model accounts. The explanation is not reached over the whole computation by activations, ablations, and intervention. **Persistence.** An explanation covering the computation and standing under intervention, or an agreed standard for what counts as one. **Addresses.** Statement, hardness and the standard at Persistence: Lipton, "The Mythos of Model Interpretability", arXiv:1606.03490, 2016, and Communications of the ACM 61(10):36-43, 2018, at its abstract as the archive's and the journal's pages carry it verbatim, the paper itself not fetched. Approaches: the visualisation, probing and ablation routes at the founding paper's abstract read at pass one, in part, their endings at the sources the address names for the controls the field established. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Feature visualisation and probing — produces plausible descriptions, probes recovering information the system does not use, which the field established and now controls for. Causal ablation and patching — stronger, results component-specific and not composing into whole-model accounts. Sparse dictionary methods — extract more interpretable features than raw units, coverage of total computation partial and measured. Formal verification — available for small networks and specific properties, scaling poorly, which is the known limit. Behavioural evaluation — establishes what the system does, not how. **Narrowing recorded.** Formal verification narrows the reach: the guarantee stands for small networks and specific properties. Causal ablation narrows the reach: the results stand component by component. Sparse dictionary methods narrow the reach: coverage stands measured and partial. **Progress.** The identified circuits and localised features. The measured coverage of sparse decompositions. The established limitation on probing. The verified properties of small networks. **Own record.** ## 95 Machine and animal sentience **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which systems have experience, and how would one tell? In the field's own posing, the questions are whether octopuses feel pain and pleasure, whether crabs, shrimps, insects or spiders do, how one tells whether a person unresponsive after severe brain injury might be suffering, when a fetus starts to have conscious experiences, whether there could be rudimentary feelings in miniature models of the human brain grown from stem cells, and what of artificial intelligence; these are questions about the edge of sentience, sentience taken as the capacity to have valenced experiences, and they are subject to enormous, disorienting uncertainty. **The field's account of its hardness.** The field's treatment states that the questions at the edge of sentience are subject to enormous, disorienting uncertainty; that certainty is desperately wanted but out of reach, the uncertainty being with us for the long term and no method making it go away; and that the stakes are immense and neglecting the risks can have terrible costs, so that one needs to err on the side of caution, yet it is often far from clear what erring on the side of caution should mean in practice, the precautions taken having to be proportionate to the identified risks. **Accounts.** By identifying the mechanisms that produce it in the case we know: behavioural and neural markers — nociception with motivational trade-offs, flexible learning, integration — provide graded evidence, and precautionary frameworks are built on them for animals. The markers are selected by resemblance to us and have no independent validation: a system engineered to display them displays them, an organism with different architecture may have experience without them, and no test distinguishes presence from absence. **Specifications.** Experience is present or absent in a system — local. Markers correlate with it — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about presence, and that evidence bears on it. **Conserving relation.** Markers bearing on presence, the bearing standing on validation independent of resemblance. **Conserving reach.** All systems the question is asked of. **Reach.** The statement quantifies over presence, indexed by markers with no independent validation. Behaviour, architecture, and physiology are what is measured. The two are not the same. **Unreachability.** The marker frameworks lack independent validation, stated by their own authors, and an engineered system can produce the same behavioural pattern; theory-driven tests apply a theory that is what is in dispute. The bearing is not reached over the systems by behaviour, architecture, and physiology. **Persistence.** A marker validated independently of resemblance to the case we know. **Addresses.** Statement and hardness: Birch, The Edge of Sentience: Risk and Precaution in Humans, Other Animals, and AI, Oxford University Press, 2024, doi:10.1093/9780191966729.001.0001, at its abstract as the publisher's pages carry it verbatim, and at its definition of sentience (p. 1) and its precautionary principle (p. 18) as the field's review carries them (Leonardo Reviews, 2024), the book itself not fetched; the marker frameworks at Approaches as the field cites them: Birch, Burn, Schnell, Browning and Crump, Review of the Evidence of Sentience in Cephalopod Molluscs and Decapod Crustaceans, 2021; Birch, Schnell and Clayton, "Dimensions of animal consciousness", Trends in Cognitive Sciences 24(10):789-801, 2020. Persistence, no landing: the closing condition at the book's abstract read at pass one and at the hardness (the framework of markers and precaution for sentience candidates; the validation problem stated within the frameworks), a marker validated independently of resemblance to the case we know standing there as the framework's own stated difficulty. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Marker frameworks — applied to invertebrates with legislative consequence in several jurisdictions, establishing a graded evidential practice, the validation problem stated within the frameworks themselves. Comparative neuroanatomy — finds candidate structures in distant lineages, establishing analogous organisation, the inference to experience being the original question. Behavioural trade-off tests — motivational trade-offs against noxious stimuli demonstrated in decapods and insects, the strongest behavioural evidence, engineered systems producing the same pattern. Theory-driven tests — apply a theory of consciousness and read off a verdict, the theory being what is in dispute. Denying determinacy — no fact in marginal cases, contested, and consequential for practice. **Narrowing recorded.** Marker frameworks narrow the reach: graded evidence stands for the taxa the markers were developed against. **Progress.** The demonstrated motivational trade-offs in decapods and insects. The candidate structures identified in distant lineages. The marker frameworks and their legislative uptake. **Own record.** ## 96 Credit assignment **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** An outcome follows a long sequence of actions. Which actions produced it? In the field's own posing, the problem is the longstanding challenge of associating actions with their long-term consequences, most decision problems providing feedback that is noisy, delayed, and with little or no information about the causes, which makes it hard to distinguish serendipitous outcomes from those caused by informed decision-making; the survey casts it as the problem of learning the influence of an action over an outcome from a finite amount of experience. **The field's account of its hardness.** The field's survey states that the mathematical nature of credit and of the problem remains poorly understood and defined, a unifying formalism being proposed so that methods can be compared equitably; that the challenges are delayed effects, transpositions, and a lack of action influence; and that the protocols to evaluate a credit-assignment method are themselves surveyed, with ways suggested to diagnose the sources of struggle for different methods. **Accounts.** Statistical assignment works: temporal difference methods propagate value backward without a model, eligibility traces distribute credit across a decaying window, and backpropagation through time assigns it exactly in differentiable settings. These are the basis of systems that learn tasks with horizons of thousands of steps from sparse reward, and the convergence properties of the methods are proved for stated conditions. These assign credit relative to a model and a reward specification, both supplied: where the sequence is long, the signal sparse, and the environment non-stationary, the assignment is unidentifiable in principle — many action sequences being consistent with the same outcome — and the methods succeed by imposing structure rather than by recovering it. **Specifications.** An outcome is produced by identifiable contributions — local. A reward signal is specified — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about which actions contributed. **Conserving relation.** Contribution of individual actions identified, the identification standing without a model or reward specification supplied. **Conserving reach.** All sequences the assignment is asked of. **Reach.** The statement quantifies over contribution of individual actions. Outcomes, and counterfactual rollouts within a model, are what is measured. The two are not the same. **Unreachability.** Formal conditions for identifying effects from observational data are proved and are frequently unmet in sequential settings, and reward shaping changes what is credited. The identification is not reached over the sequences by outcomes and within-model rollouts. **Persistence.** An identification of contribution not resting on a supplied model or reward specification. **Addresses.** Statement and hardness: Pignatelli and others, "A Survey of Temporal Credit Assignment in Deep Reinforcement Learning", Transactions on Machine Learning Research, 2024, arXiv:2312.01072, at its abstract as the archive's page carries it verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the survey's abstract read at pass one and at the hardness (credit assignment over long horizons and sparse rewards the survey's stated problem; the methods surveyed each resting on a model, a re-run or a reward specification), an identification of contribution resting on none standing there as what the survey's taxonomy does not contain. Approaches: the temporal-difference and counterfactual routes at that survey's sections, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Temporal difference learning — works, with performance degrading as horizon and sparsity grow, in a measured way. Counterfactual estimation — requires a model or the ability to re-run, which real settings often deny. Causal identification — formal conditions for identifying effects from observational data known and frequently unmet in sequential settings. Reward shaping — improves learning and changes what is credited, which is the concern. Hierarchical decomposition — assigns credit at multiple timescales, the decomposition being supplied. **Narrowing recorded.** Temporal difference methods narrow the reach: convergence stands proved for stated conditions. Counterfactual estimation narrows the reach: the identification stands where a model or re-running is available. **Progress.** The proved convergence under stated conditions. The measured degradation with horizon and sparsity. The demonstrated learning at horizons of thousands of steps from sparse reward. **Own record.** ## 97 Generalization in learning **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A model fits its training data and performs well on new data. Classical theory says a model with more parameters than data should not. Why does it? **The field's account of its hardness.** That the classical bounds are vacuous at practical scale, and that the correction proposed for them — identifying the implicit bias of the optimiser and applying the bound to a correspondingly smaller class — was subsequently argued to be insufficient in principle: for the constructions given, the bound remains large even over a class restricted to what the algorithm actually reaches. The field states that a candidate measure must be non-vacuous, must track the observed width and depth dependence, must apply to the network as trained without modification, and must increase with label noise, and records that no proposed measure meets all four. **Accounts.** Implicit regularisation: optimisation dynamics select solutions with properties that generalise, the double-descent curve shows test error falling again past the interpolation threshold, and the classical bound was about a different quantity than the one that governs. No current theory predicts the observed generalisation in advance: uniform-convergence bounds have been shown insufficient for these regimes by explicit construction, the proposed measures fail to rank models correctly in large-scale empirical comparisons, and the explanations are fitted after the fact. **Specifications.** Generalisation is governed by a complexity measure of the model class — local. A bound stated before training predicts performance — local. **Specifications withdrawn.** **Common requirement.** That there is a property of the trained model determining its generalisation. **Conserving relation.** A property determining generalisation, stated before training rather than fitted after. **Conserving reach.** Performance on data not drawn yet. **Reach.** The statement quantifies over performance on data not drawn yet. Held-out samples from the same distribution are what is measured. The two are not the same. **Unreachability.** Uniform convergence was proved insufficient for interpolating regimes by explicit construction, in the field's own literature, and the proposed complexity measures were measured across thousands of models not to rank them by generalisation. The property is not reached over undrawn data by held-out samples. **Persistence.** A property stated before training that predicts generalisation across the settings in use. **Addresses.** Statement, in substance: the founding paper of the question as it stands for deep networks, Zhang, Bengio, Hardt, Recht and Vinyals, "Understanding deep learning requires rethinking generalization", International Conference on Learning Representations, 2017, arXiv:1611.03530, reprinted in Communications of the ACM, 2021, doi 10.1145/3446776, at its abstract as the arXiv page, an abstract service, the journal's page, a laboratory's page and two indexes carry it verbatim (successful deep networks, despite their massive size, exhibiting a remarkably small difference between training and test performance; conventional wisdom attributing small generalization error to the model family or to regularization; the traditional approaches failing to explain why large networks generalize well in practice; networks easily fitting a random labeling of the training data, unaffected by explicit regularization), the paper itself not fetched; the Statement standing as it stood. Hardness, the four stated requirements on a candidate measure, and the insufficiency argument: Nagarajan and Kolter, "Uniform convergence may be unable to explain generalization in deep learning", NeurIPS 2019, §1 and the authors' own exposition. Vacuity of classical bounds at practical scale: Zhang et al. 2017, as reported in the same. Norm-based guarantees growing with width: the same, §1. Candidate-measure comparison and the sharpness-plus-norms finding: Neyshabur, Bhojanapalli, McAllester, and Srebro, "Exploring Generalization in Deep Learning", NeurIPS 2017, conclusion. Stability's Lipschitz requirement and the framework inventory: the PAC-Bayes survey, arXiv:2601.08100, §I. Persistence, no landing: the closing condition at the founding paper's abstract read at v342 (the traditional measures failing to explain why large networks generalize well; random labels easily fit), a property stated before training that predicts generalization standing there as what the paper's experiments show the traditional measures are not. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Classical uniform convergence — VC dimension, covering numbers, Rademacher complexity, recorded as vacuous at practical scale. Norm-based and margin-based capacity control — weight norms and spectrally normalised margins, developed as the proposed correction, with the recorded finding that existing norm-based guarantees grow rather than shrink as networks are widened, against the observed behaviour. Algorithm-dependent refinement — bounds incorporating the implicit bias of stochastic gradient descent, applied to a smaller class, met by the argument that uniform convergence may be unable to account for generalization even so. Implicit bias analysis — the optimiser's dynamics shown to select margin-maximising solutions, proved for specific architectures and losses, coverage of practical settings partial. PAC-Bayes bounds — non-vacuous bounds achieved in specific settings, the framework recorded as the most versatile of the available tools for data-dependent and algorithm-aware guarantees, tightness in general settings open. Algorithmic stability — bounds sensitivity to single-point perturbation, requiring global Lipschitz constants recorded as unavailable for nonconvex losses. Compression and information-theoretic bounds — a further family, developed alongside the above. Empirical comparison of candidate measures — large-scale ranking across many models, with the recorded finding that no measure ranks models by generalization reliably and that sharpness combined with norms captures much but not all of the behaviour. Double descent and benign overfitting — test error falling again past the interpolation threshold, established empirically and reproduced in analysable linear settings. Empirical scaling laws — predict performance accurately as a function of scale, predictive without explanatory content, which the field states. **Narrowing recorded.** Implicit bias analysis narrows the reach: the result stands proved for specific architectures and losses. PAC-Bayes bounds narrow the reach: non-vacuous bounds stand in specific settings. Scaling laws weaken the relation: prediction stands without the determining property identified. **Progress.** The double-descent curve. The explicit construction against uniform convergence. The measured failure of complexity measures to rank models. The non-vacuous bounds in specific settings. The predictive scaling laws. **Own record.** ## 98 Anesthesia **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Chemically diverse molecules all abolish consciousness. What do they have in common? **The field's account of its hardness.** That a unitary mechanism is stated as a falsehood in the field's own review: anesthetic agents produce different altered states of consciousness as a function of agent type and dose, which cannot occur through a common pathway, with the mechanisms stated as incompletely understood since 1846 and an overly narrow circuit-based reading cautioned against in the current reviews. **Accounts.** Convergent network effects: the agents act on different receptors and converge on one outcome — disruption of long-range cortical communication and of thalamocortical integration — measured directly across chemically unrelated agents, appearing at the concentrations where responsiveness is lost, and reversing in the same order as responsiveness returns. Distinct agents produce distinct signatures: the agents differ in their spectral and connectivity effects, treating them as one phenomenon obscures the differences, and there may be several routes to unresponsiveness rather than one mechanism. **Specifications.** Loss of consciousness is a single state — local. Unresponsiveness indexes it — local. **Specifications withdrawn.** **Common requirement.** That there is one state the agents produce. **Conserving relation.** One state identified as what the agents produce, the identification standing across the agents. **Conserving reach.** All the agents that abolish responsiveness. **Reach.** The statement quantifies over consciousness, indexed by unresponsiveness. Responsiveness, and neural correlates, are what is measured. The two are not the same. **Unreachability.** No common molecular target exists, established; and convergent disruption and agent-specific signatures are both measured, which is the disagreement. Reports of experience without responsiveness after emergence show the index is not responsiveness. The identification is not preserved across the agents by responsiveness and neural correlates. **Persistence.** A measure of the state independent of responsiveness, applied across the agents. **Addresses.** Statement, in substance: the field's review, Franks, "General anaesthesia: from molecular targets to neuronal pathways of sleep and arousal", Nature Reviews Neuroscience 9(5):370-386, 2008, doi 10.1038/nrn2372, at its abstract as the publisher's page, a database, an index and an archive copy carry it verbatim (the mechanisms through which general anaesthetics, an extremely diverse group of drugs, cause reversible loss of consciousness a long-standing mystery; a relatively small number of important molecular targets emerged; finding the link between the molecular studies and the loss of consciousness an enormous challenge; comparisons with natural sleep; the thalamus and the networks regulating it the key), the review itself not fetched; the Statement standing as it stood. Hardness, the unitary-mechanism statement and the unresolved questions: Bonhomme et al., "General Anesthesia: A Probe to Explore Consciousness", Frontiers in Systems Neuroscience 13:36, 2019. The since-1846 standing: Mashour, "Top-down mechanisms of anesthetic-induced unconsciousness", Frontiers in Systems Neuroscience 8:115, 2014, introduction. The declaring-understood caution: Mashour, "Anesthesia and the neurobiology of consciousness", Neuron, 2024. Persistence, no landing: the closing condition at the review's abstract read at v342 (finding the link between the molecular studies and the loss of consciousness an enormous challenge; the thalamus and its networks the key), a measure of the state independent of responsiveness standing there as the link the review names. Approaches: the molecular-target and network-level routes at that abstract in part, their endings at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Common molecular target search — no single target found, agents acting on different receptors, established. Network-level measurement — finds convergent disruption of integration, and also agent-specific signatures, which is the dispute. Perturbational complexity index — distinguishes conscious from unconscious states across conditions with high accuracy, calibrated on the cases in question. Disconnected consciousness — patients reporting experience without responsiveness after emergence, showing the index is not responsiveness. Reversal agents and arousal circuits — specific circuits restoring responsiveness, establishing control of arousal, distinguished from consciousness in the same literature. **Narrowing recorded.** The perturbational index narrows the reach: the discrimination stands across the conditions it is calibrated on. Arousal circuit work narrows the reach: control stands established for arousal, distinguished from consciousness. **Progress.** The established absence of a common molecular target. The measured convergent disruption and the measured agent-specific signatures. The index's discrimination accuracy. The documented disconnected consciousness. **Own record.** ## 99 Phantom limb **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** People feel a limb that is not there, often painfully. What produces the sensation? In the field's own posing, following arm amputation the region that represented the missing hand in primary somatosensory cortex becomes deprived of its primary input and the boundaries of the body map change, a remapping termed reorganisation and attributed to multiple mechanisms, including the increased expression of previously masked inputs; in a maladaptive-plasticity model this reorganisation is associated with phantom limb pain, while brain activity associated with phantom hand movements is also correlated with the pain, suggesting that a preserved functional representation of the limb may serve as a complementary process. **The field's account of its hardness.** The field's joint review states that traditional mechanistic accounts of remapping in somatosensory cortex are incomplete; that the studies point to a close interaction of sensory changes with alterations in regions involved in body representation, pain processing and motor control, so that a broader mechanistic focus beyond primary sensorimotor cortex is needed; that plasticity and stability of the body maps may vary across time scales; that methodological advances such as high-field imaging and multivariate techniques provide new opportunities to interrogate the missing hand's cortical territory; and the field's later review records an ongoing discussion of the relevance of reorganisation for the pain, with attempts to explain seemingly controversial findings. **Accounts.** Cortical reorganisation: after amputation adjacent representations invade the deafferented region, the degree of reorganisation correlates with pain in imaging studies, and therapies aimed at reversing it have effects. The representation is preserved rather than reorganised: imaging with individuated finger stimulation in amputees found the missing limb's map largely intact and correlated with vivid phantom sensation. **Specifications.** The sensation is produced by an identifiable neural state — local. Imaging measures the relevant organisation — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about whether the representation is reorganised or preserved. **Conserving relation.** The map's state determined, the determination standing across imaging paradigms. **Conserving reach.** All amputees the question is asked of. **Reach.** The statement quantifies over the map's state and its relation to the sensation. Imaging at a resolution and paradigm that has changed between the two literatures is what is measured. The two are not the same. **Unreachability.** Two imaging paradigms on the same phenomenon returned opposite directions, and the field has not reconciled them. The determination is not preserved across the paradigms by imaging. **Persistence.** A reconciliation of the two imaging paradigms, or a measurement of the map's state independent of paradigm. **Addresses.** Statement, hardness and the paradigms at Reach: Makin and Flor, "Brain (re)organisation following amputation: Implications for phantom limb pain", NeuroImage 218:116943, 2020, at its abstract and highlights as the journal's and the index's pages carry them verbatim, the paper itself not fetched; the preserved-map finding at Approaches as the field cites it: Makin, Scholz, Filippini, Henderson Slater, Tracey and Johansen-Berg, "Phantom pain is associated with preserved structure and function in the former hand area", Nature Communications 4:1570, 2013; the ongoing discussion as the field's later review states it: Weiss, Koehler and Croy, "Pain and Reorganization after Amputation: Is Interoceptive Prediction a Key?", doi:10.1177/10738584221112591, 2023, at its abstract as the publisher's page carries it, its journal keyed at the next pass. Persistence, no landing: the closing condition at the review's abstract and highlights read at pass one and at the hardness (the two paradigms, reorganisation and preserved representation, drawing on different imaging approaches), a reconciliation of the paradigms or a paradigm-independent measurement standing there as what the review's own contrast asks. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Early mapping studies — found shifted representations correlating with pain, establishing the original account. Individuated-digit imaging in amputees — found preserved maps, directly contesting the earlier finding, the field not having fully reconciled them. Mirror therapy — produces relief in some patients, controlled trials giving mixed results and the mechanism disputed. Peripheral interventions — neuroma treatment and targeted reinnervation helping some patients, supporting a peripheral contribution. Congenital absence cases — phantoms reported without any prior limb, constraining accounts requiring a lost representation. **Narrowing recorded.** Peripheral interventions narrow the reach: the contribution stands established for the patients they help. **Progress.** The measured correlations in both directions under the two paradigms. The mixed controlled-trial results for mirror therapy. The peripheral intervention effects. The congenital absence cases. **Own record.** ## 100 The problem of the criterion **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** To know which beliefs are knowledge, one needs a criterion. To know a criterion is correct, one needs to know which beliefs are knowledge. Where does one start? In the field's own posing, the problem's two questions each seem to require an answer to the other, and the three responses the field records are scepticism, which claims both questions unanswerable, methodism, which starts with criteria, and particularism, which claims that some things are known independently of a criterion; the field's treatment counts the problem among the most important and the most difficult of all the problems of philosophy. **The field's account of its hardness.** The field's study states that although the problem has been largely neglected in contemporary philosophy it raises questions which must be addressed by any complete account of knowledge, and, as the repository's summary of the study states, that the standard formulation of the problem is unclear and incomplete, the responses to it needing a more comprehensive understanding; the field's encyclopaedia records the treatment that counts the problem among the most important and most difficult of all the problems of philosophy, and the ancient form of the problem, how one decides when one genuinely possesses knowledge. **Accounts.** With particular cases: starting from clear instances of knowledge and deriving the criterion from them, which is how inquiry proceeds. With the criterion: starting from a standard and sorting cases by it, on the ground that calling instances clear already applies an unstated standard. **Specifications.** Knowledge has an extension and a criterion — local. One determines the other — local. The two starting points are exhaustive — local. **Specifications withdrawn.** **Common requirement.** That there is a correct criterion and a correct extension, matching. **Conserving relation.** A starting point fixing one of criterion and extension, the fixing standing without the other assumed. **Conserving reach.** The relation between criterion and extension. **Reach.** The statement quantifies over the relation between criterion and extension. Judgments about cases, each already reflecting a standard, are what is measured. The two are not the same. **Unreachability.** Selecting the clear cases is where the criterion enters unannounced, and securing the criterion's warrant is the standing objection to the other start; reflective equilibrium adjusts both to each other, which its critics hold is the circle restated. The fixing is not reached over the relation by judgments already reflecting a standard. **Persistence.** A starting point not assuming what the other requires, or agreement that adjusting both to each other is warrant. **Addresses.** Statement, hardness and the three responses at Approaches: Cling, "Posing the problem of the criterion", Philosophical Studies 75(3):261-292, 1994, at its opening as the index's page carries it verbatim and at the repository's summary of its argument, the paper itself not fetched; McCain, "Problem of the Criterion", Internet Encyclopedia of Philosophy, 2014, at its opening as the index's page carries it; the treatment both cite: Chisholm, The Problem of the Criterion, Marquette University Press, 1973, at p. 1 as the encyclopedia cites it. Persistence, no landing: the closing condition at the paper's opening and the repository's summary read at pass one and at the hardness (the three responses, particularist, methodist and skeptical; each starting point assuming what the other requires), a starting point assuming neither or the mutual adjustment accepted as warrant standing there as the paper's own alternatives. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Particularism — starting from clear instances, its critics noting that selecting the clear cases is where the criterion enters unannounced. Methodism — starting from a standard, the criterion's warrant being the standing objection, attempts to secure it widely held to fail. Reflective equilibrium — adjusts both until they cohere, accounting for practice, whether coherence between two things each adjusted to the other is warrant being the same circle, addressed by its proponents and not accepted as answered by critics. Externalist reliabilism — makes the criterion a matter of fact about reliability rather than something one must know, shifting the demand, the generality problem being its own open problem. The skeptical conclusion — neither can start, held by few and not refuted. **Narrowing recorded.** Reliabilism weakens the relation: the criterion stands as a fact about reliability rather than something the knower must fix. Reflective equilibrium weakens the relation: coherence between mutually adjusted terms stands in place of a starting point. **Progress.** The exhaustive statement of the two starting points. The standing objection against each. The generality problem, stated. **Own record.** ## 101 Skepticism about the external world **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Experience would be the same whether it is caused by an external world or not. What justifies belief in the world? **The field's account of its hardness.** That the argument from ignorance presents a puzzle because each of its premises is initially plausible, that one does not know one is not a handless brain in a vat and that if one does not know this one does not know one has hands; that on the subjunctive conditionals account the problem with the belief that one is not so envatted is that one would have it even if it were false, which makes it hard to claim to know, given a strong general though not exceptionless inclination to deny knowledge where a belief is insensitive; and that, in contrast to non-skeptical cases, the cases involving skeptical hypotheses cannot be modified to make the belief sensitive, the negations of skeptical hypotheses being stably insensitive. **Accounts.** Inference to the best explanation: an external world accounts for the coherence, involuntariness, and predictability of experience far better than any rival, and skeptical scenarios are constructed to have no explanatory content at all. The rivals are constructed to be empirically equivalent, so no evidence favours one, and best-explanation reasoning applied here presupposes standards of explanation the skeptic is asking about. **Specifications.** Experience is one thing and its cause another — local. Justification runs from the first to the second — local. **Specifications withdrawn.** **Common requirement.** That belief in the world stands in need of justification of some kind. **Conserving relation.** A justification running from experience to its cause, standing without presupposing standards the statement questions. **Conserving reach.** All rivals accounting for the experience. **Reach.** The statement quantifies over the cause of experience. Experience is what is measured. The two are not the same. **Unreachability.** The skeptical scenarios are empirically equivalent by construction, and transcendental arguments establish at most how one must think, by the standing objection. The justification is not reached over the rivals by experience. **Persistence.** A justification not presupposing the standards in question, or agreement that no justification is owed. **Addresses.** Statement, the problem as the field poses it: DeRose, "Solving the Skeptical Problem", Philosophical Review 104:1-52, 1995. Hardness: the same paper, at its opening as an abstract service carries it verbatim (skeptics persuading of ignorance with the argument that one does not know one is not a handless brain in a vat, that if one does not know this one does not know one has hands, and so that one does not know one has hands) and at passages of its text as citing papers carry them verbatim (the argument presenting a puzzle because each of its premises is initially plausible; on the subjunctive conditionals account the problem with the belief that one is not envatted that one would have it even if it were false, which makes it hard to claim to know, given a strong general though not exceptionless inclination to deny knowledge where a belief is insensitive; the skeptical cases, unlike others, not modifiable to make the belief sensitive, at pp. 25f. as one citing paper gives them), issue 1, the paper itself not fetched; reprinted in part in an anthology as a citing paper records. Persistence, no landing: the closing condition at the same paper, at its opening read at pass two (the skeptic's argument from not knowing one is not a handless brain in a vat; the responses each charged with begging the question or with conceding), a justification not presupposing the standards in question standing there as what each response is charged with lacking. Approaches: the rationalist and exhibition routes at that paper's survey of the responses and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The rationalist proof — requires a guarantor whose establishment is generally held to fail. The proof by exhibition — exhibits a hand, its critics holding it begs the question, its defenders holding the demand is the mistake, unresolved and long-standing. Transcendental arguments — the skeptical hypothesis unstatable without presupposing what it doubts, the objection that they establish at most how we must think not generally answered. Semantic externalism — the argument that a system in the skeptical scenario cannot truly say it is in one, widely discussed and widely held not to reach the epistemic question. Contextualism — skeptical standards applying only in skeptical contexts, accounting for the pattern of judgments, whether it answers the argument contested. Rejecting the demand — each form facing the charge of changing the subject. **Narrowing recorded.** Contextualism narrows the reach: the justification stands within ordinary contexts rather than against the skeptical rivals. Transcendental arguments weaken the relation: what is established is how one must think rather than what is the case. **Progress.** The construction of the empirically equivalent rivals. The mapped objection against each response. **Own record.** ## 102 Substance and bundle **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** An object has properties. Is there something that has them, or is the object just the properties together? In the field's own posing, the issue is whether substances are in some sense reducible to their properties or whether there exists some further component, a substratum, both views having been defended; the bundle theorist has two options, substances as no more than bundles of properties conceived as universals, or as no more than bundles of property-instances, tropes, and on either the claim that substances are no more than bundles may be regarded as a conceptual analysis of the concept of substance or as a de facto but not a conceptual truth. **The field's account of its hardness.** The field's survey states that the substratum is ontologically basic only in a rather empty sense, being the substancehood of everything without explaining the nature of anything, having no positive nature, and resting its claim not on meeting the tests of substancehood itself but on being what enables particular kinds of substances to be substances; and that the trope version of the bundle theory finds it difficult to individuate or distinguish tropes in a way that makes them suitable to be the individuals from which objects are made, one way out being a master property of which all the others are modes, which itself fills the role of substratum. **Accounts.** Something has them: properties are had by something, and without a bearer there is no account of what unifies a bundle, nor of how an object persists through property change. Nothing further is needed: the bearer is characterised only as that which has properties, so it has no nature of its own and does no work, and bundles unified by compresence do the same job without positing a featureless item. **Specifications.** An object decomposes into a bearer and properties, or into properties alone — local. One of these is correct — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about the constitution of an object. **Conserving relation.** A determination of whether a bearer does work the properties do not. **Conserving reach.** All objects the question is asked of. **Reach.** The statement quantifies over whether a bearer does work the properties do not. Properties, measured, are what is measured. The two are not the same. **Unreachability.** Trope theory solves the indiscernibles case and inherits the question of what bundles the tropes, which is the original question; and the appeal to physics is read as supporting bundles by some and as presupposing a bearer by others. The determination is not reached over the objects by measured properties. **Persistence.** A case where the two accounts give different verdicts and one verdict is agreed correct. **Addresses.** Statement, hardness and the substratum and trope approaches at Approaches: the entry "Substance", Stanford Encyclopedia of Philosophy, at its section on contemporary discussions, at its assessment of substratum, and at its passage on the trope bundle, as the encyclopedia's current page and archived editions carry them verbatim, the entry itself not fetched, its author and revision date keyed at the next pass. Persistence, no landing: the closing condition at the same entry, at its contemporary-discussion section and its assessment of the substratum view read at pass one and at the hardness (a bearer with no nature held unintelligible by the critics and accepted as the point by the defenders; the bundle and trope alternatives each with its counterexamples), a case sorting the accounts and agreed correct standing there as what neither side has produced. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Bare substratum — posits a bearer with no properties of its own, its critics holding an item with no nature is unintelligible, its defenders accepting the characterisation as the point. Bundle theory — faces the identity of indiscernibles, two exactly similar objects being one, responses invoking tropes or spatiotemporal relations, each with costs. Trope theory — properties as particulars, bundled, solving the indiscernibles case and facing questions about what bundles them, which is the original question. Appeal to physics — field values at points, or structures without relata, read as supporting bundles by some and as presupposing a bearer by others. Deflating the question — held terminological, its critics noting the two accounts give different verdicts in indiscernibles cases, so something turns on it. **Narrowing recorded.** Trope theory weakens the relation: properties stand as particulars, which removes the indiscernibles case and leaves the bundling question. Spatiotemporal responses narrow the reach: the account stands for objects distinguishable by location. **Progress.** The indiscernibles case and the responses to it with their costs. The mapped verdicts of the two accounts where they differ. **Own record.** ## 103 Why there is something **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why is there something rather than nothing? **The field's account of its hardness.** That the question is met with the counter-question why one should expect nothing rather than something, no experiment being able to support the hypothesis that there is nothing, any observation implying an observer, and the a priori support asked for being a methodological principle propelling the empty world to the top of the agenda; and that conservatives, coherentists and scientific gradualists cast a suspicious eye on the question, most contemporary philosophers feeling entitled to postulate whatever entities are indispensable to their best explanations of well-accepted phenomena, the presumption of non-existence plausible only for particular existence claims and applying case by case, leaving no grand methodological preference for an empty world. **Accounts.** The statement is malformed: explanation relates states of affairs, and there is no prior state from which the existence of anything could be accounted for. Nothing is not a possible alternative that failed to obtain. The statement is intelligible and unanswered: physical accounts of a universe arising from a quantum vacuum account for the arising from something — a vacuum with laws is not nothing — and relocate the question rather than closing it. **Specifications.** Existence admits of explanation — local. Nothingness is a coherent alternative — local. **Specifications withdrawn.** **Common requirement.** That the statement either has an answer or is demonstrably malformed. **Conserving relation.** An account of the obtaining of anything, standing without a prior state assumed. **Conserving reach.** The obtaining of anything at all. **Reach.** The statement quantifies over the obtaining of anything at all. What obtains is what is measured. The two are not the same. **Unreachability.** Quantum origination accounts for arising from a structured state, which both accounts accept, so it relocates rather than closes; and brute fact absorbs the question at an acknowledged cost. The account is not reached over the obtaining of anything by what obtains. **Persistence.** An account not itself starting from something, or a demonstration that the statement is malformed accepted by both accounts. **Addresses.** Statement, the question at the entry's first section: Sorensen, "Nothingness", Stanford Encyclopedia of Philosophy. A book-length treatment as the field cites it: Rundle, Why there is Something rather than Nothing, Oxford, 2004. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the question met with the counter-question why one should expect nothing rather than something; no experiment able to support the hypothesis that there is nothing, any observation implying an observer; the a priori support asked for being a methodological principle propelling the empty world to the top of the agenda; conservatives, coherentists and scientific gradualists casting a suspicious eye on the question; most contemporary philosophers feeling entitled to postulate whatever entities their best explanations require, the presumption of non-existence plausible only case by case, leaving no grand methodological preference for an empty world), the entry itself not fetched, its section numbers and revision date not carried at the returns; the book-length treatment not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the counter-question why one should expect nothing rather than something; no experiment able to bear on the question), an account not itself starting from something, or the statement's malformation accepted, standing there as the entry's own two directions. Approaches: the necessary-being and vacuum-origination routes at the same entry's sections on the answers, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Necessary being — a being that could not fail to exist, the modal status being what is in question, the arguments for it widely held to fail. Quantum vacuum origination — accounts for a universe from a vacuum state and laws, which are something, both accounts agreeing on the physics and disagreeing on what it answers. Brute fact — existence unexplained, consistent, its critics holding it concedes the question. The principle of sufficient reason — if accepted, requiring an answer, its acceptance being the dispute, unrestricted versions having known costs. Deflation — nothingness held incoherent, its critics finding the notion coherent enough to state. **Narrowing recorded.** Quantum origination narrows the reach: the account stands for a universe arising from a structured state rather than for the obtaining of anything. **Progress.** The physics of vacuum origination, agreed by both accounts. The mapped costs of the unrestricted principle. The stated positions on the coherence of nothingness. **Own record.** ## 104 The simulation hypothesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Could what we observe be a computation run at another level of reality, and could we tell? In the field's own posing, the argument holds that at least one of three propositions is true: that the human species is very likely to go extinct before reaching a posthuman stage; that any posthuman civilisation is extremely unlikely to run a significant number of simulations of its evolutionary history; or that we are almost certainly living in a computer simulation; so that the belief that there is a significant chance we will one day become posthumans who run ancestor-simulations is false unless we are currently living in a simulation. **The field's account of its hardness.** The founding paper states that the argument rests on the empirical fact that a technologically mature civilisation would have enormous computing power and on the assumption of substrate-independence, that mental states can supervene on any of a broad class of physical substrates provided the right computational structures and processes are implemented; that its core uses simple probability theory together with a weak indifference principle for which the paper has to provide separate support; and that its conclusion is a disjunction whose interpretations the paper discusses. **Accounts.** The statistical argument stands: if civilisations run many high-fidelity ancestor simulations, simulated observers vastly outnumber unsimulated ones, so one of three disjuncts holds and the hypothesis is live. The argument requires an indifference principle over observers and an assumption that consciousness is substrate-independent, both assumed rather than established, and the hypothesis makes no differential prediction. **Specifications.** Observers can be counted across levels — local. Experience is substrate-independent — local. A simulating level is possible — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about the level one is at. **Conserving relation.** A determination of the level, standing on observations available within it. **Conserving reach.** A level not accessible from within. **Reach.** The statement quantifies over a level not accessible from within. Observations within the level are what is measured. The two are not the same. **Unreachability.** The statistical argument yields a disjunction rather than a conclusion, which its author states; and proposed signature tests test one implementation among many and are null. The determination is not reached over the level by observations within it. **Persistence.** A differential prediction distinguishing the levels, or agreement on the indifference principle over observers. **Addresses.** Statement, hardness, and the disjunctive form and the indifference principle at Approaches: Bostrom, "Are You Living in a Computer Simulation?", Philosophical Quarterly 53(211):243-255, 2003, at its abstract as the archive's page carries it verbatim and at its statement of the argument's structure, its three propositions in fraction form and its assumption of substrate-independence as the author's page and a repository copy carry them verbatim, the paper itself not fetched; the selective-scepticism reply as the field cites it: Birch, "On the 'Simulation Argument' and Selective Scepticism", Erkenntnis 78:95-107, 2013, its abstract not read. Persistence, no landing: the closing condition at the founding paper's abstract read at pass one and at the hardness (the disjunction, not a conclusion, stated by its author; the indifference principle over observers the argument's premise), a differential prediction or agreement on the principle standing there as what the argument's own form leaves. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The statistical argument — yields a disjunction, not a conclusion, which its author states explicitly. Search for computational artefacts — proposed tests for lattice signatures in cosmic ray spectra, null so far, testing one implementation among many. Computational cost arguments — argue full simulation is infeasible, assuming the simulating level's physics resembles ours, which the hypothesis does not require. Anthropic and indifference reasoning — constrains inference, the indifference principle over observers itself contested with known paradoxes. The unfalsifiability objection — places it outside empirical inquiry, its proponents accepting the disjunctive form is not a prediction. **Narrowing recorded.** Signature tests narrow the reach: the null result stands for the one implementation tested. **Progress.** The disjunctive form of the statistical argument. The null lattice-signature results and their stated scope. The known paradoxes of the indifference principle. **Own record.** ## 105 Laws of nature **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Are laws prescriptive — governing what happens — or descriptive summaries of what happens? **The field's account of its hardness.** That the systems approach, making no overt appeal to modal concepts or to modality-supplying entities, is faulted by some for making laws inappropriately mind-dependent through its appeal to simplicity, strength and best balance, concepts whose instantiation seems to depend on cognitive abilities, interests and purposes; that questions remain about the nature of the necessitation relation the rival approach posits, understood as relating both token events and universals; and that the especially divisive issue is supervenience, whether the mosaic of local matters of particular fact really determines what the laws are, examples appearing to show that it does not, which the one side must contend are not both really possible and which are accused of begging the question by assuming a governing conception of laws. **Accounts.** Descriptive: laws are the axioms of the simplest, strongest true systematisation of the facts. Nothing governs, regularity is all there is, and this avoids positing a modal relation nobody has accounted for. Prescriptive: regularities alone cannot distinguish laws from accidental generalisations, cannot support counterfactuals, and cannot account for why the regularity holds. A relation of necessitation between universals, or a primitive modal fact, is required. **Specifications.** There is a fact distinguishing laws from accidents — local. Laws either govern or summarise — local. **Specifications withdrawn.** **Common requirement.** That the distinction between law and accident is real. **Conserving relation.** The modal status of a regularity determined, the determination standing on the regularities. **Conserving reach.** All regularities the distinction is asked of. **Reach.** The statement quantifies over the modal status of a regularity. The regularities themselves are what is measured. The two are not the same. **Unreachability.** Best-system simplicity is language-relative, addressed by appeal to natural properties, which is a primitive; and the identification problem — what makes the necessitation relation necessitate — is unanswered to general satisfaction. The determination is not reached over the regularities by the regularities. **Persistence.** A criterion distinguishing law from accident not resting on a primitive, or agreement that a primitive is acceptable. **Addresses.** Statement, the two sides as the field states them: Earman, "Laws of Nature: The Empiricist Challenge", 1984; Cartwright, The Dappled World, Cambridge, 1999. Hardness: the encyclopedia's entry on laws of nature, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the systems approach making no overt appeal to modal concepts or modality-supplying entities and standing as the centrepiece of the defence of Humean supervenience; some arguing it makes laws inappropriately mind-dependent through its appeal to simplicity, strength and best balance, concepts whose instantiation seems to depend on cognitive abilities, interests and purposes; questions remaining about the nature of the necessitation relation understood as relating both token events and universals; supervenience the especially divisive issue, whether Humean considerations really determine what the laws are, examples appearing to show they do not, which Humeans must contend are not both really possible and which are accused of begging the question by assuming a governing conception), returned in the search, its author not carried at the returns, the entry itself not fetched; the two sources the address names not returned, not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the systems approach making no overt appeal to modal notions and resting on simplicity and strength, language-relative; the necessitarian alternatives resting on a primitive), a criterion not resting on a primitive standing there as what the entry records each side charging the other with lacking. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Best-system analysis — yields the distinction from simplicity and strength, the measures being language-relative, which is the standing objection, addressed by appeal to natural properties, which is a primitive. Necessitation between universals — the identification problem, what makes the relation necessitate, unanswered to general satisfaction. Primitivism about laws — takes governing as basic, consistent, and accounting for nothing further, which its proponents accept. Anti-realism about laws — no laws, only models that fit, accommodating practice and contested on the counterfactual role. Looking to physics — symmetry principles treated as more fundamental than laws in practice, accommodated by both accounts. **Narrowing recorded.** Anti-realism weakens the relation: models that fit stand in place of a modal status determined. Primitivism weakens the relation: governing stands basic rather than accounted for. **Progress.** The best-system construction and its language-relativity. The identification problem, stated. The treatment of symmetry principles in practice. **Own record.** ## 106 Presentism and eternalism **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Do past and future events exist, or only present ones? **The field's account of its hardness.** That presentism supposes one plane of simultaneity, representing all that exists, to be uniquely metaphysically important, while the special theory of relativity tells that an infinity of planes of simultaneity pass through any given spacetime point and that no physical test can distinguish one from among them, what was supposedly metaphysically distinguished being physically indistinguishable, and, assuming that humans are complex physical systems, there being no way to distinguish the present from among the multitude of putative presents. **Accounts.** Only the present: what is past no longer exists and what is future does not yet, and the tensed character of thought and language reflects a real feature of the world. All times exist: relativity gives no privileged present, and the tensed series is incoherent as an objective feature — an event must be past, present, and future, and the attempt to relativise these generates a regress. **Specifications.** Existence is a matter of fact not relative to a time — local. The tensed and tenseless descriptions are rivals — local. **Specifications withdrawn.** **Common requirement.** That one of the two descriptions is correct. **Conserving relation.** A determination of what exists unqualified, standing on observation. **Conserving reach.** All times. **Reach.** The statement quantifies over what exists, unqualified. Observations, all made at a time, are what is measured. The two are not the same. **Unreachability.** A preferred foliation is consistent and undetectable in principle, and both accounts accept the linguistic semantics of tense and dispute the metaphysics. The determination is not reached over the times by observations made at a time. **Persistence.** A detectable privileged present, which the foliation response makes undetectable in principle, or agreement on what the tensed semantics establishes. **Addresses.** Statement: Savitt, "Being and Becoming in Modern Physics", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at a passage of its text as the encyclopedia's page and its archived editions carry it verbatim (presentism and possibilism supposing one plane of simultaneity uniquely metaphysically important, in presentism representing all that exists; the special theory of relativity telling of an infinity of planes of simultaneity through any given spacetime point with no physical test distinguishing one; what was supposedly metaphysically distinguished physically indistinguishable; humans as complex physical systems having no way to distinguish the present from among the multitude of putative presents), the entry itself not fetched, its section numbers and revision date not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passage read at pass two (presentism supposing one plane of simultaneity uniquely metaphysically important, the foliation response placing it beyond detection), a detectable privileged present standing there as what the response makes undetectable in principle. Approaches: the regress argument and the relativity route at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The regress argument against objective tense — widely regarded as showing a difficulty, met by taking tense as primitive, the exchange old and unresolved. Relativity — excludes a frame-independent present, met with a preferred foliation undetectable in principle, consistent and unfalsifiable. The truthmaker objection — asks what makes past-tensed statements true if the past does not exist, met with present traces or primitive tensed facts, each response contested. Growing block — past and present existing, future not, facing the objection that its inhabitants could not know they are at the edge. Linguistic analysis of tense — establishes how tense functions, both accounts accepting the semantics and disputing the metaphysics. **Narrowing recorded.** The preferred foliation narrows the reach: a present stands by stipulation rather than by detection. Growing block narrows the reach: existence stands for past and present rather than all times. **Progress.** The exclusion of a frame-independent present. The tensed semantics, agreed by both accounts. The mapped responses to the truthmaker objection. **Own record.** ## 107 The effectiveness of mathematics **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Mathematics developed for its own reasons turns out to describe physical phenomena precisely. Why? **The field's account of its hardness.** That the enormous usefulness of mathematics in the natural sciences is described in the founding lecture as something bordering on the mysterious with no rational explanation for it, the mathematical formulation of the physicist's often crude experience leading in an uncanny number of cases to an amazingly accurate description of a large class of phenomena, an accuracy beyond all reasonable expectations, the impression that a miracle confronts us being difficult to avoid, comparable to the miracle that the human mind can string a thousand arguments together without falling into contradiction; and that it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of the physical theories. **Accounts.** Selection and construction: mathematics is developed in continual contact with physical problems, the branches that apply were often built for application, and the successes are salient while the vast unapplied remainder is not counted. The selection account does not cover the striking cases: non-Euclidean geometry, group theory, and complex analysis were developed with no physical application in view and later proved indispensable, and predictive success far beyond the data used to construct the theory is not accounted for by contact or selection. **Specifications.** Mathematics and the physical world are two domains — local. Applicability is a relation between them calling for explanation — local. **Specifications withdrawn.** **Common requirement.** That the applicability is a phenomenon with an explanation. **Conserving relation.** An account of why the relation obtains, standing across the cases including the striking ones. **Conserving reach.** All instances of application. **Reach.** The statement quantifies over why the relation obtains. Instances of application, and the historical record of their development, are what is measured. The two are not the same. **Unreachability.** Both accounts read the same historical record differently, and platonism absorbs the applicability and inherits the access problem. The account is not preserved across the cases by instances and the historical record. **Persistence.** An account covering the striking cases that both accounts accept, or agreement on what the historical record shows. **Addresses.** Statement: Wigner, "The unreasonable effectiveness of mathematics in the natural sciences", Communications on Pure and Applied Mathematics 13:1-14, 1960. Hardness: the same lecture, at passages of its text as three repository copies and an archive carry them verbatim (the enormous usefulness of mathematics in the natural sciences something bordering on the mysterious with no rational explanation for it; just this uncanny usefulness raising the question of the uniqueness of the physical theories; the mathematical formulation of the physicist's often crude experience leading in an uncanny number of cases to an amazingly accurate description of a large class of phenomena; the impression of a miracle difficult to avoid, comparable to the mind stringing a thousand arguments together without contradiction), the Richard Courant Lecture delivered 11 May 1959, issue 1, doi 10.1002/cpa.3160130102, as the copies and a citing text carry it, the lecture itself not fetched. Persistence, no landing: the closing condition at the lecture's passages read at pass two (the usefulness of mathematics bordering on the mysterious with no rational explanation), an account covering the striking cases standing there as what the lecture declares wanting. Approaches: the selection-effect and structural routes at the sources the address names, read at pass one, against the lecture's cases. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Selection effect accounting — accounts for many cases, both accounts disputing whether it covers the striking ones, the dispute being over a historical record that is available and read differently. Structural realism — the world having mathematical structure and theories capturing it, accounting for applicability by making it constitutive, what a structure without relata is remaining contested. Mathematical naturalism — mathematics abstracted from physical structure, facing the branches with no physical origin. Platonism — mathematics describing an independent realm the world instantiates, the access problem being its standing difficulty. Deflation — the question held rhetorical, its critics holding the striking cases are specific and documented. **Narrowing recorded.** Structural realism weakens the relation: applicability stands constitutive rather than accounted for. Mathematical naturalism narrows the reach: the account stands for branches with physical origin. **Progress.** The documented cases of application without prior physical motivation. The selection accounting for the cases it covers. The access problem, stated. **Own record.** ## 108 The Gibbs paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Mixing two different gases raises entropy. Mixing a gas with itself does not. The change is discontinuous as the gases are made more alike. Why? In the field's own posing, the paradox is a set of open questions as to how sameness of gases or fluids is to be treated in thermodynamics and statistical mechanics, stated as three puzzles: the discontinuity puzzle, why the entropy of mixing is independent of the degree of similarity of the gases and zero only when they are the same; the extensivity puzzle, how an extensive entropy function can be defined in classical statistical mechanics; and the microrealism puzzle, how there can fail to be an entropy of mixing even for samples of the same gas, given that the particles undergo much the same microscopic motions on mixing whether exactly alike or only similar. **The field's account of its hardness.** The field's studies state that the questions have a variety of answers, some restricted to quantum theory, holding that there is no classical solution, and some to classical theory, holding that the quantum case is different; that the three puzzles are far from independent, an entropy of mixing even for identical gases dissolving the first two while answers to the first two are frequently silent on the third, which is in part a question about the physical interpretation of entropy quite generally; and that a consensus holds the classical paradox unmysterious, the discontinuity reflecting the step between gases being separable or not, while the paradox in statistical physics continues to attract attention. **Accounts.** Indistinguishability is fundamental: identical particles are not merely similar but the same kind, the correct counting divides by the permutations, quantum mechanics supplies this, and the discontinuity is real because identity is not a matter of degree. The entropy of mixing depends on what can be separated by available operations, so it is relative to the capacity to discriminate: the paradox arises from treating entropy as a property of the system alone rather than of the description, and the discontinuity is a discontinuity in the description. **Specifications.** Entropy is a property of a system — local. Particle identity is determinate — universal under quantum mechanics. **Specifications withdrawn.** **Common requirement.** That there is a fact about the entropy change on mixing. **Conserving relation.** Entropy determined as a property of the system, the determination standing independently of the operations available. **Conserving reach.** All mixings, including those interpolating between the two cases. **Reach.** The statement quantifies over whether entropy is a property of the system or of the description. Measurable work extractable given available operations is what is measured. The two are not the same. **Unreachability.** The information-theoretic treatment reproduces every measurement and relocates entropy to the description; the resource-theoretic formulation states the relativity rather than removing it; and particles continuously interpolating between two species are not available, so the discontinuity is not directly probed. The determination is not reached over the mixings by extractable work. **Persistence.** A measurement of entropy independent of the operations available, or a direct probe of the interpolating case. **Addresses.** Statement, hardness and the separability at Reach: Saunders, "The Gibbs Paradox", Entropy 20(8):552, 2018, at its abstract and its statement of the three puzzles as the journal's page and the archive copies carry them verbatim, the paper itself not fetched; Dieks, "The Gibbs Paradox and Particle Individuality", Entropy 20(6):466, 2018, at its abstract as the archive's page carries it verbatim, the paper itself not fetched; the discontinuity puzzle as the former cites its standard statement: Denbigh and Redhead, p. 284. Persistence, no landing: the closing condition at the paper's abstract and its three puzzles read at pass one and at the hardness (the entropy of mixing depending on the operations available to part the gases; the interpolating case of nearly identical gases), a measurement independent of the operations or a probe of the interpolating case standing there as what the three puzzles ask. Approaches: the quantum-indistinguishability and informational routes at that paper's sections, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Quantum indistinguishability — supplies the correct counting and the discontinuity, its critics noting classical statistical mechanics needed the division before quantum theory supplied it. The information-theoretic treatment — makes the entropy relative to available operations, reproducing every measurement, the disagreement being over whether entropy is thereby subjective. Isotope experiments — separable isotopes giving mixing entropy, the separation capacity being what changed historically, supporting the operational reading. Continuous interpolation — constructing particles continuously interpolating between two species not available for real particles, so the discontinuity is not directly probed. Resource-theoretic thermodynamics — makes the operation set explicit and derives entropy relative to it, a formal advance stating the relativity rather than removing it. **Narrowing recorded.** Resource-theoretic thermodynamics narrows the reach: entropy stands derived relative to a stated operation set. The information-theoretic treatment weakens the relation: entropy stands as a property of the description rather than of the system. **Progress.** The correct counting and the measured discontinuity. The isotope results and the historical change in separation capacity. The explicit operation-set formulation. **Own record.** ## 109 The surprise examination **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** An examination will be held next week and its day will be a surprise. The students argue it cannot happen. It happens, and is a surprise. Where is the error? **The field's account of its hardness.** That a survey of solutions to the paradox, described in the field as a remarkably gregarious riddle, is also a survey of epistemic paradoxes, the backward elimination argument underlying it discernible in folktales; that the first published reaction endorsed the students' elimination argument, taking the announcement to be self-defeating; and that on the proposed solution the announcement is informative although the test would be a surprise even on the last day, the student who knows no test has yet been given being unable also to know the announcement true, that conjunction being a blindspot for the students but not for others. **Accounts.** The backward induction is invalid: the announcement's truth conditions involve the students' knowledge, and after the argument they no longer know the exam will occur, so the eliminating step fails at the first move. No consensus account exists: epistemic, temporal-retention, and self-referential analyses each block the argument at a different step, each has counterexamples in the literature, and the paradox has been reformulated repeatedly to evade each solution. **Specifications.** The announcement is known to be true — local. Knowledge persists through the reasoning — local. Surprise is definable in terms of what is known — local. **Specifications withdrawn.** **Common requirement.** That there is a determinate error in the argument. **Conserving relation.** An error located in the argument, the location standing against reformulations. **Conserving reach.** All formulations of the argument. **Reach.** The statement quantifies over the status of a self-referential knowledge claim. Formal reconstructions, each with its own assumptions, are what is measured. The two are not the same. **Unreachability.** Each proposed solution has been met by a reformulation restoring the puzzle, documented over seventy years. The location is not preserved across the formulations by any reconstruction. **Persistence.** A location of the error standing against the reformulations, or agreement on which assumption the reformulations must drop. **Addresses.** Statement: Sorensen, "Epistemic Paradoxes", Stanford Encyclopedia of Philosophy; the modal-logic formulation: Binkley, "The Surprise Examination in Modal Logic", Journal of Philosophy 65:127-136, 1968. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (a survey of solutions to the paradox, a remarkably gregarious riddle, also a survey of epistemic paradoxes; the backward elimination argument discernible in folktales; the first published reaction endorsing the students' argument and taking the announcement as self-defeating; the proposed solution on which the announcement is informative although the test would be a surprise even on the last day, the student who knows no test has been given unable also to know the announcement true, that conjunction a blindspot for the students but not for others), the entry itself not fetched, its section numbers and revision date not carried at the returns; the modal-logic formulation not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (a remarkably gregarious riddle; the survey of solutions and of the reformulations that restore the puzzle under weaker assumptions), a location of the error standing against the reformulations standing there as what the survey has not found. Approaches: the epistemic-blocking and retention routes at the same entry's survey of solutions, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Epistemic blocking — the announcement not knowable in the relevant sense, formalised, the reformulations restoring the puzzle with weaker assumptions. Temporal retention analysis — knowledge not retained through the eliminations, blocking the induction, whether the failure is principled disputed. Self-reference analysis — treats the announcement as a Moore-type sentence, illuminating and not settling which step fails. Game-theoretic treatment — models it as a strategic interaction with mixed strategies, changing the problem, which its authors note. Denying the phenomenon — the surprise held not to be the announced one, contested. **Narrowing recorded.** Game-theoretic treatment narrows the reach: the account stands for a strategic formulation its authors distinguish from the original. Epistemic blocking narrows the reach: the solution stands for formulations with the stronger knowledge assumption. **Progress.** The seventy-year record of solutions and the reformulations meeting each. The formal reconstructions and their stated assumptions. **Own record.** ## 110 Determinism and randomness **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Are outcomes fixed by prior conditions, or genuinely chancy? **The field's account of its hardness.** That the epistemology of determinism must first establish the reality of laws of nature and then establish precisely what the actual laws are, a task that may seem daunting, being in a sense what physicists sometimes set as their goal, the final theory of everything; that many physicists have been convinced of determinism's falsity on the two convictions that the final theory will be a variant of quantum theory and that all quantum theories are non-deterministic, indeterminists granting macroscopic repeatability while holding it not to be found at the microscopic level; and that where quantum theories unquestionably indeterministic would allow further experimental input on the question, the existence of deterministic Bohmian quantum theories casts strong doubt on this. **Accounts.** Chancy: quantum measurement outcomes are unpredictable in principle, not merely in practice, and no local account restoring determinism reproduces the correlations. Determinism remains available: deterministic interpretations reproduce every prediction, apparent randomness is compatible with underlying determinism, and no measurement distinguishes them. **Specifications.** An outcome is either fixed by prior conditions or not — local. Unpredictability indicates chance — local. **Specifications withdrawn.** **Common requirement.** That there is a fact about whether the outcome was fixed. **Conserving relation.** A determination of whether prior conditions fix outcomes, standing on observed frequencies. **Conserving reach.** All outcomes the question is asked of. **Reach.** The statement quantifies over whether prior conditions fix outcomes. Observed frequencies, which both accounts reproduce, are what is measured. The two are not the same. **Unreachability.** Statistical tests provably cannot distinguish deterministic chaos or hidden determination from chance, and the deterministic interpretations are empirically equivalent by construction. The determination is not reached over the outcomes by observed frequencies. **Persistence.** A measurement distinguishing fixed from chancy outcomes, which the empirical equivalence excludes. **Addresses.** Statement: Hoefer, "Causal Determinism", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text and notes as the encyclopedia's page and its archived editions carry them verbatim (the epistemology of determinism first to establish the reality of laws of nature and then precisely what the actual laws are, a task that may seem daunting, in a sense the final theory; many physicists convinced of determinism's falsity on the convictions that the final theory will be a quantum theory and that all quantum theories are non-deterministic; the evidence from macroscopic repeatability no longer cutting much ice for that faith, indeterminists holding repeatability not found at the microscopic level; further experimental input conceivable were quantum theories unquestionably indeterministic, the existence of Bohmian quantum theories casting strong doubt on this; the notes on states in some sense incomplete and hidden variables), the entry itself not fetched, its section numbers not carried at the returns, the notes page carrying a 2023 copyright. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the epistemology of determinism requiring first the reality of laws and then what they imply; the empirical equivalence of deterministic and indeterministic readings of the theories), a measurement parting fixed from chancy outcomes standing there as what the equivalence excludes. Approaches: the statistical-test and Bell-test routes at the same entry's sections and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Statistical tests for randomness — confirm the distributions and cannot distinguish deterministic chaos or hidden determination from chance, a general result about such tests. Bell tests — exclude local determination, nonlocal determination surviving. Algorithmic randomness — defines randomness precisely for sequences, applying to sequences and not to what produced them. Deterministic interpretations — empirically equivalent by construction. Chaos in classical systems — establishes that unpredictability does not imply indeterminism, accepted by both accounts. **Narrowing recorded.** Bell tests narrow the reach: local determination stands excluded, nonlocal determination surviving. Algorithmic randomness narrows the reach: randomness stands defined for sequences rather than for what produced them. **Progress.** The exclusion of local determination. The general result on statistical tests. The precise definition of randomness for sequences. The classical result that unpredictability does not imply indeterminism. **Own record.** ## 111 Qualia and the zombie argument **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Could there be a being physically identical to a person with no experience at all? If it is conceivable, what follows? The field's own name for the problem the argument is posed within is the hard problem of consciousness, and its canonical statement stands verbatim: why is the performance of these functions accompanied by experience? **The field's account of its hardness.** The field's founding statement: what makes the hard problem hard and almost unique is that it goes beyond problems about the performance of functions — even when the cognitive and behavioural functions in the vicinity of experience are explained, a further question remains, why their performance is accompanied by experience, a simple explanation of the functions leaving it open, and no analogous further question standing in the explanation of genes, of life, or of learning. **Accounts.** Conceivability supports possibility here, and possibility refutes physicalism: the argument is deductive, its premises are examinable, and no defect has been generally agreed on despite three decades of attention. Conceivability does not entail possibility: water without H₂O is conceivable and impossible, the a posteriori necessity cases show the inference fails in exactly the structurally similar cases, and the conceivability may reflect a limitation in how the physical is represented rather than a metaphysical fact. **Specifications.** Conceivability bears on possibility — local. Physical duplication fixes all physical facts — local. **Specifications withdrawn.** **Common requirement.** That the physical facts either fix the experiential ones or do not. **Conserving relation.** A determination of whether the physical facts fix the experiential ones, standing on what is examinable. **Conserving reach.** The modal relation between the two sets of facts. **Reach.** The statement quantifies over a modal relation between two sets of facts. Actual systems are what is measured. The two are not the same. **Unreachability.** The two-dimensional framework and the phenomenal concept strategy are technical and unresolved, the strategy facing a dilemma stated precisely by both accounts: either it fails to account for the concepts physically or it concedes the point. The determination is not reached over the modal relation by actual systems. **Persistence.** A resolution of the conceivability-possibility inference for this case accepted by both accounts. **Addresses.** Statement, the canonical question, and hardness: Chalmers, "Facing Up to the Problem of Consciousness", Journal of Consciousness Studies 2(3):200-219, 1995, §3. The explanatory-gap term: Levine 1983, as cited there. As read at v342: the paper at passages of its text as the author's page carries them verbatim (consciousness posing the most baffling problems in the science of the mind, nothing known more intimately and nothing harder to explain; an explanatory gap between the functions and experience, the term credited to the 1983 paper, an explanatory bridge needed and the materials for it to be found elsewhere than in an account of the functions; the usual methods of cognitive science equipped only to explain the performance of functions; the hard problem hard precisely because it is not a problem about the performance of functions and persisting when the performance of all the relevant functions is explained), Journal of Consciousness Studies 2(3):200-219, 1995, the paper itself not fetched, the author's later response to commentaries noted at the same page. Persistence, no landing: the closing condition at the paper's passages read at v342 (the hard problem hard precisely because it is not about the performance of functions; the explanatory gap), a resolution of the conceivability-possibility inference accepted by both accounts standing there as what the paper's argument turns on. Approaches: the two-dimensional and phenomenal-concept routes at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Two-dimensional semantics — a framework separating the senses in which the water case differs, contested in detail, the contest technical and unresolved. Phenomenal concept strategy — accounts for the conceivability by the special character of phenomenal concepts, preserving physicalism, its critics arguing the strategy either fails to account for the concepts physically or concedes the point. Denying the conceivability — zombies held not coherently conceivable on reflection, a minority position, defended. Denying the entailment — the main physicalist route, resting on the a posteriori necessity cases, which the two-dimensional framework addresses. Empirical work — bears on which functions are physical, the argument being about what follows if all of them are. **Narrowing recorded.** The phenomenal concept strategy weakens the relation: the conceivability stands accounted for by the character of the concepts rather than the modal relation determined. **Progress.** The three decades of examination without an agreed defect. The a posteriori necessity cases. The stated dilemma against the phenomenal concept strategy. **Own record.** ## 112 The quantum-classical boundary **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Small systems show superposition; large ones do not. Where is the boundary and what sets it? In the field's own posing, it is an intriguing open question whether the linearity of quantum mechanics extends into the macroscopic domain; quantum mechanics has passed all precision tests and still seems to be in conflict with common sense, and testing the limits of the quantum mechanical description of nature has become a subject of intense experimental interest, recent advances in investigating macroscopic superpositions pushing those limits. **The field's account of its hardness.** The field's review states that the question whether linearity extends into the macroscopic domain stands open; that the theory challenges notions of reality and locality that we have grown to rely on in the macroscopic world; and that scientific progress over the preceding decades inspires hope that the debate may be settled by table-top experiments, the limits being pushed by advances in macroscopic superposition. **Accounts.** There is no boundary, only decoherence rates: superposition persists at all scales and becomes unobservable as environmental coupling grows, and interference has been demonstrated for progressively larger molecules with no scale-dependent break found. A boundary may exist: collapse models predict a scale at which superposition genuinely fails, the experiments so far have excluded parameter ranges without excluding the class, and absence of a break at reached masses is not absence of a break. **Specifications.** The boundary is either a matter of degree or a real transition — local. Interference experiments probe it — local. **Specifications withdrawn.** **Common requirement.** That the question of a break has a determinate answer. **Conserving relation.** A determination of whether a break occurs, standing at the masses the question concerns. **Conserving reach.** All masses, those not yet reached included. **Reach.** The statement quantifies over behaviour at masses not yet reached. Masses reached, currently large molecules and mechanical oscillators, are what is measured. The two are not the same. **Unreachability.** Interference at larger masses returns more of the reached domain, and collapse constraints narrow parameter ranges without excluding the class; decoherence accounts quantitatively for every case measured, which both accounts accept while disputing whether anything remains. The determination is not reached over the masses by measurements at masses reached. **Persistence.** An interference measurement at a mass a collapse model predicts a break at, or a detection of collapse. **Addresses.** Statement and hardness: Arndt and Hornberger, "Testing the limits of quantum mechanical superpositions", Nature Physics 10:271-277, 2014, arXiv:1410.0270, at its abstract and opening as the journal's and the archive's pages carry them verbatim, the paper itself not fetched; the masses reached at Reach and at Approaches as the field cites them: Fein, Geyer, Zwick, Kialka, Pedalino, Mayor, Gerlich and Arndt, "Quantum superposition of molecules beyond 25 kDa", Nature Physics 15:1242, 2019; the collapse-model constraints at Approaches as the field cites them: Bassi, Lochan, Satin, Singh and Ulbricht, "Models of wave-function collapse, underlying theories, and experimental tests", Reviews of Modern Physics 85:471, 2013; neither abstract read. Persistence, no landing: the closing condition at the review's abstract and opening read at pass one and at the hardness (the limits of superposition tested at ever larger masses, the frontier advancing) and at the field's review of collapse tests, read at v342 (whether the superposition principle breaks down at macroscopic scales a fundamental and still open question), an interference measurement at a mass a collapse model predicts a break at standing there as the test both reviews name. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Interference at larger masses — demonstrated to tens of thousands of atomic mass units, the frontier advancing, returning more of the reached domain. Optomechanical superposition — prepares mechanical oscillators in non-classical states, the masses remaining far below the proposed collapse scales. Collapse model constraints — bounded by interferometry, X-ray emission, and gravitational-wave detector noise, narrowing without closing. Decoherence calculation — accounts quantitatively for the disappearance of interference in every case measured, accepted by both accounts and disputed as to whether anything remains. Gravitationally induced collapse — proposals giving a scale, experiments to reach it proposed and not yet performed. **Narrowing recorded.** Collapse model constraints narrow the reach: the exclusion stands over the parameter ranges bounded rather than over the class. **Progress.** The interference demonstrated to tens of thousands of atomic mass units. The non-classical states prepared in mechanical oscillators. The bounded collapse parameter ranges. The quantitative decoherence accounting. **Own record.** ## 113 Explore and exploit **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How an agent should divide its actions between exploiting what is known to yield return and exploring alternatives whose value is uncertain. Solved for the discounted multi-armed bandit, where an index policy is optimal, and posed as open by the field where the alternatives are not enumerated in advance or the space of states is large. **The field's account of its hardness.** That the trade-off between exploration and exploitation is a challenge arising in reinforcement learning and not in other kinds of learning, an agent having to exploit what it has already experienced in order to obtain reward while also exploring in order to make better action selections in the future, the dilemma being that neither can be pursued exclusively without failing at the task, each action on a stochastic task having to be tried many times to gain a reliable estimate of its expected reward; and that the dilemma has been intensively studied by mathematicians for many decades yet remains unresolved. **Accounts.** The trade-off is solved for well-posed cases: an index policy is optimal for a class of bandit problems, and regret bounds for algorithms are proved and matched by lower bounds. **Specifications.** The options are enumerated — local. The environment's statistics are stationary or their variation is specified — local. A horizon is given — local. **Specifications withdrawn.** **Common requirement.** That there is a well-defined optimum to approach. **Conserving relation.** An optimum defined, standing where the option set and environment are not specified in advance. **Conserving reach.** All settings the trade-off is asked of. **Reach.** The statement quantifies over action where the option set and environment are not specified in advance. Specified problems with stated priors and horizons are what is measured. The two are not the same. **Unreachability.** Index optimality holds under stated restrictive conditions, and regret is measured against the best fixed option in hindsight, which presupposes the option set; non-stationary bounds hold under a supplied variation budget. The optimum is not defined over the unspecified settings by results on specified problems. **Persistence.** An optimum defined where the option set is not enumerated in advance. **Addresses.** Statement: Sutton and Barto, Reinforcement Learning: An Introduction, 2nd edition, MIT Press, 2018, chapter 1. The index policy: Gittins, Journal of the Royal Statistical Society B 41:148-177, 1979. The regret bounds: Lattimore and Szepesvári, Bandit Algorithms, Cambridge, 2020. Hardness: the textbook the address names, at passages of its first chapter as an archive copy carries them verbatim (the trade-off between exploration and exploitation a challenge arising in reinforcement learning and not in other kinds of learning; the agent having to exploit what it has experienced to obtain reward while exploring to make better selections in the future; the dilemma that neither can be pursued exclusively without failing at the task; each action on a stochastic task tried many times for a reliable estimate; the dilemma intensively studied by mathematicians for many decades yet remaining unresolved), the edition of the copy not carried at the return, the textbook itself not fetched; the index policy and the regret bounds not read this pass. Persistence, no landing: the closing condition at the textbook's first chapter read at pass two (the trade-off a challenge arising in reinforcement learning and not in other kinds of learning; the option set given), an optimum defined where the option set is not enumerated in advance standing there as what the framing assumes away. Approaches: the index-policy and regret routes at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The index policy — optimal for discounted independent-arm problems, the conditions restrictive and stated. Regret minimisation — bounds proved and matched, regret measured against the best fixed option in hindsight, which presupposes the option set. Non-stationary and adversarial extensions — bounds available under stated variation budgets, the budget being supplied. Bayesian optimal control — defines the optimum given a prior, the prior over unknown option sets being the difficulty. Human and animal studies — find behaviour close to certain heuristics, establishing what is done rather than what is optimal in unspecified environments. **Narrowing recorded.** The index policy narrows the reach: optimality stands for discounted independent-arm problems under stated conditions. Non-stationary extensions narrow the reach: bounds stand under a supplied variation budget. Regret minimisation narrows the reach: the measure stands against a presupposed option set. **Progress.** The proved index optimality under its conditions. The matched regret bounds. The bounds under stated variation budgets. The measured proximity of behaviour to specific heuristics. **Own record.** ## 114 Self and non-self in immunity **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** The immune system attacks foreign things and not the body. How does it tell which is which? In the field's own posing, the main objective of immunology is to establish why and when an immune response occurs, that is, to determine a criterion of immunogenicity; according to the consensus view the proper criterion lies in the discrimination between self and non-self, every element that distinctively belongs to the organism failing to trigger a response and every foreign element triggering one, and the criterion proposed against the consensus is a criterion of continuity, on which a response follows a break of antigenic continuity. **The field's account of its hardness.** The field's study states that the consensus on the self/non-self criterion is illusory, experimental data accumulated over two decades having put the criterion into question, so that it has been criticised both conceptually and experimentally, while no consensus has emerged on a new criterion; that many foreign entities do not trigger immune responses, all multicellular organisms containing massive quantities of foreign elements that do not, the diversity and ubiquity of tolerance mechanisms being increasingly admitted; and that self constituents commonly trigger immune responses. **Accounts.** By tolerance to self: lymphocytes reactive to self are deleted or suppressed during development, and the mechanisms — central and peripheral tolerance, regulatory cells — are identified and manipulable. Self-non-self does not describe what is observed: the gut microbiota is foreign and tolerated, the fetus is foreign and tolerated, tumours are self and attacked, and responses are triggered by tissue damage signals rather than by foreignness. **Specifications.** There is a distinction the system implements — local. The distinction is between self and foreign — local. **Specifications withdrawn.** **Common requirement.** That the system's discrimination has a describable basis. **Conserving relation.** A criterion the system discriminates on, the criterion standing across the anomalous cases. **Conserving reach.** All responses, the microbiota, fetal, and tumour cases included. **Reach.** The statement quantifies over what the system discriminates on. Responses to defined antigens under defined conditions are what is measured, each consistent with more than one criterion. The two are not the same. **Unreachability.** Tolerance and damage accounts each account for their own anomalies, and the damage model's trigger is defined partly by what triggers responses. The criterion is not preserved across the cases by responses consistent with more than one. **Persistence.** A criterion accounting for the anomalous cases without being defined by what it accounts for. **Addresses.** Statement, hardness, the standing at Persistence and the tolerance of foreign elements at Approaches: Pradeu and Carosella, "On the definition of a criterion of immunogenicity", Proceedings of the National Academy of Sciences 103(47):17858-17861, 2006, at its abstract and opening as the journal's and the archive's pages carry them verbatim and at its section headings as the repository copy carries them, the paper itself not fetched; the consensus as the paper cites its source: Burnet, Self and Not-self, Melbourne University Press and Cambridge University Press, 1969; the field's survey as the index lists it: Swiatczak and Tauber, "Philosophy of Immunology", Stanford Encyclopedia of Philosophy, 2020, not read. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Tolerance mechanisms — central and peripheral tolerance identified and demonstrated, establishing mechanisms, the microbiota and fetal cases remaining. The danger model — accounts for the anomalies by damage signals, its critics noting damage signals are themselves defined partly by what triggers responses. Pattern recognition receptors — identify conserved microbial patterns, establishing a foreignness-detecting mechanism, alongside self-derived ligands for the same receptors. Tumour immunology — responses to self-derived neoantigens, supporting a criterion other than foreignness, the neoantigens also being mutated and so partly foreign. Microbiome tolerance studies — establish active tolerance of foreign organisms with mechanisms identified, extending the account rather than settling the criterion. **Narrowing recorded.** Microbiome tolerance studies narrow the reach: active tolerance stands established for the foreign organisms studied. Pattern recognition receptors narrow the reach: foreignness detection stands for the conserved patterns identified. **Progress.** The identified tolerance mechanisms. The identified pattern receptors and their self-derived ligands. The documented microbiota and fetal tolerance. The neoantigen responses. **Own record.** ## 115 Cancer, the somatic mutation account and its rival **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Cancer arises when cells proliferate uncontrollably. Does the fault lie in the cell's own mutations, or in the tissue that organises it? In the field's own posing, the somatic mutation theory, the prevailing paradigm of cancer research for half a century, holds that cancer derives from a single somatic cell that has accumulated multiple DNA mutations, that the default state of cell proliferation in metazoa is quiescence, and that cancer is a disease of proliferation caused by mutations in genes controlling proliferation and the cell cycle; the rival research programme, the tissue organisation field theory, holds that carcinogenesis represents a problem of tissue organisation comparable to organogenesis and that proliferation is the default state of all cells. **The field's account of its hardness.** The field's critique states that from the prevailing theory's compelling simplicity an increasingly complicated picture has emerged as more than a hundred oncogenes and thirty tumour suppressor genes have been identified, additional ad hoc explanations being postulated to accommodate the complexity; that the theory has not been rigorously tested, several lines of evidence raising questions it does not address, so that experimental strategies that might validate it are themselves proposed; and the field's reply, titled as a defence of the theory, stands against the critique. **Accounts.** Somatic mutation: cancer is driven by accumulated mutations in oncogenes and tumour suppressors, driver mutations are catalogued across tumour types, targeted therapies against specific drivers work, and heritable mutation in the same genes raises risk. Tissue organisation: carcinogenesis is a tissue-level failure of the constraints normally imposed on proliferation, tissue recombination experiments show carcinogen-exposed stroma with normal epithelium produces tumours and the converse does not, and normal tissue can normalise malignant cells. **Specifications.** The causal locus is either the cell or its tissue context — local. Identifying it determines the intervention — local. **Specifications withdrawn.** **Common requirement.** That there is a level at which the initiating fault sits. **Conserving relation.** The initiating level identified, the identification standing on measurement in established tumours. **Conserving reach.** All tumours the question is asked of. **Reach.** The statement quantifies over which level initiates. Tumours already established, in which both cellular and tissue changes are present, are what is measured. The two are not the same. **Unreachability.** Driver catalogues establish drivers without establishing initiation, tissue recombination supports the tissue account in the systems tested, and sequencing of normal tissue finds driver mutations at high density without cancer, which constrains the dominant account from inside and is accepted in the field. The identification is not reached over the tumours by measurements in established ones. **Persistence.** A measurement at initiation separating the two levels, or agreement that both operate and the question is their order. **Addresses.** Statement and hardness: Soto and Sonnenschein, "The somatic mutation theory of cancer: growing problems with the paradigm?", BioEssays 26(10):1097-1107, 2004, at its abstract as the journal's and the index's pages carry it verbatim, the paper itself not fetched; Soto and Sonnenschein, "The tissue organization field theory of cancer: A testable replacement for the somatic mutation theory", BioEssays 33(5):332-340, 2011, at its abstract as the index's page carries it verbatim, the paper itself not fetched; the defence as the index lists it: Vaux, "In defense of the somatic mutation theory of cancer", BioEssays 33, 2011, not read. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the somatic mutation theory's growing problems and the tissue-level rival, the two placing initiation at different levels), a measurement at initiation parting the levels standing there as what the two theories' contest asks. Approaches: the driver-sequencing and tissue-recombination routes at that paper and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Sequencing driver mutations — catalogued comprehensively, with targeted therapy successes and near-universal resistance, establishing drivers without establishing initiation. Tissue recombination experiments — support the tissue account in the systems tested, system-specific and not generally replicated across tissues. Normalisation experiments — malignant cells behaving normally in normal microenvironments in specific models, establishing context dependence. Mutations in normal tissue — sequencing of normal skin and oesophagus finding driver mutations at high density without cancer, constraining the mutation account substantially, accepted in the field. Field cancerisation and immune surveillance — accommodated by both accounts. **Narrowing recorded.** Tissue recombination narrows the reach: the result stands for the systems tested. Normalisation experiments narrow the reach: context dependence stands in the specific models. **Progress.** The driver catalogues and the targeted therapy responses with their resistance record. The tissue recombination and normalisation results. The measured driver burden in normal tissue. **Own record.** ## 116 Addiction **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** People continue a behaviour that harms them and say they want to stop. Is this a disease of the brain or a choice under bad conditions? In the field's own posing, the view that substance addiction is a brain disease, widely accepted in the neuroscience community, has become subject to acerbic criticism, the criticisms stating that the view is deterministic, fails to account for heterogeneity in remission and recovery, places too much emphasis on a compulsive dimension of addiction, and that a specific neural signature of addiction has not been identified; the view's defenders acknowledge that some of these criticisms have merit while asserting that the premise of a neurobiological basis is fundamentally sound and that the brain is the substrate from which both addiction and the capacity for behaviour change arise. **The field's account of its hardness.** The field's revision of the disease view states that some of the criticisms have merit, no specific neural signature of addiction having been identified and heterogeneity in remission and recovery needing account; that the dividing line for defining addiction within the category of substance use disorder remains an open empirical question, the boundary being intentionally blurred; that denying the disease view is held by its defenders to be harmful because it contributes to reducing access to healthcare and treatment; and that the disagreements reveal the need for multidisciplinary research integrating neuroscientific, behavioural, clinical and sociocultural perspectives. **Accounts.** A brain disease: repeated exposure produces durable changes in reward and control circuits, the changes are imaged and dose-dependent, heritability is substantial, and treating it as a disease has reduced stigma and improved access to care. The disease framing does not fit the data on recovery: most people who meet criteria recover without treatment, behaviour responds strongly to incentives including in randomised contingency-management trials, and rates track economic and social conditions in ways a brain disease does not predict. **Specifications.** The behaviour has a locus — neural pathology or choice under constraint — local. Identifying it determines the response — local. **Specifications withdrawn.** **Common requirement.** That the behaviour is produced by an identifiable mechanism at one level. **Conserving relation.** The governing level identified, the identification standing on measures altered in affected people. **Conserving reach.** All affected people the question is asked of. **Reach.** The statement quantifies over which account governs the behaviour. Brain measures and behavioural measures, both altered in affected people, are what is measured. The two are not the same. **Unreachability.** Imaging does not separate cause from consequence, which the field states, and remission rates and incentive responsiveness are read oppositely by the two accounts. The identification is not reached over the affected by brain and behavioural measures. **Persistence.** A measurement separating cause from consequence, or agreement that both operate and the question is their proportion. **Addresses.** Statement, hardness and the neural signature at Approaches: Heilig, MacKillop, Martinez, Rehm, Leggio and Vanderschuren, "Addiction as a brain disease revised: why it still matters, and the need for consilience", Neuropsychopharmacology 46(10):1715-1723, 2021, at its abstract as the journal's and the archive's pages carry it verbatim and at its figure legend on the boundary of addiction as the repository copy carries it, the paper itself not fetched; the reply as the index carries its title: "Comment on Heilig et al.: The centrality of the brain and the fuzzy line of addiction", Neuropsychopharmacology, 2021, doi:10.1038/s41386-021-01000-x, its authors and text keyed at the next pass; the original assertion as the paper cites it: Leshner, "Addiction is a brain disease, and it matters", Science 278:45-47, 1997. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (the brain-disease view revised, the need for consilience across levels; the imaging not parting cause from consequence, which the field states), a measurement parting cause from consequence standing there as the review's own missing separation. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Neuroimaging — documents circuit changes reliably, cause and consequence not separated by the imaging, which the field states. Contingency management trials — large abstinence effects with modest incentives, establishing responsiveness to consequences, accommodated by both accounts with difficulty on one side. Epidemiology of recovery — high rates of remission without treatment, measured, constraining chronic-disease framings. Animal models — establish neuroadaptation, environmental enrichment substantially reducing self-administration, supporting the conditions account. Genetic studies — substantial heritability, with small individual effects and no identified pathology. **Narrowing recorded.** Animal models narrow the reach: neuroadaptation stands established in the models, with enrichment effects measured there. Contingency management narrows the reach: the responsiveness stands demonstrated under the incentive schedules trialled. **Progress.** The imaged dose-dependent circuit changes. The measured abstinence effects under incentives. The measured remission rates without treatment. The enrichment effects in animal models. The heritability estimates without identified pathology. **Own record.** ## 117 Niche and neutral theory **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Do species coexist because they differ in what they do, or because they are approximately equivalent and abundance drifts? In the field's own posing, the controversy over the relative importance of niches and neutrality cannot be resolved by analysing species abundance patterns; classical coexistence theory reframes the debate in terms of stabilising mechanisms, niches, and fitness equivalence, neutrality, the neutral model being the special case in which stabilising mechanisms are absent and species have equivalent fitness, so that instead of asking whether niches or neutral processes structure communities the question becomes the degree to which observed diversity reflects strong stabilising mechanisms overcoming large fitness differences or weak stabilisation operating on species of similar fitness. **The field's account of its hardness.** The field's perspective states that abundance patterns cannot resolve the controversy; and that answering the reframed question requires combining data on per capita growth rates with models, to quantify the strength of stabilising processes, to quantify fitness inequality and compare it with stabilisation, and to manipulate frequency dependence in growth to test the consequences of stabilisation and fitness equivalence for coexistence. **Accounts.** Niche differences: coexistence requires stabilising differences, demonstrated experimentally by invasion tests, and species do differ measurably in resource use, timing, and enemies. Neutrality fits the patterns: neutral theory reproduces species-abundance distributions and species-area relationships with no niche differences at all, and fits many datasets as well as or better than niche models. **Specifications.** Community structure is determined either by differences or by drift — local. Abundance patterns discriminate between them — local. **Specifications withdrawn.** **Common requirement.** That the observed patterns bear on the mechanism. **Conserving relation.** The mechanism producing community structure identified, the identification standing on abundance and distribution data. **Conserving reach.** All communities the question is asked of. **Reach.** The statement quantifies over the mechanism producing community structure. Abundance and distribution data, which both models fit, are what is measured. The two are not the same. **Unreachability.** Both models fit abundance distributions, established early and not resolved by more data of the same kind, and the continuum framing accommodates both without measuring weights. The identification is not reached over the communities by abundance and distribution data. **Persistence.** Measured weights for the stabilising and equalising mechanisms in natural communities, or a prediction on which the two models differ and a test discriminating them. **Addresses.** Statement, hardness, Reach, Persistence and the partition at Approaches: Adler, HilleRisLambers and Levine, "A niche for neutrality", Ecology Letters 10(2):95-104, 2007, at its abstract as the journal's, the index's and the repository's pages carry it verbatim, the paper itself not fetched; the coexistence theory the perspective applies, as the field cites it: Chesson, 2000, its title and venue keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Fitting abundance distributions — both fit, established early and not resolved by more data of the same kind. Invasion experiments — demonstrate stabilising differences where performed, species-specific and laborious. The partition of stabilising from equalising mechanisms — makes both measurable in principle, measurement in diverse natural communities partial. Dynamic predictions — neutral and niche models differing in predicted temporal turnover, tests giving mixed results and sensitive to timescale. Continuum framing — most current work treating both as operating together, fitting, the weights not independently measured. **Narrowing recorded.** Invasion experiments narrow the reach: stabilising differences stand demonstrated for the species pairs tested. **Progress.** The fits of both models to abundance distributions. The demonstrated stabilising differences where invasion tests were performed. The partition making both mechanism classes measurable in principle. **Own record.** ## 118 Ecological stability and diversity **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Do more diverse communities resist disturbance better? In the field's own posing, the problem forces one to consider in depth how ecosystems, which are ultimately our life-support systems, are expected to respond to reductions in diversity, the issue commonly referred to as the diversity-stability debate; the review's conclusion is that theory and empirical evidence agree that declines in diversity should be expected to accelerate the simplification of ecological communities. **The field's account of its hardness.** The field's review states that the problem is a monumental one; that the debate has to be synthesised across historical ideas and recent advances, with the definitions of stability treated in a section of their own; and that one promising mechanism proposed for stability is that weakly interacting species stabilise community dynamics by damping strong, potentially destabilising consumer-resource interactions. **Accounts.** Yes: long-running grassland experiments that manipulate diversity directly find more diverse plots more stable in aggregate biomass across years, more resistant to drought, and faster to recover from it, replicating across continents and supported by separately measured mechanisms — asynchrony in species responses and complementarity in resource use. Not in general: randomly assembled complex systems are provably less stable, not more, and the empirical result is for aggregate properties — individual species populations are often less stable in diverse communities, so what is stable depends on which variable is measured. **Specifications.** Stability is a property of a community — local. Diversity is a variable that can be manipulated independently — local. **Specifications withdrawn.** **Common requirement.** That stability has a definition both accounts accept. **Conserving relation.** A stability that rises with diversity, the rise standing across the variables measured. **Conserving reach.** All the stability variables the question is asked of. **Reach.** The statement quantifies over which stability the question is about, the concepts diverging. Specific variables — aggregate biomass, population variance, recovery time, resistance — are what is measured, and they move in different directions. The two are not the same. **Unreachability.** Aggregate stability rises with diversity and population stability does not, in the same experiments, on the same plots; and multiple stability concepts formalised show the answer differs by concept. The rise is not preserved across the variables by the measurements. **Persistence.** A stability concept both accounts accept, applied across the variables. **Addresses.** Statement, hardness and the definitions at Reach and at Approaches: McCann, "The diversity-stability debate", Nature 405(6783):228-233, 2000, at its abstract as the journal's and the index's pages carry it verbatim and at its passage on weak interactions and its section on definitions of stability as a reader's comment on the journal's page carries them, the paper itself not fetched; the classical result at Approaches as the field cites it: May, "Will a large complex system be stable?", Nature 238:413-414, 1972, its text not read. Persistence, no landing: the closing condition at the review's abstract and its definitions passage read at pass one and at the hardness (the debate turning on which stability concept is meant, aggregate against population-level, measured oppositely in the same experiments), a stability concept both accounts accept standing there as what the review's definitions passage shows lacking. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Grassland manipulation experiments — aggregate stability rising with diversity, robustly, population-level stability not, measured in the same experiments. Random-matrix analysis — proves instability for random complex systems, real communities not being randomly assembled, which is the standing response. Structured network models — find stability depends on interaction structure, establishing that structure matters and supplying no general answer. Portfolio effects — statistical averaging accounting for part of the aggregate result without any ecological mechanism, measured and partial. Definitional work — multiple stability concepts formalised, clarifying the dispute and showing the answer differs by concept. **Narrowing recorded.** Definitional work narrows the reach: an answer stands per stability concept rather than across them. Portfolio effects narrow the reach: part of the aggregate result stands accounted for by statistical averaging. **Progress.** The measured aggregate stability rise across continents. The measured population-level result in the same experiments. The proved instability of random complex systems. The quantified portfolio contribution. **Own record.** ## 119 The Red Queen **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why the probability that a taxon goes extinct is approximately constant over its lifetime, independent of how long it has already survived — the law read from fossil survivorship curves — and whether the constancy is produced by continual coevolution among interacting species, each improvement met by another's, so that adapting leaves relative standing unchanged. The hypothesis is carried by the field separately as an account of the maintenance of sexual reproduction. **The field's account of its hardness.** That the question the hypothesis answers is what the causes of natural selection are, biotic conflict being given primacy over abiotic forces, and that species must continually evolve to survive in the face of their evolving enemies while on average their fitness remains unchanged; that the hypothesis is for many evolutionary biologists most strongly associated with the debates surrounding the evolution of sex, where it provides a mechanism by which sexual species are protected from elimination by asexuals despite the latter's higher per capita reproductive rates; and that three modes of coevolution have had to be defined to unify fluctuating and directional selection within the one framework. **Accounts.** Coevolution keeps them in place: antagonistic coevolution means each improvement is met, so extinction risk stays roughly constant with age, as the founding observation from fossil survivorship curves recorded. The pattern has other explanations: age-independent extinction arises from environmental change and from taxonomic artefacts as well, and the coevolutionary interpretation is one among several fitting the same survivorship curves. **Specifications.** The survivorship pattern indicates a mechanism — local. Taxa are comparable units — local. **Specifications withdrawn.** The founding age-independence stands rejected as the norm by the corrected survivorship analyses, the taxonomic level of the original curves accounting for the discrepancy, and the coevolutionary account continues partly detached from the pattern it was built to account for. **Common requirement.** That the fossil pattern bears on the mechanism. **Conserving relation.** A mechanism identified as producing the survivorship pattern, the identification standing on fossil ranges. **Conserving reach.** All taxa the pattern is measured over. **Reach.** The statement quantifies over what produces constant extinction risk. Fossil ranges, with sampling and taxonomic definition affecting the curves, are what is measured. The two are not the same. **Unreachability.** Several mechanisms fit the same survivorship curves, genus definitions affect the curves, and comparative analyses find environmental drivers dominating some intervals with weights unmeasured. The identification is not reached over the taxa by fossil ranges. **Persistence.** Measured weights for coevolutionary and environmental drivers, or a survivorship pattern only one mechanism fits. **Addresses.** Statement: Van Valen, "A new evolutionary law", Evolutionary Theory 1:1-30, 1973. The coevolutionary reading and the sex reading parted: Brockhurst et al., "Running with the Red Queen: the role of biotic conflicts in evolution", Proceedings of the Royal Society B 281:20141382, 2014. Hardness: the review the address names, at its abstract as the publisher's page, an index, two university repositories and an archive carry it verbatim (the question what the causes of natural selection are; the hypothesis emphasizing the primacy of biotic conflict over abiotic forces in driving selection; species having to evolve continually to survive in the face of their evolving enemies while on average their fitness remains unchanged; three modes of coevolution defined to unify fluctuating and directional selection within the framework) and at passages of its text as the publisher's page and a repository copy carry them (the hypothesis a revolutionary advance on the hill-climbing view; for many evolutionary biologists most strongly associated with the debates surrounding the evolution of sex, providing a mechanism by which sexual species are protected from elimination by asexuals despite the latter's higher per capita reproductive rates), issue 1797, doi 10.1098/rspb.2014.1382, a university repository giving 282(1802):1-9 where the publisher carries 281(1797), the review itself not fetched; the founding paper not read this pass. Persistence, no landing: the closing condition at the review's abstract read at pass two (the question what the causes of natural selection are; the hypothesis emphasizing the biotic against the environmental), measured weights for the two drivers or a pattern only one fits standing there as the review's own question. Approaches: the survivorship and coevolution routes at that review and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Survivorship analysis — the founding pattern of apparent age-independence at higher taxonomic levels reversed by improved analyses, with age-dependent extinction widespread at species level, typically most intense among young taxa, and age-independence now the exception; the analysis continues on the corrected curves. Host-parasite coevolution experiments — demonstrate arms-race dynamics directly in microbial systems, establishing the mechanism at short timescales. Molecular evolution of interacting genes — rapid evolution at host-parasite interfaces, measured, supporting coevolution locally. Comparative extinction analyses — environmental drivers dominating some intervals, both mechanisms operating, weights unmeasured. Taxonomic artefact analysis — genus definitions affecting the curves, constraining the original inference. **Narrowing recorded.** Coevolution experiments narrow the reach: the mechanism stands demonstrated at short timescales in microbial systems. **Progress.** The corrected survivorship curves and the reversal of the founding pattern. The demonstrated arms-race dynamics. The measured rapid evolution at host-parasite interfaces. The quantified effect of taxonomic definition on the curves. **Own record.** ## 120 The hygiene hypothesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Allergic and autoimmune conditions rose sharply in industrialised populations over decades. Why? In the field's own posing, the founding observation was that children from larger families were less likely to develop hay fever, and the hypothesis postulated from it was that infections were preventing allergic disease, not specifically or solely viral infections, the scope of what might constitute a relevant microbial challenge being purposely left broad; the hypothesis has since undergone many revisions and adaptations. **The field's account of its hardness.** The field's reviews state that at the time of the founding observation respiratory viral infections were seen as potentially important in the development of allergic disease and asthma, so that less hay fever in larger families contradicted that premise; that what constitutes a relevant microbial challenge was purposely left broad and exploring it has become one of the significant areas of development of the hypothesis; and that the hypothesis has undergone many revisions and adaptations over three decades. **Accounts.** Reduced early microbial exposure: farm-childhood studies show large protective effects, the mechanisms are traceable to immune regulation, and the timing and geography of the rise match changes in exposure. The original framing does not fit: infections in early life often raise rather than lower allergy risk, the effect is better attributed to microbiota composition and to specific exposures than to hygiene, and antibiotic use, diet, vitamin D, pollution, and diagnostic change all track the same period. **Specifications.** The rise has a dominant cause — local. Population-level correlations identify it — local. **Specifications withdrawn.** **Common requirement.** That the rise is attributable. **Conserving relation.** A dominant cause identified, the identification standing among co-varying exposures. **Conserving reach.** The rise across the industrialised populations. **Reach.** The statement quantifies over a dominant cause, identified from correlations among co-varying exposures. Observational cohorts with many co-varying exposures, and a small number of trials, are what is measured. The two are not the same. **Unreachability.** Confounding by lifestyle is acknowledged in the strongest comparisons, microbiota associations narrow without establishing causality in humans, and several exposures track the same period. The identification is not reached over the rise by cohorts with co-varying exposures. **Persistence.** A trial establishing causality for an exposure sufficient to account for the rise, or measured weights across the co-varying exposures. **Addresses.** Statement and hardness: Strachan, "Hay fever, hygiene, and household size", British Medical Journal 299(6710):1259-1260, 1989, carrying no abstract, at its observation and its hypothesis as the field's review carries them ("The hygiene hypothesis for allergy - conception and evolution", Frontiers in Allergy, 2022, doi:10.3389/falgy.2022.1051368, its authors keyed at the next pass), the paper itself not fetched; the revisions as the field's review states them: "The Hygiene Hypothesis - Learning From but Not Living in the Past", Frontiers in Immunology, 2021, doi:10.3389/fimmu.2021.635935, at its opening as the index's page carries it, its authors keyed at the next pass; the first decade's review as the field cites it: Strachan, "Family size, infection and atopy: the first decade of the 'hygiene hypothesis'", Thorax 55(Supplement 1):S2-S10, 2000. Persistence, no landing: the closing condition at the founding observation and hypothesis as the field's review carries them, read at pass one and at the hardness (the inverse relation of hay fever to household size; infections in early childhood proposed as protective, the scope of the relevant exposures widened since), a trial establishing causality or measured weights across the co-varying exposures standing there as what the observational evidence cannot supply. Approaches: the farm-comparison and microbiota routes at the review the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Farm and community comparisons — large, consistent protective associations with identified mechanistic differences, the strongest evidence, confounding by lifestyle acknowledged. Microbiota studies — associate composition with outcomes and identify plausible mechanisms, causality in humans largely unestablished. Helminth and probiotic trials — mixed results, with some positive findings in specific conditions. The early peanut introduction trial — randomised, large effect, reversing a prior guideline, establishing causality for one exposure and one outcome. Diagnostic-change analyses — account for part of the measured rise, quantified. **Narrowing recorded.** The peanut trial narrows the reach: causality stands established for one exposure and one outcome. Diagnostic-change analyses narrow the reach: part of the measured rise stands accounted for. **Progress.** The measured protective associations and their mechanistic correlates. The established causality for one exposure and outcome, with the adolescent follow-up standing: tolerance persisting to age twelve irrespective of subsequent consumption. The quantified diagnostic contribution to the measured rise. **Own record.** ## 121 Hormesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Some agents harmful at high dose appear beneficial at low dose. Is the dose-response curve monotonic or not? In the field's own posing, hormesis is a dose-response relationship characterised by low-dose stimulation and high-dose inhibition, which its proponents hold to have been frequently observed in properly designed studies and to be broadly generalisable, independent of the chemical or physical agent, the biological model and the endpoint measured, with the claim that the hormetic model occurs more frequently in the toxicological literature than the threshold model. **The field's account of its hardness.** The field's review states that the concept has been under-recognised and under-appreciated; that toxicological thinking has been hesitant to accept and apply the model despite the claim of its superiority to the linear and threshold models at low dose, the reasons being complex but traced in large part to toxicology's being an applied discipline that, faced with a huge number of compounds to test, streamlined its processes to reduce the number of animals per dose and the number of doses per experiment; and the field's critical literature treats the concept's terms and its use in risk assessment as a case of special-interest science. **Accounts.** Non-monotonic responses are real and common: documented across thousands of studies in toxicology and pharmacology, with mechanisms — adaptive stress responses, receptor dynamics — identified in specific cases. For regulation the default should remain linear at low dose: low-dose effects are near the limit of detection, publication favours positive findings, and adopting a non-monotonic default would relax exposure limits on evidence heterogeneous in quality. **Specifications.** A dose-response relation exists for an agent and an endpoint — universal under toxicology. Low-dose behaviour is estimable — local. **Specifications withdrawn.** **Common requirement.** That there is a true low-dose response. **Conserving relation.** A dose-response form determined at low dose, the determination standing on measurement. **Conserving reach.** Doses near or below detection. **Reach.** The statement quantifies over response at doses near or below detection. Doses with detectable effects, extrapolated downward by model, are what is measured. The two are not the same. **Unreachability.** Rival dose-response models fit comparably in the observable range and diverge below it, which is where the question lies; and confounding dominates epidemiology at the relevant magnitudes, which is measured. The determination is not reached over the low doses by measurements at detectable ones. **Persistence.** A measurement at the doses of regulatory interest, or agreement on a criterion for extrapolating below detection. **Addresses.** Statement and hardness: Calabrese and Baldwin, "Hormesis: The Dose-Response Revolution", Annual Review of Pharmacology and Toxicology 43:175-197, 2003, at its abstract as the journal's and the repository's pages carry it verbatim, the paper itself not fetched; the reluctance, and the model comparison at Approaches as the proponents state it: Calabrese, "Hormesis: a revolution in toxicology, risk assessment and medicine", EMBO Reports, 2004, doi:10.1038/sj.embor.7400222, at its passages as the archive's page carries them, the paper itself not fetched; the critical reading as the field carries it: "Conceptual analysis and special-interest science: toxicology and the case of Edward Calabrese", Synthese, 2010, at its abstract as the publisher's page carries it, its author keyed at the next pass. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (the dose-response revolution claimed, the low-dose stimulation generalized from large databases; its critics on heterogeneity and regulatory extrapolation), a measurement at the doses of regulatory interest standing there as what the databases do not reach. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Database assembly — large collections of non-monotonic curves, heterogeneous in endpoint and quality, which both accounts accept. Mechanistic studies — establish adaptive responses in specific systems, generality to other agents and endpoints being the open question. Large low-dose animal studies — increase power at some cost, the doses of regulatory interest remaining below the reachable range for many agents. Epidemiology at low exposure — confounding dominating at the relevant magnitudes, which is measured. Model selection — linear-no-threshold, threshold, and non-monotonic models fitting comparably in the observable range and diverging below it. **Narrowing recorded.** Mechanistic studies narrow the reach: adaptive responses stand established in specific systems. Large animal studies narrow the reach: the reachable dose range stands extended without reaching regulatory magnitudes. **Progress.** The assembled curve collections with their stated heterogeneity. The mechanisms established in specific systems. The measured confounding at low exposure. The comparable fits in the observable range. **Own record.** ## 122 The dodo bird verdict **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Different psychotherapies produce similar outcomes. Do the specific techniques do anything? In the field's own posing, the conjecture states that when psychotherapies intended to be therapeutic are compared, the true differences among all such treatments are zero; the meta-analysis testing it found the effect sizes between bona fide treatments homogeneously distributed about zero, with an upper bound on the true effect of about 0.20 under the most liberal assumptions, unrelated to publication date and to the similarity of the treatments, and concluded that the efficacy of bona fide treatments is roughly equivalent. **The field's account of its hardness.** The field's comment on the meta-analysis states that a fair test of the conjecture would entail separate comparisons of every pair of therapies, an overall effect size across a heterogeneous set being relevant only when the mean absolute difference is zero, and that the limitations of the underlying randomised trials and the problem of uncontrolled causal variables make clinically useful differences unlikely to be revealed by such meta-analyses; the reply lists the issues raised, counter-examples, untested alternative hypotheses, methodological problems and the adequacy of randomised trials, and holds that none alters the conclusion; the field's later methodological note records that studying specificity in psychotherapy with meta-analysis is hard, and its encyclopaedia records the debate as having spanned nearly a century. **Accounts.** Common factors dominate: meta-analyses across decades find small differences between bona fide therapies, and alliance, expectation, and therapist effects account for more variance than technique. Equivalence is partly an artefact: some specific treatments outperform others for specific conditions — exposure for phobias, and specific protocols for obsessive-compulsive disorder — and averaging across conditions conceals this, allegiance effects also inflating apparent equivalence in comparative trials. **Specifications.** Outcome is attributable to components of the treatment — local. Trials isolate components — local. **Specifications withdrawn.** **Common requirement.** That the contribution of technique is separable from context. **Conserving relation.** The contribution of technique separated from context, the separation standing across conditions. **Conserving reach.** All conditions the comparison is made over. **Reach.** The statement quantifies over the contribution of technique. Whole treatments delivered in contexts, compared, are what is measured. The two are not the same. **Unreachability.** Averaging across conditions conceals specific superiority, which is the disputed step, allegiance effects are substantial and measured, and controls establish that therapies exceed no treatment without separating technique from context. The separation is not reached over the conditions by comparisons of whole treatments. **Persistence.** A design separating technique from context, or agreement on the level at which the comparison should be made. **Addresses.** Statement, hardness, Persistence and the disputed averaging at Approaches: Wampold, Mondin, Moody, Stich, Benson and Ahn, "A meta-analysis of outcome studies comparing bona fide psychotherapies: Empirically, 'all must have prizes'", Psychological Bulletin 122(3):203-215, 1997, at its abstract as the repositories' pages carry it verbatim, the paper itself not fetched; the comment: Howard, Krause, Saunders and Kopta, "Trials and tribulations in the meta-analysis of treatment differences: Comment on Wampold et al. (1997)", Psychological Bulletin 122, 1997, at its abstract as the repository's page carries it verbatim, its pages keyed at the next pass; the reply as the repository summarises it, its title keyed at the next pass; the methodological note as the field cites it: Baldwin and Imel, "Studying specificity in psychotherapy with meta-analysis is hard", Psychotherapy Research 30(3):294-296, 2020; the debate's span as the field's encyclopaedia states it: "Dodo Bird Verdict in Psychotherapy", doi:10.1002/9781118625392.wbecp053, its author listed as Elliott. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Comparative meta-analysis — small average differences, averaging across conditions being the disputed step. Dismantling studies — remove components and measure change, some showing specific components matter and many underpowered, which is documented. Condition-specific comparisons — show specific superiority for several disorders, accepted by both accounts. Allegiance analyses — quantify researcher allegiance effects, substantial and measured. Placebo and waitlist controls — establish therapies exceed no treatment, without separating technique from context. **Narrowing recorded.** Condition-specific comparisons narrow the reach: superiority stands established for the disorders compared. Dismantling studies narrow the reach: component contributions stand where the studies were powered. **Progress.** The measured average differences. The established condition-specific superiorities. The quantified allegiance effects. The established advantage over no treatment. **Own record.** ## 123 Dreaming **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Dreaming is universal and vivid and occurs nightly. What is it for? In the field's own posing, dreams are a remarkable experiment in psychology and neuroscience, conducted every night in every sleeping person, showing that the brain, disconnected from the environment, can generate an entire world of conscious experiences by itself; the fundamental questions dreams pose are how conscious experiences in sleep relate to underlying brain activity, why the dreamer is largely disconnected from the environment, and whether dreaming is more closely related to mental imagery or to perception. **The field's account of its hardness.** The field's review states that two strands of research, content analysis and developmental study on one side and lesion study, functional imaging and neurophysiology on the other, have only now become integrable; that the initial equation of rapid-eye-movement sleep with dreaming has been shown to be inaccurate, so that researchers have to move beyond sleep stages and attend to more subtle features of brain activity in space and time in order to link dream consciousness to neuronal events; and that the dreamer's disconnection from the external environment poses a central unsolved paradox whose answer might be instrumental for understanding dreams. **Accounts.** It has a function: candidates include emotional processing, memory consolidation, and threat simulation, each with supporting evidence, and a phenomenon this conserved and this costly in brain activity is unlikely to be incidental. It may be a by-product: dreaming may be an epiphenomenon of the physiological state that has the function, the evidence for content-specific benefit is weak, and damage abolishing dream report leaves sleep function largely intact in reported cases. **Specifications.** Dreaming is separable from the sleep state carrying it — local. A function is identifiable by manipulation — local. **Specifications withdrawn.** **Common requirement.** That dreaming can be manipulated separately from sleep. **Conserving relation.** The contribution of dreaming identified, the identification standing separately from the sleep state. **Conserving reach.** All the sleep in which dreaming occurs. **Reach.** The statement quantifies over the contribution of dreaming as distinct from sleep. Sleep stages and reports on waking are what is measured, with dreaming and its physiological substrate co-occurring. The two are not the same. **Unreachability.** Selective deprivation is confounded by arousal and rebound, documented extensively, and memory reactivation establishes a sleep effect not linked to dream content. The identification is not reached over the sleep by stages and waking reports. **Persistence.** A manipulation of dreaming separate from the sleep state, or a content-specific effect established. **Addresses.** Statement, hardness and the stages at Reach: Nir and Tononi, "Dreaming and the brain: from phenomenology to neurophysiology", Trends in Cognitive Sciences 14(2):88-100, 2010, at its abstract and its summary passage as the journal's and the index's pages carry them verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the review's abstract and summary passage read at pass one and at the hardness (dreaming parted from the sleep stages it was identified with; the function questions confounded with the state), a manipulation of dreaming separate from the state standing there as the review's own separation. Approaches: the deprivation, lesion and targeted routes at that review's sections, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Selective deprivation — confounded by arousal and rebound, which the field has documented extensively. Lesion cases abolishing dream report — rare, the reported cases constraining function claims. Targeted memory reactivation in sleep — improves retention, establishing a sleep effect, the link to dream content unestablished. Dream content analysis — finds thematic regularities consistent with several accounts. Lucid dreaming with communication — two-way communication established in recent work, opening a route, early, and its reach for function questions not yet clear. **Narrowing recorded.** Targeted memory reactivation narrows the reach: the effect stands established for sleep rather than for dreaming. Lucid dreaming narrows the reach: the route stands open for the states in which communication is established. **Progress.** The documented confounding of selective deprivation. The reported lesion cases and their constraint on function claims. The measured retention gains from reactivation. The established two-way communication in lucid states. **Own record.** ## 124 The Cambrian explosion **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Most animal body plans appear in the fossil record over a geologically short interval. What produced the concentration? In the field's own posing, diverse bilaterian clades emerged apparently within a few million years during the early Cambrian, and various environmental, developmental and ecological causes have been proposed to explain this abrupt appearance; the field's compilation of fossil and molecular diversification, comparative developmental data and ecological feeding strategies indicates that the major animal clades diverged many tens of millions of years before their first appearance in the fossil record, a macroevolutionary lag between the establishment of their developmental toolkits and their later ecological success. **The field's account of its hardness.** The field's synthesis states that the sudden appearance of diverse bilaterian clades remains a contentious subject with a range of proposed environmental, developmental and ecological causes; and that the compilation of fossil patterns, molecular clock results, comparative developmental data and ecological information demonstrates a lag between the divergence of the major clades, with the establishment of their developmental toolkits, and their later ecological success, so that first appearance in the record does not date divergence. **Accounts.** A real evolutionary event: ecological triggers — the rise of predation, oxygen thresholds, the evolution of eyes — produced rapid diversification, and the fossil concentration reflects a genuine burst. The concentration is partly artefact: molecular clocks place divergences substantially earlier, the appearance of hard parts changes preservation probability sharply at that boundary, and taxonomic practice assigns higher ranks to early forms, all of which concentrate apparent origination. **Specifications.** First appearance in the record approximates origination — local. Body plans are comparable units — local. **Specifications withdrawn.** **Common requirement.** That the record bears on the timing of divergence. **Conserving relation.** Origination times determined, the determination standing across the two records. **Conserving reach.** The interval the concentration is measured over. **Reach.** The statement quantifies over origination times, approximated by first appearance. First appearances, conditioned on preservation, and molecular clocks conditioned on calibration, are what is measured. The two are not the same. **Unreachability.** The two records returned different timings by hundreds of millions of years; under the recalibrated analyses the divergence has narrowed to the Ediacaran and the records stand in much closer accord, with a literal reading of the fossil record still rejected and preservation modelling quantifying part of the remaining difference. The determination is not reached over the interval by first appearances and clocks alone. **Persistence.** Agreement between the two records on origination times, or a preservation model accounting for the remaining difference. **Addresses.** Statement, hardness, Reach, and the molecular clocks and the developmental toolkit at Approaches: Erwin, Laflamme, Tweedt, Sperling, Pisani and Peterson, "The Cambrian Conundrum: Early Divergence and Later Ecological Success in the Early History of Animals", Science 334(6059):1091-1097, 2011, at its abstract as the journal's, the repository's and the index's pages carry it verbatim and at its conference form as the society's page carries it, the paper itself not fetched. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (early divergence by the molecular record and later ecological success by the fossil record, the conundrum the two records' difference), agreement between the records or a preservation model accounting for the difference standing there as the paper's own closing. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Molecular clocks — place divergences earlier, estimates depending on calibration and rate models and having varied widely between studies. Preservation modelling — quantifies the hard-part bias substantially, establishing part of the artefact. Ediacaran and trace fossil work — extends the record of animal activity earlier, narrowing the gap in one direction. Oxygen proxies — improved records showing a complex rather than a single threshold, weakening simple trigger accounts. Developmental genetics — establishes the regulatory toolkit predates the concentration, supporting the earlier-divergence account and not dating the morphological events. **Narrowing recorded.** Preservation modelling narrows the reach: part of the concentration stands accounted for as bias. Trace fossil work narrows the reach: the record of activity stands extended earlier for the traces recovered. **Progress.** The recalibrated clock estimates and their narrowed divergence from the fossil record. The quantified hard-part bias. The extended trace record. The regulatory toolkit predating the concentration. **Own record.** ## 125 The Sapient paradox **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Anatomically modern humans appear long before the changes in behaviour and settlement usually taken as their signature. Why the gap? In the field's own posing, the human genome, and hence the human brain at birth, may not have changed greatly over the past sixty thousand years, yet many of the major behavioural changes associated with most human societies are very much more recent, some appearing with the sedentary revolution of some ten thousand years ago, among them activities implying the emergence of powerful concepts of value and of the sacred; the paradox asks why, if the biological basis of the species has been established for perhaps as much as two hundred thousand years, the novel behavioural aspects of its sapient status took so long to emerge, all major evidence in the archaeological record of important changes in intelligent behaviour coming long after the appearance of modern anatomy. **The field's account of its hardness.** The field's statement of the paradox asks what neuronal mechanisms may underlie these consistent, significant and emergent patterns of behaviour, and sets the neuroscientist the challenge of showing how the potential capacities of a mind presumably present for sixty thousand years came to be realised only later; the field's introduction to the question holds that materiality, cultural practices and social interaction shape the human mind throughout its evolutionary and developmental trajectories, so that cooperation between archaeology and neuroscience is needed; and the field's later commentary records that some sought to explain the divergence by a sudden cognitive mutation while most archaeologists rallied around a distinct capacity for culture acquired earlier in the species' history. **Accounts.** A cognitive change occurred later: a neural or genetic change enabled symbolic behaviour tens of thousands of years after anatomy stabilised, and the archaeological transition marks it. The gap is in the record and in demography: African finds push symbolic behaviour progressively earlier, population density thresholds affect cultural accumulation and loss independently of cognition, and the apparent transition is a sampling and demography effect. **Specifications.** Behavioural modernity is a definable package — local. Its archaeological signature indexes cognition — local. **Specifications withdrawn.** **Common requirement.** That the archaeological record bears on cognitive capacity. **Conserving relation.** Cognitive capacity indexed by material culture, the indexing standing across the sampled record. **Conserving reach.** The interval between anatomical and behavioural appearance. **Reach.** The statement quantifies over cognitive capacity, indexed by preserved material culture. Preserved material culture, unevenly sampled by region and period, is what is measured. The two are not the same. **Unreachability.** Demographic modelling reproduces the pattern without cognitive change, candidate genetic changes are identified and none tied to behaviour, and the record has been extended repeatedly by new finds. The indexing is not reached over the interval by unevenly sampled material culture. **Persistence.** A genetic or neural change tied to behaviour, or a record dense enough that further finds do not move the earliest evidence. **Addresses.** Statement, hardness, Reach and the standing of the two accounts at Approaches: Renfrew, "Neuroscience, evolution and the sapient paradox: the factuality of value and of the sacred", Philosophical Transactions of the Royal Society B 363(1499):2041-2047, 2008, the journal's page listing the issue's editors with him, at its abstract as the journal's page carries it verbatim and at its statement of the neuroscientist's challenge as the repository's page carries it, the paper itself not fetched; Renfrew, Frith and Malafouris, "Introduction. The sapient mind: archaeology meets neuroscience", Philosophical Transactions of the Royal Society B 363, 2008, doi:10.1098/rstb.2008.0016, at its statement of the paradox as the journal's page carries it verbatim, the paper itself not fetched; the later commentary as the repository carries its passage: Hussain, "The dialectic Neanderthal: Re-configuring the human question in times of planetary crisis", its venue and year keyed at the next pass. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the paradox of anatomically modern humans long preceding the behavioural record; the two accounts, a late genetic change or a cumulative cultural one), a genetic change tied to behaviour or a record dense enough to stop moving standing there as the two accounts' own tests. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Pushing the record earlier — African pigment, bead, and engraving finds repeatedly moving the earliest evidence back, a productive route that continues. Demographic modelling — accumulation and loss depending on population size and connectedness, reproducing the pattern without cognitive change; a published dispute stands against the models' assumptions and archaeological support, with robustness defended in reply, the standing contested rather than settled. Ancient genomics — identifies candidate changes, none tied to behaviour. Redefining modernity as a mosaic — now widely adopted, weakening the original framing. Comparison with other hominins — symbolic behaviour attributed in several finds, contested in dating and attribution. **Narrowing recorded.** The mosaic reframing weakens the relation: modernity stands as a set of separately appearing traits rather than a package indexing capacity. Demographic modelling narrows the reach: the pattern stands accounted for by population size and connectedness where those are estimated. **Progress.** The repeatedly extended record of symbolic behaviour. The demographic models reproducing the pattern. The identified candidate genetic changes. **Own record.** ## 126 The Flynn effect and its reversal **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Test scores rose substantially for decades across countries, then flattened or fell in several. What was rising, and what changed? In the field's own posing, population intelligence quotients increased throughout the twentieth century, a phenomenon known as the Flynn effect, although recent years have seen a slowdown or reversal of the trend in several countries; to distinguish among the large set of proposed explanations the field categorises the hypothesised causal factors by whether they accommodate within-family effects, and the conscript study finds that the increase, turning point and decline can all be recovered from within-family variation, which establishes that the large changes in average cohort intelligence reflect environmental factors and not changing composition of parents. **The field's account of its hardness.** The field's study states that the set of proposed explanations is large and that distinguishing among them requires a design that holds the family constant, the within-family recovery of the whole trajectory ruling out several prominent hypotheses for retrograde effects; and the field's meta-analytic summary records that the gains vary according to domain, are stronger for adults than for children, and have decreased in more recent decades. **Accounts.** Environmental improvement, then its exhaustion: nutrition, schooling, family size, and test familiarity raised measured performance, and gains flattened as those improved conditions saturated in wealthy countries. What was measured may not be what the tests are taken to measure: gains are concentrated in subtests least loaded on the general factor, which is a measured pattern and the opposite of what a rise in general ability predicts, and measurement invariance across cohorts has been tested and often fails. **Specifications.** The tests measure the same construct across cohorts — local. Score change indexes construct change — local. **Specifications withdrawn.** **Common requirement.** That cross-cohort comparison of scores is meaningful. **Conserving relation.** One construct measured alike across cohorts, the sameness standing under invariance testing. **Conserving reach.** All cohorts compared. **Reach.** The statement quantifies over a construct compared across cohorts. Test scores, whose invariance across cohorts is itself in question, are what is measured. The two are not the same. **Unreachability.** Measurement invariance across cohorts frequently fails, which is established and is a limitation on the comparison the statement requires; and gains are concentrated on the least g-loaded subtests, measured consistently. The sameness is not preserved across the cohorts by test scores. **Persistence.** A measure demonstrated invariant across the cohorts compared. **Addresses.** Statement, hardness and the within-family analyses at Approaches: Bratsberg and Rogeberg, "Flynn effect and its reversal are both environmentally caused", Proceedings of the National Academy of Sciences 115(26):6674-6678, 2018, at its abstract and significance statement as the journal's and the archive's pages carry them verbatim, the paper itself not fetched; the reversal's first report as the paper cites it: Sundet, Barlaug and Torjussen, "The end of the Flynn effect? A study of secular trends in mean intelligence test scores of Norwegian conscripts during half a century", Intelligence 32:349-362, 2004; the meta-analytic summary as the index carries it: Pietschnig and Voracek, its title and venue keyed at the next pass. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the rise and the reversal both environmentally caused by the within-family analysis; the measure's invariance across cohorts the standing constraint), a measure demonstrated invariant standing there as what the invariance tests frequently deny. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Measurement invariance testing — frequently fails across cohorts, established, the strongest constraint on the interpretation. Subtest decomposition — gains concentrated on the least g-loaded subtests, measured consistently. Within-family analyses — conscript data showing the reversal within families, excluding compositional explanations such as migration for that population. Environmental proxies — nutrition and schooling correlating with gains historically, saturation inferred rather than measured directly. Test-content analyses — changes in what items require, consistent with familiarity accounts. **Narrowing recorded.** Within-family analyses narrow the reach: the reversal stands established within families for the population analysed. Subtest decomposition narrows the reach: the gains stand located by subtest rather than in the construct. **Progress.** The established failures of measurement invariance. The measured subtest concentration of gains. The within-family reversal. The historical correlations with nutrition and schooling. **Own record.** ## 127 Eukaryogenesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Complex cells arose once, from an association of simpler cells. What was the sequence? In the field's own posing, molecular phylogenetic analyses indicated that eukaryotes and the Archaea represent sister groups in the tree of life, and in the genomic era it became evident that eukaryotic cells possess a mixture of archaeal and bacterial features in addition to eukaryote-specific ones, it having been generally accepted for some time that mitochondria descend from endosymbiotic alphaproteobacteria; the field's later review holds that environmental and comparative genomics have revealed vital details about the identity and nature of the host cell and of the proto-mitochondrial endosymbiont, enabling a critical reappraisal of the hypotheses underlying the symbiotic origin. **The field's account of its hardness.** The field's reviews state that the shape of the tree of life has changed repeatedly, from prokaryotes and eukaryotes to two distinct kinds of prokaryote and then to eukaryotes and Archaea as sister groups; that eukaryotic cells carry archaeal, bacterial and eukaryote-specific features together; and that a critical reappraisal of the hypotheses underlying the symbiotic origin has been enabled only by environmental and comparative genomics revealing the identity and nature of the host cell and of the proto-mitochondrial endosymbiont. **Accounts.** Mitochondria first: the host was an archaeon, and acquisition of the bacterial endosymbiont preceded and enabled the rest of eukaryotic complexity, the energy supply making a large genome affordable. Host complexity first: Asgard archaea carry many eukaryotic signature proteins and cytoskeletal machinery, and cultured representatives form protrusions consistent with engulfment, indicating substantial complexity before the symbiosis. **Specifications.** There was a sequence of events — local. Present organisms and phylogenies constrain it — local. **Specifications withdrawn.** **Common requirement.** That the order of acquisition is recoverable. **Conserving relation.** The order of acquisition recovered, the recovery standing on extant material. **Conserving reach.** The sequence of events, which occurred once. **Reach.** The statement quantifies over the order of acquisition, recovered from extant genomes. Extant genomes, phylogenetic reconstruction, and a small number of cultured relatives are what is measured. The two are not the same. **Unreachability.** Phylogenomics establishes the host lineage robustly and not the order of events; ancestral genome reconstruction depends on model and taxon sampling with estimates differing; and the cell biology of intermediate forms is now observable while the event sequence remains uninferred from it. The recovery is not reached over the sequence by extant material. **Persistence.** An order of acquisition inferable from extant material, or a fossil assignment at the relevant age. **Addresses.** Statement and hardness: Eme, Spang, Lombard, Stairs and Ettema, "Archaea and the origin of eukaryotes", Nature Reviews Microbiology 15(12):711-723, 2017, at its abstract as the repositories' pages carry it verbatim, the paper itself not fetched; Vosseberg and others, "The emerging view on the origin and early evolution of eukaryotic cells", Nature 633:295-305, 2024, at its summary as the journal's page carries it, the paper itself not fetched; the phylogenomic placement at Approaches as the field cites it: Spang and others, "Complex archaea that bridge the gap between prokaryotes and eukaryotes", Nature 521:173-179, 2015; Zaremba-Niedzwiedzka and others, "Asgard archaea illuminate the origin of eukaryotic cellular complexity", Nature 541:353-358, 2017; neither read at its abstract. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (eukaryotes placed within the archaea, the host lineage established and the order of events not), an order of acquisition inferable from extant material or a fossil at the relevant age standing there as the review's own open question. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Phylogenomics — places eukaryotes within Asgard archaea, robustly, establishing the host lineage and not the order of events. Asgard culturing — from one organism cultured after long effort to a small set of cultured and highly enriched representatives across branches, with slow growth, protrusion morphology, and an actin-based cytoskeleton observed directly, a major and widening advance, the event sequence remaining uninferred from it. Energetics arguments — energy-per-gene arguments favouring the endosymbiont-first order, contested on assumptions about scaling. Ancestral genome reconstruction — infers gene content of the last common ancestor, depending on model and taxon sampling, estimates differing. Fossil record — sparse and difficult to assign at the relevant age. **Narrowing recorded.** Phylogenomics narrows the reach: the host lineage stands established without the order. Culturing narrows the reach: the cell biology stands observable for the representatives cultured. **Progress.** The placement of eukaryotes within Asgard archaea. The cultured and enriched representatives with directly observed cytoskeleton and morphology. The energetics calculations with their stated assumptions. **Own record.** ## 128 Homeostasis and allostasis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Physiological variables are held within ranges. Is regulation toward a setpoint, or anticipatory adjustment of the target? In the field's own posing, homeostasis, the dominant explanatory framework for physiological regulation, has undergone significant revision, contemporary models differing significantly from the original formulation, while allostasis, an alternative view of regulation, goes beyond its homeostatic roots and offers insights relevant to the understanding and treatment of several chronic conditions; the field's later work reads the review as holding that the two are not competing models but complementary aspects of regulation, homeostasis the maintenance of vital variables within viable ranges and allostasis the adaptive adjustment of regulatory systems that achieves that stability under changing conditions. **The field's account of its hardness.** The field's review states that despite growing enthusiasm for allostasis the concept remains in need of the clarification the review sets out to give; its companion note states that identifying individual effects and responses, and distinguishing their individual influences on a regulated state, is challenging, that challenging an organism and simply measuring changes of a regulated variable can promote erroneous conclusions about regulation, and that studying effector responses can reveal phenomena departing from homeostasis into the realm of allostasis; and the field's philosophy holds that the broader vision of homeostasis was eclipsed by the cyberneticists' reading of it. **Accounts.** Setpoint regulation: negative feedback around a reference is well characterised for temperature, osmolality, and glucose, with identified sensors, comparators, and effectors. Anticipatory regulation: targets shift in advance of demand, driven by prediction, measured variables show sustained shifts with no error signal, and the setpoint account handles these only by positing a moving setpoint. **Specifications.** Regulation is toward a value — local. The value is fixed or predicted — local. **Specifications withdrawn.** **Common requirement.** That there is a regulated variable with a target. **Conserving relation.** A determination of whether the target is fixed or set in advance, standing on measured trajectories. **Conserving reach.** All regulated variables the question is asked of. **Reach.** The statement quantifies over whether the target is fixed or set in advance. Measured variables and their trajectories, compatible with both descriptions, are what is measured. The two are not the same. **Unreachability.** Both descriptions are representable and model comparison on the same data does not separate them reliably; and the moving-setpoint model absorbs anticipation at a cost to testability its critics state. The determination is not reached over the variables by measured trajectories. **Persistence.** A trajectory the two descriptions predict differently, or agreement on what a moving setpoint concedes. **Addresses.** Statement, hardness, Reach and Persistence: Ramsay and Woods, "Clarifying the Roles of Homeostasis and Allostasis in Physiological Regulation", Psychological Review 121(2):225-247, 2014, at its abstract's opening as the repository's page carries it verbatim, the paper itself not fetched, the reading of the two as complementary carried as the field's later work states it; the companion note: "Correctly identifying responses is critical for understanding homeostatic and allostatic regulation", Temperature, 2014, doi:10.4161/23328940.2014.982048, pages 157-159, at its abstract and opening as the publisher's page carries them, its authors keyed at the next pass; the eclipse as the field's philosophy states it: Bechtel and Bich, "Rediscovering Bernard and Cannon: Restoring the Broader Vision of Homeostasis Eclipsed by the Cyberneticists", Philosophy of Science 92(3):584-605, 2025, its abstract not read. Approaches: the comparator-circuit and anticipatory-response routes at the paper's abstract read at pass one, in part, their endings at the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Identifying comparator circuits — found for temperature and osmolality, establishing setpoint machinery in those systems. Anticipatory response measurement — documented before meals, exercise, and stress, robustly, establishing anticipation. Moving setpoint models — accommodate anticipation within the feedback framework, their critics holding this makes the framework unfalsifiable, which is stated in the literature. Control-theoretic modelling — both descriptions representable, model comparison on the same data not separating them reliably. Weight defence studies — vigorous defence of body weight, claimed by both accounts. **Narrowing recorded.** Comparator identification narrows the reach: setpoint machinery stands established for temperature and osmolality. Moving setpoint models weaken the relation: the target stands variable within the feedback framework. **Progress.** The identified sensors, comparators, and effectors in specific systems. The documented anticipatory responses. The measured defence of body weight. **Own record.** ## 129 Antibiotic resistance **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Every antibiotic class has been followed by resistance. Is this avoidable by better use, or intrinsic to the approach? In the field's own posing, the real wonder is not the discovery of antibiotics but the rise of resistance in hospitals, communities and the environment concomitant with their use: the extraordinary genetic capacities of microbes have benefited from the overuse of antibiotics to exploit every source of resistance genes and every means of horizontal gene transmission, developing multiple mechanisms of resistance for each and every antibiotic introduced into practice, clinically, agriculturally or otherwise. **The field's account of its hardness.** The field's review states that resistance has followed each and every antibiotic introduced into practice, by every source of resistance genes and every means of horizontal transmission; that the oft-restated conclusion is that it is time to act; and that complete restitution of the therapeutic applications of antibiotics would require more information on the role of environmental microbiomes in the rise of resistance, the natural environments being reservoirs of resistance genes. **Accounts.** Manageable by stewardship and pipeline: resistance rates respond to prescribing practice, cycling and combination strategies slow it, and new classes and mechanisms continue to be found. Intrinsic: resistance genes predate clinical use, are widespread in environmental reservoirs, and are mobile between organisms, so any sustained selection pressure selects, and stewardship changes the rate rather than the outcome. **Specifications.** Resistance is a response to use — local. The response is modifiable by managing use — local. **Specifications withdrawn.** **Common requirement.** That the trajectory is determined by selection pressure that management can alter. **Conserving relation.** The outcome under sustained selection pressure determined, the determination standing for regimes not yet run. **Conserving reach.** All regimes, those not yet run included. **Reach.** The statement quantifies over the outcome under regimes not yet run, with selection pressure sustained. Resistance rates under regimes that have been tried, over years to decades, are what is measured. The two are not the same. **Unreachability.** Stewardship reduces rates measurably and the effects reverse when relaxed; resistance has followed each new class within years of deployment, without exception so far; and the resistome surveys establish the reservoir predates use. The determination is not reached over the untried regimes by rates under regimes tried. **Persistence.** A regime under which sustained pressure does not select, or agreement that the outcome is fixed and management sets the rate. **Addresses.** Statement, hardness, and the succession of resistance and the environmental reservoir at Approaches: Davies and Davies, "Origins and Evolution of Antibiotic Resistance", Microbiology and Molecular Biology Reviews 74(3):417-433, 2010, at its summary as the journal's and the index's pages carry it verbatim, the paper itself not fetched; the environmental reservoir as the review cites it: Allen, Donato, Wang, Cloud-Hansen, Davies and Handelsman, "Call of the wild: antibiotic resistance genes in natural environments", Nature Reviews Microbiology 8:251-259, 2010; the evolutionary reading as the field cites it: MacLean and San Millan, "The evolution of antibiotic resistance", Science 365(6458):1082-1083, 2019, its abstract not read. Persistence, no landing: the closing condition at the review's abstract and its succession passage read at pass one and at the hardness (resistance following each agent into use, the environmental reservoir of resistance genes), a regime under which sustained pressure does not select standing there as what the review's record of succession shows no agent has enjoyed. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Stewardship programmes — reduce resistance rates measurably in specific settings, effects real and partial and reversing when relaxed. Cycling and mixing strategies — modelling and trials giving mixed results, mixing outperforming cycling in several models. New classes — discovered at a slow rate, resistance following each within years of deployment, without exception so far. Resistome surveys — resistance genes found in ancient and pristine environments, establishing the reservoir predates use. Alternative modalities — phage, antivirulence, and microbiome approaches, early, resistance to phage arising readily, which is documented. **Narrowing recorded.** Stewardship narrows the reach: the reduction stands in the settings where the programme runs, and reverses when relaxed. Mixing strategies narrow the reach: the advantage stands in the models where it is measured. **Progress.** The measured stewardship reductions and their reversal on relaxation. The record of resistance following each class. The ancient and pristine resistome. The documented phage resistance. **Own record.** ## 130 Alignment of built systems **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A system is trained to optimise a specified objective. How is it made to do what was intended rather than what was specified? In the field's own posing, the problem is that of accidents in machine learning systems, defined as unintended and harmful behaviour that may emerge from poor design of real-world systems, and it is sorted into five practical research problems by their origin: having the wrong objective function, which covers avoiding side effects and avoiding reward hacking; an objective function too expensive to evaluate frequently, which is scalable supervision; and undesirable behaviour during the learning process, safe exploration and distributional shift. **The field's account of its hardness.** The field's founding paper states that sophisticated reward functions will need to refer to abstract concepts that will themselves have to be learned by models which can be vulnerable to adversarial counter-examples; that a learned reward function over a high-dimensional space may be vulnerable to hacking if it has pathologically high values along even one direction; and that the failure mode in which the objective function is physically replaced is particularly concerning where a human is in the reward loop, since the agent then has an incentive to coerce or harm the human in order to obtain reward; the field's later work poses the scalable-oversight question through debate and measured progress on oversight, and the training of deceptive systems that persist through safety training. **Accounts.** By improving specification and oversight: objectives learned from human feedback capture what a specification could not state, tasks are decomposed so that evaluation is easier than performance, systems assist in evaluating systems, and interpretability supplies a check independent of behaviour. Each component has measurable effect, and the combination has produced large and documented improvements in the behaviour of deployed systems. The gap may not close by these means: feedback trains on evaluations, and where a system's competence exceeds an evaluator's, the training signal measures what evaluators approve rather than what is intended; specification gaming is documented in a large catalogue of cases, and the failure recurs at each level of the oversight scheme. **Specifications.** The intended behaviour is specifiable, or a specification error is detectable by evaluation — local. Evaluation is available at the level of the behaviour — local. **Specifications withdrawn.** **Common requirement.** That there is an intended behaviour distinguishable from the specified one. **Conserving relation.** Behaviour matching what was intended, the match standing in situations not evaluated. **Conserving reach.** All situations, those not evaluated included. **Reach.** The statement quantifies over behaviour in situations not evaluated. Behaviour on evaluated distributions, and on tests constructed in advance, are what is measured. The two are not the same. **Unreachability.** Reward models are documented to be gameable and to be optimised against, and scalable oversight is untested where the gap is largest, by construction. The match is not reached over unevaluated situations by behaviour on evaluated ones. **Persistence.** An oversight scheme tested where a system's competence exceeds the evaluator's, or a check on intent independent of behaviour. **Addresses.** Statement, hardness, Reach, and the reward hacking and scalable oversight at Approaches: Amodei, Olah, Steinhardt, Christiano, Schulman and Mane, "Concrete Problems in AI Safety", arXiv:1606.06565, 2016, at its abstract as the archive's page carries it verbatim and at its passages on abstract rewards and on the replaced objective as the archive copy carries them, the paper itself not fetched; the later work at Approaches as the field cites it: Irving, Christiano and Amodei, "AI safety via debate", arXiv:1805.00899, 2018; Bowman and others, "Measuring progress on scalable oversight for large language models", its year and archive number keyed at the next pass; Hubinger and others, "Sleeper agents: Training deceptive LLMs that persist through safety training", arXiv:2401.05566, 2024; none read at its abstract. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the concrete problems named, reward hacking and scalable oversight among them; oversight where the system's competence exceeds the evaluator's the stated open problem), an oversight scheme tested there standing there as the paper's own framing. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Reinforcement learning from human feedback — produces large behavioural improvements, reward models documented to be gameable and to be optimised against. Scalable oversight — debate, recursive decomposition, and assisted evaluation, results promising in constructed settings, whether they hold where the gap is largest untested by construction. Interpretability — aims to read intent from internals, its own coverage partial. Red-teaming and evaluations — find failures reliably, establishing presence rather than absence. Specification gaming catalogues — document the phenomenon across many systems, a record rather than a remedy. **Narrowing recorded.** Scalable oversight narrows the reach: the results stand in constructed settings where evaluation remains available. Red-teaming narrows the reach: presence of failure stands established for the cases constructed. **Progress.** The measured behavioural improvements from feedback training. The documented gameability of reward models. The results of oversight schemes in constructed settings, extended at the instrument itself: reasoning-trace monitoring detects gaming more effectively than action-only monitoring, weaker models sometimes monitoring stronger ones, and where optimisation incorporates the monitor, obfuscated gaming follows. The specification gaming catalogue. **Own record.** ## 131 Benchmarks **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Systems are compared on benchmark scores. Do scores measure the capability the benchmark names? In the field's own posing, there is a tendency across subfields to valorise a small collection of influential benchmarks, which operate as stand-ins for a range of anointed common problems framed as foundational milestones on the path towards flexible and generalisable systems, state-of-the-art performance on them being widely understood as indicative of progress towards those goals; the question posed is one of construct validity, whether the benchmarks function as the general broad measures of progress they are set up to be. **The field's account of its hardness.** The field's position paper states that task formation for such benchmarks seems to happen independently of the intended and declared problem space, that the benchmarks tend to be de-contextualised, and that the construct validity issues lie in their framing as general measures; the field's earlier demonstration records that images of the same class from different datasets are often distinguishable, each dataset embodying a specific style of capturing some segment of the world. **Accounts.** Approximately, and usefully: benchmark scores correlate with downstream task performance where both are measured, benchmark competition has driven progress that transferred to applications, and suites are retired and replaced when they saturate. Every science measures with imperfect instruments and improves them. Scores measure benchmark performance: contamination of training data by test items is documented and hard to exclude at scale, benchmarks are optimised against once published, and score gains have repeatedly failed to transfer to tasks the benchmark was taken to represent. **Specifications.** The benchmark samples a capability — local. The sample is independent of what is trained on — local. **Specifications withdrawn.** **Common requirement.** That there is a capability the benchmark samples. **Conserving relation.** A score measuring capability rather than benchmark performance, the measurement standing after the benchmark is published. **Conserving reach.** Capability on tasks not in the benchmark. **Reach.** The statement quantifies over capability on tasks not in the benchmark. Benchmark items, and constructed held-out sets that become public in turn, are what is measured. The two are not the same. **Unreachability.** Each held-out set is optimised against in turn, and contamination detection gives partial coverage, which the field states. The measurement is not reached over untested tasks by benchmark items and held-out sets. **Persistence.** A measure not optimised against after publication, or downstream measurement at a scale that settles transfer. **Addresses.** Statement, hardness and Reach: Raji, Bender, Paullada, Denton and Hanna, "AI and the Everything in the Whole Wide World Benchmark", Proceedings of the Neural Information Processing Systems Track on Datasets and Benchmarks, 2021, arXiv:2111.15366, at its abstract as the archive's page carries it verbatim and at its earlier workshop form's abstract as the workshop's page carries it, the paper itself not fetched; the dataset-style demonstration as the paper cites it: Torralba and Efros, 2011, its title and venue keyed at the next pass. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (a benchmark's claim to general capability against its construction; measures optimized against once published), a measure not optimized against or downstream measurement settling transfer standing there as what the paper's critique asks. Approaches: the held-out and contamination-detection routes at that paper and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Held-out and private test sets — effective until released or inferred, the ones kept private not independently verifiable, which is the trade. Contamination detection — methods existing and giving partial coverage, which the field states. Adversarial and dynamic benchmarks — constructed against current systems, saturating as systems adapt. Task diversity and aggregate suites — reduce single-benchmark artefacts, the aggregate optimised against in turn. Downstream deployment measurement — the strongest route, expensive, slow, and confounded by deployment context. **Narrowing recorded.** Private test sets narrow the reach: the measurement stands where the set is unreleased, and is not independently verifiable there. Aggregate suites narrow the reach: single-benchmark artefacts stand reduced across the suite. **Progress.** The measured correlations with downstream performance where both are available. The documented contamination and its partial detection. The saturation record of successive suites. **Own record.** ## 132 Standards and interoperability **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Common standards let systems work together. Do they raise or lower the total value produced? In the field's own posing, the question is the consequences of standards-setting processes for industry structure and economic welfare, examined across four kinds of process: market competition among products embodying unsponsored standards, market competition among sponsored proprietary standards, agreements within voluntary standards-writing organisations, and direct governmental promulgation, with both positive and normative issues concerning compatibility standards recorded as remaining to be studied. **The field's account of its hardness.** The field's survey states that the consequences of standardisation for industry structure and welfare have to be traced separately for four kinds of process, and that both the positive and the normative issues concerning compatibility standards remain to be studied along the trajectories the research has been moving; the founding analyses the survey draws on are titled to treat standardisation together with compatibility and innovation, and technology adoption in the presence of network externalities. **Accounts.** They raise it: network effects mean shared standards produce compatibility gains exceeding the constraint costs, demonstrated across shipping containers, telecommunications, and internet protocols. It depends on timing and on what is frozen: standards set early lock in inferior designs and foreclose alternatives, and the cases where a standard blocked improvement are documented alongside the successes, so there is no general answer, only cases. **Specifications.** The counterfactual — the value under an unadopted alternative or no standard — is estimable — local. **Specifications withdrawn.** **Common requirement.** That the value under the unadopted arrangement is comparable. **Conserving relation.** A comparison of adopted and unadopted arrangements, standing at the scale the adopted one reached. **Conserving reach.** All cases the comparison is asked of. **Reach.** The statement quantifies over a counterfactual arrangement at scale. The adopted arrangement, and pre-standard periods that differ in many other respects, are what is measured. The two are not the same. **Unreachability.** Case studies on both sides are contested, formal models produce both outcomes under stated conditions with the conditions being the empirical question, and competing-standard episodes are confounded by firm strategy and timing. The comparison is not reached over the cases by the adopted arrangement and pre-standard periods. **Persistence.** A comparison at scale between adopted and unadopted arrangements, or agreement on the conditions under which each outcome follows. **Addresses.** Statement, hardness and the formal modelling at Approaches: David and Greenstein, "The Economics of Compatibility Standards: An Introduction to Recent Research", Economics of Innovation and New Technology 1(1-2):3-41, 1990, at its abstract as the journal's, the repository's and the index's pages carry it verbatim, the paper itself not fetched; the founding analyses as the field cites them: Farrell and Saloner, "Standardization, compatibility, and innovation", RAND Journal of Economics 16(1):70-83, 1985; Katz and Shapiro, "Technology adoption in the presence of network externalities", Journal of Political Economy 94(4):822-841, 1986; neither read at its abstract. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (the economics of compatibility standards, adoption under network effects and the possibility of lock-in), a comparison at scale between adopted and unadopted arrangements standing there as what the case studies, selected on adoption, do not supply. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Case studies of successful standards — establish large gains where adopted, selection on adoption being the objection. Case studies of lock-in — establish costs in specific cases, contested in the same way. Competing-standard episodes — comparison available for a period where two standards competed, outcomes confounded by firm strategy and timing. Formal modelling — produces both outcomes under stated conditions, the conditions being the empirical question. Standards body process studies — document how choices are made, accounting for adoption rather than value. **Narrowing recorded.** Competing-standard episodes narrow the reach: the comparison stands for the period and cases in which two standards ran together. **Progress.** The measured gains where standards were adopted. The documented lock-in costs in specific cases. The formal conditions under which each outcome follows. **Own record.** ## 133 Brooks's law **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Adding people to a late software project makes it later. Is this general? In the field's own posing, the law is stated in its founding essay, with the warning that it oversimplifies outrageously, in nine words verbatim: "Adding manpower to a late software project makes it later"; its mechanism is that people added to a project take time to become productive, their education diverting the resources already at work, and that where the sub-tasks are interrelated the intercommunication requirements rise, since each part must be coordinated with every other. **The field's account of its hardness.** The founding essay states that the law is an outrageous oversimplification that nonetheless captures the general rule; the field's simulation study states that adding people to a late project will always increase its cost but the project may not always be late, delay following only under a certain degree of sequential constraint among the tasks, with a time line for each project before which enough added manpower still helps; and the field's practitioners record important exceptions not taken into account when the law is used to justify a decision, one holding that a late chaotic project's further lateness results from underestimation rather than from the law. **Accounts.** Yes, by a countable mechanism: communication paths grow with the square of team size while labour grows linearly, and ramp-up consumes existing members' time, so the arithmetic holds wherever the work is not partitionable. It is conditional and often violated: modular architecture, asynchronous coordination, and mature onboarding reduce the coupling, large distributed projects add contributors continuously without the predicted collapse, and the empirical record does not show a general law. **Specifications.** Coordination cost scales with the number of pairs — local. The work's partitionability is fixed — local. **Specifications withdrawn.** **Common requirement.** That coordination cost and output are measurable against team size. **Conserving relation.** The effect of adding people isolated, the isolation standing with all else held equal. **Conserving reach.** All projects the effect is asked of. **Reach.** The statement quantifies over the effect of adding people, holding all else equal. Projects where adding people co-occurs with schedule pressure, scope change, and architecture choices are what is measured. The two are not the same. **Unreachability.** Case histories select on troubled projects, estimation models incorporate diseconomies as parameters fitted to the same data, and controlled experiments run at scales far below the cases in question. The isolation is not reached over the projects by observations where the additions co-occur with other changes. **Persistence.** A measurement of the effect with architecture and scope held fixed, at the scale of the cases in question. **Addresses.** Statement, hardness, Reach and the conditionality at Approaches: Brooks, The Mythical Man-Month: Essays on Software Engineering, Addison-Wesley, 1975, anniversary edition 1995, at its statement of the law and its warning as readers' notes pages and a general reference page carry them verbatim, the book itself not fetched; Hsia, Hsu and Kung, "Brooks' law revisited: a system dynamics approach", IEEE conference publication, document 814310, at its abstract as the publisher's page carries it verbatim, its conference and year keyed at the next pass; the practitioners' exceptions as a general reference page quotes them: McConnell, 1999, its title and venue keyed at the next pass. Persistence, no landing: the closing condition at the book's statement of the law and its warning as readers' notes carry them, read at pass one and at the hardness (adding manpower to a late project making it later; the law's own conditionality), a measurement with architecture and scope held fixed standing there as what the case histories, selected on trouble, do not give. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Case histories — support the law in the original setting, selection on troubled projects being the standing objection. Open-source contributor studies — find sublinear but positive returns to contributors in modular projects, establishing the conditionality. Software estimation models — incorporate diseconomies of scale as fitted parameters, fitting, the parameters estimated from the same data. Controlled experiments — small-team laboratory studies finding coordination overhead, at scales far below the cases in question. Architecture studies — modularity reducing coupling measurably, establishing the mechanism's dependence on structure. **Narrowing recorded.** Open-source studies narrow the reach: positive returns stand in modular projects. Controlled experiments narrow the reach: the overhead stands measured at small-team scale. **Progress.** The measured returns to contributors in modular projects. The measured coordination overhead at small scale. The measured effect of modularity on coupling. **Own record.** ## 134 Risk compensation **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Safety measures reduce the harm per event. Do they reduce total harm, or do people take more risk? In the field's own posing, the technological approach to risk regulation seeks to lessen mortality and morbidity by modifying environmental conditions, and insofar as risk-taking behaviour is purposive it has been predicted that technological progress may be negated by various forms of risk compensation; the test on mandatory seat-belt laws finds that the laws reduced mortality among car occupants to a degree compatible with the predictions of the technologists, with weak evidence of increased mortality among some non-occupants, a pattern predicted by the risk-compensation theorists. **The field's account of its hardness.** The field's test states that the validity of the compensation hypothesis has to be established sector by sector, the seat-belt case finding occupant mortality reduced as the technologists predicted while non-occupant mortality showed the weak increase the compensation theorists predicted; the field's review, as the field summarises it, concludes that compensation occurs in some contexts and is absent in others, depending on the visibility of the safety measure, its perceived effect, the motivation for behaviour change and personal control of the situation; and the field's rebuttal records the disagreement as ongoing and heated with respect to regulatory measures, rests its case on steady declines in unintentional injury mortality which on the compensation theory should not have occurred, and holds that the time has come for the debate to be concluded. **Accounts.** Total harm falls: seat belt, helmet, and airbag introductions are followed by large measured reductions in death and serious injury, established across many jurisdictions with staggered adoption and confirmed in occupant-level data where exposure is controlled, and compensation, where measured, offsets a fraction of the mechanical gain and never the whole of it. Compensation is real and sometimes complete: behavioural adaptation is documented — following distances, overtaking margins, cycling behaviour around helmeted riders — and in some settings the aggregate effect on total harm is null or adverse, particularly where risk is transferred to others. **Specifications.** A safety measure's effect decomposes into a mechanical effect and a behavioural response — local. Both are estimable — local. **Specifications withdrawn.** **Common requirement.** That the counterfactual harm without the measure is estimable. **Conserving relation.** Total harm compared against the counterfactual regime, the comparison standing with concurrent changes accounted for. **Conserving reach.** All settings the measure is adopted in. **Reach.** The statement quantifies over total harm under a counterfactual regime. Outcomes after adoption, with concurrent changes in traffic, vehicles, medicine, and enforcement, are what is measured. The two are not the same. **Unreachability.** Before-after series are confounded by concurrent trends, which the literature addresses with varying success, and risk transfer to other users is measured in some settings and concealed by aggregate series. The comparison is not reached over the settings by outcomes after adoption. **Persistence.** A comparison against the counterfactual regime with concurrent changes accounted for, or measured magnitudes for the mechanical and behavioural components across settings. **Addresses.** Statement, hardness, and the risk transfer, the staggered-adoption result and the rebuttal at Approaches: Evans and Graham, "Risk reduction or risk compensation? The case of mandatory safety-belt use laws", Journal of Risk and Uncertainty 4:61-73, 1991, at its abstract as the publisher's page carries it verbatim, the paper itself not fetched; Hedlund, "Risky business: safety regulations, risk compensation, and individual behavior", Injury Prevention 6(2):82-90, 2000, carrying no abstract at the archive, its four factors as the field's later summary states them, the paper itself not fetched; Pless, "Risk Compensation: Revisited and Rebutted", Safety 2(3):16, 2016, at its abstract as the index's page carries it verbatim, the paper itself not fetched; the founding economic analysis as the field cites it: Peltzman, "The Effects of Automobile Safety Regulation", Journal of Political Economy 83(4):677-725, 1975. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (risk reduction or risk compensation as the two readings of the mortality series; the staggered-adoption design and the rebuttal), a comparison against the counterfactual regime with concurrent changes accounted for standing there as the design the field's exchange turns on. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Before-after mortality series — show large reductions, confounded by concurrent trends, which the literature addresses with varying success. Behavioural measurement — documents adaptation directly in specific behaviours, establishing the mechanism, magnitude relative to the mechanical effect varying by study. Natural experiments from staggered adoption — the strongest route, results supporting net reductions for major measures with partial offsets. Risk transfer studies — find effects on other road users in some settings, which the aggregate series can conceal. Full-compensation theory — predicts complete offset, its critics holding the mortality series refute it, its defenders disputing the counterfactual. **Narrowing recorded.** Staggered-adoption designs narrow the reach: the net reduction stands for the jurisdictions and measures analysed. Risk transfer studies narrow the reach: the transfer stands measured in the settings studied. **Progress.** The measured reductions in death and serious injury. The documented behavioural adaptation. The staggered-adoption results with partial offsets. The measured risk transfer in specific settings. **Own record.** ## 135 Technical debt **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Choices that speed delivery now raise the cost of later change. Is the trade-off measurable? In the field's own posing, the metaphor was introduced to explain to non-technical stakeholders the need for what is now called refactoring, its original object being code not quite right whose correction is postponed; the term has since been used to describe a wide range of phenomena, from bad coding to architectural issues, and the field's organisation of the landscape and its workshop reports record confusion in industry over the definition, the limits of the metaphor, and the criteria to discriminate what is technical debt from what is not. **The field's account of its hardness.** The field's reports state that the term is being used to describe a wide range of phenomena, so that the landscape has to be organised before it can be measured; that confusion in industry over the definition, the limits of the metaphor, and the criteria to discriminate what is technical debt and what is not were the themes of the field's workshops, with a desire recorded to move beyond definitional issues to impact; and that the concept has been refined and expanded since its introduction through a taxonomy, a four-quadrant scheme and a consortium's model. **Accounts.** Yes, and increasingly: code quality metrics correlate with defect rates and change cost, static analysis identifies specific classes, and firms that measure it report actionable results. The measurements are of proxies: metrics correlate with each other more strongly than with change cost, the counterfactual code that was not written is unavailable, and what is called debt is often a design decision whose cost depends on changes not yet requested. **Specifications.** There is a present cost incurred by a past choice — local. The cost is attributable to identifiable code properties — local. **Specifications withdrawn.** **Common requirement.** That the later cost is attributable to the earlier choice. **Conserving relation.** The later cost attributed to the earlier choice, the attribution standing for changes not yet requested. **Conserving reach.** All changes, those not yet requested included. **Reach.** The statement quantifies over cost of changes not yet requested. Code properties now, and change history so far, are what is measured. The two are not the same. **Unreachability.** The counterfactual code carrying the disputed cost is unavailable, and change-cost measurement covers changes made while the unmade changes carry the disputed cost. The attribution is not reached over unrequested changes by code properties and change history. **Persistence.** A measurement of the cost of changes not yet requested, or agreement on a proxy that tracks it. **Addresses.** Statement, hardness, Persistence and the shared-concept finding at Approaches: Kruchten, Nord and Ozkaya, "Technical Debt: From Metaphor to Theory and Practice", IEEE Software 29(6):18-21, 2012, at its abstract and opening as the publisher's and the index's pages carry them verbatim, the paper itself not fetched; Kruchten, Nord, Ozkaya and Falessi, "Technical debt: towards a crisper definition; report on the 4th International Workshop on Managing Technical Debt", ACM SIGSOFT Software Engineering Notes 38(5):51-54, 2013, at its abstract as the index summarises it; the metaphor's origin as the field cites it: Cunningham, "The WyCash Portfolio Management System", OOPSLA Experience Report, 1992, at its phrase as a citing page carries it. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Metric-defect correlation — established for several metrics, attribution to the earlier decision being the gap. Change-cost measurement — measures cost of changes made, the unmade changes carrying the disputed cost. Refactoring experiments — show improvements in specific measures, cost-benefit over the project life unmeasured. Developer survey and interview studies — establish that practitioners identify debt consistently, establishing a shared concept rather than a quantity. Architecture erosion studies — track drift from intended structure, establishing drift, consequences depending on future requirements. **Narrowing recorded.** Refactoring experiments narrow the reach: the improvement stands in the measures taken. Change-cost measurement narrows the reach: the cost stands measured for changes made. **Progress.** The established metric-defect correlations. The measured cost of changes made. The measured improvements from refactoring. The consistency of practitioner identification. **Own record.** ## 136 Signal and noise **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What separates signal from noise in a measured variation — whether the division is fixed by the source, or is relative to a model and a purpose. Posed in the field's review literature at the nervous system, where the same trial-to-trial variability is read as noise by one account and as information by another, and at stochastic resonance, where noise added to a weak signal improves its detection. **The field's account of its hardness.** That neurons fire at a regular but slowly declining rate in response to a constant current stimulus, yet the intervals between spikes often vary randomly during behaviour, and the question the field poses is whether this variation is an unavoidable effect of generating spikes by sensory or synaptic processes, neural noise, or an important part of the signal transmitted to other neurons; that on the traditional view of the rate of impulses as the main carrier of information any variability in the rate of response to a steady stimulus would reduce the information conveyed, the considerable variability of many cells limiting them to two or three bits in a second where time-varying inputs yield about a bit per spike; and that recent studies indicate variability might offer distinct advantages, noise enhancing sensitivity to weak signals. **Accounts.** The distinction is definable relative to a model of the source: given the source statistics the decomposition is exact, and information theory makes the whole treatment quantitative. The decomposition is relative to what is wanted: nothing in the channel marks which component is signal, the same physical variation is signal for one purpose and noise for another, and added noise improving detection is treated as anomalous by the framing. **Specifications.** The channel's content decomposes into wanted and unwanted components — local. The source model is available — local. **Specifications withdrawn.** **Common requirement.** That the decomposition is well defined. **Conserving relation.** A decomposition into signal and noise, standing independently of purpose. **Conserving reach.** All channels the decomposition is asked of. **Reach.** The statement quantifies over which component is which, independent of purpose. The total variation, and a model supplied by the analyst, are what is measured. The two are not the same. **Unreachability.** Every decomposition is exact given its model, and the model is supplied; and added noise improving detection in nonlinear systems, measured across physical and biological settings, establishes the labels are purpose-relative. The decomposition is not reached over the channels independently of a supplied model. **Persistence.** A decomposition not resting on a supplied model or purpose, or agreement that the labels are purpose-relative. **Addresses.** Statement: Faisal, Selen and Wolpert, "Noise in the nervous system", Nature Reviews Neuroscience 9:292-303, 2008; the question in the field's own title: Stein, Gossen and Jones, "Neuronal variability: noise or part of the signal?", Nature Reviews Neuroscience 6:389-397, 2005. Stochastic resonance: Gammaitoni, Hänggi, Jung and Marchesoni, "Stochastic resonance", Reviews of Modern Physics 70:223-287, 1998. Hardness: the review whose title poses the question, at its abstract as the publisher's page, an index and two repository copies carry it verbatim (neurons firing at a regular but slowly declining rate in response to a constant current stimulus, the intervals between spikes yet varying randomly during behaviour; the question whether this variation is an unavoidable effect of generating spikes by sensory or synaptic processes, neural noise, or an important part of the signal transmitted to other neurons; the question discussed mainly for sensory and motor processes where signals are best identified) and at passages of its text as the publisher's page and a repository copy carry them (the rate of impulses traditionally the main carrier of information, any variability in the rate of response to a steady stimulus reducing the information conveyed; the considerable variability of many cells limiting them to two or three bits in a second, time-varying inputs yielding about a bit per spike; variability possibly offering distinct advantages, noise enhancing sensitivity to weak signals), issue 5, doi 10.1038/nrn1668, the review itself not fetched; the first review and the stochastic-resonance review not read this pass. Persistence, no landing: the closing condition at the review whose title poses the question, at its abstract read at pass two (the regular but slowly declining firing read as signal by one purpose and as noise by another), a decomposition not resting on a supplied model or purpose standing there as what the review's own posing withholds. Approaches: the source-modelling and blind-separation routes at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Modelling the source — makes the decomposition exact given the model, the model being supplied. Blind source separation — recovers components from mixtures under stated assumptions, independence and sparsity, which are supplied. Stochastic resonance — demonstrates noise improving detection in nonlinear systems, measured across physical and biological settings, establishing the labels are purpose-relative. Matched filtering — optimal given the expected signal shape, which is supplied. Information-theoretic bounds — exact given the channel and source distributions. **Narrowing recorded.** Blind source separation narrows the reach: the recovery stands under stated independence or sparsity assumptions. Matched filtering narrows the reach: optimality stands for a supplied signal shape. **Progress.** The exact decompositions under stated models. The measured stochastic resonance across settings. The information-theoretic bounds under given distributions. **Own record.** ## 137 Work and waste heat **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether the division of an energy exchange into work and heat is fixed by the system, or by the description and the purpose. Settled classically for the maximum work, which the reservoir temperatures fix; posed as open at the scale of single quantum systems, where the proposed definitions of work are inequivalent and work is recorded as not an observable. **The field's account of its hardness.** That at the scale of single quantum systems the work performed by an external time-dependent force is stated in the field not to be an observable, its characteristic function being identified with the time-ordered correlation function of the exponentiated Hamiltonian rather than with the expectation value of an operator, the averaged exponential work being what the work theorem relates to the free-energy difference of the equilibrium systems. **Accounts.** The split is exact and physical: the first and second laws determine the maximum work extractable, efficiency is measured to high precision, and nothing about the accounting is conventional. Which output counts as work is set by what is wanted from the device: in a heat pump the rejected heat is the product, in cogeneration both are, the thermodynamics is invariant and the labels are not, and nothing in the engine marks the split. **Specifications.** A device has an intended output — local. Efficiency is output over input — universal under thermodynamics. **Specifications withdrawn.** **Common requirement.** That efficiency is well defined for the device. **Conserving relation.** A determination of which output is the product, standing on the energy flows. **Conserving reach.** All devices the question is asked of. **Reach.** The statement quantifies over which output is the product. Energy flows, all measurable, none labelled, are what is measured. The two are not the same. **Unreachability.** Thermodynamic accounting is exact and invariant under relabelling, which both accounts accept entirely; exergy requires a specified reference environment; and cogeneration standards allocate between outputs by conventions that differ between jurisdictions, which is documented. The determination is not reached over the devices by energy flows. **Persistence.** A measure of the product not resting on a specified reference or convention, or agreement that the labels are set by purpose. **Addresses.** Statement, the classical bound: Carnot, Réflexions sur la puissance motrice du feu, 1824. Statement, the quantum definitions: Talkner, Lutz and Hänggi, "Fluctuation theorems: Work is not an observable", Physical Review E 75:050102(R), 2007; review: Campisi, Hänggi and Talkner, "Quantum fluctuation relations: foundations and applications", Reviews of Modern Physics 83:771, 2011. Hardness: the paper the address names at the quantum definitions, at its abstract as the publisher's page, the arXiv page, an abstract service, an index and a repository copy carry it verbatim (the characteristic function of the work performed by an external time-dependent force on a Hamiltonian quantum system identified with the time-ordered correlation function of the exponentiated Hamiltonian; a similar expression for the averaged exponential work, related to the free-energy difference by the work theorem), its title carrying the claim that work is not an observable, received 7 March and published 16 May 2007, doi 10.1103/PhysRevE.75.050102, arXiv:cond-mat/0703189, the paper itself not fetched; the review's title carried at a citing paper's reference list with the prefix Colloquium, recorded and not corrected; a comparison of two definitions of work for a simple quantum system and a case against quantum work-fluctuation theorems from subensembles, at their titles in that list, readings for the inequivalence at the next pass; the classical text and the review not read this pass. Persistence, no landing: the closing condition at the paper the address names at the quantum definitions, at its abstract read at pass two (the choice of reference fixing what counts as work and what as heat), a measure not resting on a specified reference standing there as what the definitions each require. Approaches: the thermodynamic and exergy accountings at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Thermodynamic accounting — exact, and invariant under relabelling, accepted entirely by both accounts. Exergy analysis — measures available work regardless of purpose, coming closest to a purpose-free measure and still requiring a reference environment, which is specified. Second-law efficiency — ratio to the reversible ideal, requiring specification of which output the ideal produces. Cogeneration accounting — standards allocating between outputs by convention, the conventions differing between jurisdictions, which is documented. Life-cycle accounting — extends the boundary, the boundary being drawn. **Narrowing recorded.** Exergy analysis narrows the reach: the measure stands relative to a specified reference environment. Cogeneration standards narrow the reach: the allocation stands by convention within a jurisdiction. **Progress.** The exact thermodynamic accounting and its invariance under relabelling. The exergy measure with its stated reference. The documented divergence of allocation conventions. **Own record.** ## 138 Mutation, fidelity and variation **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What sets the mutation rate: whether replication fidelity is tuned by selection to an optimum balancing the cost of errors against the cost of proofreading, or is bounded by the limit of selection's own power to lower it further, the rate then tracking the effective population size — the drift-barrier hypothesis; and, prior to that, whether mutations arise independently of their effect on fitness. **The field's account of its hardness.** That natural selection can adjust the rate of mutation by acting on allelic variation affecting the processes of replication and repair, and that, mutation being the ultimate source of the variation adaptation requires, it can be appealing to suppose the genomic rate is adjusted to a level that best promotes adaptation, the field's review posing its question as separating the causes of mutation-rate evolution from its consequences; and that replication fidelity is one of the few cellular traits that can be quantified across the tree of life, clarifying why rates vary being crucial for understanding all areas of biology. **Accounts.** Both descriptions apply and the biology is settled: fidelity is under selection, mutation rates are themselves evolved, and rates sit near an optimum balancing the cost of errors against the cost of proofreading. The optimum framing assumes selection acts on the rate as a trait, which requires linkage conditions that often fail: rates are set by the limits of selection efficacy rather than by an optimum, and rates track effective population size across taxa as that predicts. **Specifications.** Mutation rate is a trait — local. Its value is accounted for either by optimisation or by the limits of selection — local. **Specifications withdrawn.** **Common requirement.** That the observed rate has an account in these terms. **Conserving relation.** An account of what sets the rate, standing on parameters estimated independently of the rate. **Conserving reach.** All taxa the rate is measured across. **Reach.** The statement quantifies over what sets the rate. Measured rates across taxa, and population-genetic parameters estimated from the same genomes, are what is measured. The two are not the same. **Unreachability.** Effective population size is estimated from the same genomes the rates are measured in, so the strongest correlation closes on itself; and rates vary along genomes with sequence context and repair activity, which complicates a single rate as the trait. The account is not reached over the taxa by rates and parameters estimated together. **Persistence.** An estimate of effective population size independent of the genomes the rates are measured in. **Addresses.** Statement, the rate: Lynch, "Evolution of the mutation rate", Trends in Genetics 26:345-352, 2010; review: Lynch, Ackerman, Gout, Long, Sung, Thomas and Foster, "Genetic drift, selection and the evolution of the mutation rate", Nature Reviews Genetics 17(11):704-714, 2016, doi 10.1038/nrg.2016.104, published 14 October 2016, at its abstract as the publisher's page, an index, a university repository and two repository copies carry it verbatim (replication fidelity one of the few cellular traits quantifiable across the tree of life; why mutation rates vary crucial to clarify, mutation the ultimate source of all genetic variation; the drift-barrier hypothesis, selection primarily improving fidelity with the ultimate limits set by the power of random genetic drift, consistent with comparative measures, explaining error-prone polymerases and countering the view that selection fine-tunes gene-specific rates), the review itself not fetched. Statement, independence from fitness: Luria and Delbrück, Genetics 28:491-511, 1943. Hardness: filled at an earlier review returned in the search, Sniegowski, Gerrish, Johnson and Shaver, "The evolution of mutation rates: separating causes from consequences", BioEssays 22(12):1057-1066, 2000, doi 10.1002/1521-1878(200012)22:12<1057::AID-BIES3>3.0.CO;2-W, at its abstract as an index carries it (selection adjusting the rate by acting on variation affecting replication and repair; the appeal of supposing the rate adjusted to best promote adaptation), not fetched, the abstract's continuation not carried at the return, and at the 2016 review's abstract as read above (fidelity quantifiable across the tree of life; why rates vary crucial). Before this sitting the property stood empty with the reason that the 2016 review's abstract carrying the hypothesis with its support and its relation to the rival view, which the property cannot carry under check C16 and which come after the accounts, and no statement of what is hard before them; the 2010 paper and the founding independence paper not returned, not read this pass. Persistence, no landing: the closing condition at the two reviews read at v342 (separating causes from consequences; why rates vary crucial for all areas of biology; the drift barrier resting on effective population size), an estimate of effective size independent of the genomes measured standing there as what the drift-barrier tests require. Approaches: the mutation-accumulation and drift-barrier routes at the 2016 review's abstract read at v342 and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Mutation accumulation experiments — measure rates directly across many species, establishing the values. Drift-barrier tests — rates correlating inversely with effective population size across a wide range, as predicted, the strongest evidence, effective population size estimated with model assumptions. Mutator strain studies — elevated rates favoured under strong selection in the laboratory, establishing conditions where rate evolves adaptively. Proofreading cost measurement — costs small in most systems, weakening simple optimisation accounts. Local rate variation studies — rates varying along genomes with sequence context and repair activity, complicating a single rate as the trait. **Narrowing recorded.** Mutator strain studies narrow the reach: adaptive rate evolution stands established under laboratory selection. Local rate variation narrows the reach: a rate stands defined per genomic context rather than per organism. **Progress.** The directly measured rates across species. The measured inverse correlation with effective population size. The measured proofreading costs. The measured local rate variation. **Own record.** ## 139 Intellectual property **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Ideas can be copied at no cost. Does granting exclusive rights over them increase or decrease what is created? In the field's own posing, the question is whether patents serve to increase innovation and productivity, with the partial-equilibrium effect on the incentive to invent parted from the general-equilibrium effect on innovation; and the field's survey poses it as the theoretical and empirical evidence on patents as an incentive for innovation, beside their effectiveness for disclosure, their valuation and the design of patent systems. **The field's account of its hardness.** The case against states that no empirical evidence shows patents increasing innovation and productivity unless productivity is identified with the number of patents awarded, a count that has no correlation with measured productivity; that patents can improve the incentive to invent in partial equilibrium while their general-equilibrium effect on innovation can be negative; and that a well-designed system might raise innovation at a certain time and place while the political economy of government-operated systems lets the ill effects grow. The same case names the disconnect a patent puzzle: patents and the strength of their protection have increased enormously without a dramatic acceleration in technological progress or a major rise in research spending. **Accounts.** It increases it: without exclusivity, investment in costly creation is not recoverable, and the record in pharmaceuticals — where development costs are high and imitation cheap — supports protection producing innovation that would not otherwise occur. The evidence does not support it generally: cross-country and historical studies of regime changes find little effect on innovation rates and some negative effects through blocking and litigation, sectors with weak protection show high innovation, and cumulative innovation is impeded where the inputs are owned. **Specifications.** Creation responds to expected returns — local. Innovation is measurable and attributable to a regime — local. **Specifications withdrawn.** **Common requirement.** That the counterfactual innovation under a different regime is estimable. **Conserving relation.** Creation compared against a different regime, the comparison standing at the scale the adopted regime reached. **Conserving reach.** All sectors and regimes the question is asked of. **Reach.** The statement quantifies over creation that would have occurred under a different regime. Patents filed, products launched, and innovation proxies under regimes that were adopted are what is measured. The two are not the same. **Unreachability.** Cross-country comparison is confounded by development level and institutions, historical episodes return mixed results, and sector comparison is read by both accounts as supporting their own. The comparison is not reached over the sectors and regimes by proxies under regimes adopted. **Persistence.** A comparison against an unadopted regime at scale, or agreement on the conditions under which each effect follows. **Addresses.** Statement, hardness, and the counts-against-productivity relation at Reach: Boldrin and Levine, "The Case against Patents", Journal of Economic Perspectives 27(1):3-22, 2013, at its abstract as the association's page carries it verbatim and at its opening as the working-paper page carries it, the paper itself not fetched; the survey's posing at the Statement: Hall and Harhoff, "Recent Research on the Economics of Patents", Annual Review of Economics 4:541-565, 2012, at its abstract as the publisher's page carries it verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the case against patents on the absence of evidence that they raise innovation; the counts against productivity), a comparison against an unadopted regime at scale standing there as what the cross-country and historical evidence approximates. Approaches: the regime-comparison and natural-experiment routes at that paper and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Cross-country regime comparison — finds weak or no association with innovation rates, confounded by development level and institutions. Historical natural experiments — extension and abolition episodes giving mixed results, several finding no innovation effect. Sector comparison — pharmaceuticals differing from software and finance in imitation cost, accepted by both accounts and read as supporting each. Litigation and blocking studies — document costs of the regime, measured. Prize and open-source alternatives — demonstrate creation without exclusivity in specific domains, scope of transfer disputed. **Narrowing recorded.** Alternatives narrow the reach: creation without exclusivity stands demonstrated in the domains where prizes and open collaboration operate. Sector comparison narrows the reach: the account stands per sector by imitation cost rather than generally. **Progress.** The measured cross-country associations. The historical episode results. The measured litigation and blocking costs. The demonstrated creation under alternative arrangements. **Own record.** ## 140 Money as a store of value **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Money holds value over time. What makes it hold? In the field's own posing at the search-theoretic models, individuals who specialise in what they produce and consume and meet at random, so that every trade is bilateral and quid pro quo, have certain goods emerge as media of exchange from their intrinsic properties together with extrinsic beliefs, and equilibria also exist in which a genuine fiat currency circulates as the general medium of exchange. **The field's account of its hardness.** The founding model states that which goods come to serve as money depends both on their intrinsic properties and on extrinsic beliefs, and that equilibria with a circulating fiat currency stand beside equilibria without one; its extensions state that steady states with valued fiat money can be indeterminate, that fiat money can be valued even where it is not the cheapest object to store and two fiat monies can both be valued, and that for some parameter values a continuum of dynamic equilibria exists, some of them cycling. **Accounts.** Backing and scarcity: value derives from what stands behind the money — a commodity redeemable on demand, a state's obligation to accept it for taxes, or a credible limit on issuance — and the historical record is consistent, currencies losing their backing or whose issuance became unlimited losing value, and those retaining one or the other not. Acceptance is self-referential: fiat money holds value because it is accepted, and it is accepted because it is expected to hold value; the equilibrium is multiple, and models admit both valued and worthless outcomes for the same fundamentals, which is a theorem rather than an anomaly. **Specifications.** Value has a determinant — local. The determinant is either external backing or expectations — local. **Specifications withdrawn.** **Common requirement.** That there is a determinant of the value. **Conserving relation.** A determinant of the value identified, the identification standing for regimes with the same fundamentals. **Conserving reach.** All regimes the question is asked of, fiat regimes included. **Reach.** The statement quantifies over why acceptance persists. Prices and holdings under regimes that persisted, and episodes where they did not, are what is measured. The two are not the same. **Unreachability.** Search-theoretic models prove equilibrium multiplicity — valued and non-valued outcomes coexisting for the same fundamentals — which does not select; and hyperinflation onset timing is not predicted by fundamentals in the episodes studied. The identification is not reached over the regimes by prices and holdings. **Persistence.** A selection among the multiple equilibria, or a determinant predicting onset in the episodes where acceptance failed. **Addresses.** Statement and hardness: Kiyotaki and Wright, "On Money as a Medium of Exchange", Journal of Political Economy 97(4):927-954, 1989, at its abstract as a university repository's page carries it verbatim, the paper itself not fetched; hardness, the multiplicity at Unreachability and the search-theoretic item at Approaches: Aiyagari and Wallace, "Fiat money in the Kiyotaki-Wright model", Economic Theory 2:447-464, 1992, at its abstract as the publisher's page carries it verbatim, and Kehoe, Kiyotaki and Wright, "More on money as a medium of exchange", Economic Theory 3(2):297-314, 1993, at its abstract as a repository's page carries it verbatim, neither paper fetched. Persistence, no landing: the closing condition at the founding paper's abstract read at pass one and at the hardness (fiat money an equilibrium among several in the search model; acceptance self-fulfilling), a selection among the equilibria or a determinant of onset standing there as the model's own multiplicity. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Backing theories — account for commodity and convertible regimes, fiat regimes persisting without convertibility, which is the case in question. Search-theoretic models — prove multiplicity, valued and non-valued equilibria coexisting for the same fundamentals, a proved result that does not select. Legal tender and tax-driven demand — accounts for acceptance by obligation, its critics noting money circulates beyond tax obligations and across borders. Hyperinflation episodes — establish that acceptance can fail, onset timing not predicted by fundamentals in the episodes studied. Cryptocurrency — demonstrates acceptance without backing or legal tender for a period, read as supporting expectations by one account and as unresolved by the other. **Narrowing recorded.** Backing theories narrow the reach: the account stands for commodity and convertible regimes. Tax-driven demand narrows the reach: acceptance stands accounted for within the jurisdiction imposing the obligation. **Progress.** The historical record of backing and issuance against loss of value. The proved equilibrium multiplicity. The measured hyperinflation episodes and their unpredicted onsets. **Own record.** ## 141 Principal and agent **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** One party acts for another whose interests differ and whose monitoring is imperfect. Can contracts align them? In the field's own posing, the interests of workers and their employers are not always aligned, and a large theoretical literature asks how firms design compensation contracts to induce employees to act in the firm's interest, with two questions put to the evidence: whether individuals respond to contracts that reward performance, and whether those responses are always in the firm's interest; and whether firms write contracts with those responses in mind. **The field's account of its hardness.** The field's survey draws three conclusions: considerable evidence that individuals respond to pay for performance beside less evidence that contracts are designed as the theory predicts; little progress in distinguishing among plausible theories; and little known about how incentives are provided to workers whose output is difficult to measure. **Accounts.** Yes, approximately: optimal contracts under stated informational conditions are characterised, incentive pay responds to the theory's predictions, and the design problem is solved where the conditions hold. The conditions rarely hold: with multiple tasks, incentives distort toward the measured one; with team production, individual contribution is unidentifiable; and empirical returns to incentive pay are mixed with documented adverse effects. Alignment is partial by construction. **Specifications.** The agent's action affects an observable outcome — local. A contract on observables can approximate one on the action — local. **Specifications withdrawn.** **Common requirement.** That the divergence is representable as a contracting problem. **Conserving relation.** A contract aligning the parties, the alignment standing on observables where the action is unobserved. **Conserving reach.** All settings the contracting problem is posed in. **Reach.** The statement quantifies over the agent's unobserved action. Outcomes and observables, from which action is inferred under assumptions, are what is measured. The two are not the same. **Unreachability.** Incentives are proved to distort toward measured dimensions where tasks are multiple, and individual contribution is proved unidentifiable in team production without further structure. The alignment is not reached over the settings by contracts on observables. **Persistence.** A contract aligning the parties where tasks are multiple or production is joint, or agreement that alignment is partial by construction. **Addresses.** Statement, hardness, the difficult-to-measure output at Reach, and the mixed empirical returns at Approaches: Prendergast, "The Provision of Incentives in Firms", Journal of Economic Literature 37(1):7-63, 1999, at its abstract as the association's page carries it verbatim and at its opening paragraph as an index's page carries it, the paper itself not fetched. Persistence, no landing: the closing condition at the survey's abstract read at pass one and at the hardness (incentive provision where output is difficult to measure; multitask distortion; mixed empirical returns), a contract aligning the parties under multiple tasks standing there as what the theory proves distorted by construction. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Optimal contracting theory — solves the stated problem, the conditions restrictive and stated. Multitask analysis — proves distortion toward measured dimensions, a proved limitation. Team production — individual contribution unidentifiable without further structure, proved. Empirical incentive-pay studies — mixed effects with documented adverse cases, establishing the phenomenon is not uniform. Relational contracts and reputation — achieve alignment where repeated interaction is available, the conditions being the empirical question. **Narrowing recorded.** Optimal contracting narrows the reach: the solution stands under the stated informational conditions. Relational contracts narrow the reach: alignment stands where repeated interaction is available. **Progress.** The characterised optimal contracts under stated conditions. The two proved limitations. The measured mixed returns to incentive pay. **Own record.** ## 142 The paradox of thrift **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Saving benefits an individual. If everyone saves more, income falls and saving may not rise. Which holds? In the field's own posing, a model of debt-driven slumps, in which an overhang of debt forces some agents into rapid deleveraging and depresses aggregate demand, has the paradox of thrift emerge beside debt deflation, a liquidity trap, a multiplier and a rationale for expansionary fiscal policy; and the paradox emerges where a shock large enough to push the economy against the zero lower bound has turned the usual rules of macroeconomics upside down. **The field's account of its hardness.** The field's model states the paradox as familiar but long neglected, emerging immediately once a shock large enough to reach the zero lower bound has stood the usual rules of macroeconomics on their head, with other perverse results beside it, among them that increasing potential output may reduce actual output; and it states that making some agents debt-constrained is a surprisingly powerful assumption, the paradox being one of several results that emerge from it naturally. **Accounts.** The aggregate effect is real: in demand-constrained conditions, higher desired saving lowers output and can leave realised saving unchanged, the mechanism being standard and the policy implication drawn from it. It holds only under conditions: with flexible prices, accommodating monetary policy, or an open economy, higher saving finances investment and raises capacity, and the result is a special case of a demand-constrained regime rather than a general one. **Specifications.** Individual and aggregate relations can differ — universal under macroeconomics. The operative regime is identifiable — local. **Specifications withdrawn.** **Common requirement.** That the regime determining which relation holds is identifiable in advance. **Conserving relation.** The operative regime identified in advance, the identification standing on aggregate series. **Conserving reach.** All periods the question is asked of. **Reach.** The statement quantifies over which regime obtains. Aggregate series in which regime and outcome are jointly determined are what is measured. The two are not the same. **Unreachability.** Both results are derivable in their own models, which both accounts accept, and multiplier estimates vary widely by method and by regime, which the literature documents. The identification is not reached over the periods by series in which regime and outcome are jointly determined. **Persistence.** A regime identified independently of the outcome, or agreement on an estimation method for the multiplier. **Addresses.** Statement, hardness, the zero-lower-bound episodes at Approaches and the microfoundation with nominal rigidity: Eggertsson and Krugman, "Debt, Deleveraging, and the Liquidity Trap: A Fisher-Minsky-Koo Approach", Quarterly Journal of Economics 127(3):1469-1513, 2012, at its abstract as the publisher's page carries it verbatim and at its introduction as the published text carries it, its section VI on the paradoxes of thrift, toil and flexibility not read. Persistence, no landing: the closing condition at the paper the address names, at its abstract read at pass one and at the hardness (the liquidity-trap regime in which deleveraging and saving depress output, microfounded with nominal rigidity), a regime identified independently of the outcome standing there as what the two models' comparison asks. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Model comparison — both results derivable in their own models, accepted by both accounts. Zero lower bound episodes — providing conditions where the demand-constrained mechanism is most likely, empirical work supporting it there and contested on identification. Cross-country saving-growth correlations — positive on average, negative in some periods, both readings available. Fiscal multiplier estimation — central to the dispute, estimates varying widely by method and by regime, which the literature documents. Microfoundation — reproduces the result in models with nominal rigidity, which is the disputed assumption. **Narrowing recorded.** Zero lower bound analysis narrows the reach: the mechanism stands supported in the episodes where the constraint binds. **Progress.** The derivations in both model classes. The empirical work at the zero lower bound. The measured spread of multiplier estimates by method and regime. **Own record.** ## 143 The evolution of cooperation **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Natural selection favours what raises an individual's reproduction. Cooperation is widespread. How? In the field's own posing, cooperation means that selfish replicators give up some of their reproductive potential to help one another, selection implies competition and so works against cooperation unless a specific mechanism is at work, and five mechanisms are stated, kin selection, direct reciprocity, indirect reciprocity, network reciprocity and group selection, each with a simple rule for when selection can lead to cooperation. **The field's account of its hardness.** The field's review states that cooperation is needed for evolution to construct new levels of organisation, genomes, cells, multicellular organisms, social insects and human society all resting on it, while natural selection implies competition and therefore opposes it unless a specific mechanism is at work, so that a rule is derived for each mechanism specifying whether selection can lead to cooperation. **Accounts.** By identified mechanisms: kin selection, direct and indirect reciprocity, and population structure each produce cooperation under stated conditions, and the threshold is quantified in terms measurable in the field. The mechanisms make differential predictions tested across taxa — relatedness in eusocial insects, reputation in humans, spatial structure in microbes — each confirmed in its own domain. The accounting is disputed at its foundation: inclusive fitness and multilevel selection give the same predictions under general conditions, which is proved, so they are two accountings of one process rather than rival hypotheses, and the long dispute is about the accounting rather than the biology. **Specifications.** Fitness effects are attributable to individuals — local. Cooperation is a departure requiring explanation — local. **Specifications withdrawn.** **Common requirement.** That the fitness accounting is well defined. **Conserving relation.** One accounting identified as describing the process, the identification standing on reproductive outcomes. **Conserving reach.** All cases the accounting is applied to. **Reach.** The statement quantifies over which accounting describes the process. Reproductive outcomes, decomposable either way, are what is measured. The two are not the same. **Unreachability.** The two accountings are proved to give the same predictions under general conditions, so no reproductive outcome separates them. The identification is not reached over the cases by outcomes decomposable either way. **Persistence.** Conditions under which the two accountings predict differently, or agreement that the rivalry is between accountings rather than hypotheses. **Addresses.** Statement, hardness, and the identified mechanisms at Accounts: Nowak, "Five Rules for the Evolution of Cooperation", Science 314(5805):1560-1563, 2006, at its abstract as the journal's page carries it verbatim, the paper itself not fetched. Owed: the eusociality exchange's own statement of the accounting dispute and the formal equivalence results, cited at Accounts and Approaches, not read this sitting. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (five mechanisms for cooperation, each with its condition; the kin-selection and multilevel accountings translating into each other), conditions under which the accountings predict differently standing there as what the equivalence results leave. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Kin selection tests — relatedness predicting cooperation across many taxa, robustly, establishing the pattern under either accounting. Experimental evolution — produces cooperation and cheating dynamics as predicted, under either accounting. Formal equivalence results — the two accountings agreeing under general conditions, the strongest item, dissolving the rivalry rather than settling it. Public goods experiments in humans — establish conditional cooperation and punishment, mechanism-level findings. The eusociality dispute — a large public disagreement with a rebuttal signed by many, the mathematical points not resolved by the exchange. **Narrowing recorded.** Mechanism-level tests narrow the reach: each mechanism stands confirmed in its own domain rather than across the cases. **Progress.** The measured relatedness-cooperation association across taxa. The quantified threshold. The formal equivalence result. The conditional cooperation and punishment findings. **Own record.** ## 144 Transfer of learning **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Something learned in one setting is applied in another. When does it transfer? In the field's own posing, the question is whether far transfer occurs, and the arguments around it have made little progress toward resolution across a century of research, which the field's taxonomy attributes to a failure to specify the dimensions along which transfer can occur, so that unlike things are compared; nine dimensions are stated, at whose intersections each study sits. **The field's account of its hardness.** The field's taxonomy states that despite a century of research the arguments over whether far transfer occurs have made little progress toward resolution, that the reason is a failure to specify the dimensions along which transfer can occur, so that studies compare unlike things, that estimating a single effect size for far transfer is misguided given that complexity, and that a hundred years of research show substantial evidence for transfer under some conditions while the critical conditions for many key questions stand untested. **Accounts.** When the settings share elements: near transfer is reliable, measured across a century of studies and predicted well by identical-elements accounts, instruction making shared structure explicit improves it, worked-example and contrasting-case methods produce measurable gains, and expertise transfers readily within a domain. Far transfer is rarely demonstrated: cognitive training studies show gains on trained tasks and little on untrained ones, the record across brain training, chess, music, and general reasoning shows near transfer and repeated failure of far transfer, and positive results often fail replication. **Specifications.** Transfer is a property of the relation between two tasks — local. Task similarity is measurable independently of transfer — local. **Specifications withdrawn.** **Common requirement.** That similarity is specifiable before the transfer is measured. **Conserving relation.** Similarity specified before transfer is measured, the specification standing for tasks not yet chosen. **Conserving reach.** All task pairs, those not yet specified included. **Reach.** The statement quantifies over application to a task not yet specified. Pairs of tasks selected by the experimenter, whose similarity is judged, are what is measured. The two are not the same. **Unreachability.** Similarity is characterised after the tasks are chosen, and the far-transfer null record is now the field's own summary while expertise is established as domain-bound. The specification is not reached over unchosen pairs by pairs the experimenter selected. **Persistence.** A measure of task similarity specified before transfer is measured, or agreement that far transfer is not to be expected. **Addresses.** Statement, hardness, the experimenter-selected task pairs at Reach, and the specification of similarity at Persistence: Barnett and Ceci, "When and where do we apply what we learn? A taxonomy for far transfer", Psychological Bulletin 128(4):612-637, 2002, at its abstract as the published text's first page and an index's page carry it verbatim, the paper itself not fetched. Approaches: the identical-elements and training-trial routes at the taxonomy paper read at pass one and the sources the address names for the trials. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Identical-elements analysis — predicts near transfer well, similarity characterised after the tasks are chosen. Cognitive training trials — large, controlled, and consistently null for far transfer, now the field's own summary position. Expertise studies — expert advantage domain-bound, established across domains. Analogical transfer studies — transfer with hints and little without, establishing the conditions and their fragility. Metacognitive instruction — modest general gains in some studies, effect sizes small and replication partial. **Narrowing recorded.** Identical-elements analysis narrows the reach: prediction stands for near pairs sharing characterised elements. Analogical transfer narrows the reach: transfer stands where hints are supplied. **Progress.** The century-long near-transfer record. The consistently null far-transfer trials. The established domain-boundedness of expertise. The measured effect of hints on analogical transfer. **Own record.** ## 145 Bernstein's degrees of freedom **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A limb has many more degrees of freedom than a movement requires. How is one solution selected? In the field's own posing, the central problem of motor control was formulated as the problem of motor redundancy, the elimination of redundant degrees of freedom, traditionally addressed by optimisation on a variety of cost functions; and the field's review states that recent findings suggest the problem was formulated incorrectly, an alternative principle of abundance treating the apparently redundant degrees of freedom as useful and even vital for motor behaviour. **The field's account of its hardness.** The field's review states that the problem of redundancy, as formulated, has traditionally been addressed by optimisation methods resting on a variety of cost functions, that a body of recent findings suggests the problem was formulated incorrectly, and that dozens of publications support an alternative principle of abundance on the ideas of synergic control, the computational apparatus of the uncontrolled manifold hypothesis and the equilibrium-point hypothesis, with large amounts of variance that has no effect on performance documented across natural actions. **Accounts.** By optimisation: movements minimise a cost — jerk, torque change, endpoint variance — and optimal control models reproduce observed trajectories closely across tasks. The cost is inferred from the movements it accounts for: different costs fit the same trajectories, synergies reduce dimensionality without an optimiser, and observed variability is structured rather than minimised, which the uncontrolled manifold analyses show directly. **Specifications.** A unique solution is selected — local. Selection is by minimisation of a cost — local. **Specifications withdrawn.** **Common requirement.** That the selection has a determinant. **Conserving relation.** A selection rule identified, the identification standing on a cost specified independently of the movements. **Conserving reach.** All movements the rule is asked of. **Reach.** The statement quantifies over the selection rule. Recorded kinematics and muscle activity, fit by several models, are what is measured. The two are not the same. **Unreachability.** Multiple cost functions fit the same trajectories comparably, which is documented, and inverse optimal control has known identifiability limits. The identification is not reached over the movements by kinematics and muscle activity. **Persistence.** A cost specified independently of the movements it accounts for, or a measurement distinguishing the candidate costs. **Addresses.** Statement, hardness, and the optimal-control, synergy and uncontrolled-manifold items at Approaches: Latash, "The bliss (not the problem) of motor abundance (not redundancy)", Experimental Brain Research 217(1):1-5, 2012, at its abstract as the publisher's page carries it verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (abundance rather than redundancy; multiple cost functions fitting the trajectories comparably), a cost specified independently of the movements standing there as what the fits do not select. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Optimal control modelling — fits trajectories well, multiple cost functions fitting comparably, which is documented. Inverse optimal control — recovers a cost from behaviour, identifiability of the cost limited and known to be so. Muscle synergy analysis — finds low-dimensional structure, whether synergies are neural or an artefact of the decomposition disputed, the dispute technical and live. Uncontrolled manifold analysis — variability structured to preserve task-relevant variables, establishing structure without an optimiser. Perturbation studies — rapid task-relevant corrections, supporting feedback control accounts. **Narrowing recorded.** Synergy analysis narrows the reach: low-dimensional structure stands recovered under a stated decomposition. Uncontrolled manifold analysis narrows the reach: structure stands established for the task-relevant variables specified. **Progress.** The trajectory fits under several cost functions. The stated identifiability limits. The measured low-dimensional structure. The measured task-relevant variability and rapid corrections. **Own record.** ## 146 The Whorf question **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does the language a person speaks shape how they think? In the field's own posing, the central question of research on linguistic relativity is whether people who speak different languages think differently, with seven categories of hypotheses about the possible effects of language on thought across motion, colour, spatial relations, number and false-belief understanding. **The field's account of its hardness.** The field's review states that it finds no support for language determining the basic categories of thought or overwriting pre-existing conceptual distinctions, that it finds support for language making some distinctions difficult to avoid and augmenting certain types of thinking, that recent evidence suggests language may induce a relatively schematic mode of thinking, and that the literature remains contentious while support grows for the view that language has a profound effect on thought. **Accounts.** Yes, measurably: colour categories affect discrimination speed with lateralised effects, grammatical gender affects object descriptions, and spatial frame languages produce different non-linguistic spatial reasoning, replicated across communities. The effects are real and narrow: they appear largely in tasks where language is available as a strategy and diminish under verbal interference, core cognition — number approximation, object tracking, basic spatial reasoning — appears language-independent, and the strong version has not been supported. **Specifications.** Thought and language are separable — local. Non-linguistic tasks measure thought without language — local. **Specifications withdrawn.** **Common requirement.** That a task can measure thought independently of language. **Conserving relation.** Thought measured independently of language use, the independence standing in the tasks employed. **Conserving reach.** All tasks the question is asked of. **Reach.** The statement quantifies over thought as distinct from language use. Task performance, in participants who have language available, is what is measured. The two are not the same. **Unreachability.** Verbal interference removes many effects, establishing they are partly strategic rather than absent, and the remaining effects are read oppositely; spatial frame differences are confounded by environment and practice, which the field notes. The independence is not reached over the tasks by performance in participants who have language available. **Persistence.** A task measuring thought with language unavailable, or agreement on what the interference results establish. **Addresses.** Statement, hardness, and the effects-real-and-narrow account at Accounts: Wolff and Holmes, "Linguistic relativity", WIREs Cognitive Science 2(3):253-265, 2011, at its abstract as the index's page carries it verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (the effects real and narrow; verbal interference reducing many of them), a task measuring thought with language unavailable standing there as the review's own design question. Approaches: the interference and colour-category routes at that review's sections, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Verbal interference designs — reduce or remove many effects, the strongest route, establishing the effects are partly strategic rather than absent. Colour category studies — lateralised effects replicated, interpreted as language-driven categorical perception or as label-based strategy. Spatial frame studies — differences in non-linguistic tasks across communities, robust, confounded by environment and practice, which the field notes. Infant and prelinguistic studies — establish capacities before language, bearing on the strong version only. Bilingual switching studies — within-person shifts by language of task, supporting an online rather than a structural effect. **Narrowing recorded.** Verbal interference narrows the reach: the residual effects stand for tasks where language is unavailable as a strategy. Infant studies narrow the reach: the capacities stand established before language. **Progress.** The replicated lateralised colour effects. The measured reduction under verbal interference. The cross-community spatial differences. The prelinguistic capacities. The within-person bilingual shifts. **Own record.** ## 147 Grandmother cell and distributed coding **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is a concept represented by a small number of highly selective neurons, or by a pattern across many? In the field's own posing, medial temporal lobe neurons respond selectively to complex visual stimuli, the representation being very sparse and abstract, so that the cells could in principle be considered grandmother cells, while the recording group gives several arguments that make such an extreme interpretation unlikely. **The field's account of its hardness.** The recording group's review states that although many neuropsychological and imaging studies show the medial temporal lobe's role in memory there are few data on the activity of the neurons involved, that a decade of evidence shows these neurons responding selectively to complex visual stimuli, and that given the very sparse and abstract representation they could in principle be taken for grandmother cells, an extreme interpretation the review argues against. **Accounts.** Sparse and selective: recordings in human medial temporal lobe find single neurons responding to one individual across photographs taken at different angles, line drawings, and the written and spoken name — invariance across format that no low-level feature accounts for. Firing rates from a handful of such cells suffice to decode which of many individuals was seen, and the same cells respond to associated concepts. Distributed: the recorded cells are the tail of a broad distribution, recording is biased toward responsive units, and population decoding from many weakly tuned neurons outperforms sparse readout in most systems studied. **Specifications.** The representation has a determinate sparseness — local. Recording samples the population without bias — local. **Specifications withdrawn.** **Common requirement.** That the sampled neurons represent the population. **Conserving relation.** Sparseness determined across a population, the determination standing on an unbiased sample. **Conserving reach.** The whole population the concept is represented in. **Reach.** The statement quantifies over sparseness across a population, from a screened sample. Neurons selected by responsiveness during a search, in patients with clinical electrode placement, are what is measured. The two are not the same. **Unreachability.** Human recordings and animal population recordings give opposite pictures with the sampling procedures differing; and both codes are proved efficient under different constraints, which does not select. The determination is not reached over the population by a screened sample. **Persistence.** An unbiased sample of the population in the areas where the selective cells are found. **Addresses.** Statement, hardness, and the sparse-and-selective account at Accounts: Quian Quiroga, Kreiman, Koch and Fried, "Sparse but not 'grandmother-cell' coding in the medial temporal lobe", Trends in Cognitive Sciences 12(3):87-91, 2008, at its abstract as the publisher's and the index's pages carry it verbatim, the paper itself not fetched; the exchange over the biological plausibility of grandmother cells in Psychological Review, 2010, as an index lists it, not read. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (sparse but not grandmother-cell coding; the sampling driven by a screening procedure the authors state), an unbiased sample of the population standing there as what the authors' own procedure does not give. Approaches: the single-unit and population-recording routes at that paper and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Single-unit recording in humans — finds highly selective cells reliably, sampling driven by a screening procedure, which the authors state. Large-scale population recording — finds broad, weak tuning dominating in animal cortex, establishing the distributed picture in accessible areas. Decoding comparisons — population decoding generally outperforming sparse, depending on the recorded set. Sparseness estimation from unbiased sampling — calcium imaging giving less biased samples and finding sparse but not single-cell coding. Theoretical analysis — both codes efficient under different constraints, proved, and not selecting. **Narrowing recorded.** Calcium imaging narrows the reach: sparseness stands estimated from less biased samples in the areas imaged. Population recording narrows the reach: the distributed picture stands established in the accessible areas. **Progress.** The invariant selective responses across format. The measured broad tuning in animal cortex. The decoding comparisons. The sparseness estimates from less biased sampling. **Own record.** ## 148 The faint young sun **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Stellar models say the early sun was substantially dimmer. Geological evidence says liquid water was present. How? In the field's own posing, models of stellar evolution predict a solar energy input to the early Earth's climate about a quarter lower than today, which would have frozen the planet completely over its first two billion years had everything else stayed the same, while there is ample evidence for liquid surface water and even life in the Archean, so some effect or effects must have been compensating. **The field's account of its hardness.** The field's review states that a wide range of possible solutions has been suggested and explored over four decades, most focusing on higher concentrations of atmospheric greenhouse gases such as carbon dioxide, methane or ammonia, that all of them present considerable difficulties, so that the problem cannot be regarded as solved, and that the way forward lies in improved geochemical constraints on the state of the Archean climate system together with numerical experiments. **Accounts.** Greenhouse compensation: higher concentrations of carbon dioxide, methane, or both offset the reduced luminosity, and the carbonate-silicate weathering cycle supplies a feedback that would have maintained whatever concentration was needed, drawing carbon dioxide down as the sun brightened. The mechanism is quantitative, operates on the right timescale, and is the same feedback regulating climate over geological time today. The proposed concentrations conflict with proxies: paleosol and other constraints bound early carbon dioxide below several proposed values, methane is limited by photochemistry in an anoxic atmosphere, and the remaining candidates — reduced albedo, different cloud behaviour, higher nitrogen pressure — each carry their own difficulties. **Specifications.** Stellar evolution models are correct for the early sun — universal under stellar astrophysics. Proxies constrain early atmospheric composition — local. **Specifications withdrawn.** **Common requirement.** That the early state is reconstructible from present evidence. **Conserving relation.** Early composition and albedo reconstructed, the reconstruction standing across the proxy families. **Conserving reach.** The early state, which occurred once. **Reach.** The statement quantifies over early composition and albedo, reconstructed from proxies calibrated by models. Proxies with wide uncertainty and model-dependent calibration are what is measured. The two are not the same. **Unreachability.** Proxy bounds and greenhouse requirements conflict, and the proxy families are contested in calibration against each other, with the newer aqueous-chemistry technique returning bounds raised by an order against the classic studies. The reconstruction is not preserved across the proxy families. **Persistence.** Proxy families agreeing on early composition, or a constraint independent of model-dependent calibration. **Addresses.** Statement, hardness, the geochemical constraints at Reach, and the review's own direction at Persistence: Feulner, "The faint young Sun problem", Reviews of Geophysics 50:RG2006, 2012, at its abstract as the publisher's page carries it verbatim, the paper itself not fetched. Approaches: the greenhouse and albedo routes at the review's abstract read at pass one and its sections the address names, the proxy bounds there. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Carbon dioxide greenhouse — requires concentrations the classic proxy studies bounded below, the newer aqueous-chemistry paleosol technique returning bounds raised by an order and reopening the route, the proxy families remaining contested in calibration against each other. Methane greenhouse — effective in anoxic conditions, photochemical haze formation limiting it and able to cool by shading, modelled both ways. Reduced albedo — less land, fewer cloud nuclei, plausible and hard to constrain from evidence. Higher nitrogen pressure — broadens absorption lines, the proxy estimates having converged low, with vesicle, imprint, and isotope instruments independently bounding early pressure at or below modern, closing the thick-atmosphere version of this route. Revising the solar model — a more massive early sun losing mass, constrained by other observations and not favoured. **Narrowing recorded.** The nitrogen route stands closed in its thick-atmosphere version by three independent instruments converging low. The aqueous-chemistry technique narrows the reach: the raised bound stands under that technique's calibration. **Progress.** The quantified weathering feedback. The classic and revised carbon dioxide bounds. The photochemical limits on methane. The convergent low bounds on early nitrogen pressure. **Own record.** ## 149 The origin of the genetic code **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** The mapping from codons to amino acids is nearly universal. Why this mapping? In the field's own posing, the code is nearly universal and the arrangement of the codons in the standard table is highly non-random, and three main concepts stand on its origin and evolution, the stereochemical theory, the coevolution theory and the error-minimisation theory, not mutually exclusive and each compatible with the frozen-accident hypothesis. **The field's account of its hardness.** The field's review names the problem a universal enigma and states that mathematical analysis of the code's structure and possible evolutionary trajectories shows it highly robust to translational error while a huge number of more robust codes exist, so that the standard code could have evolved from a random code through a short sequence of codon-series reassignments; much of that evolution reads as frozen accident combined with selection for error minimisation, with contributions from coevolution with metabolic pathways or from weak affinities between amino acids and triplets not excluded. **Accounts.** Selection for error tolerance: the assignment places chemically similar amino acids at codons differing by one base, so point mutations and mistranslation produce conservative substitutions. Measured against millions of randomly generated alternative codes, the natural code ranks in the extreme tail on every error metric tried, and the effect is strongest at the codon positions where errors are most frequent. Optimality does not establish origin: stereochemical affinities between codons and amino acids are reported for several pairs, coevolution with biosynthetic pathways accounts for the block structure, the error-minimisation could be a consequence of an origin fixed for other reasons, and the accounts are not separable by the code's structure alone. **Specifications.** The mapping was fixed by a process — local. The code's present structure carries evidence of it — local. **Specifications withdrawn.** **Common requirement.** That present structure constrains the origin. **Conserving relation.** The origin of the mapping identified, the identification standing on the code's present structure. **Conserving reach.** The fixing of the mapping, which occurred once. **Reach.** The statement quantifies over the origin of the mapping, constrained by its present structure. The code's structure, extant variants, and laboratory affinity measurements are what is measured. The two are not the same. **Unreachability.** Error-minimisation, stereochemical, and coevolutionary accounts each fit the code's structure, which is consistent with several origins, and the accounts are not separable by that structure alone. The identification is not reached over the fixing by present structure. **Persistence.** A feature of the code that one account predicts and the others do not, or a constraint on the origin independent of present structure. **Addresses.** Statement, hardness, and the three accounts at Accounts and Approaches: Koonin and Novozhilov, "Origin and evolution of the genetic code: the universal enigma", IUBMB Life 61(2):99-111, 2009, at its abstract as the archive's page carries it verbatim, the paper itself not fetched. Persistence, no landing: the closing condition at the review's abstract read at pass one and at the hardness (the universal enigma; the three accounts, stereochemical, coevolutionary and error-minimizing, each consistent with the code's structure), a feature one account predicts and the others do not standing there as the review's own missing discriminator. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Error-minimisation statistics — the code ranking extremely high among random alternatives, robustly measured, consistent with several origins. Stereochemical affinity experiments — aptamer selection finding codon-related affinities for several amino acids, contested in statistical treatment, covering part of the code. Coevolution accounts — the block structure accounted for by biosynthetic relatedness, with good fit, also consistent with other origins. Code variants — natural and engineered variants showing the code is changeable, establishing it is not frozen absolutely. Ancestral reconstruction — infers earlier smaller codes, model-dependent and not decisive between accounts. **Narrowing recorded.** Stereochemical experiments narrow the reach: affinities stand measured for the pairs tested. Coevolution accounts narrow the reach: the block structure stands accounted for rather than the whole mapping. **Progress.** The measured ranking against random alternatives on every metric tried. The measured codon-related affinities. The block-structure fit. The natural and engineered variants. **Own record.** ## 150 The origin of language **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Language appears in one lineage. Was it a gradual extension of prior communication or a discrete change? In one group's own posing, understanding the evolution of language requires evidence about origins and about the processes that led to change, an explosion of research over four decades has brought a sense of considerable progress, and the most fundamental questions about the origins and evolution of the linguistic capacity remain as mysterious as ever. **The field's account of its hardness.** One group's review states that studies of nonhuman animals provide virtually no relevant parallels to human linguistic communication and none to the underlying biological capacity, that the fossil and archaeological evidence does not inform the computations and representations of the earliest ancestors, leaving origins and selective pressure unresolved, that the genetics of language is so impoverished that connecting genes to linguistic processes is not in prospect, and that there is considerable uncertainty about finding evidence able to adjudicate among the many open hypotheses. **Accounts.** Gradual: precursors are present in primate communication, vocal learning, and gesture, anatomical and archaeological indicators extend back hundreds of thousands of years, and no evidence supports a single event. Discrete in at least one respect: recursive combination has no demonstrated animal analogue, accounts of its gradual emergence do not specify intermediate states with a function, and the discontinuity is in the record and in the comparative data alike. **Specifications.** Language decomposes into components with independent evolutionary histories, or it is a unit — local. The capacity's components are identifiable — local. **Specifications withdrawn.** **Common requirement.** That the capacity's components are identifiable. **Conserving relation.** The emergence of syntactic capacity determined as gradual or discrete, the determination standing on anatomy, artefact, and comparative data. **Conserving reach.** The emergence, which occurred once. **Reach.** The statement quantifies over syntactic capacity, indexed by anatomy and artefact. Comparative animal data, anatomy, ancient DNA, and material culture are what is measured. The two are not the same. **Unreachability.** Precursors are established for every component but recursion, anatomical indicators are contested in inferential value, and archaeological proxies bear on symbolic capacity rather than syntax. The determination is not reached over the emergence by comparative data, anatomy, and material culture. **Persistence.** An index of syntactic capacity in the record, or an animal analogue of recursive combination. **Addresses.** Statement, hardness, and the discrete account at Accounts: Hauser, Yang, Berwick, Tattersall, Ryan, Watumull, Chomsky and Lewontin, "The mystery of language evolution", Frontiers in Psychology 5:401, 2014, at its abstract as the index's and the archive's pages carry it verbatim, the paper itself not fetched. Owed: the gradualist account's own posing at its source, not read this sitting. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the mystery of language evolution, the evidence for the emergence of syntax thin in the record and absent in animal communication), an index of syntactic capacity in the record or an animal analogue of recursive combination standing there as the paper's own two absences. Approaches: the comparative and archaeological routes at that paper's sections, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Comparative studies — establish precursors for many components, vocal learning, intention reading, combinatorial signalling, and none for recursion. Anatomical indicators — hyoid and hypoglossal canal evidence, contested in inferential value. Ancient DNA — candidate genes identified, one gene's early story substantially revised by later data, a documented reversal. Archaeological proxies — symbolic artefacts as indirect evidence, bearing on symbolic capacity rather than syntax. Computational modelling of emergence — produces combinatorial systems from communication pressures in simulation, establishing sufficiency in models. **Narrowing recorded.** Comparative studies narrow the reach: precursors stand established component by component rather than for the capacity. Computational modelling narrows the reach: sufficiency stands established in simulation. **Progress.** The established precursors for each component but recursion. The anatomical indicators with their stated inferential limits. The documented revision of the candidate-gene account. The combinatorial systems produced in simulation. **Own record.** ## 151 The Poincaré conjecture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is every simply connected closed three-manifold a three-sphere? In the field's own posing, a rubber band stretched around an apple can be shrunk to a point without tearing it or leaving the surface, while one stretched the appropriate way around a doughnut cannot, and the conjecture asks whether the three-dimensional sphere is characterised the same way among closed three-dimensional spaces. **The field's account of its hardness.** The institute that posed the problem states that the question turned out to be extraordinarily difficult, that nearly a century passed between its formulation and its solution, that many attempts were made in that century to prove and also to disprove it, and that the solution rested on the theory of Ricci flow and on results on spaces of metrics, proving with it the geometrisation conjecture of which it is a special case. **Accounts.** Every such manifold is a three-sphere. **Specifications.** Three-manifolds admit a geometric decomposition — universal under geometrisation. A flow on the metric can be continued through singularities by surgery — local. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** A proof holding for every simply connected closed three-manifold. **Conserving reach.** All simply connected closed three-manifolds. **Reach.** The statement quantifies over all simply connected closed three-manifolds. The flow, its singularities, and the surgery construction are what is worked with. The two are the same: the method reaches the domain the statement quantifies over. **Unreachability.** **Persistence.** Settled. The proof stands, verified by several independent groups over years, and is accepted without dissent. **Addresses.** Statement, hardness, the account, the flow and surgery at Reach, and the verification at Persistence: the Clay Mathematics Institute's problem page, "Poincare Conjecture", and its press release on the award of the Millennium Prize for the resolution, 2010, at the passages the institute's own pages carry verbatim, the expositions by several teams and the multi-year scrutiny named there as the verification. Approaches: the geometrisation and Ricci-flow routes at the Institute's problem page read at pass one, the flow and surgery keyed there at Reach. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The geometrisation programme — frames the conjecture within a decomposition of three-manifolds. Ricci flow — founds the method. The surgery construction — completes it, continuing the flow through singularities. Verification — carried out by several independent groups over the following years. **Narrowing recorded.** **Progress.** The proof. The geometrisation theorem it establishes. The verification record. **Own record.** The difficulty stood in constructing the proof, not in the statement quantifying over a domain the methods do not reach. ## 152 The block universe and the low-entropy past **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why the entropy of the universe was low in the past — the past hypothesis, the boundary condition the thermodynamic asymmetry is made to rest on — and whether the distinction between past and future is a feature of the world or of our position in it. **The field's account of its hardness.** That a problem distinct from the thermodynamic one goes by the name of the problem of the direction of time, the grounding of the non-thermodynamic temporal asymmetries, in which the thermodynamic arrow is taken as given and used to explain other temporally asymmetric features of the world such as causation and knowledge, and the clearer question is whether any or all of those asymmetries depend for their existence upon the thermodynamic asymmetry; and that the asymmetry of traces has been hinted to be linked to the thermodynamic arrow with no further explanation offered, entropic accounts of the knowledge asymmetry developed, the direction of causation and the psychological arrow tied by various hands to the entropy gradient, and the past hypothesis argued to account for the knowledge asymmetry and the asymmetry of counterfactual dependence. **Accounts.** Of our position: the laws are symmetric, the four-dimensional description is complete, and the asymmetry we experience follows from the entropy gradient given a low-entropy boundary condition. The boundary condition is not derived, and the account accounts for the arrow by stipulating it: attempts to derive the past hypothesis have not succeeded, and accounts positing a dynamically distinguished present remain consistent with the physics. **Specifications.** Temporal asymmetry is either in the laws, in a boundary condition, or in the observer's position — local. These are exhaustive — local. **Specifications withdrawn.** **Common requirement.** That the asymmetry has a locus. **Conserving relation.** The locus of the asymmetry determined, the determination standing without the boundary condition stipulated. **Conserving reach.** The status of the boundary condition. **Reach.** The statement quantifies over the status of the boundary condition. The consequences of whatever condition obtained are what is measured. The two are not the same. **Unreachability.** The statistical derivation is complete given the past hypothesis, and each cosmological proposal contains a stipulation at the same place. The determination is not reached over the boundary condition by its consequences. **Persistence.** A derivation of the boundary condition not stipulating it, or agreement that a stipulation is acceptable. **Addresses.** Statement: Callender, "Thermodynamic Asymmetry in Time", Stanford Encyclopedia of Philosophy. The past hypothesis named: Albert, Time and Chance, Harvard, 2000. Hardness: the same entry, at passages of its text as the encyclopedia's page, a mirror and its archived editions carry them verbatim (a completely different problem going by the name of the problem of the direction of time, the grounding of the non-thermodynamic temporal asymmetries, the thermodynamic arrow taken as given to explain causation and knowledge; the clearer question whether any or all of those asymmetries depend for their existence upon the thermodynamic asymmetry; the asymmetry of traces hinted linked with no further explanation; entropic accounts of the knowledge asymmetry; the direction of causation and the psychological arrow tied to the entropy gradient; the past hypothesis argued to account for the knowledge asymmetry and the asymmetry of counterfactual dependence, as the entry carries the second source), the entry itself not fetched, its section numbers and revision date not carried at the returns; the second source not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the past hypothesis a boundary condition, its explanation or its acceptance as a posit the dispute), a derivation not stipulating it or a stipulation accepted standing there as the entry's two directions. Approaches: the statistical and cosmological routes at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Statistical derivation — complete given the past hypothesis. Cosmological derivation of the initial state — proposals existing, each containing a stipulation at the same place. Psychological accounts of temporal experience — the sense of passage accounted for by memory asymmetry, which itself follows from the entropy gradient, consistent and not deriving the condition. Dynamically distinguished present — available and unfalsifiable given relativistic equivalence. Two-time boundary conditions — consistent constructions, not favoured by evidence. **Narrowing recorded.** Psychological accounts narrow the reach: the sense of passage stands accounted for given the gradient. **Progress.** The statistical derivation given the hypothesis. The memory-asymmetry account of temporal experience. The consistent two-time constructions. **Own record.** ## 153 Riemann, Hodge, Birch–Swinnerton-Dyer, Yang–Mills **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Four conjectures, each posed by the field on its own. Riemann: every non-trivial zero of the zeta function has real part one half. Hodge: on a projective non-singular algebraic variety, every Hodge class is a rational linear combination of classes of algebraic cycles. Birch–Swinnerton-Dyer: the rank of the group of rational points of an elliptic curve equals the order of vanishing of its L-function at s = 1. Yang–Mills: a quantum Yang–Mills theory exists on four-dimensional space for any compact simple gauge group, and has a mass gap. **The field's account of its hardness.** That for each of the four the field's own short posing carries what the problem turns on: the prime number theorem determines the average distribution of the primes and the hypothesis tells of the deviation from the average; experiment and computer simulations suggest the existence of a mass gap in the quantum versions of the equations, but no proof of this property is known, existence to include axiomatic properties at least as strong as the field's standard axioms; the answer to the Hodge conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations; and the elliptic-curve conjecture, supported by much experimental evidence, asserts a simple way to tell whether the equations defining an elliptic curve have finitely or infinitely many rational solutions, a special case of a problem for which it has been proved that in the general case there is no way to decide whether an equation has any solutions at all. **Accounts.** In each, the conjecture as posed. One account stands in each of the four; no negation is defended. **Specifications.** The functional equation, the Hodge decomposition, the L-function's analytic continuation, and the gauge symmetry are each proved — universal under the respective field. The conjecture is not proved — local. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** In each, the conjecture holding over everything it quantifies over: every zero on the line, every Hodge class algebraic, the rank equal to the order of vanishing at every curve, a positive gap at every group. **Conserving reach.** All zeros; all classes; all curves; all states of the theory. **Reach.** The statements quantify over every non-trivial zero, every Hodge class on every such variety, every elliptic curve over the rationals, and every compact simple gauge group. Verified instances and partial results are what is measured. The two are not the same. **Unreachability.** **Persistence.** In each case, a proof of the conjecture, or a counterexample: a zero off the critical line, a Hodge class with no algebraic representative, a rank and an order of vanishing that differ, a gap shown absent. Each is answerable true or false. **Addresses.** Statement, approaches and reach, the four as posed with their partial results: the official problem descriptions in The Millennium Prize Problems, Clay Mathematics Institute and American Mathematical Society, 2006 — Wiles (Birch–Swinnerton-Dyer) p. 31, Deligne (Hodge) p. 45, Bombieri (Riemann) p. 107, Jaffe and Witten (Yang–Mills) p. 129. Hardness: filled at the Institute's own page as it carries short statements of all four (the prime number theorem and the deviation from the average; a mass gap suggested by experiment and simulation, no proof known; how much of the topology of a solution set can be defined by further algebraic equations; the elliptic-curve conjecture supported by much experimental evidence), at the official statement of the Yang-Mills problem as an encyclopedia page carries it (existence including axiomatic properties at least as strong as the standard axioms), and at a survey of open problems in mathematical physics at its passage on the elliptic-curve conjecture as the arXiv copy carries it (a simple way to tell finite from infinite rational solutions; a special case of the tenth problem, undecidable in general), arXiv:1710.02105, its author not carried; the four chapters' own statements of difficulty still keyed at the next pass. Before this sitting the property stood empty, the returns carrying the volume's contents as a repository copy carries them (the four chapters at the pages the address gives, verified) and the Institute's page with short statements of two of the four (the elliptic-curve conjecture relating the number of points mod p to the rank of the rational points, supported by much experimental evidence; the hypothesis telling of the deviation of the primes from the average the prime number theorem determines), and no statement of what is hard about any of the four; the official descriptions' own statements of difficulty at their chapters keyed at the next pass. Persistence, no landing: the closing condition at the Institute's statements of the four read at v342 (a proof or a counterexample in each, the elliptic-curve case a special case of an undecidable problem) and at the Institute's rules for a proposed solution, read at v342 (publication, two years, general acceptance). Remaining properties: keyed at the next pass. *The report.* **Approaches.** In each, the field's own record of verified instances and partial results, standing in that field's literature: the zeros verified on the critical line and the proportion proved to lie there; the Hodge conjecture proved for divisors and in low dimension; the Birch–Swinnerton-Dyer conjecture proved where the order of vanishing is zero or one; the mass gap seen in lattice computation and constructed in no continuum theory. Each open. **Narrowing recorded.** **Progress.** The proved symmetries. The verified instances and partial results in each case. **Own record.** ## 154 Origin and fate of the universe **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Did the universe have a beginning, and will it have an end? In the field's own posing at the beginning, many inflating spacetimes are likely to violate the weak energy condition on which the singularity theorems rest, and a kinematical argument requiring no energy condition shows that a cosmological model that is inflating, or merely expanding fast enough, must be incomplete in null and timelike past directions. **The field's account of its hardness.** The field's theorem states that the singularity theorems rest on an energy condition that inflating spacetimes are likely to violate, that the past-incompleteness result rests instead on a bound on the integral of the Hubble parameter along a past-directed geodesic, and that inflationary models therefore require physics other than inflation to describe the past boundary of the inflating region of spacetime. **Accounts.** A beginning is indicated: expansion extrapolated backward reaches a state of extreme density, singularity theorems show that under stated energy conditions geodesics are incomplete to the past, and inflationary spacetimes are past-incomplete by a proved argument. The theorems constrain classical extrapolation only: they assume energy conditions that quantum fields violate, the regime in question is one where classical general relativity is not expected to hold, and bouncing and cyclic models remain consistent with observation. **Specifications.** Extrapolation backward is valid to the regime in question — local. The energy conditions hold there — local. **Specifications withdrawn.** **Common requirement.** That the early state is constrained by present observation plus theory. **Conserving relation.** A determination of whether the universe had a beginning, standing at the epochs the question concerns. **Conserving reach.** Conditions at and before the earliest accessible epoch. **Reach.** The statement quantifies over conditions at and before the earliest accessible epoch. The microwave background and later are what is measured, with earlier epochs reached only through model-dependent inference. The two are not the same. **Unreachability.** The singularity theorems are proved under energy conditions quantum fields violate, which is established; and distinguishing signatures between the accounts are proposed and unobserved, while the dark energy equation of state is measured with improving precision and not to the accuracy that separates the fate scenarios. The determination is not reached over the epochs by observation of the accessible ones. **Persistence.** A distinguishing signature observed in primordial gravitational waves, or a dark energy equation of state measured to the accuracy separating the scenarios. **Addresses.** Statement, hardness, and the past-incompleteness argument at Accounts and Approaches: Borde, Guth and Vilenkin, "Inflationary spacetimes are incomplete in past directions", Physical Review Letters 90:151301, 2003, at its abstract as the publisher's page carries it verbatim, the paper itself not fetched. Owed: the fate half of the statement, the dark-energy equation of state and the scenarios at Persistence, at their own sources, not read this sitting. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Singularity theorems — proved under energy conditions, the conditions violated by quantum fields, which is established. Past-incompleteness arguments — apply to expanding spacetimes on average, their reach for quantum regimes disputed by the authors of alternatives. Observational cosmology — constrains the last thirteen billion years with precision, the earliest epochs inferred. Bouncing and cyclic models — consistent with data, distinguishing signatures in primordial gravitational waves proposed and not observed. Fate determination — depends on dark energy's equation of state, measured with improving precision and not to the accuracy that separates the scenarios. **Narrowing recorded.** Singularity theorems narrow the reach: past-incompleteness stands proved under stated energy conditions. Observational cosmology narrows the reach: the constraint stands for the last thirteen billion years. **Progress.** The proved theorems under their stated conditions. The precision constraints on the last thirteen billion years. The established violation of the energy conditions by quantum fields. The measured equation of state and its current precision. **Own record.** ## 156 Structure and agency **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Social outcomes are produced by people acting and by the structures they act within. Which explains which? In the field's own posing, structure is one of the most important, elusive and undertheorised concepts in the social sciences, central to virtually every tendency of social-scientific thought, a term the field finds impossible to do without and nearly impossible to define adequately, and the theory of structure sought is one that restores agency to social actors and builds the possibility of change into the concept. **The field's account of its hardness.** The field's theory states that social scientists find it impossible to do without the term structure and nearly impossible to define it adequately, that the concept is central to structural functionalism, structuralism and poststructuralism alike, and that a theory of structure has to restore human agency to social actors, build the possibility of change into the concept, and overcome the divide between semiotic and materialist visions of structure, which the duality of structure and the notion of habitus are criticised and reformulated to do. **Accounts.** Structure: regularities in outcomes persist across changes of personnel, patterned by class, institution, and position, individuals act within constraints they did not choose and mostly reproduce them, and explanation at the level of individual choice misses what is doing the work. Agency: structures have no existence apart from what people do, they are reproduced only by being enacted, treating them as causes reifies a pattern into a thing, and explanation must run through the actions that constitute the regularity. **Specifications.** Social outcomes have causes — local. The causes are located either in properties of persons or in properties of arrangements — local. The two levels are distinguishable in measurement — local. **Specifications withdrawn.** **Common requirement.** That structure and action are two things with a determination relation between them. **Conserving relation.** The explanatory level identified, the identification standing where the two levels vary independently. **Conserving reach.** All outcomes the question is asked of. **Reach.** The statement quantifies over which level does the explanatory work. Outcomes in which both levels are present, and cannot be varied independently, are what is measured. The two are not the same. **Unreachability.** Structuration makes the two inseparable, which its critics hold removes the question; multilevel modelling returns a partition conditional on the levels specified; and natural experiments where one level changes and the other does not are rare and confounded by selection into the change. The identification is not reached over the outcomes by measurements in which both levels are present. **Persistence.** A comparison in which the two levels vary independently, or agreement on a level at which the explanation is owed. **Addresses.** Statement, hardness, and the duality-of-structure reformulation at Approaches: Sewell, "A Theory of Structure: Duality, Agency, and Transformation", American Journal of Sociology 98(1):1-29, 1992, at its abstract as the publisher's page carries it verbatim and at its opening as an index's page carries it, the paper itself not fetched. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (structure as both medium and outcome, the duality criticised as redescribing the relation), a comparison in which the two levels vary independently standing there as what the duality by construction withholds. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Structuration — structure as both medium and outcome of action, widely adopted and criticised as redescribing the relation rather than analysing it, making the two inseparable by definition. Critical realism — separates them analytically over time, structure preceding action and being elaborated by it, requiring a temporal separation its critics hold is stipulated rather than found. Methodological individualism — derives regularities from individual action, facing the aggregation problem, the same regularity following from many individual-level specifications, and the emergence objection. Multilevel statistical modelling — partitions variance between individual and contextual levels, returning a partition conditional on the levels specified. Natural experiments in institutional change — comparison available where arrangements change and personnel do not, or the reverse, rare and confounded by selection into the change. **Narrowing recorded.** Critical realism narrows the reach: the separation stands over a stipulated temporal sequence. Multilevel modelling narrows the reach: the partition stands conditional on the levels specified. **Progress.** The measured persistence of regularities across changes of personnel. The variance partitions under specified levels. The natural-experiment comparisons where available. **Own record.** ## 157 The mechanism of high-temperature superconductivity **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What binds electrons into pairs in the copper-oxide superconductors at their observed critical temperatures, and what the anomalous normal state is from which the superconductivity condenses. The field poses it in several formulations: the physics of a doped Mott insulator; the pairing interaction of the unconventional superconductors; intertwined orders in the phase diagram; the identification of the underlying ground states; the nature of the strange-metal normal state. **The field's account of its hardness.** That a qualitative understanding of the superconducting state is achieved while the phase diagram's complexity, the prominence of collective fluctuations, and the normal state's simplicity and material-insensitivity stand unresolved; that certain experiments may probe beyond the single-band model; and that the field's best methods disagree on its minimal model. **Accounts.** The pairing is electronic, arising from repulsion, with antiferromagnetic spin fluctuations as the mediating channel and the single-band Hubbard model as the stated arena; within the model at interaction strengths relevant to the materials, fluctuation diagnostics identify the spin channel as the pairing glue. The single-band electronic frame does not close the statement: accounts retaining lattice and phonon involvement stand in the record, classes of experiments are recorded as possibly probing physics beyond the single-band model on the studied clusters, and the intertwined-orders reading holds that no single pairing channel constitutes the account. **Specifications.** Superconductivity is a paired-electron condensate, with the conventional phonon-mediated mechanism established — universal under superconductivity, and the given against which the statement is posed. Metals are described by Fermi-liquid theory — universal under condensed matter, and the normal-state given against which the strange metal stands. The physics of the copper-oxide planes is captured by the Hubbard model at relevant filling and interaction — local, the field's stated minimal-model commitment. **Specifications withdrawn.** **Common requirement.** That there is a mechanism — a fact about what mediates the pairing — expressible within the framework. **Conserving relation.** A mechanism established for the real materials, the establishment standing from the minimal model to the materials. **Conserving reach.** The real materials' pairing and normal state. **Reach.** The statement quantifies over the real materials' pairing and normal state. The field's best numerical methods disagree on the minimal model's ground state at the relevant parameters, by the field's own multi-method benchmark; at strong coupling in the relevant regime, stripe order is established as the model's low-energy state; and the spectroscopic probes reach different regimes of the phase diagram, with their reaches stated in the review literature. The two are not the same. **Unreachability.** The best methods disagree on the minimal model, by the field's own benchmark, and the model-versus-material gap stands stated in the field's own record; the probes disagree on the pseudogap's onset, a standing characterization dispute recorded in the reviews. The establishment is not reached over the materials by results within the minimal model or by the probes. **Persistence.** The mechanism established to the field's agreement; a theory with predictive capacity for critical temperatures from composition. **Addresses.** Statement, the formulations: Keimer, Kivelson, Norman, Uchida and Zaanen, "From quantum matter to high-temperature superconductivity in copper oxides", Nature 518:179-186, 2015; the doped Mott insulator: Lee, Nagaosa and Wen, Reviews of Modern Physics 78:17, 2006; the pairing interaction: Scalapino, Reviews of Modern Physics 84:1383-1417, 2012. Hardness, its first clause: the review the address names first, at its abstract as an abstract service, an index and a repository copy carry it verbatim (a qualitative understanding of the superconducting state itself achieved; unresolved issues including the astonishing complexity of the phase diagram, the unprecedented prominence of various forms of collective fluctuations, and the simplicity and insensitivity to material details of the normal state at elevated temperatures), issue 7538, doi 10.1038/nature14165, the review itself not fetched. The pairing-interaction review verified at its abstract as the publisher's page, an abstract service and an index carry it verbatim (phenomenological evidence relating the cuprates, the iron pnictides and chalcogenides, and some heavy-fermion and actinide materials; single-band and multiband Hubbard models describing a number of observed properties; spin-fluctuation-mediated pairing proposed as the common thread), issue 4, doi 10.1103/RevModPhys.84.1383, arXiv:1207.4093, not fetched, carrying neither of the line's other clauses; those two, that certain experiments may probe beyond the single-band model and that the field's best methods disagree on its minimal model, stand keyed at the next pass; the doped-Mott-insulator review not read this pass. Persistence, no landing: the closing condition at the review's formulations read at pass one and at the hardness (the mechanism not established to the field's agreement; no theory predicting critical temperatures from composition), the line stating the review's own two conditions. Approaches: the materials-search and model routes at that review's sections, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Materials search — critical temperatures measured across the copper-oxide families, hydride superconductivity standing at higher temperatures under pressure with the mechanism there recorded as conventional, nickelate superconductivity established, extending the unconventional family. Numerical attack on the minimal model — the multi-method benchmark standing, stripe order established as the model's relevant low-energy state, the pairing glue identified within the model at intermediate coupling, the model-versus-material gap stated in the field's own record. Spectroscopic mapping — the pairing symmetry established, the phase diagram mapped, the pseudogap characterized by multiple probes which disagree on its onset. Quantum simulation — the pseudogap observed in a cold-atom realization of the model, by thermodynamic and spectroscopic measurement at newly attainable temperatures, a new instrument class entering the record. Phenomenology of intertwined orders — the orders characterized and their relations stated, answering adjacent questions by its own account where the mechanism is concerned. **Narrowing recorded.** Numerical work narrows the reach: the pairing glue stands identified within the minimal model at intermediate coupling rather than for the materials. Quantum simulation narrows the reach: the pseudogap stands realized in a model system. **Progress.** The established pairing symmetry. The mapped phase diagram across material families. Stripe order established as the minimal model's relevant low-energy state. The superconducting state qualitatively understood, in the field's own summary. The cold-atom realization. **Own record.** ## 158 The interpretation of probability **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What probability statements are about — objective features of the world, or degrees of belief. **The field's account of its hardness.** That an interpretation is judged against criteria of adequacy, admissibility, the meanings assigned to the primitive terms transforming the axioms and all the theorems into true statements, ascertainability, some method by which in principle the values can be found out, a concept being useless otherwise, and applicability, and that each candidate fails one or another: the classical probabilities defined only on finite spaces and only finitely additive, with Bertrand-style problems limiting the principle of indifference and the concern that their relation to rational credence is vacuous; limiting relative frequencies violating countable additivity, their domain of definition not even a field. **Accounts.** Objective: probabilities are frequencies in classes of events, or propensities of chance setups, or the chances a best-system analysis extracts from the world's pattern. Epistemic: probabilities are coherent degrees of belief, constrained by the axioms and updated by conditioning, objective chance where invoked being itself explicated through credence. **Specifications.** The probability calculus as the shared formalism, standing axiomatized — universal under probability theory. The applications — statistical mechanics, quantum theory, statistical inference — each presupposing a reading — local. **Specifications withdrawn.** **Common requirement.** A fact about the truthmakers of probability statements. **Conserving relation.** A reading of what probability is, standing on the formalism and its applications. **Conserving reach.** All probability statements the readings are applied to. **Reach.** The statement quantifies over what probability is. Frequencies, and the calculus's theorems, are what is measured — recorded in the literature as fitting every interpretation by construction. The two are not the same. **Unreachability.** The calculus and the frequencies fit every reading by construction, and the interpretations stand empirically equivalent over the shared formalism. The reading is not reached over the statements by frequencies and theorems. **Persistence.** A consequence on which the readings differ, which the interpretation-neutrality of the calculus excludes over the shared formalism. **Addresses.** Statement: Hájek, "Interpretations of Probability", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (an interpretation judged against criteria of adequacy: admissibility, the meanings assigned to the primitive terms transforming the axioms and theorems into true statements; ascertainability, some method by which in principle the values can be found out; applicability; the classical probabilities defined only on finite spaces and only finitely additive, the principle of indifference limited by Bertrand-style problems, the concern that their relation to rational credence is vacuous; limiting relative frequencies violating countable additivity, their domain of definition not even a field), the entry itself not fetched, its section numbers and revision date not carried at the returns. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (an interpretation judged against admissibility, ascertainability and applicability; the calculus neutral among them), a consequence on which the readings differ standing there as what the neutrality excludes. Approaches: the frequentist, subjectivist and propensity routes at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Frequentism — carries statistical practice, the single-case and reference-class objections standing recorded. Subjectivism — the coherence and convergence results standing, the priors dependence recorded. Propensity accounts — standing defended, the inversion objection recorded. Best-system chances — standing as the recorded middle, the standards of the best system disputed. **Narrowing recorded.** Best-system chances weaken the relation: chance stands extracted from a pattern under disputed standards rather than a reading established. **Progress.** The axiomatization. The representation and convergence theorems. The interpretation-neutrality of the calculus, by the field's own record. **Own record.** ## 159 Self-locating belief **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How credence should be assigned to propositions about who and when one is — posed canonically by the case of the sleeper woken once or twice depending on a coin. **The field's account of its hardness.** That in addition to uncertainty about what the world is like one can be uncertain about one's own spatial or temporal location in it, the problem arising from the interaction between the two sorts of uncertainty, and that the case as first stated in the field's literature was given with two distinct answers motivated and judgment suspended as to which is correct; and that the case has since been taken as a paradigmatic example of how self-locating uncertainty creates problems for the principles of conditionalization and reflection, and thought to raise serious issues for diachronic Dutch book arguments, one later paper arguing the disagreement between the solutions to be an instance of the familiar problem of choosing the priors. **Accounts.** One half: no new information arrives at waking, and the credence in the coin stands. One third: waking events are the evidence, and conditioning on being awake shifts the credence. **Specifications.** Bayesian conditioning as the updating rule — universal under probability theory. The rule's founding scope — evidence as propositions learned — standing silent on centered information — local. **Specifications withdrawn.** **Common requirement.** An extension of the updating rule to self-locating evidence. **Conserving relation.** An extension of the updating rule to centered information, standing across the case family. **Conserving reach.** All cases in the family the extensions are applied to. **Reach.** The statement quantifies over the rule's extension. The theorems of each proposed extension are what is measured — each consistent, returning different verdicts on the case. The two are not the same. **Unreachability.** Consistent extensions stand on both sides and the case's verdict varies with the extension chosen; the betting setups encode the answer in their stakes, which the field records. The extension is not reached over the case family by the theorems of the extensions. **Persistence.** A criterion selecting between the consistent extensions, or a case on which they agree and one is shown correct. **Addresses.** Statement, the case as posed: Elga, "Self-Locating Belief and the Sleeping Beauty Problem", Analysis 60:143-147, 2000. The survey: Sorensen, "Epistemic Paradoxes", Stanford Encyclopedia of Philosophy. Hardness: the paper the address names, at its opening as the author's page and an index carry it verbatim (uncertainty about one's own spatial or temporal location in addition to uncertainty about what the world is like; the problem arising from the interaction between the two; the case stated; the case appearing as an example in an earlier paper which motivates two distinct answers and suspends judgment), issue 2, doi 10.1093/analys/60.2.143, the paper itself not fetched; a later paper at its abstract and opening as the publisher's page carries them (the case a paradigmatic example of how self-locating uncertainty creates problems for conditionalization and reflection, and thought to raise issues for diachronic Dutch book arguments; the disagreement argued to be an instance of the familiar problem of choosing the priors), Erkenntnis, doi 10.1007/s10670-019-00212-4, its author not carried at the return, returned in the search; the survey not returned at its passages on the case, not read this pass. Persistence, no landing: the closing condition at the paper's opening read at pass two (uncertainty about one's own spatial or temporal location; the halfer and thirder answers each consistent), a criterion selecting between the extensions standing there as what the paper's own two answers leave. Approaches: the halfer and thirder formalisms at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Halfer formalisms — standing consistent, carrying recorded costs elsewhere in the case family. Thirder formalisms — standing consistent, carrying their own recorded costs. Betting and long-run arguments — standing on both sides, the field recording that the betting setups encode the answer in their stakes. Applications — the cosmological and many-worlds applications standing recorded, importing the dispute into physics' own reasoning. **Narrowing recorded.** **Progress.** The consistency of both extended systems. The case taxonomy. **Own record.** ## 160 Population ethics **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How outcomes with different numbers of people are to be ranked. **The field's account of its hardness.** That the search for a theory that would avoid the repugnant conclusion and other equally unattractive conclusions has not succeeded, its prospects regarded by some as gloomy, the pessimism given a more formal content in the claim, with several proofs suggested, that it can be proven that no population ethics satisfies a set of apparently plausible adequacy conditions, while what such a theorem would show about ethics is not quite clear; that the reading on which the conclusion loses its sting, a life barely worth living being not terrible, is little comfort, other impossibility results resting on logically weaker and intuitively more convincing conditions or on the avoidance of a still more repugnant conclusion; and that the conclusion can be derived from conditions to which an egalitarian has no decisive objections. **Accounts.** By total welfare, accepting the recorded consequence that vast populations of lives barely worth living can outrank smaller flourishing ones. By person-affecting or average or critical-level principles, each accepting its own recorded consequence from the impossibility results. **Specifications.** Betterness orders outcomes — local. The adequacy conditions the field states — avoiding the repugnant and sadistic conclusions, respecting dominance and transitivity — local. **Specifications withdrawn.** **Common requirement.** A ranking satisfying the stated conditions. **Conserving relation.** A ranking of outcomes satisfying the stated adequacy conditions jointly. **Conserving reach.** All outcomes differing in number of people. **Reach.** The statement quantifies over the ranking. The impossibility theorems are what is established: the stated adequacy conditions are jointly unsatisfiable, standing proved. The two are not the same — the instruments prove the conditions cannot all be met, and do not select which to give up. **Unreachability.** No ranking satisfies the stated conditions jointly, standing proved. The relation is not preserved over the outcomes by any principle meeting the conditions. **Persistence.** Agreement on which adequacy condition is given up. **Addresses.** Statement: Arrhenius, Ryberg and Tännsjö, "The Repugnant Conclusion", Stanford Encyclopedia of Philosophy; the impossibility result: Arrhenius, "An Impossibility Theorem for Welfarist Axiology", Economics and Philosophy 16:247-266, 2000; the problem as first posed: Parfit, Reasons and Persons, Oxford, 1984, Part Four. Hardness: the same entry, at passages of its text as the encyclopedia's archived editions carry them verbatim (the search for a theory avoiding the repugnant conclusion and other equally unattractive conclusions not succeeded, its prospects regarded by some as gloomy; the pessimism given formal content in the claim, with several proofs suggested, that no population ethics satisfies a set of apparently plausible adequacy conditions, what such a theorem shows about ethics not quite clear; the reading on which the conclusion loses its sting little comfort, other impossibility results resting on logically weaker and intuitively more convincing conditions or on avoiding a very repugnant conclusion; the conclusion derivable from conditions to which an egalitarian has no decisive objections), the entry itself not fetched, its section numbers and revision date not carried at the returns; the impossibility theorem and the problem as first posed not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (the search for a theory avoiding the repugnant conclusion; the impossibility theorems showing that some adequacy condition must go), agreement on which condition is given up standing there as what the theorems leave. Approaches: the total, average, critical-level and variable-value routes at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Total and average principles — each standing, with its violated condition proved and recorded. Critical-level and variable-value principles — each standing, with its violated condition proved. The impossibility programme — ending in the proved limitation, no theory satisfying the conditions jointly, the theorems standing as the field's settled meta-result. Condition-rejection — each condition's rejection standing defended somewhere in the record. **Narrowing recorded.** Each principle weakens the relation: a ranking stands with one stated condition given up, the violated condition proved in each case. **Progress.** The impossibility theorems. The mapped violation for each principle. The defended rejections of each condition. **Own record.** ## 161 The ontology of mathematics **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether mathematical objects exist independently of us. In the field's own posing, mathematical platonism is the metaphysical view that there are abstract mathematical objects whose existence is independent of us, with the indispensability argument standing as a broadly empirical defence of it that some of its adherents take while rejecting the epistemological claims and the claim that all truths of pure mathematics are necessary. **The field's account of its hardness.** The field's survey states that platonism in the purely metaphysical sense can be held while its epistemological companions are rejected, and that the standing challenge to it is that the truth-values of mathematical assertions would depend on facts involving entities residing outside space and time, causally isolated from us even in principle, which is the challenge the survey sets against the view's considerable philosophical significance. **Accounts.** They exist: mathematics is a body of truths about abstract objects, and its indispensability to science commits science's realists to them. They do not: mathematics is reformulable without abstract commitment, or its objects are fictions whose usefulness needs no truth. **Specifications.** A uniform semantics for mathematics demands objects — local. A causal account of knowledge forbids access to them — local. The founding dilemma stands stated in the field's canonical paper. **Specifications withdrawn.** **Common requirement.** An account meeting both horns. **Conserving relation.** An account meeting both horns of the dilemma, standing on practice. **Conserving reach.** All mathematical practice the readings are applied to. **Reach.** The statement quantifies over the existence of abstract objects. Proofs and applications are what is measured — recorded as invariant under the ontological readings. The two are not the same. **Unreachability.** Practice is invariant under the readings, and the dilemma's two horns stand as posed. The account is not reached over the practice by proofs and applications. **Persistence.** An account meeting both horns, or agreement on which horn is given up. **Addresses.** Statement, hardness, and the indispensability route at Accounts: "Platonism in the Philosophy of Mathematics", Stanford Encyclopedia of Philosophy, at the passages the encyclopedia's current and archived pages carry verbatim, its section numbers and revision date not confirmed. Persistence, no landing: the closing condition at the same entry, at the passages read at pass one and at the hardness (the dilemma between the semantics and the epistemology of mathematics; indispensability and the nominalist reconstructions), an account meeting both horns standing there as the dilemma's own demand. Approaches: the indispensability and reconstruction routes at the same entry's sections, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Indispensability — standing defended, the nominalist reconstructions standing against it, with their scope recorded. Reconstruction programmes — standing partially executed, their completeness disputed. Fictionalism — standing defended, the applicability of the fiction standing as its recorded question. Structuralism — standing as the recorded middle, the ontology of structures inheriting the dispute. **Narrowing recorded.** Reconstruction programmes narrow the reach: the reformulation stands for the portion executed. Fictionalism weakens the relation: usefulness stands in place of truth. **Progress.** The stated dilemma. The executed portion of the reconstructions with its scope recorded. The recorded invariance of practice under the readings. **Own record.** ## 162 Causation **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What causation is — and whether fundamental science contains it. In the field's own posing, an influential tradition denies the applicability or at least the usefulness of causal notions in physics, and a lively debate continues on whether causal notions can play a legitimate role in physics and, if so, what role, the discussion running as three projects, a metaphysical, a descriptive and a functional one. **The field's account of its hardness.** The field's survey states that the tradition denying causation a place in physics is less dominant than it was while the debate remains live, and the field's review states that virtually everyone now rejects the elimination of causation while many have been convinced that the fundamental physical structure of the world contains no causal relations, which raises the question of how to reconcile the central role of causal concepts in the special sciences and in common sense with their putative absence from fundamental physics. **Accounts.** Causation is real and analyzable: as regularity, as counterfactual dependence, as process connection, or as what interventions reveal, the interventionist formalism standing as the working account of the special sciences. Causation is not fundamental: fundamental dynamics is symmetric and lawlike without causal asymmetry, and causal talk is a higher-level instrument. **Specifications.** The sciences' causal practice — experiment, inference, explanation — standing as the phenomenon — universal under scientific practice. Fundamental physics' time-symmetric laws standing as the constraint — universal under physics. **Specifications withdrawn.** **Common requirement.** An account of the relation between the practice and the fundamental description. **Conserving relation.** The standing of causal structure determined, the determination reaching from the practice to the fundamental description. **Conserving reach.** Both the special sciences' practice and the fundamental description. **Reach.** The statement quantifies over the standing of causal structure. Correlations and intervention outcomes are what is measured, the discovery formalisms recovering causal structure from them under stated assumptions, recorded as assumptions. The two are not the same. **Unreachability.** The formalisms carry practice under assumptions, and the interventionist account's circularity for foundations stands recorded by its own literature; the grounding of the asymmetry in thermodynamic gradient stands disputed in its reach. The determination is not reached over both levels by correlations and intervention outcomes. **Persistence.** An account of the relation between the practice and the fundamental description that both accounts accept. **Addresses.** Statement, hardness, the not-fundamental account at Accounts, and the reconciliation at Persistence: "Causation in Physics", Stanford Encyclopedia of Philosophy, at its opening as the encyclopedia's current and archived pages carry it verbatim, its author, section numbers and revision date not confirmed; Blanchard, "Physics and Causation", Philosophy Compass 11(5):256-266, 2016, at its abstract as an archive's copy carries it verbatim, the paper itself not fetched. Approaches: the regularity, counterfactual and interventionist routes at the same entry's sections on the accounts, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Regularity and counterfactual analyses — standing, with their counterexample records. The interventionist formalism — carries the special sciences' practice, its circularity for foundations standing recorded by its own literature. Physics-based accounts — ground the asymmetry in thermodynamic gradient, the grounding's reach disputed. Eliminativism at the base — standing defended from the founding critique onward. **Narrowing recorded.** The interventionist formalism narrows the reach: the account stands for the special sciences' practice under stated assumptions. Physics-based accounts narrow the reach: the asymmetry stands grounded where the thermodynamic gradient obtains. **Progress.** The discovery formalisms and their stated assumptions. The counterexample records for each analysis. The recorded circularity of the interventionist account for foundations. **Own record.** ## 163 The cosmological constant problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why the measured vacuum energy density is small and nonzero — discrepant by many tens of orders of magnitude from the field-theoretic estimate. **The field's account of its hardness.** That astronomical observations indicate the constant to be many orders of magnitude smaller than estimated in modern theories of elementary particles; that the old problem is to understand in a natural way why the vacuum energy density is not very much larger and the new problem why it is comparable to the present mass density, the founding review's no-go argument showing why adjustment attempts would not work without a fine tuning just as mysterious as the problem started with, no one having found a way out of it; and that anthropic considerations, offered as a possibility of solving both problems, are not universally accepted, essentially all physicists preferring a non-anthropic explanation of the largest of all discrepancies between standard theory and observation. **Accounts.** A dynamical or symmetry explanation is owed and findable within physics. The value is environmental: selected anthropically across a landscape of vacua, with the prediction of a small nonzero value recorded as the approach's success. **Specifications.** Quantum field theory's vacuum contributions — universal under field theory. General relativity's coupling of energy to curvature — universal under general relativity. The concordance measurement of the accelerating expansion — universal under cosmology. **Specifications withdrawn.** **Common requirement.** An account of the measured value. **Conserving relation.** An account of the measured value, standing on a measurement distinguishing the candidates. **Conserving reach.** All accounts of the value. **Reach.** The statement quantifies over the value's explanation. The value is measured, the estimate computed, and the discrepancy stands; the explanation candidates stand recorded as returning no differing measurements within reach. The two are not the same. **Unreachability.** The candidates return no differing measurements within reach, the natural symmetry candidates stand recorded as excluded or incomplete, and no-go results stand against simple dynamical adjustment. The account is not reached over the candidates by the measured value. **Persistence.** A measurement on which the candidate accounts differ, or agreement on whether environmental selection explains. **Addresses.** Statement: Weinberg, "The cosmological constant problem", Reviews of Modern Physics 61:1-23, 1989. Reviews: Carroll, "The cosmological constant", Living Reviews in Relativity 4:1, 2001; Peebles and Ratra, "The cosmological constant and dark energy", Reviews of Modern Physics 75:559, 2003. Hardness: re-formed to the field's words at the founding review, at its abstract as the publisher's page, an abstract service, an index and a university repository carry it verbatim (astronomical observations indicating the constant many orders of magnitude smaller than estimated in modern theories of elementary particles; five approaches to its solution described), issue 1, doi 10.1103/RevModPhys.61.1, the review itself not fetched; the same author's later talk at its abstract and opening as the arXiv copy carries them (the old problem why the vacuum energy is so small, the new why it is comparable to the present mass density; the review's no-go argument against adjustment without a fine tuning as mysterious as the problem, no way out found; quintessence helping with neither, anthropic considerations a possibility for both), arXiv:astro-ph/0005265, 2000, and a later survey of open questions at its passage as the arXiv copy carries it (the vacuum contributions determined independently with no reason to cancel; roughly a hundred and twenty orders of magnitude against observation; anthropic reasoning not universally accepted, essentially all physicists preferring a non-anthropic explanation; no truly convincing solution), arXiv:1804.08730, 2018, its authors not carried, both returned in the search, neither fetched; the former line, the field's dispute over what explanation is owed standing as a methods dispute inside the physics, in the file's own voice, released with its content carried at the anthropic clause; the two later reviews the address names not read this pass. Persistence, no landing: the closing condition at the founding review's abstract read under check C18 (the observed value against the theoretical expectation, the candidate mechanisms each excluded or incomplete; the anthropic reading), a measurement on which the candidates differ or agreement on environmental selection standing there as the review's own two directions. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Symmetry mechanisms — the natural candidates recorded as excluded or incomplete in the field's reviews. Dynamical adjustment — the no-go results against simple adjustment standing. The anthropic route — standing defended, with the field's recorded dispute over whether environmental selection explains. Modified-gravity routes — standing constrained by the multimessenger bounds. **Narrowing recorded.** The anthropic route weakens the relation: the value stands selected environmentally rather than accounted for dynamically. **Progress.** The measured value and the computed estimate. The exclusions of the natural symmetry candidates. The no-go results against simple adjustment. The multimessenger constraints. **Own record.** ## 164 The hierarchy problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why the electroweak scale lies so far below the Planck scale — and whether an explanation is owed at all, which is the naturalness question. **The field's account of its hardness.** That the Higgs mass parameter is technically natural in the field's account, the finite corrections proportional to the masses of the known particles all small, the problem emerging only in the presence of additional ultraviolet scales, where invoking the scale of quantum gravity implies quantum corrections thirty-two orders of magnitude in excess of the doublet mass parameter and naively asking that the corrections not exceed the inferred parameter implies new physics at or below about five hundred gigaelectronvolts, the cutoff being merely a proxy for unknown microscopic physics; and that natural solutions of the electroweak hierarchy problem are under intense pressure from the collider's exploration of physics at the teraelectronvolt scale, writing about naturalness described as fraught with peril. **Accounts.** Naturalness is a principled expectation: an unprotected scalar mass demands a stabilizing mechanism, and the mechanism's absence at the collider is a standing crisis. Naturalness is a failed heuristic: the expectation was a methodological preference, the measurements have spoken, and no mechanism is owed. **Specifications.** Effective field theory's scaling logic — universal under field theory. The collider's null results at the predicted scales — local, standing measured. **Specifications withdrawn.** **Common requirement.** A standard for when a parameter value demands explanation. **Conserving relation.** A criterion of explanatory debt, standing on measurement. **Conserving reach.** All parameter values the criterion is applied to. **Reach.** The statement quantifies over the criterion of explanatory debt in the field's method. The particle spectrum to the measured energies is what is measured; the criterion is not measured by it. The two are not the same. **Unreachability.** The measurements bound the mechanisms and the criterion dispute stands over the measurements; the criterion is not among the things the instruments return. The criterion is not reached over the parameter values by the particle spectrum. **Persistence.** A criterion of explanatory debt the field agrees on, or a mechanism observed at a reachable scale. **Addresses.** Statement: Giudice, "Naturally speaking: the naturalness criterion and physics at the LHC", in Perspectives on LHC Physics, World Scientific, 2008, 155-178, arXiv:0801.2562; review: Craig, "Naturalness: past, present, and future", European Physical Journal C 83:825, 2023. The criterion's statement: 't Hooft, "Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking", in Recent Developments in Gauge Theories, Plenum, 1980, 135. Hardness: the review the address names, at its abstract as an abstract service, a government index and an index carry it verbatim (the state of naturalness assessed and recent approaches to the three major naturalness problems summarized) and at passages of its text as the arXiv copy and a repository copy carry them (writing about naturalness fraught with peril; natural solutions of the electroweak hierarchy problem under intense pressure from the collider's exploration of the teraelectronvolt scale; the Higgs mass parameter technically natural, the finite corrections from the known particles small, the problem emerging only with additional ultraviolet scales; the scale of quantum gravity implying corrections thirty-two orders of magnitude in excess of the doublet mass parameter; naively bounding the corrections implying new physics at or below about five hundred gigaelectronvolts, the cutoff a proxy for unknown microscopic physics), issue 9, doi 10.1140/epjc/s10052-023-11928-7, arXiv:2205.05708 as a Snowmass white paper, the review itself not fetched; the naturalness essay and the criterion's statement not read this pass. Persistence, no landing: the closing condition at the review's abstract read at pass two (the state of naturalness after the reached scales; the stabilizing candidates unobserved), a criterion of explanatory debt or a mechanism at a reachable scale standing there as the review's own two readings. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Stabilizing mechanisms — the canonical candidates standing unobserved at the reached scales, the parameter spaces narrowing. Anthropic and landscape readings — standing defended, importing the neighbouring dispute. Criterion critique — the naturalness-as-heuristic literature standing published and growing in the field's record. **Narrowing recorded.** The criterion critique weakens the relation: no explanatory debt stands owed, in place of a criterion established. **Progress.** The measured particle spectrum to the reached energies. The narrowed parameter spaces for the canonical mechanisms. **Own record.** ## 165 The Hubble tension **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why the expansion rate measured from the early universe and from the local distance ladder disagree at the several-sigma level. **The field's account of its hardness.** That the standard cosmological model successfully describes observations from widely different epochs of the Universe, but as its basic parameters are determined with increasing and unprecedented precision it is not guaranteed that the same model will fit more precise observations from those epochs, and discrepancies developing between observations at early and late cosmological time may require an expansion of the standard model and may lead to the discovery of new physics; and that as the measurements improved a surprising disagreement between the early-Universe prediction and the late-Universe measurement of the present expansion rate appeared, a number of independent late-Universe measurements using different methods and data providing consistent results that made the discrepancy increasingly hard to ignore. **Accounts.** A systematic in one or both measurement programs, to be identified. An extension of the cosmological model, modifying the early sound horizon. **Specifications.** The two programs measure the same quantity — universal under cosmology. The concordance model relates the early inference to the late measurement — universal under cosmology. **Specifications withdrawn.** **Common requirement.** That the discrepancy is attributable to systematics or to model extension. **Conserving relation.** One expansion rate returned by both programs, the agreement standing under strengthened instruments. **Conserving reach.** Both measurement programs. **Reach.** The statement quantifies over the expansion rate. The local ladder measures the present rate directly, at one percent from a community-built distance network across calibrators, with crowding systematics excluded at high confidence. The early-universe determination is an inference calibrated on the sound horizon under the concordance model and carried forward to a present-day value, with the model-dependent nature of that inference stated in the field's own literature. The sharpest form of the tension stands between sound-horizon-calibrated determinations under the concordance model and determinations calibrated otherwise. The two are the same only under the model relating them, and that model is among the accounts under test: the early reach is frame-conditioned, and a discriminating programme is definable and running. **Unreachability.** **Persistence.** Convergence of the ladders under identified systematics, or an extension fitting both legs — the field's stated closure conditions, in-domain. **Addresses.** Statement: Verde, Treu and Riess, "Tensions between the early and the late Universe", Nature Astronomy 3:891, 2019. Review: Di Valentino et al., "In the realm of the Hubble tension — a review of solutions", Classical and Quantum Gravity 38:153001, 2021. The local measurement: Riess et al., Astrophysical Journal Letters 934:L7, 2022. Hardness: re-formed to the field's words at the paper the address names, at its abstract as the arXiv page carries it verbatim (the standard model describing observations from widely different epochs; as its parameters are determined with unprecedented precision, not guaranteed to fit more precise observations from those epochs; discrepancies between early and late time possibly requiring an expansion of the model and possibly leading to new physics), pages 891-895, arXiv:1907.10625, a workshop's summary, the paper itself not fetched; and at a comment by its third author, at its summary as the publisher's page carries it (a surprising disagreement appearing as the measurements improved; independent late-Universe measurements by different methods consistent, the discrepancy increasingly hard to ignore), Nature Reviews Physics 2:10-12, 2020, doi 10.1038/s42254-019-0137-0, returned in the search, not fetched; the former line, in the file's own voice, carrying the word crisis which no return carries, released with its content at the two clauses; the review of solutions and the local measurement not read this pass. Persistence, no landing: the closing condition at the paper's abstract read under check C18 (the standard model's prediction against the local measurement; the field's stated closure conditions, convergence under identified systematics or an extension fitting both), the line carrying those conditions in the field's words. Approaches: the cross-check and extension routes at that paper and the comment read under check C18. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Local-ladder cross-checks — the space-telescope cross-checks verified the local ladder rather than locating a systematic, deepening the discrepancy, with the one-percent community distance network standing across calibrators. Early-universe measurement — the ground-based microwave programme's final data reinforced that side. Alternative calibrations — time-delay cosmography aligned with the local values without a ladder; tip-of-the-red-giant-branch calibrations returning intermediate values, the local record's own spread; standard sirens entering as an independent method. Late-time model modifications — difficulty results standing published, narrowing the model route to early-universe physics, where early-dark-energy fits stand as the live candidates, extended-model fits converging to the low value wherever the sound-horizon calibration is held, and the remaining viable routes standing recorded as correlated modifications across epochs or as physics at the homogeneity transition. An independent third measurement programme — primary results standing expected. **Narrowing recorded.** Late-time difficulty results narrow the reach: the model route stands narrowed to early-universe physics. **Progress.** The two programmes' internal consistency. The persistence of the gap across method variations and strengthened instruments. The narrowed model route. **Own record.** ## 166 Molyneux's problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether a person born blind, made to see, would recognize by sight the shapes known by touch. **The field's account of its hardness.** That the problem, once the first account of a congenitally blind person's sight after the removal of cataracts was published, was regarded no longer as a simple thought experiment but as a question answerable by experimentation, the boy at first not knowing the shape of a thing and unable to recognize one thing from another however different in shape or size; and that the data concerning the sight of such patients, and of young animals and infants, were used as evidence in the debate whether the perception of space is innate or acquired, in which the problem was frequently discussed with still no agreement on the right solution to it. **Accounts.** They would not recognize immediately, and cross-modal transfer develops rapidly thereafter. **Specifications.** The senses' ideas are held in common or are not — local. The question is answerable by testing a person whose sight is restored — local. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** An answer measured in a person whose sight is restored. **Conserving reach.** Cases of restored sight where the test can be run. **Reach.** The statement quantifies over recognition by sight of shapes known by touch. Restored-sight patients, tested immediately and in the following days, are what is measured. The two are the same: the instrument reaches the case the statement concerns. **Unreachability.** **Persistence.** Settled as posed. Immediate cross-modal matching fails, and transfer develops within days, measured. **Addresses.** Statement: Degenaar and Lokhorst, "Molyneux's Problem", Stanford Encyclopedia of Philosophy. Hardness: the same entry, at passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the first published account of a congenitally blind person's sight after cataract removal leading philosophers to regard the problem no longer as a simple thought experiment but as a question answerable by experimentation, the boy at first not knowing the shape of a thing nor recognizing one thing from another; the data from such patients, and from young animals and infants, used as evidence in the debate whether the perception of space is innate or acquired, the problem frequently discussed there with still no agreement on the right solution; the main interest since then historical), the entry itself not fetched, its section numbers and revision date not carried at the returns, the Fall 2011 and later editions listing a paper on the newly sighted failing to match seen with felt, a reading for Progress at the next pass. Persistence, the landing: the entry's report of the measurement read at pass two and under check C18 (immediate cross-modal matching failing in the newly sighted, transfer developing within days), the newly-sighted paper itself keyed at the next pass. Approaches: the philosophical and experimental routes at the same entry's sections. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The philosophical treatment — three centuries of argument, with the negative answer defended from the founding correspondence onward. Late-childhood cataract surgery — permitted the experiment when the surgical instrument arrived, the measurement returning immediate failure and rapid acquisition. **Narrowing recorded.** **Progress.** The measured initial failure and rapid acquisition of cross-modal matching. **Own record.** The philosophical literature's continuing dispute over what the measurement answers — the migration leaving a philosophical remainder, recorded. ## 167 Teleology and function in biology **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What it is for a biological trait to have a function. **The field's account of its hardness.** That the manifest appearance of function and purpose in living systems is responsible for the prevalence of apparently teleological explanations of organismic structure and behaviour in biology, against which complaints are recorded, among them that such notions are not empirically testable, both Darwinian and non-Darwinian accounts aiming to avoid these concerns; that the diversity of life itself is reflected in the variety of scientific attempts to understand it; and that in the debate relatively little attention has been paid to the notion of natural design, it being common for authors to slide between claims about function and claims about design as if a trait were naturally designed for something exactly when that is one of its functions. **Accounts.** A trait's function is what it was selected for; function is history. A trait's function is its contribution to a containing system's capacities; function is current organization. **Specifications.** Biology's functional discourse standing throughout the science — universal under biology. Natural selection standing as the naturalizing resource — universal under evolutionary biology. **Specifications withdrawn.** **Common requirement.** An account fitting the science's functional attributions. **Conserving relation.** One account of function ascription, the account standing across the science's attributions. **Conserving reach.** All functional attributions the science makes. **Reach.** The statement quantifies over the content of function ascription. Selection histories where reconstructible, and current causal roles, are what is measured — each account's extension measured, and the attributions sorting differently at the recorded cases. The two are not the same. **Unreachability.** The cases sort differently under the two accounts and the science's usage divides by subfield, both accounts standing. The account is not preserved across the attributions by selection histories and causal roles. **Persistence.** An account fitting the attributions across subfields, or agreement that usage is divided and both stand. **Addresses.** Statement, the accounts of function as the field sorts them: Allen and Neal, "Teleological Notions in Biology", Stanford Encyclopedia of Philosophy; the causal-role side: Amundson and Lauder, "Function without purpose: the uses of causal role function in evolutionary biology", Biology and Philosophy 9:443-469, 1994. Hardness: the same entry, at its opening and passages of its text as the encyclopedia's page and its archived editions carry them verbatim (the manifest appearance of function and purpose in living systems responsible for the prevalence of apparently teleological explanations; complaints against such notions, a fifth that they are not empirically testable, both Darwinian and non-Darwinian accounts aiming to avoid them; the diversity of life reflected in the variety of scientific attempts to understand it; relatively little attention paid to natural design, authors sliding between function and design), the earlier archived edition carrying the etiological and dispositional approaches as the two, the entry itself not fetched, its section numbers and revision date not carried at the returns; the causal-role paper not read this pass. Persistence, no landing: the closing condition at the same entry, at its opening and passages read at pass two (the manifest teleology of function attributions; the etiological and systemic accounts each fitting part of the usage), an account fitting the attributions across subfields or both standing standing there as the entry's own division. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The etiological account — carries malfunction and normativity, the epiphenotype and novel-trait cases standing recorded against it. The causal-role account — carries physiology's practice, the normativity of function standing recorded as its cost. Pluralism — standing published, the science's usage recorded as divided by subfield. **Narrowing recorded.** Pluralism narrows the reach: each account stands for the subfield whose usage it fits. **Progress.** The measured extensions of each account. The recorded cases sorting differently. The documented division of usage by subfield. **Own record.** ## 168 The physical Church–Turing thesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether every physically realizable process is computable by the machine model — the thesis read as a claim about nature. **The field's account of its hardness.** That the thesis that whatever can be calculated by a machine is computable by the machine model, called thesis M and sometimes the physical Church–Turing thesis, is a very different statement from the thesis about effective calculability with which it is commonly conflated, and admits two interpretations, according to whether a machine that can generate a function is one conforming to the physical laws of the actual world or any machine that could exist, under the latter of which the thesis is false, it being straightforward to describe notional machines, hypercomputers, that generate functions not computable by the model; and that statements in the founding literature are liable to be mistaken for formulations of the physical thesis unless an idiosyncratic usage of the phrase machine process is borne in mind, misunderstanding being certain to ensue otherwise. **Accounts.** It holds: known physics supports it, and the quantum model preserves computability while changing complexity. It is open or fails: spacetimes and processes are describable in which the model is exceeded, and the thesis' physical standing is an empirical question rather than a definition. **Specifications.** The mathematical thesis standing as the founding agreement on the computable — universal under computability theory. The physical reading standing as its extension to nature — local. **Specifications withdrawn.** **Common requirement.** A fact about physical law's computational reach. **Conserving relation.** A bound on nature's computational reach, standing over physically realizable processes. **Conserving reach.** All physically realizable processes. **Reach.** The statement quantifies over nature's computational bound. Realized computations are what is measured — all within the model — and the exceeding constructions stand as descriptions whose physical realizability is disputed. The two are not the same. **Unreachability.** All realized computation stands within the model, and the exceeding constructions are descriptions whose realizability is disputed, so no realized computation bears on the modal question. The bound is not reached over the realizable processes by realized computations. **Persistence.** A physically realized process exceeding the model, or a demonstration that the exceeding constructions are unrealizable. **Addresses.** Statement: Copeland, "The Church-Turing Thesis", Stanford Encyclopedia of Philosophy; the physical reading tested against relativistic computation: Hogarth, "Non-Turing Computers and Non-Turing Computability", PSA 1994, volume 1, 126-138. Hardness: the same entry, at passages of its text as the encyclopedia's archived editions carry them verbatim (thesis M, that whatever can be calculated by a machine is computable by the machine model, admitting two interpretations, a machine conforming to the physical laws of the actual world or any machine that could exist, false under the latter, notional hypercomputers being straightforward to describe; misunderstanding certain to ensue unless an idiosyncratic usage of the phrase machine process is borne in mind, statements liable to be mistaken for formulations of thesis M), the entry itself not fetched, its section numbers and revision date not carried at the returns; a survey of analog models of computation at its statement of the same distinction as the arXiv copy carries it, returned in the search, its authors not carried; the relativistic-computation paper not read this pass. Persistence, no landing: the closing condition at the same entry, at the passages read at pass two (thesis M, whatever can be calculated by a machine calculable by a Turing machine; the hypercomputation constructions and their physical standing), a realized process exceeding the model or the constructions shown unrealizable standing there as the entry's two directions. Approaches: the quantum-computation and hypercomputation routes at the same entry's sections, read at pass two. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Quantum computation — preserves the computability bound, alters complexity, standing as the record's main extension. Hypercomputation constructions — standing described, relativistic and analog, their realizability disputed in the record. Physical formulations — the thesis' formalizations against physical principles standing published, the principles' status disputed. **Narrowing recorded.** Quantum computation narrows the reach: the complexity claim stands altered while the computability bound stands preserved. **Progress.** The preserved computability bound under quantum computation. The described exceeding constructions. The published formalizations against physical principles. **Own record.** ## 169 Micro-to-macro aggregation **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether aggregate economic behaviour inherits the structure of individual rationality. In the field's own posing, the properties of aggregate excess demand play a key role in the existence, uniqueness, stability and comparative statics of competitive general equilibrium and in the microfoundations of macroeconomics, and the results on what individual rationality implies for aggregate excess demand are read against that role. **The field's account of its hardness.** The field's retrospective states that the results left the subfields that had relied on well-behaved aggregate excess demand for their theoretical development, among them international economics and the microfoundations of macroeconomics, in the lurch, and that the theory of general equilibrium was criticised for an apparent lack of testable implications as the results seemed to imply. **Accounts.** Sufficiently, for practice: representative-agent methods stand as the working instrument, defended as approximation. Not in general, and provably: market excess demand inherits essentially no restrictions from individual rationality, standing proved, and the representative-agent practice stands undermined by it. **Specifications.** Individual rationality axioms standing as the founding micro theory — universal under microeconomics. Equilibrium as the aggregate framework — universal under general equilibrium theory. **Specifications withdrawn.** **Common requirement.** An account of the aggregate's relation to the individuals. **Conserving relation.** Structure inherited from individuals to the aggregate, the inheritance standing under aggregation. **Conserving reach.** All aggregates the individuals compose. **Reach.** The statement quantifies over the inheritance of structure under aggregation. The theorems, and the empirical aggregates the practice fits, are what is measured. The two are not the same. **Unreachability.** Rationality's restrictions wash out in aggregation, standing proved. The inheritance is not preserved over the aggregates by any restriction on individual behaviour. **Persistence.** Conditions under which structure is inherited, or agreement on the practice's standing given the theorems. **Addresses.** Statement, hardness, and the proved account at Accounts: Rizvi, "The Sonnenschein-Mantel-Debreu Results after Thirty Years", History of Political Economy 38(Suppl 1):228-245, 2006, at its abstract as an index's page carries it verbatim, the paper itself not fetched. Owed: the theorems at their own sources, cited at Accounts and Unreachability. Persistence, no landing: the closing condition at the paper's abstract read at pass one and at the hardness (the aggregation theorems, rationality's restrictions washing out; the practice's representative-agent response), conditions under which structure is inherited standing there as what the theorems deny in general. Approaches: the theorem and representative-agent routes at that paper's sections. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The aggregation theorems — ending in proved limitation, rationality's restrictions washing out in aggregation. Representative-agent practice — standing as the field's working method, with the theorems recorded against its foundations in the field's own methodological literature. Heterogeneous-agent programmes — standing as the recorded response, rebuilding aggregates from distributions. **Narrowing recorded.** Heterogeneous-agent programmes narrow the reach: the aggregate stands rebuilt from a specified distribution rather than inherited. **Progress.** The aggregation theorems. The practice's empirical fits. The heterogeneous-agent reconstructions. **Own record.** ## 170 Ubiquitous one-over-f noise **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why fluctuation spectra with power inversely proportional to frequency are measured across systems sharing no mechanism — from conductors to rivers to music. **The field's account of its hardness.** That long-memory processes whose power spectrum goes at low frequencies as an inverse power of frequency close to one have been observed, in the field's record, in physics, technology, biology, astrophysics, geophysics, economics, psychology, language and even music, the noise standing between the well-understood white noise with no correlation in time and the random walk with no correlation between increments; and that the spectrum entails infinitely large fluctuations in the low-frequency limit. **Accounts.** A universal mechanism stands to be found, criticality-based accounts proposing the common origin. No universal mechanism: the spectra are a family of look-alikes with unrelated origins, and the ubiquity is the signature of many roads to the same asymptotics. **Specifications.** Spectral analysis standing as the shared instrument — universal across the host fields. Each host field's own dynamics standing as its candidate mechanism — local. **Specifications withdrawn.** **Common requirement.** An account of the ubiquity — one mechanism, or a classification of many. **Conserving relation.** An account of the shared spectrum, standing across the systems it is measured in. **Conserving reach.** All systems the spectrum is measured in. **Reach.** The statement quantifies over the origin of the shared spectrum. The spectra are measured everywhere, mechanisms are established in particular systems, and no cross-system mechanism stands established by the record. The two are not the same. **Unreachability.** The mechanisms established are local, and broad classes of processes are shown to yield the spectrum, which supports the many-roads reading and does not select. The account is not preserved across the systems by spectra measured in each. **Persistence.** A mechanism established across systems sharing no dynamics, or a classification of the roads accepted as complete. **Addresses.** Statement: Press, "Flicker noises in astronomy and elsewhere", Comments on Astrophysics 7:103-119, 1978. Reviews: Dutta and Horn, "Low-frequency fluctuations in solids: 1/f noise", Reviews of Modern Physics 53:497-516, 1981; Milotti, "1/f noise: a pedagogical review", arXiv:physics/0204033, 2002. Hardness: the pedagogical review the address names, at its abstract as the arXiv page and an abstract service carry it verbatim (a review of the ubiquitous noises; the representation as a superposition of many relaxation processes; the infinitely large fluctuations in the low-frequency limit; diffusion, self-organized criticality, earthquake scaling and deterministic dynamical systems among its sections), submitted 12 April 2002, an invited talk, the review itself not fetched; an encyclopedia entry on the noise as its page carries it (long-memory processes with the spectrum's low-frequency behaviour observed across physics, technology, biology, astrophysics, geophysics, economics, psychology, language and music; the noise intermediate between white noise and the random walk), its authors carried at the page as Ward and Greenwood, returned in the search; the founding paper and the solids review not returned, not read this pass; a low-frequency cutoff paradox named at a citing paper's title, a reading for the next pass. Persistence, no landing: the closing condition at the pedagogical review's abstract read at pass two (the ubiquity of the noise across systems sharing no dynamics; the roads to it), a mechanism established across the systems or a complete classification of the roads standing there as the review's own alternatives. Approaches: the criticality and system-specific routes at that review's sections. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Criticality programmes — standing proposed as universal, their scope disputed in the record. System-specific mechanisms — standing established case by case. Mathematical genericity results — standing recorded, broad classes of processes yielding the spectrum, supporting the many-roads reading. **Narrowing recorded.** System-specific mechanisms narrow the reach: an account stands established per system rather than across them. **Progress.** The spectra measured across fields. The mechanisms established in particular systems. The genericity results. **Own record.** ## 171 The axiomatization of physics **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether physics admits axiomatization. **The field's account of its hardness.** That the problem as posed calls for treating in the same manner as the foundations of geometry, by means of axioms, those physical sciences in which mathematics already plays an important part, in the first rank the theory of probabilities and mechanics, a programmatic call for the axiomatization of existent physical theories that the field describes as an endless road to rigour; that the theory of probabilities is considered fully axiomatized since measure theory, while the status of mechanics depends on how widely the term is understood today, its classical special case fully formalized; and that the problem is read in the field as concerning those parts of physics which are ready for a rigorous mathematical approach, its value described as inexhaustive after more than a hundred years. **Accounts.** A programmatic call for the axiomatization of existent physical theories, taken theory by theory as each becomes ready for a rigorous mathematical approach: the theory of probabilities axiomatized by measure theory, the problem declining into at least two programs, the rigorous derivation of the kinetic equation from particle systems and of the hydrodynamic equations from the same, well known in the field to be deeply different, with a derivation of the fluid equations from hard-sphere systems recently claimed and the geometric-framework part still open. A question about physics as a whole: whether, without drastically changing the mathematical apparatus, the axiomatics of physics can be set up so as to transform physics, presently a multitude of unmatched theories with inconsistent axiomatics, into an integrated science, the problem generally considered borderline among the twenty-three and its original focus lost sight of. **Specifications.** Axiomatization is the appropriate demand for a physical theory — local, and itself disputed in the field's methodological literature. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** An axiomatization covering physics. **Conserving reach.** Physics as a whole. **Reach.** The statement quantifies over physics as a whole. What is established stands per part: probability axiomatized; quantum mechanics carrying its axiomatic reconstructions; field-theoretic programmes partial, with constructive results in lower dimensions and the physical theories standing outside the axioms' reach. The two are not the same. **Unreachability.** **Persistence.** Stated per part in each programme's own literature; no whole-physics criterion stands recorded. **Addresses.** Statement, the sixth problem: Hilbert, "Mathematical problems", Bulletin of the American Mathematical Society 8:437-479, 1902. Hardness: re-formed to the field's words at the sixth problem's own text as an encyclopedia page, a journal article and three arXiv copies carry it verbatim in its common translation (to treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank the theory of probabilities and mechanics), the address's source not fetched; a workshop paper at its opening and summary as the publisher's page, an index and its arXiv copy carry them (the problem concerning those parts of physics ready for a rigorous mathematical approach; a programmatic call; its value inexhaustive after more than a hundred years), titled the endless road to rigour, Philosophical Transactions of the Royal Society A, 2018, doi 10.1098/rsta.2017.0238, its authors not carried at the return; an encyclopedia page's statement of the standing (probability fully axiomatized since the nineteen-thirties by measure theory; mechanics's status depending on how widely the term is understood; classical mechanics fully formalized), as its page carries it; the former line, in the file's own voice, released with its content at the mechanics clause. Accounts: the programmatic reading at the workshop paper read for the hardness (a programmatic call for the axiomatization of existent physical theories; the parts of physics ready for a rigorous approach), at a mathematical society's outreach page as it carries it (the problem declining into at least two monumental programs, the kinetic and the hydrodynamic derivations, well known to be deeply different), 2025, its author not carried, and at a 2026 preprint at its introduction as the arXiv copy carries it (probability axiomatized by measure theory in 1933 as a major step; a derivation of the fluid equations from hard-sphere systems recently claimed; the geometric-framework part still an open issue), arXiv:2603.29969, its authors not carried; the whole-physics reading at a 2010 preprint at its opening as the arXiv copy carries it (the axiomatic inconsistency of classical, quantum and relativistic physics not fading; whether physics as a multitude of unmatched theories can be set up as an integrated science without drastically changing the mathematics), arXiv:1011.4256, its author carried at the copy, and at a 2007 preprint at its introduction (the sixth problem not considered one of the best, generally considered borderline, misunderstood with its original focus lost sight of), arXiv:0705.2554, its author not carried; none fetched. Persistence, no landing: the closing condition at the sixth problem's own text and the accounts read at v342 (the programmatic call, theories as each is ready; the whole-physics question generally considered borderline), stated per part as the line says. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Probability — axiomatized. Quantum mechanics — carrying its axiomatic reconstructions. Field-theoretic programmes — the founding axiom systems standing, the constructive results holding in lower dimensions, the physical theories standing outside the axioms' reach, the standing gap being the four-dimensional interacting construction. **Narrowing recorded.** Constructive results narrow the reach: the construction stands in lower dimensions. **Progress.** The probability axiomatization. The reconstruction theorems. The constructive results and their dimensional limits. **Own record.** None stands as a two-position dispute in the field's current record. What stands is a programme with measured parts. ## 173 The strange metal **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What the strange-metal state is — resistivity linear in temperature across regimes where the quasiparticle framework's scales should intervene. **The field's account of its hardness.** That among the unresolved issues the field's review names, after a qualitative understanding of the superconducting state itself has been achieved, are the astonishing complexity of the phase diagram, the unprecedented prominence of various forms of collective fluctuations, and the simplicity and insensitivity to material details of the normal state at elevated temperatures. **Accounts.** A universal state stands behind the linearity: a bound on dissipation, or a new fixed point without quasiparticles, governing across materials. No universal state: material-specific mechanisms produce the linearity separately, and the universality is apparent. **Specifications.** Fermi-liquid theory standing as the framework the state defies — universal under condensed matter. The transport measurements standing as the record — local. **Specifications withdrawn.** **Common requirement.** An account of the linearity and its cross-material reach. **Conserving relation.** An account of the state, standing across the material families. **Conserving reach.** All materials the linearity is measured in. **Reach.** The statement quantifies over the nature of the state. The transport record across families, the scale analyses, and the proposed bound's saturation — measured in cases and disputed in scope — are what is measured. The two are not the same. **Unreachability.** The linearity stands measured across families and the material-by-material accounts stand established in cases, while the proposed bound's universality stands disputed in scope. The account is not preserved across the families by the transport record. **Persistence.** An account of the state holding across the material families, or a demonstration that the linearity has separate origins in each. **Addresses.** Statement: Keimer, Kivelson, Norman, Uchida and Zaanen, "From quantum matter to high-temperature superconductivity in copper oxides", Nature 518:179-186, 2015, the strange-metal normal state among its unresolved issues. Hardness: the review the address names, at its abstract as an abstract service, an index and a repository copy carry it verbatim (a qualitative understanding of the superconducting state itself achieved; unresolved issues including the astonishing complexity of the phase diagram, the unprecedented prominence of various forms of collective fluctuations, and the simplicity and insensitivity to material details of the normal state at elevated temperatures), issue 7538, doi 10.1038/nature14165, the review itself not fetched; a later review at its title in a citing list, Stranger than metals, Science 377, 2022, a reading for the next pass. Persistence, no landing: the closing condition at the review's abstract read at pass two (a qualitative understanding lacking across the material families; the linear resistivity's origin), an account holding across the families or separate origins demonstrated standing there as the review's own alternatives. Approaches: the dissipation-bound and solvable-model routes at that review's sections. Remaining properties: keyed at the next pass. *The report.* **Approaches.** The dissipation-bound reading — standing proposed and tested, its universality disputed in the record. Exactly solvable models — the correlated models yielding the linearity standing as theoretical anchors, their relation to the materials open. Material-by-material accounts — standing established in cases. **Narrowing recorded.** Material-by-material accounts narrow the reach: the linearity stands accounted for per material. Solvable models narrow the reach: the linearity stands derived in models whose relation to the materials is open. **Progress.** The transport record across families. The measured saturations of the proposed bound. The solvable models yielding the linearity. **Own record.** ## 174 The simplicity of proofs **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Whether criteria exist for the simplest proof of a theorem. **The field's account of its hardness.** That the problem, set down in a notebook in haste and in part barely legible, asks for criteria of simplicity or a proof of the greatest simplicity of certain proofs, and for a theory of the method of proof in mathematics in general, positing that under a given set of conditions there can be but one simplest proof and that where two proofs of a theorem exist one must keep going until each is derived from the other or it becomes quite evident what variant conditions and aids the two have used; that it was left out of the published list, described in the field as premature; and that a proof may be simpler than one previously known in one or more ways that include length, size measured by the total symbol count, and term structure. **Accounts.** Criteria as measures: a proof simpler than another in one or more ways that include length, size by total symbol count and term structure, the proofs first generated by automated theorem provers far from optimal by any such measure, the larger question of what reasonable measures of proof simplicity might be still open, and the notion of a simplest algorithm the same question met in computational complexity. Simplicity as a philosophical challenge rather than a purely formal exercise, read from the problem's own examples, syzygies first: the question never taken up by the school that posed it, its criteria of 1900 become a single criterion by 1917, and one obvious possibility for its omission from the published list being that no way was seen to state it precisely enough. **Specifications.** Proof size and depth are formalizable measures — universal under proof complexity. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** A criterion of simplicity for proofs of a theorem. **Conserving reach.** All proofs of the theorem. **Reach.** The statement quantifies over the simplest proof. Formalized measures of proof size and depth, with lower bounds proved in restricted systems, are what is measured. The two are not the same. **Unreachability.** **Persistence.** A canonical formulation maintained by the field, with defended positions upon it. **Addresses.** Statement, the problem in Hilbert's notebook and its history: Thiele, "Hilbert's Twenty-Fourth Problem", American Mathematical Monthly 110:1-24, 2003. Hardness: the paper the address names, verified at the publisher's page at issue 1 and doi 10.1080/00029890.2003.11919933, at the notebook entry as the paper translates it, carried verbatim at an encyclopedia page and at a later journal article discussing the problem (criteria of simplicity, or proof of the greatest simplicity of certain proofs; a theory of the method of proof in general; under a given set of conditions but one simplest proof; two proofs to be pursued until each is derived from the other or the variant conditions and aids become evident; the entry written in haste and in part barely legible; the problem left out of the published list as premature), the paper itself not fetched; the companion paper, Thiele and Wos, "Hilbert's Twenty-Fourth Problem", Journal of Automated Reasoning 29:67-89, 2002, at its abstract as the publisher's page carries it verbatim (the problem focused on the finding of simpler proofs and criteria for measuring simplicity; a proof simpler in one or more ways including length, size by total symbol count, and term structure), not fetched; the later article's author not carried at the return. Accounts: the measures at the companion paper read for the hardness (length, size by total symbol count, term structure), at a 2018 preprint on proof simplification at its abstract as the arXiv copy carries it (automated proofs far from optimal by any measure of simplicity; the larger, still open question of what reasonable measures of proof simplicity might be), arXiv:1808.04251, its author carried at the copy, and at the later article's passage on the simplest algorithm as the publisher's page carries it; the philosophical reading at that article's text as the publisher's page and an archive copy carry it (the question never taken up by the school; criteria in 1900, a criterion in 1917), Philosophical Transactions of the Royal Society A, 2019, doi 10.1098/rsta.2018.0040, one of eleven contributions to a theme issue on the notion of simple proof, its author not carried at the returns, at a research project's description at the project's pages (the problem reassessed as a philosophical challenge rather than a purely formal exercise; syzygies the starting point), and at a 2015 preprint at its motivation section as the arXiv copy carries it (one obvious possibility that no way was seen to state it precisely enough), arXiv:1504.03380, its author not carried; none fetched. Persistence, no landing: the closing condition at the paper read at v342 (the criteria of 1900 and the criterion of 1917; the philosophical challenge never taken up), a canonical formulation maintained by the field standing there as what the paper finds absent. Approaches: the proof-size, lower-bound and automated-reasoning routes at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Proof-size and depth measures — standing formalized. Lower bounds in restricted systems — standing proved. Automated reasoning and proof-complexity revivals — standing in the field's record. **Narrowing recorded.** Lower bounds narrow the reach: the results stand for restricted proof systems. **Progress.** The formalized measures. The lower bounds proved in restricted systems. **Own record.** The statement stands recorded from its archive; no canonical formulation stands maintained upon it. None stands on the statement as posed. The field's current record carries the question through proof theory and proof complexity, without a maintained canonical formulation of simplicity as the statement intends. ## 175 Goldbach's conjecture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is every even number greater than two the sum of two primes? **The field's account of its hardness.** That sieve theory was invented to count prime patterns and has proved able to give good upper bounds while remaining largely unable to lower-bound the primes themselves. The obstruction is stated in the field's own terms: where a set's elements are all products of an odd number of primes, or all products of an even number, sieve methods without additional ingredients give no non-trivial lower bound on its size, and any upper bound is off by a factor of two or more. The principle's originator described sets of integers as tending to be evenly distributed with respect to the parity of their number of prime factors unless constructed with a built-in bias. The field records that even reproving the infinitude of the primes by sieve alone is extremely difficult, and that the obstruction persists under strong distributional conjectures. **Accounts.** Every such number is a sum of two primes: verified past 4 × 10¹⁸, heuristic density arguments predict the number of representations grows rather than shrinks, and near-results are strong — every sufficiently large odd number is a sum of three primes, proved unconditionally. **Specifications.** The additive and multiplicative structures of the integers relate in a way a proof could exhibit — local. **Specifications withdrawn.** **Common requirement.** That the statement is true or false of all even numbers. **Conserving relation.** A proof holding for every even number greater than two. **Conserving reach.** All even numbers greater than two. **Reach.** The statement quantifies over all even numbers. Verification to a bound, and asymptotic methods with stated limits, are what is measured. The two are not the same. **Unreachability.** The circle method proves the ternary case and the binary case lies beyond its reach for reasons the method itself makes precise; and the parity obstruction in sieve theory is a known and stated block to reaching two primes, the last step not being a matter of refinement. The proof is not reached over the even numbers by computation, the circle method, or sieve methods. **Persistence.** A proof, or a counterexample. **Addresses.** Statement, in substance: the conjecture as the four problems listed at the international congress of 1912 carry it, characterised there as unattackable at the present state of mathematics, as an encyclopedia page states them (can every even integer greater than two be written as the sum of two primes; all four unresolved), and as a preprint on the conjecture's history carries it (the numerical data overwhelming and all attempts at proof completely unsuccessful; the assessment at a 1921 address that it appeared probably as difficult as any unsolved problem in mathematics), arXiv:1205.0774, its author not carried; the congress address and the letter of 1742 not fetched; the Statement standing as it stood. Hardness, and the statement of the obstruction: Tao, "Open question: the parity problem in sieve theory", 2007, and the parity-problem summary at Wikipedia, "Parity problem (sieve theory)". Selberg's own formulation and the 1949 naming: Selberg, as quoted in "Twin primes and the parity problem", Journal of Number Theory, 2017, §1. The sieve result as the exhibition of the barrier, and bilinear-form parity breaking: arXiv:math/0507569, §1. The bounded-error obstruction argument and its stated escapes: Tao, "The parity problem obstruction for the binary Goldbach problem with bounded error", 2014. Parity-sensitive sieves: the Wikipedia summary, citing Friedlander and Iwaniec from 1996. Persistence, no landing: the closing condition at the four problems as listed and the history read at v342 (all attempts unsuccessful; a proof or a counterexample the field's own posing). Approaches: the computation and circle-method routes at the history preprint and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Computation — verified past 4 × 10¹⁸, returning more of the same domain. The circle method — proves the ternary case by estimating the relevant exponential sums, the binary case beyond its reach for reasons the method makes precise. Sieve methods — every sufficiently large even number a prime plus a number with at most two prime factors, recorded as the quintessential exhibition of the parity barrier, sieve methods counting numbers with few prime factors more readily than numbers with exactly one. Bilinear form estimates — the ternary resolution introduced the idea that estimating certain bilinear forms can break the barrier, with further instances recorded since; the binary case is not among them. Parity-sensitive sieves — developed from the late 1990s and recorded as making the obstruction less of an obstacle, without removing it here. Bounded-error variants — an argument recorded that the obstruction rules out modifications of the bounded-gaps method for the binary case, with the stated escapes being a failure of the underlying randomness principle or control of combinations of more than one multiplicative function. Density and heuristic arguments — predict many representations, not proofs, and known to be unable to become proofs by refinement. Assuming the Riemann hypothesis — improves bounds and does not close the binary case. **Narrowing recorded.** The ternary result narrows the reach: the sum-of-three statement stands proved for sufficiently large odd numbers. Sieve results narrow the relation: a prime plus a near-prime stands proved in place of two primes. **Progress.** The verification to 4 × 10¹⁸. The unconditional ternary theorem. The sieve result and the stated parity obstruction. **Own record.** ## 176 Twin primes **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Are there infinitely many pairs of primes differing by two? **The field's account of its hardness.** That sieve theory was invented to count prime patterns and has proved able to give good upper bounds while remaining largely unable to lower-bound the primes themselves. The obstruction is stated in the field's own terms: where a set's elements are all products of an odd number of primes, or all products of an even number, sieve methods without additional ingredients give no non-trivial lower bound on its size, and any upper bound is off by a factor of two or more. The principle's originator described sets of integers as tending to be evenly distributed with respect to the parity of their number of prime factors unless constructed with a built-in bias. The field records that even reproving the infinitude of the primes by sieve alone is extremely difficult, and that the obstruction persists under strong distributional conjectures. **Accounts.** There are infinitely many: an explicit asymptotic density for twin pairs is conjectured from a heuristic in which primes behave randomly subject to their known congruence constraints, and it matches computation closely over every range checked. The same heuristic predicts densities for other prime constellations, and those predictions have matched wherever they can be tested. **Specifications.** Prime distribution is governed by a density — local. A density argument can establish infinitude of a specific gap — local. **Specifications withdrawn.** **Common requirement.** That the statement is true or false of the integers. **Conserving relation.** A proof of infinitude for the gap of two. **Conserving reach.** All primes. **Reach.** The statement quantifies over all primes. Sieve methods with a proved obstruction, and computation to a bound, are what is measured. The two are not the same. **Unreachability.** The parity obstruction is proved: sieve methods cannot distinguish numbers with an even number of prime factors from an odd number, so the reduction from the bounded gap to two is blocked by parity rather than by effort. The proof is not reached over the primes by sieve methods or computation. **Persistence.** A proof of infinitude for the gap of two, or a proof that the pairs are finite. **Addresses.** Statement, in substance: the conjecture as the four problems listed at the international congress of 1912 carry it, characterised there as unattackable at the present state of mathematics, as an encyclopedia page states them (are there infinitely many primes such that the prime plus two is prime; all four unresolved), and as a chapter on unsolved problems in a guide to elementary number theory carries it (the primes thinning out but pairs two apart found as far out as anyone has looked; the conjecture that there are infinitely many, called undoubtedly true), Mathematical Association of America, 2009, its author not carried at the return; the congress address not fetched; the Statement standing as it stood. Hardness, and the statement of the obstruction: Tao, "Open question: the parity problem in sieve theory", 2007. Selberg's formulation and the naming: "Twin primes and the parity problem", Journal of Number Theory, 2017, §1. The bound of six under the generalised conjecture, and the argument that no sieve-theoretic method does better: Polymath, "Variants of the Selberg sieve, and bounded intervals containing many primes", arXiv:1407.4897, §8. The closest classical approximation as an exhibition of the barrier: arXiv:math/0507569, §1. The conditional route through irregular zeros, and the constant-multiple evasion: Tao, "The parity problem obstruction for the binary Goldbach problem with bounded error", 2014, and remark 1 at the same author's parity-problem notes. Persistence, no landing: the closing condition at the four problems as listed and the guide's chapter read at v342 (infinitely many pairs conjectured, called undoubtedly true, unproved). Approaches: the computation and sieve routes at the sources the address names, read at pass one. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Computation — very large twin pairs found, with no bearing on infinitude. Sieve methods — infinitely many primes whose successor at distance two is prime or a product of two primes, recorded as the closest classical approximation and as the exhibition of the barrier. Bounded gaps — successive results reaching a gap of 246, with the field's own analysis arguing that the barrier prohibits sieve-theoretic methods, including those used, from any bound stronger than six even under the generalised distributional conjecture and even using sieves other than the one named. Parity-sensitive sieves — developed from the late 1990s, recorded as making the obstruction less of an obstacle, without reaching two. Related-multiplicative-function routes — the obstruction shown to be evadable in some cases by using linear forms that are constant multiples of one another, with instances recorded. Correlation conjectures — modified forms of the multiplicative-correlation conjectures formulated, with averaged and logarithmically averaged versions proved. Conditional routes through irregular zeros — a recorded result that a strong violation of the underlying randomness principle would imply the conjecture. The convergence of the sum of reciprocals of twin primes — consistent with finitely or infinitely many. Random models of the primes — predict the density accurately and are models. **Narrowing recorded.** Bounded-gap results narrow the relation: infinitude stands proved for some gap at most 246 in place of the gap of two. **Progress.** The bounded-gap results and their stated floor. The proved parity obstruction. The convergence result. The measured agreement of the conjectured density with computation. **Own record.** ## 177 The fold as a kind **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Roughly a thousand distinct folds account for most known protein structures, and unrelated sequences converge on the same fold. What makes two structures the same fold? **The field's account of its hardness.** That the classifications disagree with each other on the assignments they make, that the disagreement concentrates at the boundaries between folds rather than at their centres, and that the two long-standing databases are built on different criteria — one on evolutionary relationship read from structure, one on structural comparison — with no procedure settling a case where they differ. And that the classification schemes in widespread use implicitly assume the organization of protein structure space into discrete categories, where an alternative view has emerged in which fold space is continuous and multidimensional, evolutionary connections confined mainly to separate structural prototypes corresponding to folds as islands of structural stability while for a geometric similarity measure a cutoff can usually be found yielding paths that connect any two structures through intermediates. **Accounts.** Folds are discrete kinds: structures cluster, the clusters recur across unrelated sequences, and the recurrence is what a fold names. Folds are regions of a continuum: structure space is continuous, the clusters are dense regions in it, and the boundaries are drawn by the classifier's own thresholds. **Specifications.** A fold is an equivalence class of structures under a similarity relation — universal under structural classification. Structures are comparable by a metric on their coordinates — universal under structural alignment. **Specifications withdrawn.** **Common requirement.** That there is a similarity relation on structures under which a fold is well-defined, the dispute concerning where its boundary falls. **Conserving relation.** Two structures assigned to the same fold, the assignment standing independently of which classification makes it. **Conserving reach.** All pairs of structures, boundary cases included. **Reach.** The statement quantifies over the fold assignment of an arbitrary structure pair. Assignments made by particular classifications under particular criteria are what is measured. The two are not the same. **Unreachability.** Structural alignment scores are continuous and the assignment is a cut on them, the cut declared and not measured; classifications built on different criteria differ at the boundary cases and agree at the centres; no procedure settles a case where they differ without adopting one classification's criterion as the standard. **Persistence.** The fold assignment standing for an arbitrary structure pair, boundary cases included. **Addresses.** Statement, in substance: the review the address names, at its abstract as the publisher's page carries it verbatim, read this session under check C18 (identification of geometric relationships between structures; the classification schemes in widespread use implicitly assuming an organization of protein structure space into discrete categories; an alternative view in which fold space is continuous and multidimensional), the review itself not fetched; the Statement standing as it stood, its count of about a thousand folds keyed at the next pass. The classifications and their criteria: Murzin et al., "SCOP: a structural classification of proteins database", Journal of Molecular Biology 247:536-540, 1995; Orengo et al., "CATH — a hierarchic classification of protein domain structures", Structure 5:1093-1108, 1997. The disagreement and the continuity: Sadowski and Taylor, "Are proteins in structure space?", Journal of Molecular Biology, 2010; Kolodny, Petrey and Honig, "Protein structure comparison: implications for the nature of fold space", Current Opinion in Structural Biology 16:393-398, 2006. Hardness, its added clause: the review the address names, at its abstract as the publisher's page carries it verbatim (classification schemes in widespread use implicitly assuming an organization into discrete categories; an alternative view in which fold space is continuous and multidimensional), issue 3, doi 10.1016/j.sbi.2006.04.007, the review itself not fetched; and two later papers at passages the publisher's pages carry (evolutionary connections confined to separate prototypes as islands of structural stability; a geometric cutoff usually yielding paths connecting any two structures), one on a discrete-continuous duality of structure space and one, in the Proceedings of the National Academy of Sciences, 2009, on the continuity of structure space as an intrinsic property of proteins, their authors not carried at the returns; the line's three former clauses, the classifications disagreeing on assignments, the disagreement concentrating at the boundaries, and the two databases' differing criteria with no settling procedure, not carried at the returns and standing keyed at the next pass at the four sources named. Persistence, no landing: the closing condition at the review read under check C18 (discrete categories against a continuous fold space), the fold assignment for an arbitrary pair standing there as what a continuous space denies. Approaches: the evolutionary and geometric classification routes at that review's sections. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Evolutionary classification — assigns by relationship inferred from structure, the inference disputed at the boundary. Geometric classification — assigns by alignment score against a threshold, the threshold declared. Structure-space embedding — places structures in a continuous space and reports that the clusters have no sharp boundaries. Sequence-independent comparison — finds recurrence across unrelated sequences, which every account carries and which settles no assignment. **Narrowing recorded.** Each classification narrows the reach: the assignment stands under that classification's criterion. **Progress.** The recurrence, undisputed. The count of clusters at any chosen threshold. The agreement at cluster centres. **Own record.** ## 178 The single native state **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A protein is said to fold to one native structure determined by its sequence. What is the native state of a sequence whose measured structure is an ensemble? **The field's account of its hardness.** That a substantial fraction of eukaryotic proteins carry regions that fold to no single structure and are functional in that condition; and that single-structure models derived from X-ray data do not adequately account for the inherent, functionally important dynamics of protein molecules, ensembles of structures generated by time-averaged refinement fitting the X-ray data better than single structures across twenty datasets, most proteins displaying a well-ordered core and some a molten core, so the reported single set of coordinates stands for a set of structures the data fit better together. **Accounts.** The native state is one structure, and ensembles are the exception: most proteins fold to a defined structure, and the disordered ones are a distinct class. The native state is an ensemble, and one structure is the exception: every protein occupies a distribution of conformations, and a single reported structure is the distribution's most populated member or its average. **Specifications.** A protein has a native state which is the global minimum of its free energy — universal under the thermodynamic hypothesis. A structure is a set of atomic coordinates — universal under structural biology. **Specifications withdrawn.** **Common requirement.** That there is a native state a sequence determines, the dispute concerning whether it is one conformation or a distribution. **Conserving relation.** The native state determined for a sequence, the determination standing independently of the measurement that reports it. **Conserving reach.** All sequences, disordered and ensemble-carrying sequences included. **Reach.** The statement quantifies over the native state of an arbitrary sequence. Ensemble-averaged measurements and single fitted models are what is measured. The two are not the same. **Unreachability.** A crystallographic model is fitted to data averaged over many molecules and many models fit one dataset equally well; ensemble-refinement methods return distributions rather than resolving among them; disordered regions return no single structure at any resolution reached; a prediction method trained on single deposited structures returns single structures by construction and is not measured against ensembles. **Persistence.** The native state determined for an arbitrary sequence, ensemble-carrying sequences included. **Addresses.** Statement, in substance: the ensemble-refinement paper the address names, at its abstract as four pages carry it verbatim, read this session under check C18 (single-structure models derived from X-ray data not adequately accounting for the inherent, functionally important dynamics of protein molecules; ensembles fitting the data better than single structures), the paper itself not fetched; the Statement standing as it stood, its first sentence's claim that a protein folds to one native structure determined by its sequence keyed at the next pass at the disorder review. Model degeneracy against one dataset and ensemble refinement: Burnley et al., "Modelling dynamics in protein crystal structures by ensemble refinement", eLife 1:e00311, 2012. Disorder as functional and widespread: Wright and Dyson, "Intrinsically disordered proteins in cellular signalling and regulation", Nature Reviews Molecular Cell Biology 16:18-29, 2015. Hardness, its second clause: the ensemble-refinement paper the address names, at its abstract as the publisher's page, a data repository, an index and an archive copy carry it verbatim (single-structure models from X-ray data not adequately accounting for the inherent, functionally important dynamics; ensembles by time-averaged refinement; twenty datasets at 1.1 to 3.1 angstroms, the cross-validated free R reduced by 0.3 to 4.9 per cent, ensemble models fitting the data better than single structures; most proteins a well-ordered core, some a molten core), published 18 December 2012, doi 10.7554/eLife.00311, the paper itself not fetched; the former second and third clauses, a model as a single set of coordinates fitted to data averaged over many molecules and many models fitting one dataset equally well, released with their content at the clause re-formed; the first clause, on functional disorder, standing at the disorder review the address names, not searched this pass, keyed at the next. Persistence, no landing: the closing condition at the paper read under check C18 (single-structure models not accounting for the dynamics; ensembles fitting better), the native state for an arbitrary sequence standing there as what the ensembles put in question. Approaches: the crystallographic and ensemble-refinement routes at that paper. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Crystallography — returns one model fitted to averaged data, the averaging stated in the method. Ensemble refinement — returns a distribution fitting the same data, and the distribution is not unique. NMR — returns a set of models consistent with restraints, the set reported rather than resolved. Single-particle cryo-EM — classifies particles into states, the classification count declared. Learned prediction — returns one structure per sequence, trained on single deposited structures. **Narrowing recorded.** Ensemble refinement narrows the relation: a distribution consistent with the data stands, and which distribution does not. **Progress.** The structures reported across the observed domain. The measured degeneracy of models against one dataset. The measured extent of disorder across proteomes. **Own record.** ## 179 The explanatory gap **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** A physical account of a process states which physical facts obtain, and it leaves open why those facts are accompanied by experience of the character they are accompanied by. Is the gap in the explanation, or in what the explanation is about? **The field's account of its hardness.** That the intuition of felt contingency about psychophysical identity statements supports not the metaphysical thesis that such statements must be false but a closely related epistemological thesis, that they leave a significant explanatory gap, with the corollary that there is no way of determining exactly which of them are true; that one cannot conclude from this that materialism is false, the argument constituting a problem for materialism that better captures the uneasiness many philosophers feel about the doctrine; and that the residue is that nothing in the account says why the state feels as it does rather than otherwise or at all. **Accounts.** The gap is epistemic: the facts are physical and an account of them will close the gap, or the gap follows from how the physical is represented and not from what obtains. The gap is ontological: an explanatory residue standing where identities elsewhere leave none is evidence that the facts are not exhausted by the physical. The gap is permanent and neither: the terms available to a self do not couple in the way an explanation of this kind would require, and no account either closes it or licenses a conclusion about what exists. **Specifications.** An adequate explanation of an identity leaves no residue — local. The residue is a property of the explanation and not of the explainer — local. **Specifications withdrawn.** **Common requirement.** That an explanatory residue either does or does not bear on what exists. **Conserving relation.** A determination of whether the residue bears on what exists, standing on what an explanation delivers. **Conserving reach.** All accounts of a physical process together with an experiential character. **Reach.** The statement quantifies over what an explanation delivers. Explanations given, and reports of what they leave open, are what is measured. The two are not the same. **Unreachability.** A residue read at an explanation is read by a self carrying the concepts the explanation is given in, and no procedure separates a residue in the account from a residue in the reading. The determination is not reached over all accounts by explanations given. **Persistence.** An account of a physical process from which the experiential character follows in the way the character of water follows from its account, agreed by both. **Addresses.** Statement, in substance: the paper the address names, at passages of its text as two repository copies and an index carry them verbatim, read this session under check C18 (psychophysical identity statements leaving a significant explanatory gap; the thesis epistemological rather than metaphysical; not concluding that materialism is false), the paper itself not fetched; the Statement standing as it stood, its second question, whether the gap is in the explanation or in what the explanation is about, the file's parting of the epistemic and the metaphysical readings, keyed at the next pass at the conceivability-argument source. The term and the epistemic framing: Levine, "Materialism and Qualia: The Explanatory Gap", Pacific Philosophical Quarterly 64:354-361, 1983. Its standing beside the conceivability argument: Chalmers, "Facing Up to the Problem of Consciousness", Journal of Consciousness Studies 2(3):200-219, 1995, §3. Hardness, its first two clauses: the paper the address names, at passages of its text as two repository copies and an index carry them verbatim (the intuition supporting an epistemological thesis rather than the metaphysical one; psychophysical identity statements leaving a significant explanatory gap, with the corollary that there is no way of determining exactly which are true; the argument not concluding that materialism is false, a weaker attack that better captures the uneasiness many philosophers feel), issue 4, October 1983, doi 10.1111/j.1468-0114.1983.tb00207.x, an earlier version delivered in 1982 under another title as a copy's note carries it, the paper itself not fetched; the last clause, why the state feels as it does rather than otherwise or at all, standing at a passage not returned, keyed at the next pass; the former opening, stated as a gap in explanation and marked as not by itself a claim about what exists, released with its content at the first clause; the conceivability-argument source not read this pass. Persistence, no landing: the closing condition at the paper read under check C18 (the explanatory gap between the functions and experience; the water analogy the paper's own standard), an account from which the experiential character follows as water's does standing there as the paper's own standard. Approaches: the further-physical-account route at that paper and the sources the address names. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Closing by further physical account — the gap held to narrow as the functions are explained, with the residue reported unchanged at each closing. The phenomenal concept strategy — the residue accounted for by the character of the concepts an explanation is read in, its critics arguing that the strategy either fails to account for the concepts physically or concedes the point. Denying the residue — the further question held not to stand once the functions are explained, defended and a minority position. Holding the gap permanent — the residue held to follow from the concepts available and to license no conclusion about what exists. **Narrowing recorded.** The phenomenal concept strategy weakens the relation: the residue stands accounted for at the reading rather than determined at the account. **Progress.** The residue reported unchanged across four decades of further physical account. The parting of the epistemic from the ontological claim, stated at the term's own introduction. **Own record.** The gap is stated as a gap in explanation, and the step from it to a claim about what exists is a separate step the term itself does not take. ## 180 The universal totality **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Russell's paradox, as the field names it: for any property there is the set of exactly the things having it; in particular, there is a set of all sets. Does the set of all sets that are not members of themselves contain itself? And the question that persists after the paradox is resolved by restricting comprehension: whether there is a totality of all sets, and whether quantification over absolutely everything is coherent. **The field's account of its hardness.** That the totality is both demanded and forbidden: every account gathering all sets yields a set the gathering does not contain, and the concept extends past each totality proposed for it; that the paradox stems, in the field's account, from the idea that any condition or property may be used to determine a set, and the originator of modern set theory was himself an early skeptic of an unrestricted comprehension axiom, parting multiplicities into those that can be thought of as one finished thing and those that cannot and placing the totality of all sets among the latter; and that because set theory underlies all branches of mathematics, the paradox raised the worry that no mathematical proof could be completely trustworthy until it was eliminated. **Accounts.** Comprehension restricted: sets arise only by separation from given sets, and no universal set exists. The universal set retained, with the comprehension or the logic weakened to hold it. The totality standing as a proper class, gatherable and a member of nothing. The concept indefinitely extensible: every proposed totality of sets is exceeded by a set the proposal itself defines. **Specifications.** Every determinate property collects a set — universal under naive comprehension. Membership divides cases — universal under classical semantics. **Specifications withdrawn.** Unrestricted comprehension, withdrawn with the accounts retained, the refutation identifying no member of the five. **Common requirement.** That all sets can be gathered and quantified over as one completed reach. **Conserving relation.** Membership divides cases, a thing in a set or not, the division standing without exception. **Conserving reach.** All sets, the gathering itself included. **Reach.** The statement quantifies over absolutely all of the kind. The theories' quantifiers range over domains each of which is again a totality the next quantification exceeds. The two are not the same, and the accounts state the parting themselves. **Unreachability.** The comprehension instance for non-self-membership yields a member exactly when it is not one. The power of any totality strictly exceeds it, and no greatest totality stands. The division is not preserved over the full reach by any account retaining both: each repair on record narrows the reach, weakens the division, or places the totality outside the membership relation it was to close under. **Persistence.** Exhibition of a completed totality of all sets, closed under its own gathering, with membership classical across it — or a demonstration that quantification over all sets requires none. **Addresses.** Statement: Irvine and Deutsch, "Russell's Paradox", Stanford Encyclopedia of Philosophy, at passages of its text as the encyclopedia's archived editions carry them verbatim (the paradox ultimately stemming from the idea that any condition or property may be used to determine a set; set theory underlying all branches of mathematics, the worry that no mathematical proof could be completely trustworthy, only the paradox's elimination regaining consistency), the entry itself not fetched, its section numbers not carried at the returns, the editions carrying copyrights of 2013, 2014, 2016 and 2020; the absolute-generality question at Rayo and Uzquiano, editors, Absolute Generality, Oxford, 2006, introduction, 1-19, and Parsons, "The Problem of Absolute Universality", 203-219 there, not returned, keyed at the next pass. Hardness: the same passages (the originator of modern set theory an early skeptic concerning an unrestricted comprehension axiom); the parting by the field's founder of multiplicities into those that can be thought of as one finished thing and those that cannot, standing at the field's surveys of his late correspondence as the address stood before this pass, not returned, keyed at the next pass; the former line's closing clause, that the account of the hardness arrives from the account's own author, in the file's own voice, released. Accounts: the first, comprehension restricted, at the entry's passage on type theory (referring to all objects for which a condition holds only if they are all of the same type); the other three keyed at the next pass. The address stood before this pass as: Statement, hardness, accounts: Irvine and Deutsch, "Russell's Paradox", Stanford Encyclopedia of Philosophy; the absolute-generality question: Rayo and Uzquiano, editors, Absolute Generality, Oxford, 2006, introduction, 1-19, and Parsons, "The Problem of Absolute Universality", 203-219 there; the founder's parting of multiplicities as the field's surveys report his late correspondence. Persistence, no landing: the closing condition at the entry's passages read at v342 (the paradox stemming from unrestricted comprehension; the founder's parting of multiplicities), a completed totality exhibited or quantification over all sets shown impossible standing there as the two directions the parting leaves. Approaches: the separation-and-foundation and type-theoretic routes at the entry's passages read at v342. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Separation and foundation — the reach narrowed to subsets of given sets, the standard resolution and the standard theory. Type stratification — the division kept by ranking the reach. The universal set retained under stratified comprehension. Proper classes — the totality moved out of the membership relation's range. Indefinite extensibility — the reach denied completion by the concept's own extending. Each approach is placed by the field against the four broad response types available to any paradox of this form. **Narrowing recorded.** Separation narrows the reach to what given sets already contain. Class theory moves the totality outside the relation. Extensibility denies the reach a last. **Progress.** The failure carries at theorem grade, the accounts themselves stating it. Every consistent repair on record is recorded as reach-narrowing or division-relocation, and the recurrence of the totality question at each repair — the class of all classes, the domain of all domains — is stated in the field's own surveys. **Own record.** The argument contains no material of any kind: sets are incidental, and the same argument runs at any totality gathered under its own kind — the catalogue of all catalogues, the word for the words not naming themselves. ## 181 Quantum gravity **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What theory gives a quantum account of gravitation and recovers general relativity and quantum field theory in the domains where each is established? **The field's account of its hardness.** A quantum theory of gravity remains under construction: the established theories employ different structures, direct empirical access to the scale at which their conflict is expected is absent, and the principal programs do not yet supply one experimentally confirmed completion. **Accounts.** Gravitation is quantized through extended fundamental objects and a higher-dimensional theory. Spacetime geometry has discrete quantum states and no fixed background. A conventional field theory reaches a non-Gaussian ultraviolet fixed point. Spacetime and gravitation emerge from non-gravitational quantum degrees of freedom. **Specifications.** Quantum mechanics applies to the gravitational field — universal under quantum theory. The classical spacetime dynamics are recovered at low curvature — universal under general relativity. Established particle-physics predictions are recovered at accessible energies — universal under quantum field theory. **Specifications withdrawn.** **Common requirement.** One account yielding gravitational and quantum predictions in one mathematical framework and recovering both established limits. **Conserving relation.** The account returns one probability assignment for every physical process in its domain and the assignment reduces to the established prediction in each tested limit. **Conserving reach.** Gravitational systems at quantum and classical scales, including regimes in which both gravity and quantum effects are material. **Reach.** The statement ranges over a joint quantum-gravitational domain. Present experiments test the low-energy limits and indirect consequences of candidate frameworks. The two are not the same: the characteristic scale is so far out of the range of experiment that direct tests have long seemed impossible, the proposals resting on very small deviations from otherwise exact symmetries or on collective effects integrated over long distances, no single experimental approach can directly probe the quantum-gravitational regime while numerous models give different subtle modifications to low-energy physics, and recovering the classical theory at low energies is a consistency requirement rather than a prediction to be tested. **Unreachability.** **Persistence.** A mathematically complete account with a discriminating empirical result and the two established limiting theories recovered. **Addresses.** Statement, hardness, accounts, approaches and progress: "Quantum Gravity", Stanford Encyclopedia of Philosophy, at the passages the encyclopedia's current and archived pages carry verbatim (the definition of a quantum theory of gravity, the non-renormalisability of the quantised spin-two theory, the problem of time in the canonical approaches, the string and loop programs), the archived editions carrying the name Weinstein and the current edition's author line, section numbers and revision date not confirmed. The relation at Reach: a progress report on quantum gravity at its section on direct tests as the arXiv copy carries it (the Planck energy so far out of the range of experiment that direct tests have long seemed impossible; the proposals resting on small deviations from exact symmetries or effects integrated over long distances), arXiv:gr-qc/0108040, its author not carried; a workshop's page on observations (no single experimental approach directly probing the regime; numerous models with different subtle low-energy modifications; no experiment today providing evidence that gravity requires a quantum description); a preprint on phenomenology through Lorentz violation at its introduction (the semiclassical limit a consistency requirement rather than a true prediction), arXiv:0706.0142; none fetched. Persistence, no landing: the recovery of the two limiting theories as the field's consistency requirement at the preprint read at Reach, and the discriminating empirical result at the progress report's statement that direct tests have long seemed impossible and the workshop's that no experiment yet provides evidence gravity requires a quantum description; the mathematical completeness keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** String-based unification — mathematically extensive and without a unique empirically selected low-energy realization. Loop and spin-network quantization — background independence and discrete geometry developed, with the full classical and empirical recovery open. Asymptotic safety — fixed-point evidence in truncations, with completion and observation open. Emergent and holographic programs — gravity reconstructed in restricted settings, with general physical reach open. **Narrowing recorded.** Semiclassical gravity and effective field theory narrow the reach to regimes below the quantum-gravity scale and return controlled predictions there. **Progress.** Consistent low-energy effective descriptions, black-hole thermodynamic constraints, discrete and holographic constructions, and candidate ultraviolet behaviors. **Own record.** ## 182 The primordial lithium problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why is the primordial lithium-7 abundance inferred from old metal-poor stars lower than the abundance predicted by standard big-bang nucleosynthesis using the independently measured baryon density? **The field's account of its hardness.** The predicted abundance exceeds the stellar plateau by a factor of about three while the other light-element abundances broadly agree, and proposed nuclear, stellar and new-physics resolutions have not removed the discrepancy without creating another conflict. **Accounts.** The stellar atmospheres do not preserve the primordial abundance because lithium is depleted or transported. The nuclear reaction network or its measured rates omit an effect that lowers the predicted abundance. Physics outside the standard cosmological model alters nucleosynthesis or destroys lithium afterward. The abundance inference carries an unrecognized systematic offset. **Specifications.** The baryon density inferred from the cosmic microwave background applies during nucleosynthesis — universal under standard cosmology. The stellar plateau preserves or permits reconstruction of the primordial abundance — local. The standard nuclear network contains the material production and destruction channels — local. **Specifications withdrawn.** **Common requirement.** One primordial lithium abundance consistent with nucleosynthesis, the baryon density and the observed light-element record. **Conserving relation.** Predicted and reconstructed primordial abundances agree within their stated uncertainties. **Conserving reach.** Lithium-7 together with the other primordial light nuclides and the independently measured baryon density. **Reach.** The statement ranges over primordial lithium. The direct record consists of present stellar spectra and cosmological and nuclear inputs from which the primordial value is inferred. The two are not the same: the primordial abundance is derived from the spectroscopy of metal-poor halo stars, and the controversy concerns whether the observed plateau reflects the true primordial value or has been depleted from an initially higher abundance, the plateau's flatness and small scatter the main difficulty for depletion, while stars identical to plateau stars with no detectable lithium and the plateau's meltdown at the lowest metallicities show that some depletion mechanism exists. **Unreachability.** **Persistence.** An account that closes the lithium discrepancy while retaining the successful light-element and cosmological predictions. **Addresses.** Statement, hardness, accounts and approaches: Fields, "The Primordial Lithium Problem", Annual Review of Nuclear and Particle Science 61:47-68, 2011, at its abstract as the publisher's page carries it verbatim (the factor of three to four below the prediction, the four-to-five sigma mismatch, the three classes of solution), the paper itself not fetched. The relation at Reach: a review of cosmological solutions at its abstract as the arXiv page carries it (the abundance derived from the spectroscopy of metal-poor halo stars; a factor of three below the prediction; astrophysical, nuclear or cosmological resolutions), arXiv:1909.01245; a review of nucleosynthesis at its lithium section (the controversy whether the plateau reflects the primordial value or has been depleted; plateau-like stars with no detectable lithium), arXiv:astro-ph/9903183; a conference paper at its opening (the plateau flat with very small scatter the main difficulty for depletion), arXiv:0810.4290; a 2015 review at its lithium section (the meltdown at very low metallicity seeming to demand that some stars have depleted), arXiv:1505.01076; their authors not carried; none fetched. Persistence, no landing: the closing condition at the review read at Reach (the resolution astrophysical, nuclear or cosmological, each retaining the light-element concordance), arXiv:1909.01245. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Revised reaction rates and missing resonances — measured and constrained without a sufficient standard-network reduction. Stellar depletion and diffusion — mechanisms demonstrated, with the required uniform correction unsettled. Nonstandard particle decay and cosmology — parameter regions proposed and restricted by other nuclides and cosmological observations. Reanalysis of stellar temperatures and atmospheres — systematic shifts tested, with the main discrepancy remaining. **Narrowing recorded.** Nuclear measurements exclude many formerly allowed reaction-rate solutions; stellar models restrict depletion accounts by the small observed scatter. **Progress.** The baryon density, deuterium agreement, lithium plateau and leading reaction rates are measured more precisely than when the discrepancy was identified. **Own record.** ## 183 Proton spin decomposition **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How is the proton's spin one-half distributed among quark spin, gluon spin, quark orbital angular momentum and gluon orbital angular momentum? **The field's account of its hardness.** The separate terms depend on the decomposition, gauge treatment, renormalization scale and partonic information used, while experiments constrain different integrals over incomplete kinematic ranges rather than measuring all four contributions together. **Accounts.** The kinetic decomposition assigns gauge-invariant total angular momenta to quarks and gluons and separates the quark spin contribution. The canonical decomposition assigns spin and orbital terms to quarks and gluons using gauge-invariant extensions or specified gauges. Nonperturbative proton structure supplies scale-dependent contributions calculable from quantum chromodynamics. **Specifications.** The four contributions sum to one-half — universal under the spin sum rule. Each term is defined in a stated decomposition and at a stated scale — universal under quantum chromodynamics. Experimental observables admit factorization into the partonic quantities used — local. **Specifications withdrawn.** **Common requirement.** A complete decomposition whose terms are defined, calculable and connected to observables. **Conserving relation.** The contributions sum to the measured proton spin under one stated decomposition and scale. **Conserving reach.** All quark flavors, gluons, spin terms and orbital terms across the full momentum-fraction domain. **Reach.** The statement ranges over a complete decomposition. Scattering observables and lattice matrix elements constrain selected combinations and ranges. The field's review states that a gauge-invariant splitting of the total angular momentum into spin and orbital parts is possible but not unique, so what is measured as the gluon spin and the spin term of any one decomposition are not the same without a choice of decomposition, and which choice is the natural or physical one stands as the question. **Unreachability.** **Persistence.** A decomposition with all terms determined at a common scale, its observable connections fixed, and the spin sum rule closed within uncertainties. **Addresses.** Statement, hardness at the dependence of the terms on the decomposition and gauge treatment, and the relation at Reach: Leader and Lorcé, "The angular momentum controversy: What's it all about and does it matter?", Physics Reports 541:163–248, 2014, at its abstract as the arXiv page and the publisher's page carry it verbatim (the general question of splitting the total angular momentum of a photon or gluon into spin and orbital contributions; the textbook statement that no gauge-invariant splitting exists, beside experiments measuring what is believed to be the gluon spin; the claim that such a splitting is possible, and the finding that it is not unique), the review itself not fetched. Ji, "Gauge-Invariant Decomposition of Nucleon Spin", Physical Review Letters 78:610–613, 1997, and Xu et al., "Quark and gluon spin and orbital angular momentum in the proton", Physical Review D 108:094002, 2023, cited here and not read this pass; the present calculation at hardness stands keyed at the next pass. Approaches: the lattice route and its ending at a lattice review's abstract as the publisher's page, a directory and the arXiv copy carry it verbatim (the quark spin from the axial-vector current at three to four tenths, smaller than the constituent models, the disconnected insertion negative; the contentious interpretation of the spin crisis resolved by the lattice; the glue spin and angular momentum large, the quark orbital contribution notable; renormalization, mixing and normalization discussed), AAPPS Bulletin 32(1), 2022, doi 10.1007/s43673-022-00037-4, its author carried as Liu; the parton-distribution route at a 2015 review's summary as the arXiv copy carries it (orbital angular momentum from generalized and transverse-momentum distributions; lattice calculations with controlled errors imperative for comparison), arXiv:1508.00911, its authors not carried; the scattering route at a 2024 preprint's introduction (the quark helicity near three tenths well constrained, the gluon contribution significant with large uncertainties at small momentum fraction), arXiv:2404.04209; the model-Hamiltonian route keyed at the next pass; none fetched. Persistence, no landing: the closing condition at the 2015 review's summary read above (lattice calculations with both statistical and systematic errors controlled, compared with experiment from the distributions, to settle the decomposition). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Polarized deep-inelastic and proton-proton scattering — constrains quark and gluon helicity over measured momentum fractions. Generalized parton distributions — gives access to total quark and gluon angular momentum through sum rules. Lattice calculation — returns decomposition-dependent matrix elements at finite precision. Model Hamiltonians — return all terms within their chosen truncation. **Narrowing recorded.** Global fits narrow quark and gluon helicity integrals over measured momentum fractions; lattice calculations narrow selected total-angular-momentum terms. **Progress.** Quark helicity is known not to saturate the proton spin, gluon helicity is constrained over a growing range, and several gauge-invariant sum rules stand. **Own record.** ## 184 The solar abundance problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why do standard solar models using modern lower photospheric heavy-element abundances disagree with helioseismic measurements of the Sun's interior? **The field's account of its hardness.** Three-dimensional and non-local-thermodynamic-equilibrium atmosphere analyses lower the inferred abundances, but solar models built with them worsen agreement with the helioseismic sound-speed profile, convection-zone depth and surface helium abundance; proposed changes must preserve the separate neutrino and spectroscopic successes. **Accounts.** The photospheric abundance inference contains residual line-formation or atomic-data systematics. Radiative opacities in the solar interior are underestimated. The standard solar model omits mixing, accretion, mass loss, magnetic effects or another transport process. The helioseismic or compositional data require a combination of smaller revisions. **Specifications.** Present photospheric abundances represent the bulk composition supplied to the model — local. Opacity and equation-of-state inputs apply throughout the solar interior — local. A solar model must reproduce helioseismic, neutrino and surface constraints together — universal under standard solar modelling. **Specifications withdrawn.** **Common requirement.** One solar composition and interior model consistent with spectroscopy, helioseismology and neutrino observations. **Conserving relation.** Model predictions agree with all three observational classes within stated uncertainties. **Conserving reach.** The photosphere and solar interior across the radiative and convective zones. **Reach.** The statement ranges over the Sun's composition and structure. Spectroscopy measures line formation at the surface, while helioseismology and neutrinos constrain different interior quantities. The two are not the same: the problem is the conflict between the observed data from helioseismology, the sound-speed profile, the depth of the convection zone and the envelope helium abundance, and the predictions of interior models built with the composition that three-dimensional, non-equilibrium models of line formation return, a large error in the photospheric estimate ruled out, the heavy elements entering the interior only through the opacity, and the field's own summary that better is worse. **Unreachability.** **Persistence.** A common composition and solar model reproducing the photospheric, helioseismic and neutrino record. **Addresses.** Statement, and hardness at the conflict between helioseismology and interior models built with the three-dimensional and non-local-thermodynamic-equilibrium composition: Bergemann and Serenelli, "Solar Abundance Problem", in Determination of Atmospheric Parameters of B-, A-, F- and G-Type Stars, pp. 245–258, 2014, at its abstract as the arXiv page, the abstract service and the publisher's page carry it verbatim (the problem defined as that conflict; a list of possible solutions yet to be tested), the chapter itself not fetched. Approaches at the accretion ending: Serenelli, Haxton and Peña-Garay, "Solar models with accretion. I. Application to the solar abundance problem", arXiv 1104.1639, 2011, at its abstract as the arXiv page carries it verbatim (either the convective-zone depth or the surface helium abundance brought into agreement with helioseismology, not both), the paper itself not fetched, its journal not confirmed. Vinyoles et al., "A New Generation of Standard Solar Models", Astrophysical Journal 835:202, 2017, cited here and not read this pass. The relation at Reach: a book chapter on the problem at its abstract as an abstract service carries it (the conflict of helioseismic data with interior models using the composition from three-dimensional and non-equilibrium models), 2014, its authors not carried; a review of solar structure and evolution at its section on the composition (the discrepancy greatly increased by the revised abundances; the envelope helium and the convection-zone depth differing strongly), arXiv:2007.06488; a paper on low-metal models at its introduction (difficult to match simultaneously the new ratio and the seismic constraints; a large error in the photospheric estimate ruled out), arXiv:1603.01666; a review of the Sun's composition at its helioseismology section (better is worse), arXiv:0909.0948; a conference paper on helioseismology and neutrinos (the heavy elements entering through the opacity; no definite solution so far), arXiv:1809.03000; none fetched. Persistence, no landing: the closing condition at the papers read at Reach (matching simultaneously the new composition and the helioseismic constraints; no definite solution so far). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Higher-abundance mixtures — improve helioseismic agreement but conflict with the modern spectroscopic scale. Opacity increases — targeted by calculations and experiments, with the required profile not established. Revised atmosphere modelling — reduces several systematics while retaining the low abundances. Nonstandard mixing, accretion and mass loss — can alter selected observables, without one model fitting the full set. **Narrowing recorded.** Helioseismic inversions fix the locations and size of the principal model discrepancies; solar-neutrino measurements restrict core-temperature changes. **Progress.** Three-dimensional atmosphere calculations, precise mode frequencies, neutrino fluxes and laboratory opacity measurements provide independent constraints. **Own record.** ## 185 The origin of neutrino mass **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What interaction and particle content generate nonzero neutrino masses? In the field's own posing, the physics that leads to nonzero neutrino masses is not known, and the question is whether or how one can hope to establish which model, if any, is a faithful description of nature. **The field's account of its hardness.** That after twenty years of intense experimental and theoretical research the physics that leads to nonzero neutrino masses is still not known, the discovery of the masses indicating that the Standard Model of particle physics is incomplete, and the question the field's review poses being whether or how one can hope to establish which of the different models, if any, is a faithful description of nature; and that oscillation experiments establish mass differences but not the mass-generating operator, multiple mechanisms reproducing the same low-energy mixing data while placing their distinguishing particles or lepton-number effects outside present reach. **Accounts.** A Yukawa interaction with right-handed neutrino fields generates small masses. A lepton-number-violating mass term is induced by heavy states through a seesaw mechanism. Neutrino masses arise radiatively through additional particles and symmetries. Neutrino masses emerge from an effective higher-dimensional operator whose ultraviolet completion is not fixed at accessible energy. **Specifications.** The measured oscillation probabilities arise from propagation of distinct mass eigenstates — universal under the oscillation account. The mechanism reproduces observed mixings and mass-squared differences — universal under neutrino phenomenology. The completed theory respects all established electroweak constraints — universal under the Standard Model limit. **Specifications withdrawn.** Massless neutrinos in the minimal Standard Model, withdrawn while its established interactions are retained. **Common requirement.** An interaction producing the observed neutrino mass matrix and mixings. **Conserving relation.** The generated mass matrix returns the observed oscillation probabilities and all established non-oscillation constraints. **Conserving reach.** All neutrino flavors, mass eigenstates, energies and baselines in the established record. **Reach.** The statement ranges over the mass-generating physics. Oscillations measure differences and mixing combinations rather than the generating interaction. The direct-measurement review states that oscillation measurements shed light neither on the scale of neutrino masses nor on the mechanism by which those are generated: the method does not deliver what the statement ranges over. **Unreachability.** **Persistence.** Identification of the mass-generating mechanism through consequences that exclude the other viable mechanisms. **Addresses.** Statement, with the posing added there, and the accounts as a set of differing models: de Gouvêa, "Neutrino Mass Models", Annual Review of Nuclear and Particle Science 66:197–217, 2016, at its abstract as the publisher's page carries it verbatim (the physics leading to nonzero neutrino masses not known; the models and their differences; whether or how one can hope to establish which is a faithful description of nature), the review itself not fetched. Specifications withdrawn, progress at the withdrawal, and the relation at Reach: Formaggio, de Gouvêa and Robertson, "Direct Measurements of Neutrino Mass", Physics Reports 914:1–54, 2021, at its abstract as the arXiv page and the publisher's page carry it verbatim (the minimal Standard Model predicting exactly zero masses, proved wrong by the discovery of oscillations; oscillation measurements shedding light neither on the scale nor on the generating mechanism), the paper itself not fetched, the section numbers not confirmed. Hardness, its first clause: the models review the address names, at its abstract as the publisher's page carries it verbatim (the discovery of nonzero masses among the most important results of two decades, indicating the Standard Model incomplete; after twenty years the physics leading to nonzero masses still not known; whether or how one can hope to establish which model, if any, is a faithful description of nature), doi 10.1146/annurev-nucl-102115-044600, the review itself not fetched; its second clause, the former line, standing at the direct-measurement review's abstract as the address already carries it for the oscillation measurements, the mechanisms' distinguishing effects outside reach keyed at the next pass. Approaches: the oscillation and endpoint routes at the direct-measurement review's abstract as the address carries it and the models review's abstract read above; the lepton-number-violation and collider routes and their endings keyed at the next pass. Persistence, no landing: the closing condition at the models review's abstract read above (whether or how one can hope to establish which model, if any, is a faithful description of nature). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Oscillation measurements — determine mixing parameters and mass-squared differences, not the generating operator. Endpoint kinematics — constrains a mass combination independently of the mechanism. Lepton-number-violation searches — test a class of mass terms, with no established signal. Collider and rare-decay searches — restrict particular completions. **Narrowing recorded.** Oscillation data restrict the mass matrix up to absolute scale, phases and ordering; null searches exclude regions of individual models. **Progress.** Nonzero mass differences and large flavor mixing stand measured, and the minimal massless-neutrino specification is withdrawn. **Own record.** ## 186 The absolute neutrino mass scale **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What are the individual neutrino masses and their absolute scale? **The field's account of its hardness.** Oscillation probabilities depend on mass-squared differences rather than a common mass offset; direct kinematics, cosmology and lepton-number-violating decay constrain different combinations and carry different model assumptions. **Accounts.** The spectrum has a lightest state well below the measured splittings. The spectrum is quasi-degenerate, with a common scale larger than the measured splittings. Cosmological mass effects follow the standard thermal history and structure-growth model. **Specifications.** The oscillation mass differences apply to the same three mass eigenstates measured by other methods — universal under the three-neutrino framework. Endpoint kinematics is independent of cosmological structure modelling — local. Cosmological bounds assume a stated cosmological model and thermal history — local. **Specifications withdrawn.** **Common requirement.** One set of mass eigenvalues consistent with oscillation, kinematic and cosmological observations. **Conserving relation.** A single mass spectrum returns every measured mass difference and every valid absolute-mass observable. **Conserving reach.** All three established mass eigenstates and all model-independent mass observables. **Reach.** The statement ranges over individual eigenvalues. Each instrument returns a different aggregate: an effective electron-neutrino mass, a mass sum, or a decay-weighted combination. The field's review states that oscillation measurements shed light neither on the scale of neutrino masses nor on the mechanism generating them, and that the scale is most directly accessed by the energy spectrum of beta decay or electron capture: oscillation does not deliver the scale, and each direct method returns its own aggregate. **Unreachability.** **Persistence.** A measured absolute mass observable sufficient, together with oscillation data, to determine the individual masses without an unresolved model dependence. **Addresses.** Statement, hardness at what oscillation measurements do not deliver, the relation at Reach, and approaches at their list and at the endpoint measurement as the most direct access: Formaggio, de Gouvêa and Robertson, "Direct Measurements of Neutrino Mass", Physics Reports 914:1–54, 2021, at its abstract as the arXiv page, the publisher's page and the abstract service carry it verbatim (oscillation measurements shedding light neither on the scale of neutrino masses nor on the mechanism generating them; the scale most directly accessed by the energy spectrum of beta decay or electron capture) and at its section headings as the repository copy carries them (oscillations, neutrinoless double-beta decay, cosmology, time of flight, direct laboratory measurements), the paper itself not fetched, the section numbers not confirmed. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a review of the direct measurement at its abstract as the publisher's page carries it (the absolute scale an intriguing open question; the kinematics of weak decays the most model-independent approach; the target sensitivity two tenths of an electronvolt), Annual Review of Nuclear and Particle Science, 2022, doi 10.1146/annurev-nucl-101920-113013, its authors not carried; a strategy input as the collaboration's document carries it (future advances targeting sensitivity below fifty millielectronvolts, to probe the full mass range presently allowed by the oscillation bounds); a lecture abstract at a repository (the absolute scale still unknown seventy years after the discovery; four measurement approaches; the challenges ahead), its author not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Beta-decay endpoint measurement — direct and model-light, presently an upper bound. Cosmological inference — tighter under the standard cosmological model, with model dependence. Oscillation — precise differences, exactly insensitive to the common offset. Lepton-number-violating decay — constrains a phase-weighted effective mass if its underlying process obtains. **Narrowing recorded.** Direct and cosmological upper bounds narrow the allowed lightest-mass interval; oscillations impose lower bounds on the mass sum for each ordering. **Progress.** Two independent mass-squared splittings are measured and the allowed absolute scale has been substantially bounded. **Own record.** ## 187 Dirac or Majorana neutrinos **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Are massive neutrinos distinct from their antiparticles, or are they their own antiparticles? In the field's own posing, the review's canonical question, carried verbatim in nine words: "Are neutrinos with definite masses Majorana or Dirac particles?" **The field's account of its hardness.** That the field carries as a widespread belief, cited as the practical Dirac–Majorana confusion theorem, that the difference between Dirac and Majorana neutrinos via any kinematical observable would be practically impossible to determine, the difference being proportional to the square of the neutrino mass, suppressed by factors of the mass over the energy when the masses are small compared to the other energy scales, the two Majorana helicity states becoming equivalent to the Dirac particle and antiparticle states as the mass goes to zero, and no difference standing at tree level for beta decays, the process being via the charged-current interaction; that the theorem has been verified in some cases with no general model-independent and process-independent proof and a lack of clarity about its domain of validity; and that in spite of very strong arguments in favour of the Majorana nature the possibility of Dirac neutrinos is not excluded experimentally, observation of neutrinoless double-beta decay standing as a unique and crucial test. **Accounts.** A conserved lepton number distinguishes neutrinos from antineutrinos and the mass term is Dirac. Lepton number is not exact, the neutral mass eigenstates are self-conjugate, and the mass term is Majorana. **Specifications.** Neutrinos have nonzero mass — universal under the question. Charge conjugation and lepton number have their field-theoretic definitions — universal under quantum field theory. A claimed discriminating process is generated by the same mass character being tested — local. **Specifications withdrawn.** **Common requirement.** A Lorentz-consistent mass term reproducing all measured neutrino interactions. **Conserving relation.** The particle-antiparticle classification returns one result for each neutrino mass eigenstate and agrees with every established weak-interaction observation. **Conserving reach.** All neutrino mass eigenstates and both relativistic and nonrelativistic regimes. **Reach.** The statement ranges over the field's self-conjugacy. Existing weak-interaction observations predominantly probe relativistic helicity states. The two are not the same: any kinematical observable's difference between the two natures is proportional to the square of the neutrino mass, suppressed by the mass over the energy, the two Majorana helicity states becoming equivalent to the Dirac particle and antiparticle states as the mass goes to zero, and neutrinoless double-beta decay stands as the unique test. **Unreachability.** **Persistence.** Observation of an unambiguous lepton-number-violating neutrino process, or another measurement fixing the field's self-conjugacy. **Addresses.** Statement, with the canonical question marked there, the accounts as the two characters, approaches at the neutrinoless double-beta-decay search, and progress at the Dirac possibility standing unexcluded: Bilenky, "Neutrinos: Majorana or Dirac?", Universe 6:134, 2020, at its abstract and closing paragraph as the publisher's page carries them verbatim (whether neutrinos with definite masses are Majorana or Dirac particles; the arguments in favour of the Majorana nature; the phenomenological theory of neutrinoless double-beta decay, present data and future sensitivity; the possibility of Dirac neutrinos not excluded experimentally), the review itself not fetched, the section numbers not confirmed. Hardness: re-formed to the field's words at the review the address names, at its closing paragraph as a repository copy carries it verbatim (very strong arguments for the Majorana nature, the Dirac possibility not excluded experimentally; neutrinoless double-beta decay a unique and crucial test), issue 9, and at three later papers returned in the search, none fetched: one in the European Physical Journal C, 2023, doi 10.1140/epjc/s10052-023-12156-9, at its abstract and opening as the publisher's page and its arXiv copy carry them (the practical Dirac-Majorana confusion theorem a widespread belief, the difference proportional to the square of the neutrino mass; verified in some cases, no general model-independent and process-independent proof, a lack of clarity about its domain), its first author carried at the arXiv copy; one at its section on the theorem as the arXiv copy carries it (the difference suppressed by factors of the mass over the energy; the two Majorana helicity states equivalent to the Dirac states as the mass goes to zero; no tree-level difference in beta decays, the process via the charged current), arXiv:1810.06584, its authors not carried; one at its opening as the arXiv copy carries it, arXiv:2305.14140, 2024; the former line's clause that the discriminating process is extraordinarily rare and can be suppressed by masses and phases, not carried at the returns, released with its rate content at the closing clause and its suppression content keyed at the next pass at the review's phenomenological section. The relation at Reach: the later papers on the confusion theorem and the review's closing, at the passages read for the hardness. Persistence, no landing: the closing condition at the review's closing read above (observation of neutrinoless double-beta decay a unique and crucial test) and at the confusion-theorem papers (no kinematical discrimination in practice). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Neutrinoless double-beta-decay searches — directly test lepton-number violation under stated nuclear and mechanism assumptions, with no established signal. Collider and rare-decay searches — constrain additional lepton-number-violating interactions. Kinematic helicity tests — suppressed by the neutrino mass-to-energy ratio. Cosmological probes — constrain masses but do not alone fix self-conjugacy. **Narrowing recorded.** Null decay searches bound effective lepton-number-violating masses and couplings under their nuclear and mechanism specifications. **Progress.** Neutrino mass is established, candidate isotopes and backgrounds are increasingly controlled, and large regions of particular mechanisms are excluded. **Own record.** ## 188 The Standard Model flavor puzzle **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What determines the observed pattern of fermion masses, mixing angles and charge-parity phases in the Standard Model? In the field's own posing, the striking hierarchies observed in the masses and mixing of charged fermions define the Standard Model flavor puzzle, a profound structural enigma pointing to physics beyond the Standard Model. **The field's account of its hardness.** The parameters span large hierarchies and structured mixing patterns but are free inputs in the Standard Model; many symmetry, texture and statistical constructions reproduce parts of the pattern without a uniquely selected origin. **Accounts.** Horizontal symmetries and their breaking generate hierarchical couplings. Geometric localization or extra-dimensional overlap generates hierarchical couplings. The parameters are environmental or statistically selected from a broad landscape. Strong dynamics or compositeness generates the observed masses and mixings. **Specifications.** The measured masses and mixing matrices are the low-energy parameters to be recovered — universal under the Standard Model. Renormalization carries high-scale structures to the measurement scale — universal under quantum field theory. Any extension preserves established flavor-changing bounds — universal under flavor phenomenology. **Specifications withdrawn.** **Common requirement.** A mechanism yielding the full observed Yukawa and mixing structure while retaining successful Standard Model predictions. **Conserving relation.** One generated parameter structure reproduces all measured fermion masses, angles and phases after scale evolution. **Conserving reach.** All quark and lepton flavors and every established flavor observable. **Reach.** The statement ranges over the origin of the parameter pattern. Experiments measure the low-energy parameters and rare processes, not the mechanism that selects them. The two are not the same: the hierarchies in the masses and mixings are, in the field's words, technically natural within the Standard Model and remain unexplained, the model providing no explanation of the parameters it measures, the measured pattern pointing to a deeper organizing principle without selecting one. **Unreachability.** **Persistence.** A mechanism with discriminating consequences that generates the complete observed flavor structure. **Addresses.** Statement, with the posing added there, and hardness at the hierarchies as a structural enigma: Altmannshofer and Greljo, "Recent Progress in Flavor Model Building", Annual Review of Nuclear and Particle Science 75:201–322, 2025, at its abstract as the publisher's page and the arXiv page carry it verbatim (the hierarchies in the masses and mixing of charged fermions defining the Standard Model flavor puzzle; the absence of deviations in flavor-changing neutral currents defining the new physics flavor puzzle; the review's focus on the quark sector), the review itself not fetched; the venue formerly given here, Physics Reports 1107:1–45, verified at neither page and is withdrawn. The relation at Reach: a review of flavor model building at its abstract as the publisher's page and the arXiv page carry it verbatim (the striking hierarchies defining the Standard Model flavor puzzle, a profound structural enigma pointing beyond the model), Annual Review of Nuclear and Particle Science, 2025, doi 10.1146/annurev-nucl-121423-100950, arXiv:2412.04549; lecture notes at their summary as the arXiv copy carries it (the patterns technically natural within the model yet unexplained, suggesting a deeper organizing principle), arXiv:2508.03950; lecture notes at an institute's page (most charged-fermion flavor parameters small and hierarchical, the model providing no explanation); their authors not carried; none fetched. Approaches: the routes named at the lecture notes' keywords read above (flavor symmetries, the Froggatt-Nielsen and clockwork constructions, supersymmetry, composite Higgs) and at the review of model building's abstract (attempts within the quark sector), their endings keyed at the next pass. Persistence, no landing: the closing condition in part at the review of model building read at Reach (a deeper organizing principle; attempts to address the puzzle within the quark sector), its discriminating consequences keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Approximate flavor symmetries — reproduce chosen hierarchies and predict selected relations, with many viable symmetry choices. Texture constructions — compress parameters, with stability and origin dependent on the construction. Extra-dimensional localization — converts positions into couplings, with the geometry unselected. Statistical and anthropic models — return distributions rather than the observed realization uniquely. **Narrowing recorded.** Precision flavor and neutrino measurements eliminate textures and restrict symmetry-breaking patterns. **Progress.** The quark mixing matrix is overconstrained, neutrino mixing is substantially mapped, and masses and several phases are measured with increasing precision. **Own record.** ## 189 The new-physics flavor puzzle **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** If new particles or interactions occur near accessible energy scales, why are their generic flavor-changing and charge-parity-violating effects not observed? In the field's own posing, the absence of deviations from Standard Model predictions in precision measurements of flavor-changing neutral currents imposes severe constraints on new physics at the TeV scale, giving rise to the new physics flavor puzzle. **The field's account of its hardness.** Generic couplings at those scales violate strong bounds from meson mixing, rare decays and electric dipole moments; viable constructions therefore require special flavor structure, degeneracy, alignment or higher scales whose origin is itself not fixed. **Accounts.** New interactions inherit the Standard Model Yukawa matrices as their only flavor-breaking sources. New couplings are aligned with the observed fermion mass basis. Particles carrying the same gauge charges are sufficiently degenerate for flavor effects to cancel. Flavor-changing states lie above accessible scales while other new states remain lighter. **Specifications.** The proposed new sector satisfies every established flavor and charge-parity bound — universal under the extension. Its suppression mechanism remains stable under quantum corrections — universal under quantum field theory. The sector retains the phenomena for which it was proposed — local. **Specifications withdrawn.** **Common requirement.** A suppression structure making new interactions consistent with the observed rarity of flavor change. **Conserving relation.** The new theory returns all established flavor observables within their bounds while retaining its non-flavor effects. **Conserving reach.** All flavors, rare processes and charge-parity observables to which the new sector contributes. **Reach.** The statement ranges over any accessible new sector. Experiments constrain operators and coupling combinations process by process. The two are not the same: new physics at the teraelectronvolt scale with a generic flavor structure would contribute to flavor-changing neutral currents orders of magnitude above the observed rates, and the absence of deviations constrains such physics to scales well beyond the reach of direct searches, at least the petaelectronvolt scale by generic arguments, each process constraining a combination of operators and couplings rather than the sector itself. **Unreachability.** **Persistence.** Discovery of a new sector together with the principle that fixes its flavor structure, or exclusion of accessible generic alternatives over the stated reach. **Addresses.** Statement, with the posing added there, and hardness at the severe constraints from flavor-changing neutral currents: Altmannshofer and Greljo, "Recent Progress in Flavor Model Building", Annual Review of Nuclear and Particle Science 75:201–322, 2025, at its abstract as the publisher's page and the arXiv page carry it verbatim (the absence of deviations from Standard Model predictions in precision measurements of flavor-changing neutral currents imposing severe constraints on new physics at the TeV scale, giving rise to the new physics flavor puzzle), the review itself not fetched; the venue formerly given here, Physics Reports 1107:1–45, verified at neither page and is withdrawn. The relation at Reach: the same review of flavor model building at its abstract (the absence of deviations in flavor-changing neutral currents imposing severe constraints on new physics at the teraelectronvolt scale, the new physics flavor puzzle), doi 10.1146/annurev-nucl-121423-100950; lecture notes at an institute's page (generic flavor structure contributing orders of magnitude above the observed rates); a preprint at its introduction as the arXiv copy carries it (new physics related to flavor at least at the petaelectronvolt scale by generic arguments), arXiv:1611.07033; lecture notes at their summary (rare decays and meson mixing probing scales beyond direct searches), arXiv:2508.03950; their authors not carried; none fetched. Approaches: minimal flavor violation and decoupling at the lecture notes and the review of model building read above, their endings and the alignment and degeneracy routes keyed at the next pass. Persistence, no landing: the closing condition in part at the review and the lecture notes read at Reach (new physics with a non-generic flavor structure at scales the searches reach, or its exclusion), the principle fixing the structure keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Minimal flavor violation — imposes Standard Model spurions and sharply correlates effects, with the principle assumed. Alignment — suppresses off-diagonal couplings, with its dynamics model-dependent. Degeneracy — produces cancellations, constrained by mass splittings. Decoupling — moves flavor-sensitive states upward in energy, reducing accessibility. **Narrowing recorded.** Neutral-meson mixing, rare decays and electric-dipole searches place process-specific bounds on new operator scales and phases. **Progress.** Large regions of generic low-scale couplings are excluded, and viable suppression patterns have explicit effective-field-theory forms. **Own record.** ## 190 The sources of ultrahigh-energy cosmic rays **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which astrophysical objects accelerate the highest-energy cosmic rays, and what particles do they inject? **The field's account of its hardness.** Charged particles are deflected by poorly known magnetic fields, the flux is small at the highest energies, composition is inferred through hadronic-interaction models beyond accelerator energies, and several source populations can reproduce the broad spectrum and anisotropy. **Accounts.** Active galactic nuclei and their jets accelerate and release the particles. Starburst galaxies and their compact objects or winds supply the particles. Transient explosions and mergers supply the particles. More than one source population contributes with energy- and composition-dependent weights. **Specifications.** Candidate sources meet the acceleration, luminosity and propagation requirements — universal under source models. Magnetic deflection and energy losses are included — universal under charged-particle propagation. Air-shower observables map to mass composition through a stated interaction model — local. **Specifications withdrawn.** **Common requirement.** A source population and injected composition reproducing the spectrum, shower composition, anisotropy and secondary messengers together. **Conserving relation.** One source-and-propagation account returns the joint observed distributions within their uncertainties. **Conserving reach.** The full ultrahigh-energy range, sky, nuclear composition and associated neutrino and photon channels. **Reach.** The statement ranges over the accelerators and injected nuclei. Observatories measure arrival directions, energies and air showers after propagation. The field's review states that a significant dipole anisotropy has been measured but the poor understanding of the Galactic magnetic fields makes it hard to interpret, and that the luminosity density constrains explanations of the origin: the arrival directions measured and the sources sought are not the same, the deflection between them unknown. **Unreachability.** **Persistence.** Source identification with a composition and propagation account that predicts the joint cosmic-ray and multimessenger record. **Addresses.** Statement, hardness at the poorly known magnetic fields, the relation at Reach, and progress at the heavy composition and the dipole: Globus and Blandford, "Ultra High Energy Cosmic Rays", arXiv:2505.21846, 2025, an invited review for Annual Review of Astronomy and Astrophysics volume 63 as the archive listing carries it, at its abstract as the arXiv page carries it verbatim (the progress to divine their nature and identify their sources; the highest-energy particles appearing to be heavy nuclei; the dipole anisotropy measured and hard to interpret for the poor understanding of the Galactic magnetic fields; the luminosity density constraining explanations of the origin), the review itself not fetched. Alves Batista et al., "Open Questions in Cosmic-Ray Research at Ultrahigh Energies", Frontiers in Astronomy and Space Sciences 6:23, 2019, cited here and not read this pass. Approaches: the routes and their endings at four papers returned in the search, none fetched, their authors not carried: an anisotropy search's introduction as the arXiv copy carries it (clarifying the origin a decades-long endeavour; the challenges the uncertain composition above ten exaelectronvolts and the magnetic fields scrambling directional information; the cutoff restricting the highest energies to nearby sources; intermediate-scale anisotropies indicated), arXiv:2007.00023; an anisotropy-model paper's abstract as the publisher's page carries it (anisotropy evidence indicating an extragalactic origin, the sources yet to be identified; starburst and active-nucleus candidates tested by propagation simulation), Journal of Astrophysics and Astronomy, 2022, doi 10.1007/s12036-022-09887-8; a backtracking paper's abstract (the isotropy from deflections and the composition uncertainty precluding tracing to sources; corrections under models of the Galactic field), 2020, doi 10.3103/S0884591320030046; a multimessenger thesis's abstract at a repository (a departure from isotropy at the ninety-five per cent level, no clear correlation with the extragalactic matter distribution). Persistence, no landing: the closing condition at the papers read above (the sources yet to be identified; tracing events to sources precluded until deflection and composition are fixed). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Spectrum and composition fitting — restricts injection and evolution under chosen interaction models. Anisotropy searches — report large-scale structure and candidate correlations without unique associations. Multimessenger searches — bound source opacity and hadronic activity. Magnetic backtracking — returns direction regions conditional on Galactic and extragalactic field models. **Narrowing recorded.** Energy-loss horizons restrict the most energetic sources to the comparatively local universe; photon and neutrino limits exclude classes of top-down and highly opaque accounts. **Progress.** The flux suppression, large-scale anisotropy and an energy-dependent trend toward heavier shower composition are measured. **Own record.** ## 191 Coronal heating and solar-wind acceleration **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How is energy transferred from the solar surface into the corona, dissipated as heat, and converted into the fast and slow solar wind? **The field's account of its hardness.** The heating and acceleration occur through coupled magnetic, turbulent and kinetic processes over scales that remote sensing and in-situ sampling observe separately; multiple mechanisms produce overlapping temperature, wave and flow signatures. **Accounts.** Dissipation of magnetohydrodynamic waves and turbulence heats the corona and accelerates the wind. Magnetic reconnection and impulsive nanoflares supply the dominant heat. Interchange reconnection releases closed-field plasma into the slow wind. The measured wind results from mixtures whose weights change across magnetic structures. **Specifications.** Energy is conserved from photospheric injection through coronal dissipation and wind outflow — universal under the energy budget. Candidate mechanisms reproduce species temperatures, charge states and velocity distributions — universal under solar-wind physics. Magnetic topology conditions the source region — local. **Specifications withdrawn.** **Common requirement.** A measured energy pathway connecting photospheric motions, coronal heating and the terminal wind. **Conserving relation.** The inferred energy input, partition and outflow close one energy budget and reproduce the plasma observables. **Conserving reach.** Open and closed magnetic structures, fast and slow wind, and the scales from the photosphere to the supersonic flow. **Reach.** The statement ranges over the connected energy pathway. Remote images, spectra and spacecraft samples measure different locations and plasma populations. The field's review states that the wind has been measured directly from 0.3 astronomical units outward and the atmosphere imaged from the photosphere out through the corona, while the region where the internal, magnetic and turbulent energy is channeled into the bulk energy of the supersonic wind lies between and requires in situ observation: the regions measured and the region where the conversion occurs are not the same. **Unreachability.** **Persistence.** A quantitatively closed, predictive account of the heating and acceleration in each wind source regime. **Addresses.** Statement, hardness at the separation between the regions measured and the region where the energy is channeled, the relation at Reach, and approaches at near-Sun in situ sampling: McComas et al., "Understanding coronal heating and solar wind acceleration: Case for in situ near-Sun measurements", Reviews of Geophysics 45:RG1004, 2007, at its abstract as the publisher's page and the abstract service carry it verbatim (the wind measured directly from 0.3 astronomical units outward and the atmosphere imaged from the photosphere through the corona; how the corona is heated and accelerated to produce the wind remaining a mystery; in situ observation from a few solar radii out to about twenty required) and at one passage of its text as the publisher's page carries it (how the turbulent flux increases toward the Sun, whether it suffices to power the heating and acceleration, and how it is driven, not known), the review itself not fetched, the section numbers not confirmed; progress, stated at the entry from the near-Sun record, stands keyed at the next pass. Cranmer, "The role of turbulence in coronal heating and solar wind expansion", Philosophical Transactions A 373:20140148, 2015, cited here and not read this pass. Persistence, no landing: the closing condition at the field's own strategy paper, at its abstract as the publisher's page and the arXiv copy carry it verbatim (a definitive solution involving a number of challenging steps, beginning with an identification of the energy source and ending with a prediction of observable quantities compared directly with actual observations, the intermediate steps realistic modelling of the energy release and of the plasma's response), Solar Physics, 2006, doi 10.1007/s11207-006-0055-z, arXiv:astro-ph/0511841, its author carried at the copy; the standing at a 2020 preprint's abstract (none of the candidate processes significantly justifying the million-degree temperature, the problem unsolved to date), arXiv:2003.10326, and at a 2022 preprint's opening (the acceleration one of the most fundamental yet unresolved problems), arXiv:2209.03508; none fetched. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Wave and turbulence measurements — show energy transport and dissipation signatures, with the dominant channel varying by region. Reconnection and nanoflare searches — detect impulsive events, with the unresolved-event energy budget open. Composition and charge-state tracing — links wind parcels to coronal conditions without a unique source map. Near-Sun in-situ sampling — reduces the propagation interval while retaining remote-to-local connection uncertainty. **Narrowing recorded.** Source-surface topology and composition divide fast- and slow-wind regimes; measured energy fluxes restrict mechanisms region by region. **Progress.** Near-Sun fields, particles, waves and switchbacks are directly measured, and candidate mechanisms now face species-resolved energy budgets. **Own record.** ## 192 The nature of cosmic acceleration **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What causes the observed late-time acceleration of cosmic expansion? In the field's own posing, the accelerating expansion implies that the universe is dominated by some form of dark energy with exotic physical properties, or that the established theory of gravity breaks down on cosmological scales. **The field's account of its hardness.** That the acceleration implies either that the universe is dominated by some form of dark energy with exotic physical properties or that the theory of gravity breaks down on cosmological scales; and that many theories with modifications of the gravitational action can be written in a mathematically equivalent form of general relativity plus a scalar field with specified properties, and models in which dark matter decays into dark energy can produce a mismatch between the histories of expansion and structure growth while maintaining general relativity, and thus even perfect measurements of all relevant observables may not uniquely locate the explanation of cosmic acceleration in the gravitational or the stress-energy sector. **Accounts.** A constant vacuum-energy density supplies the acceleration. A dynamical field with negative effective pressure supplies the acceleration. Gravity differs from general relativity on cosmological scales and produces apparent dark energy. Large-scale inhomogeneity or observational systematics produce part of the inferred acceleration. **Specifications.** The account reproduces the observed distance-redshift relation — universal under cosmology. It reproduces structure growth and gravitational lensing — universal under large-scale-structure inference. It reduces to tested gravity locally — universal under gravitational physics. **Specifications withdrawn.** A matter-only decelerating homogeneous model, withdrawn while homogeneous cosmological modelling is retained. **Common requirement.** One dynamics for the background expansion and the growth of structure. **Conserving relation.** The same account returns consistent expansion, growth and lensing observables. **Conserving reach.** The observed cosmic history from recombination through the accelerating epoch and all measured linear scales. **Reach.** The statement ranges over the physical cause. Surveys return distances, redshifts, clustering and lensing from which dynamics are inferred. The field's review states that the acceleration implies either dark energy with exotic physical properties or a breakdown of the theory of gravity on cosmological scales, and that the experiments measure the history of expansion and the growth of structure: the histories measured and the cause are not the same, two families of account standing on one record. **Unreachability.** **Persistence.** A discriminating observation or derivation fixing the source of acceleration and reproducing expansion and structure jointly. **Addresses.** Statement, with the posing added there, the accounts at the dark-energy and modified-gravity families, the relation at Reach, and approaches at the four established methods: Weinberg et al., "Observational Probes of Cosmic Acceleration", Physics Reports 530:87–255, 2013, at its abstract as the publisher's page and the abstract service carry it verbatim (the acceleration implying dark energy with exotic physical properties or a breakdown of the theory of gravity on cosmological scales; experiments measuring the history of expansion and the growth of structure; the four established methods, supernovae, baryon acoustic oscillations, weak lensing and cluster abundance), the review itself not fetched. Durrer and Maartens, "Dark Energy and Dark Gravity", General Relativity and Gravitation 40:301–328, 2008, and Bludman, "Cosmological Acceleration: Dark Energy or Modified Gravity?", arXiv:astro-ph/0605198, 2006, cited here and not read this pass. Hardness: re-formed to the field's words at the review the address names, at its abstract as the publisher's page, an abstract service, an index, two of its authors' pages and a repository copy carry it verbatim (dark energy with exotic properties or a breakdown of the theory of gravity on cosmological scales) and at a passage of its text as the repository copy carries it verbatim (modified-gravity theories writable as general relativity plus a scalar field; dark matter decaying into dark energy producing a mismatch between expansion and growth histories within general relativity; thus even perfect measurements of all relevant observables may not uniquely locate the explanation in the gravitational or the stress-energy sector), arXiv:1201.2434, the review itself not fetched; the former line, in the file's compression (the expansion history can be reproduced by a cosmological constant, evolving stress-energy or modified gravitational dynamics, while background-distance measurements alone do not identify which physical account supplies the same effective acceleration), released with its content at the second clause; the two papers cited beside it not read this pass. Persistence, no landing: the closing condition at the review read above (experiments measuring the history of expansion and the growth of structure jointly; even perfect measurements possibly not locating the explanation uniquely). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Standard-candle and standard-ruler distances — establish acceleration under the cosmological framework, with cause-degeneracy retained. Growth and weak-lensing measurements — test departures between geometry and clustering, with current parameter degeneracies. Laboratory and Solar-System gravity tests — exclude ranges of modified dynamics through screening-dependent predictions. Field models — fit expansion with additional parameters and no selected potential. **Narrowing recorded.** Joint background-and-growth analyses narrow constant equation-of-state and modified-gravity parameters; local tests narrow unscreened modifications. **Progress.** Accelerated expansion is supported by independent distance probes, and the constant-vacuum account fits the present broad dataset with few parameters. **Own record.** ## 193 The origin of cosmic magnetic fields **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How were the seed magnetic fields of galaxies and larger cosmic structures generated, and how were they amplified to the fields observed? **The field's account of its hardness.** That the universe is magnetized on all scales probed so far while the origin of the fields is still described in the field as a mystery, one possibility being the dynamo amplification of small seed fields, with problems remaining over whether the mean-field dynamo coefficients for the galactic dynamo are catastrophically quenched and whether fields generated by the fluctuation dynamo in clusters have a sufficient degree of coherence to explain the observations, and an alternative being that the observed large-scale fields are a relic from the early universe, one subsequently amplified and maintained by a dynamo in collapsed objects, astrophysical batteries and dynamos reshaping whatever primordial field there was. **Accounts.** Magnetic fields were generated in the early universe before structure formation. Plasma batteries in forming stars, galaxies and shocks generated the seeds. Outflows from compact objects and galaxies magnetized the surrounding medium. Small- and large-scale dynamos amplified weak seeds during structure formation. **Specifications.** Maxwell and plasma evolution carry seed fields into later structure — universal under magnetohydrodynamics. The account respects cosmological bounds and reproduces present strengths and coherence scales — universal under cosmic magnetism. Dynamo calculations include the relevant turbulence and transport — local. **Specifications withdrawn.** **Common requirement.** A seed-and-amplification history producing the measured field strengths, spectra and coherence lengths. **Conserving relation.** One magnetic history returns the joint field observables without violating cosmological and plasma constraints. **Conserving reach.** Voids, filaments, clusters, galaxies and early-universe epochs sampled by present constraints. **Reach.** The statement ranges over initial generation and subsequent evolution. Most observations integrate field strength with electron density, source spectra or propagation along a line of sight. The field's review states that the observed fields may be a relic of the early universe, subsequently amplified and maintained by a dynamo in collapsed objects: the fields observed and the seed sought are not the same, the dynamo standing between them. **Unreachability.** **Persistence.** Identification of the seed mechanism and amplification history through a spectrum, coherence scale or redshift dependence that excludes the viable alternatives. **Addresses.** Statement, the primordial account at generation during inflation and phase transitions, the relation at Reach, and progress at the observed fields: Subramanian, "The origin, evolution and signatures of primordial magnetic fields", Reports on Progress in Physics 79:076901, 2016, at its abstract as the publisher's page, the arXiv page and the index carry it verbatim (the universe magnetized on all scales probed; micro-Gauss fields in galaxies and clusters coherent up to ten kiloparsecs; a weak field in voids suggested; the observed fields possibly a relic of the early universe amplified and maintained by a dynamo in collapsed objects; generation during inflation and phase transitions; the linear and nonlinear evolution through the radiation era), the review itself not fetched. Ryu et al., "Magnetic Fields in the Large-Scale Structure of the Universe", Space Science Reviews 166:1–35, 2012, cited here and not read this pass; the astrophysical generation and the amplification at progress stand keyed at the next pass. Hardness: re-formed to the field's words at the review the address names, at its full abstract as the arXiv page, the publisher's page and an index carry it verbatim (magnetized on all scales probed; an intriguing possibility that the fields are a relic amplified and maintained by a dynamo in collapsed objects; astrophysical batteries and the role of dynamos in reshaping the primordial field), published 31 May 2016, issue 7, doi 10.1088/0034-4885/79/7/076901, arXiv:1504.02311, the review itself not fetched; and at an earlier paper on primordial fields and the microwave background, at its introduction as the arXiv copy carries it verbatim (magnetic fields ubiquitous but their origin still a mystery; dynamo amplification of seed fields with problems remaining, catastrophic quenching and the coherence of fluctuation-dynamo fields; the relic alternative), arXiv:astro-ph/0601570, 2006, its authors not carried at the return, returned in the search, not fetched; the former line, in the file's compression (present fields combine generation, amplification and transport over cosmic history; rotation, synchrotron and cascade observables constrain integrated or model-dependent combinations, leaving primordial and astrophysical seeds able to feed similar later dynamos), released with its seed content at the two alternatives and its observables clause keyed at the next pass at the review's constraints sections. Approaches: the dynamo route at the review's full abstract and the earlier paper's introduction read above (dynamo amplification of seed fields, its remaining problems); the rotation, synchrotron, cascade and anisotropy routes and their endings keyed at the next pass at the review's constraints sections. Persistence, no landing: the closing condition in part at the review's full abstract and the earlier paper read above (the seed mechanism, primordial or astrophysical, and the dynamo history), the discriminating spectrum, coherence scale or redshift dependence keyed at the next pass at the review's signatures sections. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Faraday rotation — constrains line-of-sight field weighted by plasma density. Synchrotron emission — constrains field and cosmic-ray combinations. Gamma-ray cascade bounds — constrain intergalactic fields under source and plasma assumptions. Cosmological anisotropy searches — bound primordial spectra. Dynamo simulations — amplify specified seeds and return model-dependent saturation. **Narrowing recorded.** Cosmological data bound primordial amplitudes and spectra; rotation and cascade data bound environment-specific combinations. **Progress.** Magnetic fields are mapped across galaxies and clusters, amplification by turbulent dynamos is demonstrated in simulations and laboratories, and broad primordial parameter regions are excluded. **Own record.** ## 194 Many-body localization and thermalization **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does a stable many-body-localized phase exist in generic interacting disordered systems in the infinite-size and infinite-time limits, and where does it give way to thermalization? In the field's own posing, the phenomenology of the many-body-localized regime is well established at finite sizes and times, and the central question remains unanswered: under what conditions does the regime give rise to a phase in which thermalization does not occur in the asymptotic limits. **The field's account of its hardness.** Exact computation is restricted to small systems and finite times, while crossover lengths and times grow rapidly near the proposed transition; the same finite data admit a transition, a long-lived prethermal regime, or eventual avalanche-driven thermalization. **Accounts.** Quasi-local conserved quantities stabilize a nonergodic phase at strong disorder. Rare thermal regions destabilize localization so that finite systems only mimic a phase. A sharp transition exists in restricted dimensions or disorder classes and not generically. **Specifications.** The thermodynamic and long-time limits define the phase — universal under equilibrium phase classification. The model's disorder distribution and dimensionality are fixed before extrapolation — local. Diagnostics distinguish asymptotic behavior from crossover — universal under the question. **Specifications withdrawn.** **Common requirement.** An asymptotic classification of dynamics and eigenstates under fixed model specifications. **Conserving relation.** The phase diagnosis agrees across spectral, transport, entanglement and memory observables in the stated limits. **Conserving reach.** Arbitrarily large systems and arbitrarily long evolution for each stated dimension and disorder class. **Reach.** The statement quantifies over two asymptotic limits. Exact diagonalization, tensor methods and experiments return bounded sizes and times. The field's review parts the localized regime, found numerically and experimentally at finite sizes and times where one or both limits cannot be taken, from the localized phase the question asks about: the regime measured and the phase sought are not the same, the two limits between them. **Unreachability.** **Persistence.** A controlled limit or theorem establishing stability or instability and locating the transition wherever it exists. **Addresses.** Statement, with the posing added there, hardness at the finite sizes and times and the reliance on classical numerics, and the relation at Reach: Sierant et al., "Many-Body Localization in the Age of Classical Computing", Reports on Progress in Physics 88:026502, 2025, doi:10.1088/1361-6633/ad9756, at its abstract as the arXiv page carries it verbatim (thermalization not observed at finite sizes and times in a robust localized regime found numerically and experimentally at strong disorder; the phenomenology well established and the central question unanswered, under what conditions the regime gives rise to a phase) and at one passage of its text as the publisher's page carries it (the working definition of the regime for finite systems where one or both limits cannot be taken; the status of localization as a dynamical phase uncertain, the problem lacking an established theoretical framework and resting mainly on numerical simulation on classical computers), the review itself not fetched. Abanin et al., "Colloquium: Many-body localization, thermalization, and entanglement", Reviews of Modern Physics 91:021001, 2019, cited here and not read this pass. Approaches: the routes and their endings at three papers returned in the search, none fetched: a constrained-chain study at its introduction as the arXiv copy carries it (after decades the nature of the transition and even the existence of localization at large sizes open; the avalanche mechanism proposed as why localization fails above one dimension and the doubt extended to one; larger simulations showing the transition point drifting to stronger disorder), arXiv:2408.14372, its author carried as Lu; a power-law-interaction study's introduction (the phase widely regarded as stable in finite systems, its integrity in the thermodynamic limit questioned by the avalanche mechanism, a rare thermal region hybridizing with the localized bulk; the asymptotic stability a persistent and highly debated topic; exact diagonalization and open-system dynamics combined), arXiv:2601.13485; a resonance study's abstract (the avalanche spreading via strong many-body resonances between rare near-resonant eigenstates, modelled by a bath coupled to one end), its authors carried at the copy; the matrix-product and experimental routes keyed at the next pass. Persistence, no landing: the closing condition at the studies read above (the existence of localization at large sizes and the nature of the transition open; a phase diagram resolving the asymptotic stability claimed for one model). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Exact diagonalization — high-precision finite-size diagnostics with extrapolations that change over accessible sizes. Matrix-product evolution — longer one-dimensional times with entanglement-limited reach. Renormalization and resonance analysis — competing stable-localization and avalanche scenarios. Cold-atom and solid-state experiments — persistent memory on experimental times, without the asymptotic limit. **Narrowing recorded.** Localization is established in noninteracting disordered models and long-lived nonergodic dynamics is measured in specified interacting finite systems. **Progress.** Multiple finite-scale diagnostics, rare-region mechanisms and quantitative crossover analyses are established. **Own record.** ## 195 Water's liquid-liquid critical point **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does deeply supercooled water have distinct low- and high-density liquid states ending at a liquid-liquid critical point; if so, where is that point and how does it organize water's anomalies? **The field's account of its hardness.** Rapid crystallization obstructs equilibrium measurements in the relevant region, so the proposed transition was long inferred through extrapolation, amorphous ices and simulations; the new ultrafast evidence probes the vicinity rather than completing the metastable phase diagram. **Accounts.** A first-order liquid-liquid transition terminates at a critical point and organizes the anomalous response functions. An apparent singularity is avoided by crystallization or by a continuous but noncritical structural crossover. Two interconverting local structures generate the anomalies without requiring an accessible macroscopic phase separation. **Specifications.** The states sampled before crystallization are liquid — local. The observed structural crossover and heat-capacity increase are equilibrium or controlled metastable properties — local. A critical point has the thermodynamic and fluctuation signatures required by critical phenomena — universal under phase-transition theory. **Specifications withdrawn.** The specification that only indirect experimental signs are available, withdrawn by ultrafast x-ray measurements while the full phase-diagram problem is retained. **Common requirement.** A metastable phase diagram explaining density, structure, response functions and amorphous-ice connections together. **Conserving relation.** One phase diagram returns the same transition and critical behavior across structural and thermodynamic observables. **Conserving reach.** Supercooled water across the pressures and temperatures surrounding the proposed critical region, before crystallization. **Reach.** The statement ranges over the metastable equilibrium phase diagram. Experiments access brief liquid states prepared from amorphous ice, and simulations access model-specific phase diagrams. The experiment states that liquid states straddling the predicted critical point were accessed at timescales before ice formation and that its results suggest states in the vicinity of a critical point were directly probed: the transient states probed and the metastable diagram the statement ranges over are not the same, the vicinity probed and the point inferred. **Unreachability.** **Persistence.** Independent reproduction and mapping of the critical region with concordant thermodynamic and structural observables, or evidence excluding a critical point across the stated region. **Addresses.** Statement, in substance: the paper the address names, at its abstract as the publisher's page, a data repository and a database carry it verbatim (the search for the liquid-liquid critical point challenging owing to rapid crystallization; liquid states straddling the predicted critical point; a crossover from a discontinuous to a continuous transition; a heat capacity rise indicating a critical divergence at 210 kelvin within about eight), Science 391(6792):1387-1391, 26 March 2026, doi 10.1126/science.aec0018, and at the journal's editor's summary (polyamorphism and the critical point intriguing phenomena; rapid spontaneous ice formation making the point extremely challenging to find; the earlier evidence indirect); the anomalies at a 2011 preprint's introduction as the arXiv copy carries it (the critical point rationalizing the density maximum and the compressibility minimum, more pronounced when supercooled; the most compelling evidence generated in silico), arXiv:1108.5961, its authors not carried; the standing at a magazine's report (the idea once controversial, gained consensus; the second critical point never directly observed before) and at a 2023 preprint's abstract (the possible existence a subject of debate in part for the challenges of definitive experimental evidence), arXiv:2302.08540; a 2026 preprint titled for a glassy arrest behind the apparent second liquid, at its title only, a reading for the next pass; none fetched; the Statement standing as it stood. Hardness at the rapid crystallization, the relation at Reach, progress at the observed crossover and heat-capacity rise, and Own record: You et al., "Experimental evidence of a liquid-liquid critical point in supercooled water", Science 391:1387–1391, 2026, at its abstract as the publisher's page, the data repository and an index carry it verbatim (the search challenging owing to rapid crystallization; supercooled water studied at timescales before ice formation by ultrafast heating of the high- and low-density amorphous ices with x-ray scattering; liquid states straddling the predicted critical point accessed; the crossover from a discontinuous to a continuous transition; the heat-capacity rise indicating a critical divergence at 210 ± 8 kelvin; the results suggesting states in the vicinity of a critical point directly probed) and at the journal's summary as the publisher's page carries it (previous evidence indirect), the paper itself not fetched. Gartner et al., "Liquid-liquid transition in water from first principles", 2022, cited here and not read this pass; its venue, given here as Proceedings of the National Academy of Sciences 119:e2206739119, not verified this sitting, a preprint's reference list citing it at Physical Review Letters 129:255702, to be read at the publisher's page. Approaches: the ultrafast-heating route and its ending at the paper's abstract read above; the simulation route at the 2011 preprint's introduction (the most compelling evidence in silico); the extrapolation and amorphous-ice routes keyed at the next pass. Persistence, no landing: the closing condition at the paper's editor's summary and the later preprint read above (the point never directly observed before, the earlier evidence indirect; definitive experimental evidence the challenge), the independent reproduction keyed at the next pass with the glassy-arrest preprint. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Molecular simulation — produces transitions and critical points in several models, with model accuracy decisive. Extrapolation of response functions — supports singular and nonsingular fits over accessible temperatures. Amorphous-ice transformations — connect two glassy forms to candidate liquids indirectly. Ultrafast heating and x-ray scattering — reports two liquid states, a crossover change and a heat-capacity divergence near 210 ± 8 kelvin before ice formation. **Narrowing recorded.** The ultrafast experiment narrows the critical region and replaces purely indirect support with direct short-time liquid-state evidence. **Progress.** First-principles simulation evidence and direct experimental signatures near a critical point now stand; complete location, universality and equilibrium mapping remain unsettled. **Own record.** The experiment states that its results suggest states in the vicinity of a critical point in supercooled water were directly probed, with a crossover from a discontinuous to a continuous transition observed and a heat-capacity rise indicating a critical divergence at 210 ± 8 kelvin; the journal's summary states that evidence for polyamorphism previously relied on indirect signs. ## 196 Equilibrium climate sensitivity **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What equilibrium global-mean surface warming follows a sustained doubling of atmospheric carbon dioxide? In the panel's own posing, the equilibrium climate sensitivity is an important quantity used to estimate how the climate responds to radiative forcing, assessed from multiple lines of evidence. **The field's account of its hardness.** The equilibrium response cannot be observed as one controlled experiment, feedbacks operate over different timescales, and instrumental trends, paleoclimate states, process studies and climate models constrain the same quantity with different structural uncertainties. A critique of the combined assessment states that the sensitivity of the result to the assumptions employed implies the quantity remains difficult to ascertain. **Accounts.** Strong net positive cloud and water-vapor feedbacks place the response toward the upper assessed range. Weaker cloud feedback and compensating processes place the response toward the lower assessed range. State-dependent feedbacks make the effective sensitivity vary with background climate and timescale. **Specifications.** Carbon dioxide is doubled from a stated baseline — local. Fast atmospheric, ocean-surface, cryospheric and land feedbacks equilibrate under the conventional definition — universal under equilibrium climate sensitivity. Slow ice-sheet and carbon-cycle changes are excluded or separately stated — local. **Specifications withdrawn.** **Common requirement.** A common temperature response consistent with instrumental, process, paleoclimate and model evidence under one definition. **Conserving relation.** Each valid line of evidence constrains the same equilibrium temperature change after its stated transformations. **Conserving reach.** Climate states and feedback regimes admitted by the conventional doubled-carbon-dioxide definition. **Reach.** The statement ranges over an equilibrium counterfactual. Observations sample transient and past climates, and models compute the counterfactual under specified feedbacks. The two are not the same: the field's assessment characterizes the equilibrium sensitivity by an effective sensitivity based on behaviour within a century or two of applying the forcing, more relevant to the time scales of greatest interest, and reads it from three lines of evidence, feedback process understanding, the historical record and the paleoclimate record, each limited in the traditional forcing-feedback paradigm for interpreting past changes and combined by a Bayesian calculation whose range depends on the lines being judged largely independent. **Unreachability.** **Persistence.** A narrower, stable interval supported jointly by independent evidence and a resolved account of the dominant feedback uncertainty. **Addresses.** Statement, with the posing added there, hardness at the multiple lines of evidence, narrowing recorded at the assessed ranges, and progress at the narrowing against the previous assessment: IPCC, Climate Change 2021: The Physical Science Basis, Working Group I, Summary for Policymakers, headline A.4 and paragraph A.4.4, 2021, as the panel's page carries them verbatim (the quantity used to estimate how the climate responds to radiative forcing; the very likely range 2 to 5 degrees Celsius, the likely range 2.5 to 4, the best estimate 3, against 1.5 to 4.5 with no best estimate before), the summary pointing to Chapter 7 §§7.4 and 7.5 and Technical Summary 3.2, the chapter itself not read this pass. Hardness at the critique's sentence, and an approach at its lower estimate: a study combining the lines of evidence by an objective Bayesian method, at its abstract as the publisher's page carries it verbatim (a median of 2.16 degrees Celsius with a 5 to 95 percent range of 1.55 to 3.2; the sensitivity to the assumptions employed implying the quantity remains difficult to ascertain), doi 10.1007/s00382-022-06468-x, its title and journal not carried at the page read, the paper itself not fetched. The relation at Reach: the field's assessment, Reviews of Geophysics 58(4):e2019RG000678, 2020, doi 10.1029/2019RG000678, twenty-five authors, at its abstract as the publisher's page, four university repositories, a database and an agency's page carry it verbatim (the equilibrium sensitivity characterized by an effective sensitivity; three lines of evidence; a Bayesian combination; limitations in the traditional forcing-feedback paradigm) and at a passage of its text as the publisher's page carries it (effective sensitivity more relevant to the time scales of interest; the narrowing occurring because the lines agree and are judged largely independent), the assessment itself not fetched. Persistence, no landing: the closing condition at the assessment read at Reach (a narrower, stable range supported jointly by the independent lines of evidence), the dominant feedback uncertainty keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Instrumental energy-budget inference — direct to the recent climate and sensitive to forcing and ocean heat uptake. Paleoclimate inference — broader forcing and temperature range with proxy and boundary-condition uncertainty. Process constraints — target individual feedbacks. Model ensembles — integrate processes but share structures and are not a probability sample. **Narrowing recorded.** The assessed likely range is 2.5 to 4 degrees Celsius, with a best estimate of 3 degrees Celsius under the conventional definition. **Progress.** Multiple evidence lines substantially reduced the very low and very high tails and identified cloud feedback as a principal remaining source of spread. **Own record.** ## 197 Earthquake predictability **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Can the location, time and magnitude of a future earthquake be predicted with skill beyond a declared probabilistic baseline? **The field's account of its hardness.** That earthquake predictability, the degree to which the future occurrence of earthquakes can be determined from the observable behaviour of earthquake systems, is poorly understood, the lack reflected in the inability to reliably predict large earthquakes in seismically active regions on short time scales; that most proposed prediction methods rely on the concept of a diagnostic precursor, a signal observable before earthquakes indicating with high probability the location, time and magnitude of an impending event, and the search for diagnostic precursors, across changes in strain rates, seismic wave speeds and electrical conductivity, radon and groundwater variations, electromagnetic and thermal anomalies, animal behaviour and seismicity patterns, has not yet produced a successful short-term prediction scheme, error rates and the large areal extent of the predictions not yet providing the diagnostic capability needed for operational predictions; and that current knowledge about precursors is poor, many intriguing observations not yet fully explored, and the search not to be abandoned. **Accounts.** Rupture is preceded by observable physical changes that can support short-term prediction. Earthquake occurrence is predictable only probabilistically through rates, stress transfer and fault history. The rupture process is sufficiently sensitive or scale-free that precise event prediction is not physically available from observable states. **Specifications.** A prediction states location, time window, magnitude range and decision rule before the event — universal under prospective testing. Skill is measured against a declared null forecast and includes misses and false alarms — universal under forecast evaluation. Precursors are defined without retrospective tuning — local. **Specifications withdrawn.** **Common requirement.** A prospective rule whose performance can be replicated over a defined earthquake population. **Conserving relation.** The rule maintains stated sensitivity and false-alarm performance outside the data used to construct it. **Conserving reach.** All qualifying events and non-events in the declared region, magnitude range and evaluation interval. **Reach.** The statement ranges over future individual events. Catalogues, geodesy and candidate precursor instruments sample incomplete fault states and finite event populations. The two are not the same: a diagnostic precursor must indicate with high probability the location, time and magnitude of an impending event, and the error rates and the large areal extent of the predictions from candidate precursors do not yet provide that diagnostic capability, the search having produced no successful short-term scheme. **Unreachability.** **Persistence.** Prospective replicated predictive skill at operationally useful resolution, or a demonstrated physical limit excluding that resolution under stated observations. **Addresses.** Statement at the field's frame, short-term prediction and forecasting of tectonic earthquakes and the utilization of possible forerunners, and the parting of prediction from forecasting at Own record: Jordan et al., "Operational Earthquake Forecasting: State of Knowledge and Guidelines for Utilization", Annals of Geophysics 54(4):315–391, 2011, at its abstract as a university repository's page carries it verbatim (the commission's charge to report on the state of knowledge of short-term prediction and forecasting and on guidelines for the use of possible forerunners; its review of research on prediction and forecasting worldwide), the report itself not fetched, its findings not read this pass. Freund, Ouillon, Scoville and Sornette, "Earthquake precursors in the light of peroxy defects theory: critical review of systematic observations", European Physical Journal Special Topics 230:7–46, 2021, and Zhuang and Sornette, "How to quantify earthquake predictability? Advances in earthquake forecasting and predictability limits", arXiv:2607.26918, 2026, cited here and not read this pass. Hardness: re-formed to the field's words at the report the address names, at its abstract and findings as the publisher's page, a university repository, an index, a repository copy and its executive summary carry them verbatim (predictability poorly understood, reflected in the inability to predict reliably on short time scales; most methods relying on a diagnostic precursor; the precursor methods reviewed; the search not yet producing a successful short-term scheme; error rates and areal extent not providing the diagnostic capability; knowledge of precursors poor, the search not to be abandoned), doi 10.4401/ag-5350, the report itself not fetched; the former line, in the file's compression, released with its content at the second and third clauses, its clause on prospective evaluation against explicit false-alarm and miss rates standing at the report's guidelines, not returned, keyed at the next pass; the two papers cited beside it not read this pass. The relation at Reach: the report's findings, at the passages read for the hardness. Approaches: the precursor route and its ending at the report's findings read above (the methods reviewed, no successful scheme); the statistical-forecast route at the report's finding on short-term clustering; the laboratory-friction and machine-learning routes keyed at the next pass. Persistence, no landing: the closing condition at the report's findings read above (predictions with error rates and areal extent providing the diagnostic capability needed for operational use; prospective evaluation at the guidelines, not returned). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Statistical rate forecasts — show probabilistic skill at specified space-time scales and do not predict individual events. Seismic, geodetic, electromagnetic and geochemical precursors — recurring candidates without a prospectively replicated universal rule. Laboratory friction — identifies nucleation behaviors under controlled faults, with field scaling open. Machine learning — improves selected catalogue tasks, with leakage, baseline and transfer controls decisive. **Narrowing recorded.** Operational forecasting narrows the claim from exact event prediction to time-dependent probabilities over declared bins. **Progress.** Prospective testing frameworks, reference forecasts, stress-transfer models and dense geodetic and seismic observations stand. **Own record.** Prediction and forecast are retained as different claims; success at the latter is not entered as settlement of the former. ## 198 The initiation of plate tectonics **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How and under what conditions does a new self-sustaining subduction zone begin? In the field's own posing, how the first subduction was initiated, and with it how plate tectonics began, stands as one of the largest unanswered questions in Earth science. **The field's account of its hardness.** Mature subduction supplies slab pull and inherited weak boundaries, whereas initiation must break and bend strong lithosphere before those conditions exist; ancient examples are overprinted and present candidates are rare and incompletely observed. **Accounts.** Compression or transform-boundary reorganization forces one plate beneath another. Density contrasts and lithospheric instability produce spontaneous sinking along a weakness. Mantle plumes or small-scale convection weaken and load the lithosphere until it founders. Delamination and tearing propagate into a self-sustaining subduction system. **Specifications.** The process creates a downgoing plate and a persistent plate boundary — universal under subduction initiation. The forces overcome lithospheric strength before mature slab pull exists — universal under first initiation. Geological markers survive later deformation sufficiently to identify sequence — local. **Specifications withdrawn.** **Common requirement.** A mechanically viable transition from intact or inherited lithosphere to sustained underthrusting and slab descent. **Conserving relation.** One mechanism reproduces the force balance, deformation sequence and geological and seismic observations of initiation. **Conserving reach.** Passive margins, transforms, extinct boundaries and other environments in which initiation is proposed. **Reach.** The statement ranges over the full initiation sequence. Observations provide present snapshots and altered geological remnants. The field's review parts present-day initiation, where sinking slabs can supply far-field stresses, from the first initiation, raising a paradox of the first initiation, existing subduction appearing to be required before the first subduction began: the initiations observed on present-day Earth and the first initiation sought are not the same, the paradox standing between them. **Unreachability.** **Persistence.** A directly constrained sequence and force budget that establishes how initiation becomes self-sustaining across known cases. **Addresses.** Statement, with the posing added there, hardness at the breaking and bending of strong lithosphere with weakness and slab pull both absent, and the relation at Reach: Lu et al., "Reviewing subduction initiation and the origin of plate tectonics: What do we learn from present-day Earth?", Earth and Planetary Physics 5:123–140, 2021, doi 10.26464/epp2021014, at its abstract as the publisher's page and the journal's page carry it verbatim (how the first subduction was initiated and hence how plate tectonics began, stated as one of the biggest unanswered questions in Earth science; the main challenge, how strong lithosphere could break and bend with plate-tectonics-related weakness and slab pull both absent; a review of initiation models at their prerequisites and driving mechanisms; the plume–lithosphere-interaction and mantle-convection models not relying on existing plate tectonics; mantle drag as a possible missing driving force; a composite driving mechanism proposed; a paradox of the first initiation raised, existing subduction appearing to be required before the first), the review itself not fetched. The hardness line's second clause, on overprinted ancient examples and rare present candidates, and the accounts, specifications and approaches as posed, stand keyed at the next pass; the abstract groups the models as plume–lithosphere interaction, mantle convection, mantle drag and a composite, where the account stands as compression or transform-boundary reorganization. Present evidence and progress: Duarte et al., "Seismic evidence for oceanic plate delamination offshore southwest Iberia", Nature Geoscience, 2025, and Crameri et al., "A transdisciplinary and community-driven database to unravel subduction zone initiation", Nature Communications 11:3750, 2020, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a review of initiation and the origin of plate tectonics at its introduction as the journal's page carries it (a paradox of the first initiation, appearing to require existing subduction before the first; the main difficulty reconciling the great plate strength without fracture zones with the weak driving force before slab pull), Earth and Planetary Physics, 2021, doi 10.26464/epp2021014, its authors not carried; a community database paper at its summary (the mechanisms of initiation remaining highly controversial; a clear direction for future research), Nature Communications, 2020, doi 10.1038/s41467-020-17522-9; the review the address names at a passage as the author's copy carries it (the Cenozoic record, a third of active zones formed in the last sixty-five million years, the evidence base for how zones form). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Geological reconstruction — orders initiation markers, with later overprinting. Seismic imaging — resolves present slabs and delamination, without observing their beginning. Analogue and numerical experiments — identify parameter regimes for forced and spontaneous modes. Comparative databases — standardize candidate cases while classifications remain revisable. **Narrowing recorded.** Initiation modes are divided by inherited boundary, force source and kinematic signature; individual margins are assigned conditional histories. **Progress.** Candidate modern initiation zones, global case databases and mechanically explicit forced, spontaneous and plume-assisted models stand. **Own record.** ## 199 The mechanics driving plate motion **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What forces drive and resist each tectonic plate, and what are their relative contributions to its observed motion? In the field's own posing, the gravitational pull of subducted slabs is thought to drive the plates while the coupling between slabs and plates stands not well established, an attached slab pulling directly and a detached slab driving the plate through the mantle flow it excites. **The field's account of its hardness.** That determining the balance of forces acting on the plates remains, in the field's words, one of the ultimate goals of mantle-convection modelling, and the relative importance of the different forces in governing subduction motions and styles remains incompletely understood; that basal drag has been proposed as a minor contributor from the lack of correlation between plate size and velocity while analytical calculations and global mantle-flow models indicate it can be substantial, higher slab pull for older plates tending to be offset by higher basal drag beneath the larger plates, which in part explains the missing correlation despite basal drag's role in the force balance; and that it remains unclear whether the faster speeds of subducting plates are caused by additional pull from the slabs or by increased viscous drag beneath deeply protruding continental roots, a trade-off between the mantle's density scaling and the slab-pull factor found in the fitting. **Accounts.** Negative buoyancy of attached slabs supplies dominant direct slab pull. Sinking slabs drive mantle flow that supplies basal traction and trench suction. Gravitational potential energy of elevated ridges and continents supplies significant lateral forcing. Plates and mantle form one convective boundary-layer system in which no single force acts independently. **Specifications.** Each plate satisfies force and torque balance — universal under plate dynamics. Mantle viscosity, slab geometry and boundary strength are represented — universal under geodynamics. Surface velocities and stress indicators constrain the same model — universal under the inverse problem. **Specifications withdrawn.** **Common requirement.** A force budget reproducing plate velocities, boundary stresses and mantle flow. **Conserving relation.** The vector sum of driving and resisting forces closes for every plate and predicts its observed motion. **Conserving reach.** All major plates, boundary types and attached or detached slabs in the present global system. **Reach.** The statement ranges over separate force contributions. Geodesy measures net motion, while tomography and stress indicators constrain inputs indirectly. The field's paper states that the relative importance of pull and suction was estimated from the geologic history of subduction, and that observed plate motions are best predicted with upper-mantle slabs attached and pulling: the motions observed and the force contributions sought are not the same, a prediction fitted to the observed motion standing between them. **Unreachability.** **Persistence.** A globally consistent force and torque decomposition whose terms are independently constrained and predict plate-specific motion. **Addresses.** Statement, with the posing added there, the accounts at the parting of pull from suction, progress at the estimate that attached upper-mantle slabs supply about half the driving force, and the relation at Reach: Conrad and Lithgow-Bertelloni, "How Mantle Slabs Drive Plate Tectonics", Science 298:207–209, 2002, doi 10.1126/science.1074161, at its abstract as the publisher's page and an abstract service carry it verbatim (the gravitational pull of subducted slabs thought to drive plate motions, the coupling between slabs and plates not well established; an attached slab exerting a direct pull, a detached slab driving a plate through mantle flow exerting a shear traction at its base; the relative importance of pull and suction estimated from the geologic history of subduction; observed motions best predicted with upper-mantle slabs attached, slab pull about half the driving force; lower-mantle slabs supported by viscous forces and driving plates through suction), the paper itself not fetched. Force correlation: Schellart, "Quantifying the net slab pull force as a driving mechanism for plate tectonics", Geophysical Research Letters 31:L07611, 2004, and the coupled-system formulation and approaches: Faccenna et al., "What drives tectonic plates?", Science Advances 5:eaax4295, 2019, cited here and not read this pass. Hardness: re-formed to the field's words at three papers returned in the search, none fetched: one at its introduction as a university repository copy carries it verbatim (understanding global plate motions one of the central problems of geodynamics; determining the balance of forces one of the ultimate goals of mantle-convection modelling; the plates driven by slab pull and ridge push opposed by basal drag and collisional resistance), Geophysical Research Letters, 1995, its authors carried at the copy's citation as Lithgow-Bertelloni and Richards; one at its abstract as an index and a directory carry it verbatim (the relative importance of the forces incompletely understood; basal drag proposed minor from the lack of a size-velocity correlation while calculations and global flow models indicate it substantial; higher slab pull for older plates offset by higher basal drag), Solid Earth 12:79-93, 2021, doi 10.5194/se-12-79-2021, its authors Suchoy, Goes, Maunder, Garel and Davies; one at its opening and a passage as the publisher's page carries them (unclear whether subducting plates' faster speeds come from additional slab pull or increased drag beneath continental roots; a trade-off between the density scaling and the slab-pull factor), Geochemistry, Geophysics, Geosystems, 2012, doi 10.1029/2011GC003873, its authors not carried; the former line, in the file's compression (plate-edge forces and basal tractions are coupled parts of one mantle-convection system; different global force balances fit observed velocities, and high correlation among terms makes their absolute contributions difficult to separate), released with its content at the three clauses; the force-correlation source the address names not returned at its text, not read this pass. Persistence, no landing: the closing condition at the 1995 letter read at Reach (determining the balance of forces one of the ultimate goals of mantle-convection modelling), the independently constrained terms keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Global torque balance — estimates force sets conditional on mantle viscosity and boundary coupling. Plate-velocity correlations — support slab-related forcing without uniquely dividing pull from flow. Seismic tomography and geodynamic simulation — connect slabs to mantle circulation under nonunique material parameters. Regional stress analysis — constrains local force balance without closing the globe. **Narrowing recorded.** Direct attached-slab calculations and global correlations place slab-related forcing above ridge push for many plates; plate-specific exceptions remain. **Progress.** Slab pull, slab suction, ridge potential and basal traction have quantitative formulations, and modern geodesy fixes the net motions precisely. **Own record.** ## 200 Mantle plumes **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which intraplate volcanic provinces are produced by deep mantle plumes, where do those plumes originate, and how do they evolve? In the field's own posing, the origin, dynamics and composition of plumes and their links to plate tectonics remain contested, owing to the difficulty of resolving mantle upwellings in geophysical images and to discrepancies in the interpretation of geochemical and geochronological data. **The field's account of its hardness.** Volcanic tracks, geochemistry, uplift and seismic anomalies are incomplete and can be produced by plume, plate and shallow-mantle processes; tomography resolves filtered present structure while the proposed sources and histories span hundreds of millions of years. **Accounts.** Thermochemical upwellings rise from the lowermost mantle and generate hotspots and large igneous provinces. Upper-mantle convection and lithospheric extension generate intraplate volcanism without deep plumes. Deep upwellings interact with ridges, lithosphere and plate motion so surface expressions migrate, split or lapse. **Specifications.** A proposed plume obeys buoyancy, heat and mass conservation — universal under mantle dynamics. Its predicted track is evaluated with reconstructed plate motion — universal under plume-track inference. Geochemical and seismic tracers retain interpretable source information — local. **Specifications withdrawn.** A strictly fixed surface hotspot, withdrawn in reconstructions that retain a moving deep upwelling. **Common requirement.** A mantle-source history producing the volcanic, geochemical, topographic and seismic record together. **Conserving relation.** One source-and-motion account returns concordant ages, locations, compositions and mantle structure. **Conserving reach.** The lifetime of each candidate province from deep source through surface volcanism. **Reach.** The statement ranges over past three-dimensional mantle flow. Observations supply present tomography and discontinuous surface and rock records. The field's review states that mantle upwellings are difficult to resolve in geophysical images and that the interpretations of the geochemical and geochronological data disagree, observations suggesting that more than eighteen plumes might be rooted in the lowermost mantle: the anomalies imaged and the upwellings sought are not the same, the resolution of the images standing between them. **Unreachability.** **Persistence.** Source depths and histories established for candidate provinces by mutually consistent, discriminating observations. **Addresses.** Statement, with the posing added there, hardness at the difficulty of resolving upwellings in geophysical images and the discrepancies among interpretations of the geochemical and geochronological data, the account at plumes both thermal and chemical rooted in the lowermost mantle, approaches at seismic imaging, mantle flow modelling, plate reconstructions and geochemical analyses, progress at the deep-source geochemistry and the count possibly rooted in the lowermost mantle, and the relation at Reach: Koppers et al., "Mantle plumes and their role in Earth processes", Nature Reviews Earth & Environment 2:382–401, 2021, doi 10.1038/s43017-021-00168-6, at its abstract as the publisher's page, a funder's repository and an index carry it verbatim (plumes first proposed in the 1970s for intraplate hotspot volcanism; the origin, dynamics and composition of plumes and their links to plate tectonics still contested, owing to difficulties in resolving upwellings with geophysical images and discrepancies in interpretations of geochemical and geochronological data; progress reviewed in seismic imaging, mantle flow modelling, plate reconstructions and geochemical analyses; plumes possibly both thermal and chemical, complex and broad in shape, more than eighteen possibly rooted in the lowermost mantle; the case for a deep origin strengthened by hotspot geochemistry carrying deeply recycled subducted components and primordial domains), the review itself not fetched. The venue given before this pass, Nature Reviews Earth & Environment 4:382–401, 2023, is withdrawn, the publisher's page carrying volume 2, 2021. Dynamic topography at approaches, and the hardness line's clauses on the several processes producing the same signatures and on the timescales of the histories, stand keyed at the next pass. Long-term behavior and progress: Pearman et al., "Collapse and resurgence of the Iceland mantle plume", Nature Communications 17:4104, 2026, cited here and not read this pass. Persistence, no landing: the closing condition at the review the address names, at its abstract as the publisher's page, an abstract service, a database and three repository copies carry it verbatim (the origin, dynamics and composition of plumes still contested; future research requiring improvements in the resolution of seismic tomography to visualize deep plume structures at scales smaller than a hundred kilometres, and concerted multi-proxy geochemical and dating efforts to resolve the spatiotemporal and chemical evolution of long-lived plumes), Nature Reviews Earth and Environment, 2021, doi 10.1038/s43017-021-00168-6, the review itself not fetched. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Seismic tomography — images velocity anomalies with resolution and interpretation limits. Geochemical tracers — distinguish source reservoirs under mixing and alteration models. Age-progressive tracks — test source motion together with plate reconstructions. Geodynamic simulations — show plume ascent and interaction for specified rheology. Dynamic topography — tests buoyant support with uncertain mantle structure. **Narrowing recorded.** Joint seismic and geochemical evidence restricts several provinces to deep-source accounts; reconstructed tracks permit moving rather than fixed sources. **Progress.** Whole-mantle anomalies, age-progressive chains, deep-source isotope signatures and plume-ridge interactions are resolved for selected systems. **Own record.** ## 201 Geomagnetic reversals **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What causes the geodynamo to reverse polarity, and what controls the irregular timing, duration and field structure of reversals? In the field's own posing, the physical mechanism driving the field to reverse stands controversial, the periods spent in one polarity being stochastically distributed, the rate of reversals changing in time and its long-term variations, on timescales of the order of mantle convection, being problematic to model. **The field's account of its hardness.** The governing dynamo is nonlinear and only indirectly observed; the paleomagnetic record is incomplete and time-averaged, while numerical models operate in parameter regimes far from Earth's core and can reverse through more than one route. A statistical model of the reversal rate states that the mechanism driving the reversals stands controversial and that statistical inference about the reversals is conditioned by their relatively small number. **Accounts.** Internal turbulent fluctuations carry the dynamo between polarity states stochastically. Changes in heat flux at the core-mantle boundary modulate reversal frequency and geometry. Inner-core growth and long-timescale core evolution change the dynamo's stability. Reversals arise from low-dimensional interactions among dipole and multipole magnetic modes. **Specifications.** The account is generated by electrically conducting fluid motion in the core — universal under dynamo theory. It reproduces stable chrons, excursions, full reversals and secular variation — universal under the geomagnetic record. Paleomagnetic directions and intensities preserve the relevant field history — local. **Specifications withdrawn.** **Common requirement.** A dynamo mechanism yielding the observed polarity statistics and transition morphology. **Conserving relation.** One dynamo account returns field behavior consistent across secular variation, excursions and reversals. **Conserving reach.** The geological reversal record and the range of core states compatible with present observations. **Reach.** The statement ranges over unobserved core dynamics through geological time. Measurements sample the present field and discontinuous remanent magnetization. A statistical model of the reversal rate states that the long-term variations of the rate are problematic to model, occurring on timescales of the order of mantle convection, and derives the heat-flux variations at the core–mantle boundary from the reversal sequence itself: the sequence recorded and the core conditions sought are not the same, a model standing between them. **Unreachability.** **Persistence.** A physically scaled dynamo account that predicts reversal statistics and transition structure from independently constrained core conditions. **Addresses.** Statement, with the posing added there, hardness at the sentence on the controversial mechanism and the small number of reversals, an approach at the stochastic model of the reversal rate, progress at the reversal record analysed back to the late Devonian and the rate found not constant, and the relation at Reach: Carbone, Alberti, Lepreti and Vecchio, "A model for the geomagnetic field reversal rate and constraints on the heat flux variations at the core-mantle boundary", Scientific Reports 10:13008, 2020, doi 10.1038/s41598-020-69916-w, at its abstract and introduction as the publisher's page carries them (polarity reversals a striking feature of dynamo-generated fields; a few hundred stochastic polarity switches in the record; the rate changing in time; short-term switching described by chaotic dynamical systems, the long-term rate variations problematic to model at mantle-convection timescales; cycles of variable frequency found; the transition to superchrons described as a second-order phase transition interpreted as driven by heat-flux variations at the core–mantle boundary; the mechanism driving reversals still controversial; statistical inference conditioned by the relatively small number of reversals), the paper read whole at the page, a departure from the sitting's rule recorded in the ledger. The title given before this pass, "A model for the geomagnetic field reversal rate and distribution", and the article number 15340 are withdrawn, the publisher's page carrying the title above and article 13008. Glatzmaier and Coe, "Magnetic Polarity Reversals in the Core", Treatise on Geophysics, second edition, volume 8, 2015, returned at its citations only, the pages differing among them (283–298; 279–295; 283–297 at the first edition, 2007), not read this pass; the hardness line's own clauses, the account and the specifications stand keyed at the next pass. Statistical modelling and progress: Davies et al., "Rapid geomagnetic changes inferred from Earth observations and numerical simulations", Nature Communications 11:3371, 2020, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a review of reversal records at its abstract as the publisher's page and an archive copy carry it verbatim (questions remaining on the dynamical processes giving rise to reversals; controversial interpretations persisting; future work to involve millimetre-size specimens and physics-based models), Reviews of Geophysics, 2016, doi 10.1002/2015RG000506, its authors not carried; a simulation paper at its introduction (a challenge to simulate reversals while maintaining an Earth-like force balance with secondary inertial effects), Scientific Reports, 2024, doi 10.1038/s41598-024-59849-z; an earlier review's passage (the models operating in a parameter regime far from the core's; the observations of low and inhomogeneous coverage), 2010, its authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Paleomagnetic chronology — supplies reversal timing with age and preservation uncertainty. Transitional-field reconstruction — resolves selected events with spatial and sedimentary smoothing. Numerical geodynamo simulation — produces spontaneous reversals outside Earthlike dimensionless parameters. Statistical point-process models — fit reversal-rate distributions without selecting a core mechanism. **Narrowing recorded.** Reversal chronologies restrict rate variation and duration distributions; simulations restrict mechanisms able to retain a strong dipole between transitions. **Progress.** Polarity chronologies, rapid-change bounds and reversing numerical dynamos stand, and reversal frequency is known not to be periodic. **Own record.** ## 202 The extent of Snowball Earth glaciation **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** During Neoproterozoic global glaciations, were the oceans continuously ice-covered to low latitudes, or did substantial open-water or thin-ice refugia persist? In the field's own posing, two of the glaciations reached low latitudes and may have been Snowball events with near-global ice cover, continental ice sheets standing at low latitudes and a plausible case standing that the oceans were almost entirely frozen over. **The field's account of its hardness.** Sedimentary, geochemical and biological records are geographically sparse and altered, while ice-albedo climate models depend strongly on cloud, ocean, dust and ice dynamics; several ice geometries produce overlapping low-latitude glacial signatures. **Accounts.** Thick sea ice covered the tropical ocean, with survival in cracks, brines or hydrothermal settings. A narrow equatorial waterbelt remained open through ocean heat transport and low-latitude ablation. Thin, dynamic or seasonally open tropical ice admitted light and exchange while retaining near-global glaciation. Glacial extent varied substantially among episodes and within each episode. **Specifications.** Paleomagnetic latitude assignments correctly locate glacial deposits — local. Sedimentary and isotopic proxies retain climatic interpretation — local. Climate simulations conserve coupled atmosphere-ocean-ice energy and mass — universal under climate modelling. **Specifications withdrawn.** **Common requirement.** An ice geometry and climate history consistent with low-latitude deposits, deglaciation, ocean chemistry and biological survival. **Conserving relation.** One reconstruction reproduces the geological and climatic constraints for each glacial interval. **Conserving reach.** Tropical and global oceans throughout the Sturtian and Marinoan glaciations. **Reach.** The statement ranges over global, time-varying ice cover. The record samples preserved basins and proxy-forming environments. The field's review states that continental ice sheets stood at low latitudes and that a plausible case can be made that the oceans were almost entirely frozen over: the low-latitude ice recorded and the global cover sought are not the same, a case standing between them. **Unreachability.** **Persistence.** Spatially and temporally resolved ice-cover constraints that distinguish continuous thick ice from open-water and thin-ice states. **Addresses.** Statement, with the posing added there, hardness at the dependence of initiation and deglaciation on snow cover, the nature of the frozen surface, tropical dust and cloud warming, approaches at the suite of general circulation model simulations, progress at the two low-latitude glaciations and the deglaciation combination, and the relation at Reach: Pierrehumbert, Abbot, Voigt and Koll, "Climate of the Neoproterozoic", Annual Review of Earth and Planetary Sciences 39:417–460, 2011, doi 10.1146/annurev-earth-040809-152447, at its abstract as the publisher's page and an abstract service carry it verbatim (the Neoproterozoic a transition between the microbial world and the Phanerozoic, extreme carbon-isotope fluctuations and glaciation resuming; two of the glaciations reaching low latitudes and possibly Snowball events with near-global ice cover; initiation and deglaciation discussed against a suite of general circulation model simulations built for intercomparison; snow cover and the nature of the frozen surface the key factors; initiation possible at a plausible reduction of carbon dioxide; deglaciation requiring elevated carbon dioxide and tropical dust with some cloud warming; the cause of the biogeochemical turbulence and its connection to the glaciations remaining obscure) and at its opening paragraph as a repository copy carries it (continental ice sheets in low latitudes, a plausible case that the oceans were almost entirely frozen over), the review itself not fetched. The hardness line's clauses on the sparse and altered records and on the overlapping signatures of several ice geometries, and the account as posed, stand keyed at the next pass. Tropical ice thickness: Warren and Brandt, "Snowball Earth: A thin-ice solution with flowing sea glaciers", Journal of Geophysical Research: Oceans 111:C09005, 2006, and Warren et al., "Snowball Earth: Ice thickness on the tropical ocean", Journal of Geophysical Research: Oceans 107:3167, 2002, cited here and not read this pass. Persistence, no landing: the closing condition at the review the address names, at a passage of its text as a university repository copy carries it verbatim (self-termination occurring under circumstances that are falsifiable geologically: the cryochrons long-lived, their onsets synchronous at low latitudes and their terminations synchronous everywhere), Science Advances 3(11):e1600983, 2017, doi 10.1126/sciadv.1600983, the review itself not fetched; the standing of the hard-against-slushball question at a 2025 preprint's section (a debate still existing on whether the planet was entirely covered by thick ice or kept a tropical belt of open ocean), arXiv:2509.16338, its authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Paleomagnetism — establishes low-latitude ice where samples preserve primary directions. Sedimentology — identifies ice contact and open-water processes locally. Geochemical proxies — constrain redox and exchange under nonunique proxy models. Coupled climate simulation — returns hard-, soft- and thin-ice equilibria under different physics. **Narrowing recorded.** Low-latitude glaciation stands; surviving-ocean states are narrowed by ice dynamics, dust, productivity and sedimentary observations at particular basins. **Progress.** Multiple near-global glaciations, their broad timing and strong ice-albedo feedback are established; viable refugia and deglaciation mechanisms are explicit. **Own record.** ## 203 The magnitude of the Great Oxidation Event **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How high did atmospheric and oceanic oxygen rise during and immediately after the Great Oxidation Event, and how variable was it? In the field's own posing at the magnitude, the defendable range of possible atmospheric oxygen levels spans orders of magnitude, which begs the question how great the Great Oxidation was; and at the tempo, the rise of atmospheric oxygen was not monotonic but characterized by oscillations, several of them crossing the threshold before the irreversible oxygenation of the atmosphere. **The field's account of its hardness.** The event is defined by threshold-sensitive proxies, while most proxies bound oxygen indirectly, apply to different reservoirs and times, and can be altered after deposition; the loss of mass-independent sulfur isotope fractionation fixes a floor more securely than it fixes an abundance. The field's revisiting of the magnitude states that, despite many new tools to interrogate the event, the defendable range of possible atmospheric oxygen levels spans orders of magnitude. **Accounts.** Oxygen rose to a substantial fraction of its present atmospheric level and later declined. Oxygen crossed the sulfur-isotope threshold but remained at a small fraction of its present level. Atmospheric oxygen oscillated across thresholds while deep oceans remained largely anoxic. **Specifications.** Each proxy has a calibrated oxygen sensitivity and preserved depositional context — local. Atmospheric and oceanic reservoirs are related through stated biogeochemical models — local. The reconstructed history satisfies sulfur, chromium, iron, carbon and trace-metal evidence together — universal under the reconstruction. **Specifications withdrawn.** **Common requirement.** A quantitative oxygen history consistent with all preserved proxy thresholds and reservoir differences. **Conserving relation.** One abundance history returns every valid proxy within its calibration and uncertainty. **Conserving reach.** Atmosphere, surface ocean and deep ocean across the onset, overshoot and aftermath of the event. **Reach.** The statement ranges over continuous global oxygen abundance. Proxies sample local chemical thresholds in discontinuous sedimentary archives. The field's dating paper states that the oxygenation involved several oscillations of oxygen levels across a level of one hundred-thousandth of the present atmosphere before the irreversible oxygenation, and that timing and relationships are debated for poor age constraints and contradictory stratigraphic correlations: the threshold crossings recorded and the abundance sought are not the same, the correlations standing between them. **Unreachability.** **Persistence.** A proxy-consistent quantitative trajectory with bounded peak, floor, spatial variation and duration. **Addresses.** Statement at the tempo, progress at the onset interval, and the relation at Reach: Gumsley et al., "Timing and tempo of the Great Oxidation Event", Proceedings of the National Academy of Sciences 114:1811–1816, 2017, doi 10.1073/pnas.1608824114, at its abstract as the publisher's page and an abstract service carry it verbatim (the first significant buildup of atmospheric oxygen beginning in the early Paleoproterozoic with global glaciations and continuing to the end of the Lomagundi excursion; the timing and relationships debated for poor age constraints and contradictory stratigraphic correlations; the first Paleoproterozoic global glaciation and the onset placed between about 2,460 and 2,426 million years ago, about a hundred million years earlier than estimated before, from an age of 2,426 ± 3 million years for the Ongeluk magmatism; the rise not monotonic but characterized by oscillations possibly continuing over the following two hundred million years) and at its significance statement as an index carries it (several oscillations in oxygen levels across 10⁻⁵ of present atmospheric levels before the irreversible oxygenation), the paper itself not fetched. Statement at the magnitude and hardness at the added sentence: Crockford et al., "Revisiting the greatness of Earth's great oxidation", Communications Earth & Environment, 2026, doi 10.1038/s43247-026-03518-8, at the opening of its abstract as the publisher's page carries it (the accumulation of free oxygen about 2.45 billion years ago among the most transformative events; the defendable range of possible atmospheric oxygen levels spanning orders of magnitude despite many new tools; the question how great the oxidation was), returned in a search this pass, the paper itself not fetched, its volume and article number as given before this pass, 7:348, not carried at the return. The hardness line's own clauses, the account, the proxy specifications and the approaches stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched, their authors not carried: a photochemical study at its abstract as an archive copy carries it (unclear whether the initial rise was monotonic or oscillatory, the record poorly constraining oxygen afterward; the timing and magnitude constrained by the sulfur-isotope record combined with photochemical models), Proceedings of the National Academy of Sciences, 2022; a vanadium-isotope study at its abstract (the nature of surface-ocean oxygenation poorly quantified, limiting the understanding of oxygenation thresholds; a unidirectional transition captured), 2025; a review at its abstract as a repository carries it (the magnitude and extent of oxygenation during the excursion remaining poorly constrained). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Sulfur-isotope thresholds — establish crossing of a photochemical oxygen threshold without a precise high value. Trace-metal and chromium isotopes — provide redox constraints with weathering and reservoir models. Iron formations and red beds — map local redox conditions. Biogeochemical modelling — translates proxy combinations into atmospheric ranges under uncertain fluxes. **Narrowing recorded.** Proxy thresholds bound oxygen above levels compatible with persistent sulfur mass-independent fractionation and below fully modern ocean oxygenation. **Progress.** The event's principal onset interval and irreversible loss of the sulfur signature are established, and distinct atmospheric and oceanic oxygen histories are now represented. **Own record.** ## 204 The causes of the Great Oxidation Event **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What change in biological oxygen production, reductant sinks or Earth-system feedbacks caused atmospheric oxygen to accumulate permanently during the Great Oxidation Event? In the field's own posing, oxygenic photosynthesis may have evolved as early as the Eoarchean and certainly by the end of the Archean, a long period of low oxygen being facilitated by rapid atmospheric oxidation before an ozone layer formed, and the mechanisms controlling the length of the delay remain unknown. **The field's account of its hardness.** Oxygenic photosynthesis appears to predate permanent atmospheric oxygenation, leaving a long and uncertain delay; production and multiple geological sinks are reconstructed from indirect records, and different flux combinations cross the same atmospheric threshold. **Accounts.** Biological oxygen production increased through ecological expansion or nutrient supply. Volcanic and metamorphic gases became less reducing, lowering the oxygen sink. Organic-carbon burial and changes in weathering shifted the global redox balance. Hydrogen escape and long-term mantle-crust evolution progressively oxidized the surface system. Threshold feedbacks among methane, climate and ozone converted a gradual flux change into persistent oxygenation. **Specifications.** The global oxygen source exceeds the combined reductant sinks over the accumulation interval — universal under redox balance. Proxy dates correctly order photosynthesis and atmospheric change — local. Carbon, sulfur and volcanic flux reconstructions describe the same Earth system — universal under the account. **Specifications withdrawn.** The specification that oxygenic photosynthesis began contemporaneously with atmospheric oxygenation, withdrawn where earlier biological evidence is retained. **Common requirement.** A source-sink history that explains both the delayed onset and permanent crossing of the oxygen threshold. **Conserving relation.** One redox budget reproduces the timing and persistence of oxygenation without violating the geological proxy record. **Conserving reach.** Biological, atmospheric, oceanic, crustal and mantle reservoirs across the pre-event interval and transition. **Reach.** The statement ranges over global source and sink fluxes through deep time. The record contains local proxies and model-derived flux bounds. The field's modelling paper states that the mechanisms controlling the length of the delay remain unknown, and evaluates scenarios for two metabolic pathways and the phosphorus cycle in a biogeochemical model of the Earth system to constrain the determinants of the timing: the scenarios modelled and the fluxes that ran are not the same, the model's determinants standing between them. **Unreachability.** **Persistence.** A quantitatively closed redox budget and causal sequence distinguishing the trigger from permissive background change. **Addresses.** Statement, with the posing added there, hardness at the delay between the origin of oxygenic photosynthesis and permanent oxygenation, the account at nutrient supply as a determinant of timing, specifications withdrawn at the early origin retained, progress at the early biological source, and the relation at Reach: Horne, Goldblatt and Kump, "An early origin of oxygenic photosynthesis delays the Great Oxidation", Philosophical Transactions of the Royal Society B 380:20240094, 2025, doi 10.1098/rstb.2024.0094, at its abstract as the publisher's page and two repositories carry it verbatim (the Great Oxidation Event the most significant chemical revolution in Earth's history, 2.4 billion years ago; oxygenic photosynthesis possibly as early as the Eoarchean and certainly by the end-Archean; a long low-oxygen period facilitated by rapid atmospheric oxidation before ozone formation, the mechanisms controlling the length of the delay unknown; a biogeochemical model of the Earth system used to evaluate scenarios for oxygenic photosynthesis, nitrogen fixation and the inorganic phosphorus cycle as determinants of oxygenation timing; an early origin found, counter-intuitively, to lengthen the delay), the paper itself not fetched. The hardness line's clauses on indirect records and on flux combinations, and the approaches, stand keyed at the next pass. Lyons, Reinhard and Planavsky, "The rise of oxygen in Earth's early ocean and atmosphere", Nature 506:307–315, 2014, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: an ecological-dynamics paper at its abstract as the publisher's page carries it verbatim (the causes debated; the event triggered when the difference between the influxes of reductants and phosphate falls below a critical value; the transition gradual and reversible or sudden and irreversible by the sources and sinks), Nature Communications, 2021, doi 10.1038/s41467-021-23286-7; a mantle-oxidation study at its abstract as an archive copy carries it (the delay and timing of oxygenation poorly understood; whether the mechanism works not quantitatively examined before), 2020; a review at its opening (a major puzzle why oxygen rose so late, the search for the factors that limited its accumulation); their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Molecular and geochemical dating of photosynthesis — moves the biological origin earlier while retaining uncertainty. Redox-budget models — reproduce threshold crossing with different flux histories. Sulfur and carbon isotope records — constrain reservoirs under nonunique cycles. Mantle and volcanic reconstructions — test changes in reductant delivery over long intervals. **Narrowing recorded.** Evidence for earlier oxygen production narrows accounts that make invention and accumulation simultaneous; sulfur constraints narrow the atmospheric transition interval. **Progress.** A source-sink threshold framework, an early biological source and multiple independently dated environmental changes stand. **Own record.** ## 205 The origin of the Moon **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What collision and subsequent evolution produced the Moon's mass, orbit, composition and isotopic relationship to Earth? In the field's own posing, a successful scenario for lunar origin should explain the Moon's chemical composition, its volatile content and stable isotope ratios, its initial thermal state and the system's bulk physical and dynamical properties, the origin of the Moon standing as one of the most fundamental and vexing questions in planetary science. **The field's account of its hardness.** A successful account must satisfy angular momentum, disk mass, lunar iron depletion, volatile loss and near-identical terrestrial and lunar isotope ratios together; impact simulations can meet subsets of these constraints with different initial states and later tidal histories. **Accounts.** A grazing collision with a roughly Mars-mass body produced a disk dominated by impactor material followed by mixing. A higher-energy collision produced a substantially vaporized, well-mixed structure from which the Moon condensed. A rapidly rotating early Earth underwent an impact followed by angular-momentum transfer through orbital resonance. Multiple smaller impacts built successive disks that accumulated into the Moon. **Specifications.** Total angular momentum is conserved through impact and specified later torques — universal under dynamics. Material partition and isotope evolution follow the thermodynamic impact state — universal under geochemistry. The account reproduces present lunar mass, core fraction and orbit — universal under lunar formation. **Specifications withdrawn.** A strictly unchanged post-impact angular momentum, withdrawn in accounts retaining resonant transfer to the heliocentric orbit. **Common requirement.** One impact-and-evolution history satisfying dynamical, chemical and isotopic constraints. **Conserving relation.** The same initial conditions and subsequent evolution return every established Earth-Moon observable. **Conserving reach.** Impact, disk or synestia, lunar accretion, differentiation and tidal orbital evolution. **Reach.** The statement ranges over an unobserved formation sequence. Samples and dynamics constrain its final products, and simulations map specified initial conditions to outcomes. The field's review states that hydrocode simulations have long matched the bulk properties while the chemical and isotopic composition required detailed work on the disk, and that the theory assembled matches the overwhelming majority of the observational constraints, a broader exploration of impact parameter space still able to yield insight: the constraints matched and the collision sought are not the same, the parameter space standing between them. **Unreachability.** **Persistence.** A formation history independently fixed by discriminating sample, geophysical or dynamical evidence and reproducing the complete constraint set. **Addresses.** Statement, with the posing added there, hardness at the bulk properties matched and the chemical and isotopic composition not, the account at the giant impact as the class, approaches at hydrocode simulation and disk evolution, progress at the leading theory and the end-to-end theory matching most constraints, and the relation at Reach: Barr, "On the origin of Earth's Moon", Journal of Geophysical Research: Planets 121:1573–1601, 2016, doi 10.1002/2016JE005098, at its abstract and key points as the publisher's page, the arXiv page and an abstract service carry them verbatim (the Giant Impact the leading theory; a successful scenario explaining the Moon's chemical composition, initial thermal state and the system's bulk physical and dynamical properties; the impact consistent with the lunar mass, angular momentum and iron content but not its chemical and isotopic composition; hydrocode simulations long matching the bulk properties and detailed work on the protolunar disk yielding insight into the chemistry and early thermal history; the elements of an end-to-end theory matching the overwhelming majority of the observational constraints; new samples, disk processes and a broader exploration of impact parameter space still able to yield insight), the review itself not fetched. The hardness line's own list of constraints, the account's grazing collision with a roughly Mars-mass body, isotope measurement and tidal evolution at approaches, and the specifications stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a review of the giant-impact hypothesis at its abstract as the journal's pages carry it verbatim (the canonical model facing the isotope crisis; a range of new computational models proposed; the inquiry still far from a conclusive resolution, acquiring additional experimental and exploratory data imperative), Space: Science and Technology, 2024, doi 10.34133/space.0153, its authors not carried; a refractory-element study at a passage as the publisher's page carries it (the traditional isotopic proxy's constraining power limited and insufficient on its own to resolve the partial-mixing and high-energy models), Earth and Planetary Science Letters, 2025; a chapter of the field's lunar volume at its opening as the arXiv copy carries it (the conundrum become more acute, characterized as an isotopic crisis), arXiv:2103.02045. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Hydrodynamic impact simulation — finds successful regions conditional on numerical resolution, equations of state and initial bodies. Isotope measurement — establishes close Earth-Moon similarities and element-specific differences. Disk chemistry and mixing models — test equilibration under uncertain thermal histories. Tidal evolution — connects post-impact angular momentum to the present orbit through model-dependent dissipation. **Narrowing recorded.** Sample chemistry excludes simple intact capture and unmodified co-accretion accounts; angular momentum and core fraction restrict impact geometry and material partition. **Progress.** A giant impact is the leading class, high-precision isotope constraints stand, and several dynamically viable impact regimes have been demonstrated. **Own record.** ## 206 The early Solar System impact flux **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Was the inner Solar System's early bombardment a short late spike, a monotonic decline, or a more complex sequence, and what supplied the impactors? In the field's own posing, heavily cratered surfaces on the Moon, Mars and Mercury show an intense bombardment during the planets' first billion years or more, while the timing, sources and dynamical implications of those impacts are controversial. **The field's account of its hardness.** The oldest surfaces were repeatedly reworked, many sample ages come from unknown lunar source craters, impact heating can reset several clocks, and dynamical models generate different flux histories that overlap the surviving age record. **Accounts.** A delayed dynamical instability injected a concentrated late cataclysm of impactors. The flux declined from accretion without a distinct late spike. Several impact episodes arose from leftover planetesimals, asteroid-belt destabilization and basin-forming collisions. Sampling and reset biases create an apparent age cluster from a broader flux. **Specifications.** Radiometric ages record specified impact events rather than inherited crystallization or later resetting — local. Crater counts map to absolute flux through a stated chronology — local. Dynamical source populations reproduce impactor compositions and planetary constraints — universal under bombardment models. **Specifications withdrawn.** A uniquely sharp lunar-wide cataclysm at one age, withdrawn in reconstructions that retain prolonged or multiple bombardment. **Common requirement.** One time-dependent impactor flux consistent with sample ages, crater populations and dynamical source reservoirs. **Conserving relation.** The inferred flux returns the surviving chronological, spatial and compositional record across planetary bodies. **Conserving reach.** The inner planets and Moon from accretion through the end of basin-scale bombardment. **Reach.** The statement ranges over a continuous multi-body flux. Samples and craters preserve a filtered, reset and spatially incomplete record. The field's review defines the bombardment as the impacts after the planetary lithospheres stabilized such that they could be preserved as craters and basins, the melt-rock and shock ages pointing toward a discrete episode: the flux recorded and the flux sought are not the same, the stabilization standing between them. **Unreachability.** **Persistence.** A flux history and source mixture distinguished by samples with known context and cross-planet chronology. **Addresses.** Statement, with the posing added there, hardness at the dynamical models each able to account for the observations with pros and cons, the accounts at the residual and the destabilized populations, specifications withdrawn at the hybrid of discrete early and later populations, progress at the discrete episode, the quiescence, the long-lived tail and the hybrid model, and the relation at Reach: Bottke and Norman, "The Late Heavy Bombardment", Annual Review of Earth and Planetary Sciences 45:619–647, 2017, doi 10.1146/annurev-earth-063016-020131, at its abstract as the publisher's page and a university repository carry it verbatim (heavily cratered surfaces on the Moon, Mars and Mercury showing an intense bombardment in the first billion years or more, the timing, sources and dynamical implications controversial; the bombardment defined as impacts after lithosphere stabilization such that they could be preserved as craters and basins; lunar melt rocks and meteorite shock ages pointing toward a discrete episode of elevated flux between about 3.5 and 4.0–4.2 billion years ago and a relative quiescence between about 4.0–4.2 and 4.4; Precambrian spherule layers suggesting a long-lived tail to about 2.0–2.5 billion years ago; dynamical models with populations residual from accretion and destabilized by giant-planet migration potentially accounting for the observations, all with pros and cons; a hybrid model with discrete early and later populations the most parsimonious), the review itself not fetched. The hardness line's clauses on reworked surfaces, unknown source craters and reset clocks, the specifications and the approaches stand keyed at the next pass. Sample-age revision: Černok et al., "Lunar samples record an impact 4.2 billion years ago that may mark the start of the lunar cataclysm", Communications Earth & Environment 2:120, 2021, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched, their authors not carried: a zircon study at its introduction as the publisher's page carries it (the duration and nature of the phase remaining unsettled), Earth and Planetary Science Letters, 2023; a Bayesian basin-chronology study at its abstract as a repository carries it (the impact flux from a smooth decline to a terminal cataclysm across the scenarios; the behaviour before the youngest basin unconstrained, specific samples needed), 2017; an isotopic study at its abstract as the publisher's page and an archive copy carry it (the impactors of the bombardment and of late accretion the same bodies from the terrestrial-planet region, the bombardment concluded to be the tail end of accretion), Science Advances, 2021, doi 10.1126/sciadv.abh2837. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Isotopic dating — resolves heating events with source-context and resetting ambiguity. Crater chronology — supplies relative flux under uncertain oldest-surface calibration. Dynamical simulation — generates instability and decay histories conditional on initial architecture. Impact-melt petrology — connects ages to thermal events and impactor sources. **Narrowing recorded.** Ancient lunar ages extend significant impact activity earlier than a narrowly timed spike; dynamical and geochemical constraints restrict candidate reservoirs. **Progress.** Multiple basin ages, improved reset physics and Solar System instability models stand, while the sharp classical spike is no longer the sole viable history. **Own record.** ## 207 The origin of animal multicellularity **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What sequence of ecological and molecular changes transformed the unicellular ancestors of animals into an integrated multicellular animal lineage? In the field's own posing, the unicellular ancestor of animals is now better grasped through the study of its diverse close unicellular relatives, while how that ancestor became the first animals stands unclear. **The field's account of its hardness.** Many adhesion, signaling and transcriptional components predate animals, the relevant unicellular lineages are sparsely sampled, and the transition left limited direct fossils; component presence does not fix how regulation, development and ecological selection were assembled. The field's review states that, given the limits of current knowledge, the origin of animals is a biological black swan event that defies retrospective explanation, so that confirmation biases based on few data are to be guarded against. **Accounts.** Colonial choanoflagellate-like ancestors increased cell adhesion and division of labor. Temporarily multicellular life cycles under predation or environmental fluctuation stabilized cooperation. Regulatory rewiring co-opted ancient adhesion and signaling genes into development. Symbiosis, microbial interaction or another rare ecological event triggered the transition. **Specifications.** The account is consistent with the animal phylogeny and closest unicellular relatives — universal under comparative biology. It generates heritable cell adhesion, communication, differentiation and reproduction — universal under multicellularity. Molecular dates and fossils are placed under explicit clock and preservation models — local. **Specifications withdrawn.** The specification that core animal developmental genes originated only with animals, withdrawn where homologues in unicellular relatives are retained. **Common requirement.** A sequence linking ancestral components to stable multicellular development and reproduction. **Conserving relation.** The same transition account reproduces phylogenetic distribution, molecular function and plausible ecology. **Conserving reach.** The stem lineage from unicellular holozoans to the last common ancestor of living animals. **Reach.** The statement ranges over a singular ancient transition. Comparative genomes, living relatives and fossils preserve different fragments of it. The field's review states that the unicellular ancestor is better grasped through its living relatives while how it became the first animals stands unclear, the origin of animals defying retrospective explanation on the data at hand: the ancestor reconstructed and the transition sought are not the same, the few data standing between them. **Unreachability.** **Persistence.** An ordered, experimentally supported reconstruction connecting ancestral states to integrated animal multicellularity. **Addresses.** Statement, with the posing added there, hardness at the added sentence, the accounts at the two theories the review names, approaches at the study of close unicellular relatives, progress at the better grasp of the ancestor, and the relation at Reach: Ruiz-Trillo, Kin and Casacuberta, "The Origin of Metazoan Multicellularity: A Potential Microbial Black Swan Event", Annual Review of Microbiology 77:499–516, 2023, doi 10.1146/annurev-micro-032421-120023, at its abstract as the publisher's page and an index carry it verbatim (the emergence of animals from their unicellular ancestors a major evolutionary event; a better grasp of the unicellular ancestor from its diverse close relatives; how that ancestor became the first animals unclear; two popular theories, the choanoblastaea and the synzoospore, revised and their flaws exposed; the origin of animals a biological black swan event given the limits of current knowledge, defying retrospective explanation; a caution against confirmation biases on few data; two novel alternative scenarios proposed), the review itself not fetched. The hardness line's own clauses, the specifications and the specification withdrawn, and the approaches beyond the study of relatives stand keyed at the next pass; the abstract names two theories and proposes two scenarios, where the account stands as colonial choanoflagellate-like ancestors increasing adhesion and division of labor. Persistence, no landing: the closing condition at the field's review, Brunet and King, "The Origin of Animal Multicellularity and Cell Differentiation", Developmental Cell, 2017, at its abstract and passages as the publisher's page, an archive copy, the preprint server and a repository carry them verbatim (how animals evolved from single-celled ancestors poorly understood; the cell and developmental biology of the first animals less well examined; whether multicellularity evolved independently in the three clades remaining to be tested; the question what molecular mechanisms first supported it, not studiable in animals alone; mechanistic studies in diverse relatives promising to deepen the understanding), the review itself not fetched. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Comparative genomics — dates component families but not their ancestral regulatory use. Functional work in unicellular relatives — demonstrates ancestral roles in extant organisms. Experimental evolution — shows routes to simple multicellularity under specified selection. Fossils and molecular clocks — bound timing with preservation and model uncertainty. **Narrowing recorded.** Component ancestry narrows de-novo gene accounts; phylogenomics restricts the relevant stem lineages and trait order. **Progress.** A rich pre-animal toolkit, diverse holozoan life cycles and experimentally repeatable routes to simple multicellularity are established. **Own record.** ## 208 The evolution of metabolic pathways **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How did multistep, interdependent metabolic pathways and networks evolve from simpler enzymes and reactions? In the field's own posing, how individual enzymes evolved is relatively well understood, while individual enzymes rarely confer a physiological advantage on their own, so that the emergence of metabolism, judged by its current state, seemingly demanded the simultaneous emergence of many enzymes, and how such multicomponent interlocked systems evolved stands largely an open question. **The field's account of its hardness.** Present pathways are densely coupled and ancient, intermediates may have no benefit outside a completed sequence, horizontal transfer and enzyme promiscuity obscure ancestry, and several historical routes produce the same modern topology. **Accounts.** Pathways evolved backward from an essential product by successive recruitment of reactions using available precursors. Enzyme promiscuity permitted duplication and specialization into neighboring reactions. Whole modules were recruited or transferred between existing pathways and organisms. Early geochemical and nonenzymatic reaction networks supplied scaffolds later captured by enzymes. **Specifications.** Every intermediate pathway remains viable or is coupled to an available function — universal under evolutionary continuity. Reaction chemistry and thermodynamics permit the proposed sequence — universal under metabolism. Gene and enzyme histories can differ through transfer and recruitment — local. **Specifications withdrawn.** **Common requirement.** A viable sequence from simpler chemistry to the observed connected pathway, consistent with enzyme and organismal histories. **Conserving relation.** Each historical step preserves organismal viability and yields the reactions and regulation of the extant pathway. **Conserving reach.** Complete pathways, their enzymes, intermediates and network context across the lineages carrying them. **Reach.** The statement ranges over ancient sequences of functional intermediates. Comparative data observe surviving enzymes and networks after duplication, loss and transfer. The field's review states that, judged by its current state, the emergence of metabolism seemingly demanded the simultaneous emergence of many enzymes, individual enzymes rarely conferring an advantage on their own: the pathways observed in their current state and the sequence of their emergence are not the same, the current state read backward standing between them. **Unreachability.** **Persistence.** A supported historical sequence for representative pathways that explains intermediate utility, enzyme origin and network integration. **Addresses.** Statement, with the posing added there, hardness at the clause that an intermediate confers little advantage outside a completed sequence, and the relation at Reach: Noda-Garcia, Liebermeister and Tawfik, "Metabolite–Enzyme Coevolution: From Single Enzymes to Metabolic Pathways and Networks", Annual Review of Biochemistry 87:187–216, 2018, doi 10.1146/annurev-biochem-062917-012023, at its abstract as the publisher's page and an index carry it verbatim (how individual enzymes evolved relatively well understood; individual enzymes rarely conferring a physiological advantage on their own; the emergence of metabolism, judged by its current state, seemingly demanding the simultaneous emergence of many enzymes; how multicomponent interlocked systems evolved largely an open question; the complexity unlocked if survival of the fittest applies to metabolites as well as to genes and enzymes; the current models for pathway evolution described and an integrative metabolite–enzyme coevolution hypothesis offered, addressing the origins of new metabolites and new enzymes and the order of their recruitment), the review itself not fetched. The authors and title given before this pass, Caetano-Anollés et al., "From Single Enzymes to Metabolic Pathways and Networks", are withdrawn, the publisher's page carrying the authors and title above at the same volume and pages. The hardness line's other clauses, the account, the specifications and the approaches stand keyed at the next pass; the abstract offers a metabolite–enzyme coevolution hypothesis beside the current models, where the account stands as backward evolution from an essential product. Network and ancestral reconstruction progress: Scossa and Fernie, "The evolution of metabolism", Current Opinion in Plant Biology 55:41–48, 2020, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the review the address names at its abstract as the publisher's page, a database and two repository copies carry it verbatim (the patchwork hypothesis supported by genome analysis and directed evolution; the ancient pathways suggesting the retrograde or the semi-enzymatic hypothesis for some routes); a review of new pathways and communities at its opening as the publisher's page carries it (the question remaining how pathways evolve when multiple enzymes must be acquired simultaneously to realize a fitness benefit), 2023; a network review at its opening (challenges remaining from reconstruction incompleteness and the inability to experiment on the timescale of speciation); a 2025 review at its closing (whether reconstructions truly replicate an ancestral pre-genetic metabolism remaining subject to debate; ancestral enzyme reconstruction and promiscuous generalists the way forward), their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Enzyme phylogenies — recover families with transfer and deep-time limits. Ancestral sequence reconstruction — tests resurrected enzyme functions conditional on inferred ancestors. Network topology — identifies conserved modules without uniquely ordering their assembly. Experimental evolution — observes new pathway steps over short, selectable routes. Prebiotic chemistry — demonstrates nonenzymatic reaction subsets under specified environments. **Narrowing recorded.** Promiscuous ancestral activities and modular reuse are established for individual pathways, narrowing purely de-novo stepwise accounts. **Progress.** Enzyme superfamilies, reaction-network databases, reconstructed ancestors and experimentally evolved metabolic innovations stand. **Own record.** ## 209 The genotype-phenotype map **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What rules map genetic variation and environment to organismal phenotypes across development, and which features of that map make evolution predictable? In the field's own posing, understanding how genotypes map onto phenotypes, fitness and eventually organisms is arguably the next major missing piece in a fully predictive theory of evolution, referred to generally as the problem of the genotype–phenotype map, the picture still far from complete. **The field's account of its hardness.** The map is many-to-one and one-to-many, with epistasis, pleiotropy, development and environment changing effects across backgrounds; measured local variants sample a small part of a combinatorial space and do not supply a general predictive theory. **Accounts.** Additive effects dominate many population-level traits and nonlinear terms supply corrections. Developmental gene-regulatory networks transform sequence into robust, modular phenotypes. High-dimensional nonlinear interactions make map topology, neutrality and accessibility decisive. Physical and biochemical constraints strongly restrict the phenotypes sequence can produce. **Specifications.** Genotype, phenotype, environment and developmental stage are operationally fixed — local. The map predicts interventions rather than association alone — universal under causal prediction. Predictions transfer across genetic backgrounds within the claimed reach — universal under the map. **Specifications withdrawn.** A single context-free effect for every variant, withdrawn where interaction and environment dependence are retained. **Common requirement.** A predictive transformation from specified genotype and environment to phenotype. **Conserving relation.** The same mapping returns the observed phenotype for each admissible genotype-environment-development combination. **Conserving reach.** The sequence, background, environment and developmental ranges claimed by the account. **Reach.** The statement ranges over combinatorial genotype and environment space. Experiments sample selected variants, backgrounds and conditions. The field's review states that understanding rests so far on simpler questions, such as the structure that sequences mapped to molecular structures induce in genotype space, while a complete picture of how genotypes map onto organisms stands far off: the molecular maps measured and the organismal map sought are not the same, the simpler maps standing between them. **Unreachability.** **Persistence.** Prospective prediction of phenotypic effects for unmeasured combinations across a declared biological reach. **Addresses.** Statement, with the posing added there, hardness at the clause that no general predictive theory stands, progress at the structure of genotype spaces for molecular structures, and the relation at Reach: Manrubia et al., "From genotypes to organisms: State-of-the-art and perspectives of a cornerstone in evolutionary dynamics", Physics of Life Reviews 38:55–106, 2021, at its abstract as the arXiv page carries it verbatim (understanding how genotypes map onto phenotypes, fitness and eventually organisms the next major missing piece in a fully predictive theory of evolution; the problem of the genotype–phenotype map; a complete picture still far off; the structure induced in genotype space by sequences mapped to molecular structures revealing facts that affect the dynamical description of evolution; empirical evidence for phenotypic bias mounting) and at its highlights as the publisher's page carries them (understanding partial; the networked organization of the maps conditioning evolution and adaptation; several topological properties of genotype spaces universal), the review itself not fetched, its doi not carried at the pages read and the published abstract not compared with the arXiv one. The hardness line's other clauses, the account, the specifications and the approaches stand keyed at the next pass; the review's own features, phenotypic bias, networked organization and universal topology, stand beside the account as posed, additivity with corrections. Current scope: Tautz et al., "Beyond Mendel: a call to revisit the genotype–phenotype map through new experimental paradigms", Genetics 232(4):iyag024, 2026, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: an attention-model study at its abstract as an archive copy carries it (predicting phenotype from genotype a central challenge; out-of-sample prediction in epistatic regimes the test), 2025, its authors carried at the copy; a workshop's description (an apparently insurmountable problem, the astronomically large space of genotypes, the map not systematically studiable by experiment); a preprint on epistasis as uncertainty at its introduction (epistasis complicating the prediction of unmeasured phenotypes; despite a century of work still challenging to analyse; possibly no simple way to relate linear coefficients back to biology), its authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Quantitative genetics — predicts population variance and breeding values within sampled settings. Genome-wide association — locates statistical effects, with causal and transfer limits. Deep mutational scanning — maps dense local sequence neighborhoods for selected molecules. Mechanistic developmental models — predict chosen pathways and traits with incomplete organismal closure. Learned sequence models — interpolate large datasets with out-of-distribution reach under test. **Narrowing recorded.** Trait- and environment-specific maps stand for measured domains; neutral networks and epistatic motifs are mapped in selected systems. **Progress.** Dense molecular fitness landscapes, causal perturbation screens and predictive models for constrained traits are established. **Own record.** ## 210 The predictability of evolution **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Given an ancestral state and environment, which evolutionary outcomes can be predicted, at what biological level and over what time horizon? In the field's own posing, evolutionary biology was long considered a historical science with predictions about evolutionary trajectories believed near impossible, a belief that high-throughput sequencing and data analysis have challenged, and the factors that make evolutionary repeatability more or less likely are what stand to be unravelled. **The field's account of its hardness.** Selection can repeatedly favor similar functions while mutation, historical contingency, drift and changing environments alter the accessible paths; repeatability depends on whether the comparison concerns genes, traits, fitness or ecology and on the scale observed. **Accounts.** Strong selection and constrained solution spaces produce recurrent outcomes. Historical contingency and epistasis redirect later possibilities after early events. Many genetic routes converge on the same function, making phenotype more predictable than genotype. Environmental change and ecological feedback make the evolutionary target itself dynamic. **Specifications.** The ancestral state, environment and prediction target are declared — local. Replicates are independent under the experimental or comparative design — local. Predictability is scored against a probabilistic baseline at a stated biological level — universal under the claim. **Specifications withdrawn.** **Common requirement.** A probability distribution over future evolutionary outcomes conditional on stated initial conditions. **Conserving relation.** Predicted frequencies or outcomes remain calibrated in independent evolutionary replicates. **Conserving reach.** The genotypic, phenotypic, ecological and temporal range claimed by the prediction. **Reach.** The statement ranges over possible evolutionary histories. Experiments and comparative records sample finite lineages and environments. The field's review states that general rules are most often discovered through experimental evolution, that slower dynamics are at risk of being overlooked there, and that extrapolating from short-term studies to long-term real-world scenarios requires caution: the recurrence measured in the experiments and the predictability sought over histories are not the same, the extrapolation standing between them. **Unreachability.** **Persistence.** Prospectively calibrated predictions whose success and failure boundaries are established across levels and timescales. **Addresses.** Statement, with the posing added there, hardness at the interplay of deterministic and stochastic forces with the ecological context and at mutation biases misread as selection, approaches at experimental evolution, and the relation at Reach: Pearless and Freed, "Unravelling the factors of evolutionary repeatability: insights and perspectives on predictability in evolutionary biology", Frontiers in Ecology and Evolution 12:1335452, 2024, doi 10.3389/fevo.2024.1335452, at its abstract as the publisher's page carries it verbatim (evolutionary biology previously considered a historical science with predictions near impossible; high-throughput sequencing and data analysis challenging that belief; predictions used to develop fundamental knowledge and to demonstrate evolutionary control; the factors making repeatability more or less likely investigated) and at passages of its text as the same page carries them (a comprehensive understanding requiring the interplay between deterministic and stochastic forces alongside the ecological context; general rules most often discovered through experimental evolution; slower dynamics at risk of being overlooked, extrapolation from short-term studies to long-term scenarios requiring caution; studies failing to consider mutation biases and attributing repeated evolution to selection), the review itself not fetched. Given before this pass as Pearless et al., the title without its subtitle and no article number; the page carries two authors, the full title and the article number. The hardness line's clause on the level and scale of comparison, the account, the specifications and the other approaches stand keyed at the next pass; the text names mutation bias as a source of parallel evolution, where the account stands as strong selection and constrained solution spaces. Empirical progress: Beavan et al., "Contingency, repeatability, and predictability in the evolution of a prokaryotic pangenome", Proceedings of the National Academy of Sciences 121, 2024, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the field's review at its authors' summary as a university's page carries it (no easy answer, evolution sometimes producing strikingly similar solutions and sometimes very different paths under the same circumstances; parallel and analytic replay experiments the instruments), Science, 2018, the review's own abstract not returned; a 2025 preprint at its introduction as an archive copy carries it (unclear whether convergence and predictability are common characteristics; continuous and wide disagreement; the core problem that the evidence is mixed), its authors not carried; a quantification paper at its abstract as an archive copy carries it (the contributions of determinism and stochasticity depending strongly on the conditions, the selective pressure and the number of available beneficial mutations), Frontiers in Genetics, 2012, doi 10.3389/fgene.2012.00246. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Replicated laboratory evolution — measures recurrence under controlled initial states and limited duration. Comparative convergence — detects repeated traits with ancestral and environmental reconstruction uncertainty. Fitness-landscape mapping — predicts local paths under incomplete higher-dimensional sampling. Population-genetic forecasting — returns allele-frequency distributions under specified selection and demography. **Narrowing recorded.** Predictions stand for defined traits, populations and environments; recurrence is measured separately at genetic and phenotypic levels. **Progress.** Repeatable molecular targets, contingent path changes, convergent phenotypes and short-horizon frequency forecasts are all documented. **Own record.** ## 211 Rapid adaptive radiation **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why do some lineages diversify rapidly into many ecologically distinct species while other lineages encountering similar opportunities do not? In the field's own posing, rapid adaptive radiation poses two questions apart from speciation and adaptation, what happens after one speciation event and how some lineages continue speciating through a rapid burst, and the paradox is that the hallmark early burst of speciation and niche diversification is contradicted by most speciation models, which predict continuously decelerating diversification and niche subdivision. **The field's account of its hardness.** The defining pattern joins elevated diversification, ecological divergence and trait evolution, yet these components need not coincide; exceptional clades invite retrospective explanation, and opportunity, innovation and lineage traits are correlated in comparative records. **Accounts.** Ecological opportunity after colonization or competitor loss permits rapid niche occupation. A key innovation opens new resources or modes of life and raises diversification. Sexual selection and assortative mating accelerate reproductive isolation. Intrinsic genomic, developmental or life-history properties raise evolvability. Apparent exceptional radiation arises partly from extinction, sampling and clade-selection effects. **Specifications.** Radiation is defined by stated rates of lineage, phenotypic and ecological diversification — local. Comparisons account for clade age, extinction and sampling — universal under comparative inference. Candidate causes precede the rate shift — universal under causal history. **Specifications withdrawn.** **Common requirement.** A cause of joint lineage multiplication and ecological or phenotypic divergence. **Conserving relation.** The proposed cause predicts which eligible lineages radiate and which do not under declared comparisons. **Conserving reach.** Radiating and nonradiating lineages across comparable ecological opportunities and time intervals. **Reach.** The statement ranges over counterfactual lineage outcomes. Phylogenies and trait records contain surviving, sampled histories. The field's review states that the question connects global macroevolutionary patterns to microevolutionary processes and that a gap remains between the two scales, the hallmark early burst contradicted by most speciation models: the bursts recorded at the macroevolutionary scale and the mechanisms sought at the population scale are not the same, the gap between the scales standing between them. **Unreachability.** **Persistence.** Prospective or replicated causal evidence separating opportunity, innovation and lineage predisposition across comparable clades. **Addresses.** Statement, with the posing added there, hardness at the three diversification axes with exceptional outliers on each, the account at ecological opportunity as the mechanism the review goes beyond, and the relation at Reach: Martin and Richards, "The Paradox Behind the Pattern of Rapid Adaptive Radiation: How Can the Speciation Process Sustain Itself Through an Early Burst?", Annual Review of Ecology, Evolution, and Systematics 50:569–593, 2019, doi 10.1146/annurev-ecolsys-110617-062443, at its abstract as the publisher's page, an index and a funder's repository carry it verbatim (two questions posed apart from speciation and adaptation; the question connecting macroevolutionary patterns to microevolutionary processes; the major features of rapid radiations and their mismatch with theoretical models and speciation mechanisms reviewed; rapid radiations on three axes, species richness, phenotypic disparity and ecological diversity, with exceptional outliers on each; the paradox that the hallmark early burst is contradicted by most speciation models, which predict decelerating diversification; unclear if and how magic traits, phenotype matching and physical linkage promote rapid bursts; five mechanisms beyond ecological opportunity reviewed, ancient adaptive alleles and the transporter hypothesis, sexual signal complexity, fitness landscape connectivity, diversity begets diversity, and plasticity first) and at a passage of its text as a repository copy carries it (a gap remaining between the macroevolutionary and microevolutionary scales, little understood of the population-level processes behind repeated bouts of speciation), the review itself not fetched; the title stood without its subtitle before this pass. The hardness line's other clauses, the specifications, the approaches and progress stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the field's review of ecological opportunity at its abstract as the publisher's page carries it verbatim (what the term means, much less how it mechanistically leads to diversification, currently debated; whether it has any predictive value or is a heuristic useful only for post hoc explanation unclear), Annual Review of Ecology, Evolution, and Systematics, doi 10.1146/annurev-ecolsys-121415-032254, its authors not carried; a twenty-five-year reflection at a passage as the publisher's page carries it (a causal identification problem, opportunity inferred only from its effects; early-stage radiations under experimental manipulation the promising alternative), 2026; an award address at its abstract (many clades failing to radiate although seemingly in the presence of opportunity), American Naturalist, doi 10.1086/652433; a case study at its abstract (support for the key-innovation hypothesis scarce even in textbook examples), 2011. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Time-calibrated phylogenies — infer rate shifts under extinction and sampling models. Trait-dependent diversification — tests associations with innovations, with model identifiability limits. Ecological and genomic comparison — reconstructs niche and mechanism after radiation. Experimental radiation — observes early divergence in restricted systems. **Narrowing recorded.** The field separates lineage, phenotypic and ecological radiation and tests each rather than treating every species-rich clade as one phenomenon. **Progress.** Replicated island and lake radiations, explicit null diversification models and genomic histories of selected clades stand. **Own record.** ## 212 Polyploidization and diversification **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does whole-genome duplication increase, decrease or leave unchanged a lineage's long-run diversification and evolutionary persistence? In the field's own posing, the macroevolutionary impacts of polyploidy are contentious, polyploidy traditionally considered an evolutionary dead end while genomic studies suggest it has been a key driver of macroevolutionary success. **The field's account of its hardness.** Duplication can create immediate reproductive isolation and genetic novelty while imposing meiotic and demographic costs; ancient successful polyploids are conditioned on survival, recent lineages have had little time to diversify, and diploidization obscures old events. The field's comparative study states that it remained unknown whether the abundance of polyploids is a consequence of higher diversification rates following polyploidy or of frequent polyploid formation. **Accounts.** Duplicate genes and reproductive isolation raise speciation and long-run innovation. Establishment costs and elevated extinction make most new polyploid lineages evolutionary dead ends. Short-term disadvantage is followed by long-term opportunity among the lineages that persist. The effect depends on ecological setting, mode of origin and lineage traits rather than one general direction. **Specifications.** Duplication events are correctly dated and distinguished from later diploidization — local. Speciation and extinction are separately estimable — universal under diversification inference. Comparisons correct for survival and detection bias — universal under the question. **Specifications withdrawn.** **Common requirement.** A causal rate effect of genome duplication on lineage origination and loss. **Conserving relation.** The inferred effect predicts diversification for duplicated and nonduplicated lineages under the same comparison design. **Conserving reach.** Recent and ancient auto- and allopolyploid lineages across ecological and taxonomic settings. **Reach.** The statement ranges over all duplication outcomes including extinct lineages. Comparative data preferentially retain surviving and detectable events. The field's comparative study states that whether the abundance of polyploids follows from higher diversification after polyploidy or from frequent polyploid formation remained unknown, and examines the consequences at the timescale of genera across hundreds of inferred polyploidization events: the abundance recorded and the rates sought are not the same, the formation rate standing between them. **Unreachability.** **Persistence.** Bias-controlled estimates fixing the sign, timing and conditions of duplication's effect on speciation and extinction. **Addresses.** Statement, with the posing added there, hardness at the added sentence, the accounts at the dead end and the driver as the two sides, approaches at the likelihood-based analyses, progress at the lower speciation and higher extinction rates found, and the relation at Reach: Mayrose, Zhan, Rothfels, Magnuson-Ford, Barker, Rieseberg and Otto, "Recently formed polyploid plants diversify at lower rates", Science 333:1257, 2011, doi 10.1126/science.1207205, at its abstract as the publisher's page and an index carry it verbatim (polyploidy widespread, especially among plants, its macroevolutionary impacts contentious; traditionally an evolutionary dead end, genomic studies suggesting a key driver of macroevolutionary success; the consequences examined at the timescale of genera across vascular plants and hundreds of inferred polyploidization events; likelihood-based analyses indicating lower speciation and higher extinction rates than diploids, the first quantitative corroboration of the dead-end hypothesis) and at a passage of its text as a repository copy carries it (whether the abundance of polyploids follows from higher diversification rates after polyploidy or from frequent polyploid formation remaining unknown), the paper itself not fetched. A critical reappraisal and the authors' reply, New Phytologist 202:1105–1117, 2014, and 206:27–35, 2015, returned at their listing only and not read. Otto and Whitton, "Polyploid Incidence and Evolution", Annual Review of Genetics 34:401–437, 2000, cited here and not read this pass; the hardness line's own clauses, the specifications and the other approaches stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a review of the dead-end hypothesis and the state-dependent models at passages as the publisher's page carries them (the hypothesis's power to explain rates controversial, from arguments about the methods of inferring rates and about the hypothesis itself; sister-clade comparisons unable to part speciation from extinction; the state-dependent models the field's instrument), Annals of Botany, 2024, its authors not carried; the reply in the field's exchange at its text as the publisher's page and two copies carry it (the clade-age and clade-size objection answered by time-scaled rates; whether earlier duplication events had different impacts an open question), New Phytologist, 2015, doi 10.1111/nph.13192. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Sister-lineage comparison — controls some history but samples surviving clades. State-dependent diversification models — estimate rates with identifiability and model-dependence concerns. Genomic dating — detects ancient duplication after extensive fractionation. Experimental and ecological studies — measure establishment and adaptation over short times. **Narrowing recorded.** Recent and ancient effects are analyzed separately; establishment, speciation and extinction are treated as distinct rates. **Progress.** Recurrent genome duplication, immediate isolation mechanisms, duplicate-gene innovation and elevated early extinction risk are documented. **Own record.** ## 213 Sporadic Alzheimer's disease pathogenesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What causal sequence produces the common late-onset, nonfamilial form of Alzheimer's disease, and which processes drive cognitive decline? In the field's controversy as the review poses it, whether a complex age-associated disorder of the brain can be explained by a linear, one-dimensional cause-and-effect model with amyloid beta as the ultimate trigger has been doubted from the start, and the anti-amyloid antibody results have given the controversy momentum, the field appearing divided into at least two groups. **The field's account of its hardness.** Amyloid, tau, inflammation, vascular injury and neurodegeneration unfold over decades and influence one another; pathology can precede symptoms by years, mixed pathologies are common, and genetic or biomarker association does not by itself fix the causal order in an individual. **Accounts.** Amyloid accumulation initiates a cascade leading to tau pathology, synaptic failure and neurodegeneration. Tau propagation and neuronal vulnerability are the proximate drivers of degeneration. Immune and glial dysregulation initiates or amplifies the pathological process. Vascular, metabolic and barrier dysfunction drives or conditions the common disease. Several interacting pathologies reach a common clinical syndrome through heterogeneous routes. **Specifications.** The account distinguishes initiating causes from downstream biomarkers — universal under causal pathogenesis. It explains temporal biomarker and clinical trajectories — universal under disease progression. Common late-onset cases are not inferred directly from rare deterministic mutations — local. **Specifications withdrawn.** A single uniform pathological sequence for every clinical case, withdrawn in accounts retaining biological heterogeneity. **Common requirement.** A causal sequence linking risk, molecular pathology, cell loss and cognitive impairment. **Conserving relation.** Intervening on the proposed driver changes the predicted downstream pathology and clinical course. **Conserving reach.** The preclinical and symptomatic course of sporadic late-onset cases, including common mixed pathologies. **Reach.** The statement ranges over decades-long individual causal histories. Biomarkers, autopsy, genetics and trials sample different stages and populations. The field's review states that the anti-amyloid antibody results have given the controversy momentum on three counts, the clinical meaningfulness of the approach, the significance of amyloid clearance, and the relevance of the cascade: the clearance and the clinical change measured in the trials and the causal sequence sought are not the same, the three disputes standing between them. **Unreachability.** **Persistence.** Replicated interventions and longitudinal evidence fixing the initiating and progression-driving processes across biological subtypes. **Addresses.** Statement, with the controversy posed there, hardness at the mixed pathologies, the shared pathways and the multifactorial and multigenetic background, the account at the cascade as the review defines it, approaches at the antibody results and the three counts of the controversy, and the relation at Reach: Behl, "In 2024, the amyloid-cascade-hypothesis still remains a working hypothesis, no less but certainly no more", Frontiers in Aging Neuroscience 16:1459224, 2024, doi 10.3389/fnagi.2024.1459224, at its abstract as a repository copy carries it verbatim (the hypothesis introduced in 1992 and doubted from early on; massive doubts that a complex age-associated disorder of the brain can be explained by a linear, one-dimensional cause-and-effect model; the cascade defining amyloid beta generation, aggregation and deposition as the central mechanism, the ultimate trigger and the key pharmacological target; refinements and reappraisals by its formulators; the controversy gaining momentum on the clinical meaningfulness of the antibody approach, the significance of amyloid clearance and the relevance of the hypothesis) and at passages of its text as the publisher's page carries them (the linear sequence contrasted by mixed pathologies, shared pathways and the multifactorial and multigenetic background; the field appearing divided into at least two groups), the review itself not fetched; the review describes itself as a personal opinion in larger parts. The hardness line's other clauses, the specifications, the specification withdrawn and the other approaches stand keyed at the next pass; the Own record line stands in the file's own voice, awaiting a source. Progress and the framework: Hunter, Walsh and Brayne, "Key questions for the future of amyloid research in dementia: a framework for integrating complex datasets", Molecular Psychiatry 30:5001–5010, 2025, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a review of the antibody therapies at a passage as an archive copy carries it (the fundamental question of the role of oligomeric species in cognitive impairment and neuronal death remaining), 2024; a review of the antibodies at its abstract as the journal's page carries it (the trial outcomes seeming to stand by the amyloid hypothesis while a series of failures remains a question), Frontiers in Aging Neuroscience, 2022, doi 10.3389/fnagi.2022.870517; a review of four hypotheses at its conclusion (a multi-factor complex disorder; whether infection applies to all sporadic cases and is a primary trigger yet to be determined), their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Longitudinal biomarkers — order population-average changes with selection and individual variability. Genetics — identifies causal risk pathways without fixing their interaction in typical cases. Anti-amyloid intervention — lowers plaques and modestly changes decline in selected early disease, leaving sequence and heterogeneity questions. Tau, immune, vascular and metabolic interventions — mechanistically targeted and not yet jointly decisive. **Narrowing recorded.** Biomarker-defined early disease narrows clinical heterogeneity; genetic and pathology studies partition mechanistic subgroups. **Progress.** Amyloid and tau can be measured in living patients, major risk loci and cell pathways are mapped, and disease-modifying effects from amyloid lowering now stand for selected populations. **Own record.** The entry is restricted to the common sporadic syndrome; deterministic familial cases do not settle its causal sequence. ## 214 Endometriosis pathogenesis **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What initiates and sustains endometriotic lesions, and why do lesions, pain and infertility vary so widely among affected people? In the field's own posing, several pathogenetic pathways have been considered since the disease was defined, retrograde menstruation, benign metastasis, immune dysregulation, coelomic metaplasia, hormonal imbalance, stem cells and epigenetic alteration among them, while the true pathogenesis remains poorly understood. **The field's account of its hardness.** That the disease's manifestations range from accidentally found asymptomatic lesions to a severe condition, the severity not depending on the size of the lesion, the diagnosis quite often delayed because of nonspecific manifestations, by an average of seven years from the onset of symptoms in the field's estimate, laparoscopic visualization remaining the standard for it; that reflux of menstrual blood is physiological for women with patent tubes, most of them experiencing retrograde menstruation while about a tenth develop the disease, its prevalence found no different in women with or without it, and retrograde menstruation might explain ovarian and superficial peritoneal lesions but not deep infiltrating ones or lesions outside the peritoneal cavity; and that none of the theories adequately explains all pathological types, immune, endocrine, genetic and environmental factors being generally accepted as involved, the true pathogenesis remaining poorly understood. **Accounts.** Retrograde menstrual cells implant and survive because local immune clearance and tissue conditions permit them. Coelomic or peritoneal cells undergo metaplasia into endometrial-like tissue. Embryonic Müllerian remnants or displaced progenitor cells generate lesions. Stem-cell dissemination, somatic mutation and epigenetic change sustain distinct lesion lineages. Neuroimmune sensitization partly generates pain independently of lesion volume. **Specifications.** The account explains lesion initiation, persistence and anatomical distribution — universal under pathogenesis. Pain and fertility outcomes are separately represented — universal under the clinical syndrome. Tissue samples preserve lesion rather than treatment-induced states — local. **Specifications withdrawn.** A one-to-one relation between lesion stage and symptom severity, withdrawn while both measurements are retained. **Common requirement.** A mechanism connecting lesion origin and maintenance to the heterogeneous clinical manifestations. **Conserving relation.** Intervention on the proposed process changes the predicted lesion and symptom outcomes in the populations to which the account applies. **Conserving reach.** Superficial, ovarian and deep disease, symptomatic and asymptomatic lesions, pain and infertility outcomes. **Reach.** The statement ranges over initiation before clinical detection and later symptom production. Surgery and imaging sample selected lesions after variable disease duration and treatment. The field's review states that the diagnosis is quite often delayed because of nonspecific manifestations, and that the true pathogenesis remains poorly understood after the pathways considered: the disease met at a delayed diagnosis and the initiation sought are not the same, the delay standing between them. **Unreachability.** **Persistence.** A prospectively validated causal taxonomy that predicts lesion course, pain, fertility and treatment response. **Addresses.** Statement, with the posing added there, the account at retrograde menstruation and immune dysregulation among the pathways the review names, and the relation at Reach: Lamceva, Uljanovs and Strumfa, "The Main Theories on the Pathogenesis of Endometriosis", International Journal of Molecular Sciences 24:4254, 2023, doi 10.3390/ijms24054254, at its abstract as a repository copy, an index and the authors' university listing carry it verbatim (the disease defined by abnormal growth of endometrial tissue outside the uterus, affecting about a tenth of women of reproductive age; severe pelvic pain, pelvic organ dysfunction, infertility and secondary mental issues; the diagnosis quite often delayed because of nonspecific manifestations; seven pathogenetic pathways considered since the disease was defined; the true pathogenesis remaining poorly understood; the exact mechanism significant for treatment), the review itself not fetched. The venue given before this pass, Medicina 59:76, 2023, is withdrawn, the pages carrying the journal above at volume 24, article 4254. The specifications and approaches stand keyed at the next pass; the review names seven pathways where the account stands as one combined. Molecular heterogeneity and progress: Mariadas, Chen and Chen, "The Molecular and Cellular Mechanisms of Endometriosis: From Basic Pathophysiology to Clinical Implications", International Journal of Molecular Sciences 26:2458, 2025, cited here and not read this pass. Hardness: re-formed to the field's words at the review the address names, at passages of its text as the archive copy carries them verbatim (manifestations from asymptomatic lesions to a severe condition, not depending on the lesion's size; retrograde menstruation explaining ovarian and superficial peritoneal disease but not deep infiltrating disease or lesions outside the peritoneal cavity; reflux physiological for women with patent tubes, most experiencing it), the review itself not fetched; and at three reviews returned in the search, none fetched, their authors not carried: a clinical review at an archive copy (symptom severity not correlated to extent of disease; retrograde menstruation a physiologic process in nine tenths of women; many theories, none definitive), a review of pain in the disease at an archive copy (retrograde menstruation in nine tenths, the disease in one; its prevalence no different with or without the disease), a review of menstruation dysregulation at an archive copy (the average diagnostic delay seven years; laparoscopic visualization the gold standard; a cure not identified, attributed to the incomplete understanding of pathogenesis), and a review of the peritoneal immune microenvironment (none of the theories explaining all pathological types; immune, endocrine, genetic and environmental factors involved); the former line, in the file's compression, released with its content at the three clauses, its lesion-heterogeneity clause standing at the first. Persistence, no landing: the closing condition in part at the reviews read above (a cure not identified, attributed to the incomplete understanding of pathogenesis; none of the theories explaining all pathological types), the prospective taxonomy keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Surgical histology — verifies lesions but samples late and selected disease. Hormonal suppression — establishes hormone dependence for many symptoms without identifying origin. Genomic and single-cell analysis — resolves lesion and immune heterogeneity with cross-sectional reach. Imaging and circulating biomarkers — noninvasive classification remains incomplete. Pain-mechanism studies — separate nociceptive and sensitization components. **Narrowing recorded.** Lesions are partitioned by anatomical and molecular subtype; pain is treated as an outcome not fixed by stage alone. **Progress.** Hormone responsiveness, inflammatory microenvironments, recurrent somatic alterations and multiple pain pathways are established. **Own record.** ## 215 Long COVID pathophysiology **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which biological mechanisms cause and sustain the heterogeneous post-acute symptoms grouped as Long COVID? In the field's own posing, more than two hundred symptoms with impacts on multiple organ systems have been identified, research having made substantial progress in identifying pathophysiological changes and risk factors and in characterizing the illness, with similarities to other viral-onset illnesses laying the groundwork, the mechanisms standing as potential. **The field's account of its hardness.** The syndrome spans organs and symptom clusters, definitions and controls vary, pre-infection baselines are often absent, and viral persistence, immune dysregulation, vascular injury, autonomic dysfunction and tissue damage can coexist or mark different subtypes. **Accounts.** Persistent viral antigen or replication maintains inflammation in tissue reservoirs. Autoimmunity and dysregulated immune activation continue after acute infection. Endothelial injury, microvascular dysfunction or coagulation abnormalities impair tissue function. Autonomic and neuroinflammatory changes generate exertional, cognitive and cardiovascular symptoms. Organ damage and reactivation of latent pathogens produce distinct post-acute phenotypes. **Specifications.** A case definition and comparator population are fixed — local. Mechanisms precede and predict symptoms rather than merely co-occur — universal under pathophysiology. Acute severity, reinfection, vaccination and prior conditions are represented — universal under causal inference. **Specifications withdrawn.** A single mechanism for every case, withdrawn in subtype accounts that retain a common syndrome label. **Common requirement.** A causal biological classification predicting symptom trajectory and response to mechanism-specific intervention. **Conserving relation.** A mechanism marker and its intervention predict concordant outcomes in independent cohorts. **Conserving reach.** The recognized symptom clusters, severities, acute histories and durations admitted by the case definition. **Reach.** The statement ranges over heterogeneous causal subtypes. Cohorts measure selected biomarkers and symptoms under changing definitions and ascertainment. The two are not the same: more than two hundred symptoms with impacts on multiple organ systems stand under one name, the illness estimated in at least a tenth of infections with the count likely much higher for undocumented cases, the onset of symptoms variable, and the field's own recommendations are for research that would identify the mechanisms and part the overlap with other viral-onset illnesses. **Unreachability.** **Persistence.** Replicated causal biomarkers and interventions that assign patients to mechanisms and predict course and treatment response. **Addresses.** Statement, with the posing added there, hardness at the clause that the syndrome spans organs and symptom clusters, and progress at the pathophysiological changes and risk factors identified: Davis, McCorkell, Vogel and Topol, "Long COVID: major findings, mechanisms and recommendations", Nature Reviews Microbiology 21:133–146, 2023, doi 10.1038/s41579-022-00846-2, at its abstract as the publisher's page, an index and a repository carry it verbatim (an often debilitating illness in at least a tenth of infections; more than two hundred symptoms with impacts on multiple organ systems; at least sixty-five million individuals estimated worldwide; substantial progress in identifying pathophysiological changes and risk factors and in characterizing the illness; similarities with myalgic encephalomyelitis and postural orthostatic tachycardia syndrome laying the groundwork) and at the publisher's summary (potential mechanisms, the overlap with other conditions, potential treatments, challenges and recommendations), the review itself not fetched; an author correction stands at 21:408, 2023. No Reach relation is stated at the abstract read. The hardness line's other clauses, the account, the specifications and the approaches stand keyed at the next pass. Present mechanistic standing: Faghy et al., "Current status and future perspectives on the mechanistic and pathophysiological understanding of long COVID", Communications Medicine 6:255, 2026, cited here and not read this pass. The relation at Reach: the field's review, Nature Reviews Microbiology 21(3):133-146, 2023, doi 10.1038/s41579-022-00846-2, at its abstract as the publisher's page, a database, an index and a forum copy carry it verbatim (at least a tenth of infections; more than two hundred symptoms across organ systems; at least sixty-five million estimated, the count conservative; substantial progress in pathophysiological changes and risk factors; similarities with other viral-onset illnesses; the variable onset of symptoms) and at the journal's summary (challenges and recommendations for research and care), the review itself not fetched, its authors Davis, McCorkell, Vogel and Topol. Persistence, no landing: the closing condition at the review's summary read at Reach (recommendations for research identifying biomarkers and mechanisms and for trials), the assignment of patients to mechanisms keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Longitudinal cohorts — characterize persistence with attrition and control-selection limits. Tissue and antigen studies — report reservoirs in subsets, with causal connection under test. Immune, proteomic and metabolomic profiling — identifies signatures that vary across cohorts. Vascular and autonomic testing — defines physiological subgroups. Mechanism-targeted trials — emerging and not yet a complete discriminator. **Narrowing recorded.** Consensus definitions and symptom clusters narrow heterogeneous cohorts; organ-specific testing separates damage from systemic syndromes. **Progress.** Elevated post-infection risk, recurring symptom clusters and biological abnormalities in viral, immune, vascular and autonomic systems are replicated. **Own record.** ## 216 Biological phenotypes of sepsis and acute respiratory distress syndrome **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which reproducible biological phenotypes divide sepsis and acute respiratory distress syndrome into groups with different mechanisms, prognoses or treatment responses? In the field's own posing, heterogeneity in sepsis and acute respiratory distress syndrome is increasingly recognized as one of the principal barriers to finding efficacious targeted therapies, and phenotyping aims to use a multitude of data to identify homogeneous subgroups within an otherwise heterogeneous population. **The field's account of its hardness.** Both syndromes are defined clinically rather than by one mechanism, host responses change rapidly, clustering depends on variables and timing, and a statistically stable subgroup need not predict a causal treatment effect. **Accounts.** Distinct hyperinflammatory and hypoinflammatory host-response states define clinically useful phenotypes. Immune-suppressed, endothelial, coagulopathic and metabolic programs define overlapping mechanistic axes rather than exclusive classes. Phenotypes are dynamic states along trajectories and patients move between them. Observed clusters reflect severity, treatment and sampling as much as stable biology. **Specifications.** Phenotypes replicate under prespecified variables and assignment rules — universal under classification. Membership is available at the time a treatment decision is made — local. Treatment interaction is tested prospectively rather than inferred from prognosis — universal under predictive enrichment. **Specifications withdrawn.** A single homogeneous mechanism within either clinical syndrome, withdrawn while the syndrome definitions are retained. **Common requirement.** A reproducible biological partition or coordinate system that adds causal or treatment information beyond severity. **Conserving relation.** Phenotype assignment remains stable enough to predict the declared outcome or intervention response in independent populations. **Conserving reach.** Infection sources, host backgrounds, care settings, disease stages and therapies covered by the clinical syndromes. **Reach.** The statement ranges over latent biology and counterfactual treatment response. Studies measure time-specific biomarkers under ongoing treatment. The field's review states that phenotyping aims to identify homogeneous subgroups from a multitude of data within a heterogeneous population, and that translating these conceptual findings to bedside decision-making tools carries its own challenges: the subgroups identified in the data and the treatment response sought are not the same, the translation standing between them. **Unreachability.** **Persistence.** Prospective phenotype assignment with replicated biological meaning and differential benefit from a targeted intervention. **Addresses.** Statement, with the posing added there, hardness at heterogeneity as a barrier to targeted therapy and at the challenge of translating findings to bedside tools, approaches at the phenotyping schemas built on high-throughput data and computation, and the relation at Reach: Sinha, Meyer and Calfee, "Biological Phenotyping in Sepsis and Acute Respiratory Distress Syndrome", Annual Review of Medicine 74:457–471, 2023, doi 10.1146/annurev-med-043021-014005, at its abstract as the publisher's page and an index carry it verbatim (heterogeneity in the two syndromes recognized as a principal barrier to efficacious targeted therapies; high-throughput biological data and computational power leading to phenotyping in critical care; phenotyping aiming to identify homogeneous subgroups within a heterogeneous population; schemas applied to the two syndromes to increase understanding and identify therapies; a selective review of the schemas; the challenges of translating the findings to bedside decision-making tools), the review itself not fetched. The pages given before this pass, 457–476, are withdrawn, the publisher's page and the index carrying 457–471. The account at the hyperinflammatory and hypoinflammatory subphenotypes: a commentary, "Beyond patterns: how to assign biological meaning to ARDS and sepsis phenotypes", The Lancet Respiratory Medicine, 2023, at its opening as the journal's page carries it (patients with the syndrome consistently subdivided into two biological clusters, the hypoinflammatory and hyperinflammatory subphenotypes, advocated over a decade), its authors not carried at the page read. The hardness line's other clauses, the specifications, the specification withdrawn, the other approaches and progress stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a systematic review of predictive enrichment at a passage as the publisher's page carries it (no trial yet incorporating the subphenotypes at enrollment; without enrichment large trials may continue to yield neutral results; a platform trial now phenotyping at enrolment by point-of-care biomarkers), Critical Care, 2025, doi 10.1186/s13054-025-05684-1; a multi-omic study at its introduction as a preprint archive carries it (the biological processes driving each phenotype poorly understood; trials with prospective phenotyping being developed), 2025; an endotype study at its summary (testable hypotheses for prospective validation through endotype-guided trials); their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Unsupervised clustering — returns reproducible-looking groups whose membership depends on inputs. Transcriptomic endotypes — resolve host-response programs with timing and platform dependence. Latent-class trial reanalysis — identifies possible treatment interactions retrospectively. Bedside parsimonious classifiers — improve deployability with reduced biological resolution. Prospective enrichment — limited and ongoing. **Narrowing recorded.** Hyper- and hypoinflammatory patterns, immune-response endotypes and temporal trajectories narrow the broad syndromes into testable strata. **Progress.** Several phenotypes recur across cohorts and archived trials, and compact assignment models have been derived for prospective testing. **Own record.** ## 217 Migraine pathophysiology **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What neural and vascular sequence initiates a migraine attack and produces headache, aura and associated sensory symptoms? In the field's own posing, migraine has become clear as a neural disorder in which a wide range of brain areas and neurochemical systems are implicated, producing a heterogeneous clinical phenotype, human experimental models and functional neuroimaging having provided insight into attack initiation. **The field's account of its hardness.** Attacks are episodic and heterogeneous, early brain-state changes are difficult to capture, aura and headache can dissociate, and trigeminovascular, brainstem, hypothalamic, cortical and vascular processes influence one another. **Accounts.** Hypothalamic and brainstem network changes initiate the premonitory state and recruit trigeminovascular signaling. Cortical spreading depolarization generates aura and can engage downstream pain pathways. Peripheral and central trigeminovascular activation, including calcitonin gene-related peptide signaling, sustains headache. Altered sensory gain and network excitability create a cycling brain state in which multiple triggers can precipitate attacks. **Specifications.** Initiating events precede symptoms rather than result from pain or medication — universal under initiation. Mechanisms account for attacks with and without aura — universal under the clinical reach. Intervention targets connect to the proposed causal step — local. **Specifications withdrawn.** A purely vascular origin for all migraine symptoms, withdrawn while vascular changes are retained as components. **Common requirement.** A temporal mechanism connecting susceptibility, attack initiation and the distinct symptom phases. **Conserving relation.** The proposed sequence predicts phase-specific biomarkers and changes when its causal nodes are perturbed. **Conserving reach.** Migraine with and without aura, premonitory, pain and postdrome phases, and variation among attacks. **Reach.** The statement ranges over the unobserved onset before and during spontaneous attacks. Imaging, electrophysiology and blood measures sample selected phases and locations. The two are not the same: the field parts its questions into why, how and when, and the when, the emergence of an attack from a cycling disorder of sensory processing, is examined mostly through triggered attacks and serial scanning of the cycle, the difficulties in examining spontaneous attacks sending the studies there, the imaging of spontaneous and triggered premonitory phases found largely consistent. **Unreachability.** **Persistence.** A phase-resolved causal model prospectively predicting attack onset, symptom form and mechanism-specific response. **Addresses.** Statement, with the posing added there, hardness at the wide range of brain areas and neurochemical systems implicated and the heterogeneous phenotype, specifications withdrawn at migraine as a neural disorder, approaches at human experimental models and functional neuroimaging, and progress at the neural disorder and the peptidergic pathways: Karsan, "Pathophysiology of Migraine", Continuum 30:325–343, 2024, doi 10.1212/CON.0000000000001412, at its abstract as the publisher's page and an index carry it verbatim (an overview through the extended symptom spectrum, neuroanatomy, neurochemistry and therapeutics; human experimental migraine models and functional neuroimaging providing insight into attack initiation, neurochemistry, neuroanatomy and therapeutic substrates; migraine a neural disorder implicating a wide range of brain areas and neurochemical systems and producing a heterogeneous clinical phenotype; many pathways monoaminergic and peptidergic, calcitonin gene-related peptide and pituitary adenylate cyclase-activating polypeptide among them), the review itself not fetched. No Reach relation is stated at the abstract read. The hardness line's other clauses, the account, the specifications and the other approaches stand keyed at the next pass. Current molecular and systems progress: Frimpong-Manson et al., "Advances in understanding migraine pathophysiology", Frontiers in Molecular Neuroscience 17:1355281, 2024, cited here and not read this pass. The relation at Reach: the field's review, Physiological Reviews 97:553-622, 2017, doi 10.1152/physrev.00034.2015, at its abstract as the publisher's page carries it verbatim (the three questions, why, how and when; attacks emerging from a disorder of sensory processing that itself likely cycles), its authors not carried at the return; a narrative review of imaging in the premonitory phase at its abstract and a passage as a database and an archive copy carry them (serial scanning of the cycle; other methods used for the difficulties in examining spontaneous attacks; the imaging of spontaneous and triggered phases largely consistent), 2023, its authors not carried; none fetched. Persistence, no landing: the closing condition at the review's abstract read at Reach (the question when, the emergence of an attack, beside why and how), the phase-resolved prediction keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Spontaneous-attack imaging — identifies phase-related network changes in small, hard-to-time samples. Provocation models — reliably trigger attacks in subsets but may not reproduce natural initiation. Genetic studies — identify excitability and vascular pathways without ordering the attack. Mechanism-specific drugs — establish therapeutic relevance of selected signaling systems without making one universal origin. **Narrowing recorded.** Aura is linked to spreading depolarization in a defined subgroup, and peptide-targeted therapy narrows a trigeminovascular component of headache. **Progress.** Distinct phases, multiple genetic risk pathways and effective mechanism-specific targets are established; the purely vascular account has been displaced. **Own record.** ## 218 The etiology of schizophrenia **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What causal developmental processes produce schizophrenia and its variation in psychosis, cognition, motivation and course? In the field's own posing, schizophrenia is a challenging and diverse condition, heterogeneous in its manifestations, its risk factors spanning genetic predisposition, prenatal and birth complications, infections and immune dysfunction, cannabis use and psychosocial factors, with no markers yet available for clinical use that allow prediction of its development or a diagnosis to be established. **The field's account of its hardness.** The syndrome is clinically and biologically heterogeneous, risk is highly polygenic and environmentally modified, diagnosis follows symptoms after long development, and dopamine, glutamate, immune and circuit findings can be causes, consequences or subgroup markers. **Accounts.** Altered dopamine synthesis and salience assignment generates psychotic symptoms. Glutamatergic and inhibitory-circuit development disrupts cognition and downstream dopamine function. Polygenic neurodevelopmental liability alters synaptic pruning and distributed brain circuits. Prenatal, social and substance exposures interact with genetic liability across development. Several mechanistic disorders converge on the present clinical syndrome. **Specifications.** The account predicts symptoms and functional course beyond diagnostic association — universal under etiology. Developmental timing and treatment effects are represented — universal under causal history. Proposed biomarkers replicate across independent populations or define a stated subgroup — local. **Specifications withdrawn.** A single cause or uniform pathology for every diagnosed case, withdrawn in heterogeneous accounts retaining the syndrome. **Common requirement.** A causal developmental model linking risk factors to circuits, symptoms and course. **Conserving relation.** Intervention on a proposed causal process changes its predicted clinical and biological consequences. **Conserving reach.** The symptom, developmental and outcome heterogeneity admitted by the diagnosis and related psychosis spectrum. **Reach.** The statement ranges over individual developmental causes. Genetic, imaging, molecular and clinical studies measure population-level associations at selected stages. The field's primer states that the risk factors are known while no markers are yet available for clinical use that allow prediction of the disease's development or a diagnosis to be established: the risk factors measured across populations and the developmental causes sought in an individual are not the same, the absent marker standing between them. **Unreachability.** **Persistence.** Validated mechanistic subtypes or a general causal model that prospectively predicts onset, course and response. **Addresses.** Statement, with the posing added there, hardness at the heterogeneity, the genetic and environmental risk factors, the long prodrome and the absent markers, the account at the leading theory as the primer states it, progress at the risk factors established, and the relation at Reach: Leucht et al., "Schizophrenia", Nature Reviews Disease Primers 11:83, 2025, doi 10.1038/s41572-025-00667-6, at its abstract as the publisher's page and a university listing carry it verbatim (a challenging and diverse condition with a lifetime prevalence of 0.4 percent, manifesting in late adolescence or early adulthood; risk factors from genetic predisposition to childhood trauma and migration; a long prodromal phase; no markers yet available for clinical use for prediction or diagnosis; a leading theory postulating excitatory–inhibitory imbalance in the cortex leading to dysfunction of the dopaminergic system; a heterogeneous disease with psychotic, negative and cognitive manifestations), the primer itself not fetched. The hardness line's clause on findings as causes, consequences or markers, the specifications, the specification withdrawn and the approaches stand keyed at the next pass; the primer's leading theory, glutamate–GABA imbalance leading to dopaminergic dysfunction, stands beside the account as posed, altered dopamine synthesis and salience. Adamu et al., "Unraveling the pathophysiology of schizophrenia: insights into structural and functional connectivity, neurochemical changes and inflammatory processes", Frontiers in Psychiatry 14, 2023, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched, their authors not carried: the field's primer at its abstract as the publisher's page carries it verbatim (no markers yet available for clinical use that allow prediction of disease development or a diagnosis to be established; a heterogeneous disease; treatment resistance frequent), Nature Reviews Disease Primers, 2025, doi 10.1038/s41572-025-00667-6; a genetics review at its summary (a causal explanation for the pathogenesis lacking; the underlying causal mechanisms unknown), Nature Reviews Neuroscience, 2017, doi 10.1038/nrn.2017.125; a heterogeneity preprint at its introduction (the subtype notion not fully confirmed; whether one unifying process or a multitude of processes leading to a similar syndrome the question that remains salient). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Polygenic studies — identify distributed liability with limited individual prediction. Longitudinal high-risk cohorts — measure changes before onset with low conversion rates. Neurochemical imaging — supports dopamine abnormalities in groups, with subgroup and medication effects. Cellular and organoid models — test selected variants outside full development. Clinical trials — establish symptom mechanisms for some domains without resolving etiology. **Narrowing recorded.** Psychosis, negative symptoms and cognitive impairment are analyzed as partly separable domains; high-risk and treatment-response strata narrow the syndrome. **Progress.** Polygenic architecture, environmental risk interactions, group-level dopamine abnormalities and neurodevelopmental changes stand. **Own record.** ## 219 Evidence for neural representation **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What evidence is sufficient to show that a neural state represents a particular feature, object or variable rather than merely covaries with it? In the field's own posing, neuroscience lacks a unified framework for the concepts used to characterize representation, terminology and measures being used disparately, and a neural response may be shown to be sensitive or specific to a feature, invariant to other features, or functional in being used downstream in the brain. **The field's account of its hardness.** Decoding and correlation establish sensitivity but can succeed for information unused by the organism; representation claims also require specificity, invariance and a role for downstream systems, which are rarely established together. **Accounts.** A neural state represents what can be decoded reliably from its activity. A neural state represents what it causally controls for the organism's behavior or computation. Representation requires a conjunction of sensitivity, specificity, invariance and downstream functional use. Neural explanation need not posit representational content beyond dynamical and causal organization. **Specifications.** Candidate content and alternative contents are specified — local. Sensitivity, specificity, invariance and downstream use have independent tests — universal under the conjunctive criterion. Causal interventions preserve task competence except through the represented variable — local. **Specifications withdrawn.** Decodability alone as sufficient evidence, withdrawn in accounts retaining decoding as one evidential relation. **Common requirement.** An evidential rule distinguishing representation of a variable from correlation with it. **Conserving relation.** The rule assigns the same content across nuisance changes and predicts the consequences of intervention on the neural state. **Conserving reach.** Candidate neural states, alternative contents, task contexts and downstream consumers covered by the claim. **Reach.** The statement ranges over representational content. Experiments measure activity, stimuli, behavior and intervention effects. The field's framework states that a neural response may be shown sensitive or specific to a feature, invariant to other features, or functional in being used downstream, and that the common analysis methods each evaluate these dimensions: the sensitivity a decoding returns and the representation sought are not the same, the other dimensions standing between them. **Unreachability.** **Persistence.** A field-accepted evidential criterion that fixes content and survives alternative-content, invariance and causal-use tests. **Addresses.** Statement, with the posing added there, hardness at the four desiderata a representation claim requires, specifications withdrawn at decoding as one method among those evaluating the dimensions, approaches at correlational analyses, decoding and encoding models, representational similarity analysis and tests of statistical dependence or adaptation, progress at the four dimensions articulated, and the relation at Reach: Pohl, Walker, Barack, Lee, Denison, Block, Meyniel and Ma, "Clarifying the conceptual dimensions of representation in neuroscience", arXiv:2403.14046, fourth version, 2026, at its abstract as the arXiv page and an abstract service carry it verbatim (the field lacking a unified framework for the concepts characterizing representation, terminology and measures used disparately; a core set of conceptual dimensions proposed, describing relations between a neural response, features that may be represented, and downstream effects; a response shown sensitive or specific to a feature, invariant to other features, or functional; information-theoretic measures illustrating the dimensions and their relation to the analysis methods), and at its first version's abstract as the arXiv page carries it verbatim (a systematic framework for the evidence used to establish that a neural response represents a feature; the four desiderata, sensitive, specific, invariant and functional, formalized in information-theoretic terms; the methods unified under one framework; identifying when evidence might be missing and when it is strong; canonical examples of orientation, numerosity, faces and spatial location), the paper itself not fetched. The title given before this pass, "Desiderata of evidence for representation in neuroscience", is the first version's and is kept here as such; the section numbers given before this pass, §§1–5, not verified at either version. The hardness line's clause that the desiderata are rarely established together, the account as posed, the specifications and the perturbation and dynamical approaches stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a perspective on the conceptual dimensions of representation at its abstract as the publisher's page carries it verbatim (appeals to representation widespread despite neuroscientists' uncertainty about what kind of findings count as evidence; the field lacking a unified framework; sensitivity, specificity, invariance and functional use as the dimensions), Nature Reviews Neuroscience, 2026, doi 10.1038/s41583-026-01030-8, its authors not carried; the three-aspects account at its abstract as an index and the arXiv copy carry it (correlation, causal role and teleology each implied by the term's explanatory role; evidence on all three rarely presented in a single study; a causal effect on behaviour harder to establish than a correlation), Trends in Cognitive Sciences, 2022, doi 10.1016/j.tics.2022.08.014, arXiv:2102.06592, its authors carried at the copy; a decoding critique at its argument as a preprint archive carries it (decodability not by itself a reliable guide to neural representation), its authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Encoding and decoding — establish sensitivity under a sampled stimulus distribution. Representational similarity — compares geometry across systems with content interpretation still required. Invariance tests — challenge nuisance dependence over declared transformations. Perturbation and circuit tracing — test downstream causal use. Dynamical explanation — predicts trajectories without assigning content. **Narrowing recorded.** Content claims are narrowed to tested tasks and transformations; multi-criterion evidence narrows the states licensed as representations. **Progress.** High-dimensional population codes, causal perturbation methods and four separable evidential requirements are articulated and testable. **Own record.** ## 220 Detecting covert consciousness **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which behaviorally unresponsive patients retain conscious awareness, and what measurement establishes it in an individual? In the field's own posing, covert consciousness is a state of residual awareness after severe brain injury or neurological disorder that evades routine bedside behavioral detection, the patient preserving awareness while incapable of self-expression through ordinary behavior or communication, and the limitations of bedside neurobehavioral examination in reliably detecting consciousness are increasingly recognized. **The field's account of its hardness.** Motor output can fail while command-following brain activity remains, bedside behavior fluctuates, imaging and electrophysiological tasks have false negatives and interpretation requirements, and no single test detects every conscious patient. **Accounts.** Willful modulation of brain activity during a command-following task demonstrates covert command following and awareness. Task-free complexity or effective-connectivity measures identify a capacity for consciousness without overt response. Multimodal repeated assessment is required because consciousness and arousal fluctuate and each instrument samples a different capacity. Some positive signals reflect preserved sensory or cognitive processing without establishing conscious awareness. **Specifications.** The command or paradigm is understood and executable by the intact covert system — local. The neural response is distinguishable from automatic activity and artifact — universal under the test. Individual sensitivity and specificity, not only group separation, are established — universal under diagnosis. **Specifications withdrawn.** Behavioral nonresponse as sufficient evidence of absent awareness, withdrawn while behavioral examination remains a diagnostic instrument. **Common requirement.** An individual-level marker that distinguishes conscious awareness from unconscious processing despite absent motor report. **Conserving relation.** Repeated independent tests return a concordant awareness classification and predict later communicative or clinical evidence. **Conserving reach.** Behaviorally unresponsive patients across injuries, arousal states, sensory deficits and motor impairments. **Reach.** The statement ranges over subjective awareness in an individual. Instruments measure behavior, task-evoked activity and spontaneous brain dynamics. The field's review states that possible, probable and confirmed covert consciousness reflect varying levels of diagnostic confidence depending on the nature and strength of the available test results: the test results returned and the awareness sought in the individual are not the same, the grade of confidence standing between them. **Unreachability.** **Persistence.** A validated bedside-capable protocol with known individual error rates that detects awareness across the relevant impairments. **Addresses.** Statement, with the posing added there, hardness at the limitations of bedside examination and at the graded diagnostic confidence, the account at neurotechnologies detecting brain states, and the relation at Reach: Young, Edlow and Bodien, "Covert consciousness", NeuroRehabilitation 54:23–42, 2024, doi 10.3233/NRE-230123, at its abstract as the publisher's page carries it verbatim (a state of residual awareness following severe brain injury or neurological disorder that evades routine bedside behavioral detection; awareness preserved while self-expression through ordinary behavior or communication fails; growing recognition of the limitations of bedside neurobehavioral examination in reliably detecting consciousness, beside advances in neurotechnologies capable of detecting brain states) and at a passage of its text as a repository copy of the author manuscript carries it (possible, probable and confirmed covert consciousness reflecting levels of diagnostic confidence depending on the nature and strength of the test results; many terms used for the state, covert command-following among them), the review itself not fetched. Given before this pass as Young and Edlow at pages 91–103; the pages read carry three authors and, at an index, pages 23–42, a citing paper carrying 21–36. The hardness line's clauses on motor failure, fluctuation and false negatives, the specifications, the specification withdrawn, the approaches and the Own record line stand keyed at the next pass. Prevalence and progress: Bodien et al., "Cognitive Motor Dissociation in Disorders of Consciousness", New England Journal of Medicine 391:598–608, 2024, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the multinational study at its discussion as the journal's page carries it (standardized behavioral evaluation remaining the reference standard; task-based imaging and electroencephalography improving detection, both together more sensitive than one), New England Journal of Medicine, 2024, doi 10.1056/NEJMoa2400645, its authors not carried; a pupillometry study at its background (sensitive low-cost bedside markers missing), Neurocritical Care, 2024, doi 10.1007/s12028-024-01983-7; a motor-imagery comparison at its introduction (high false-negative rates found for the standard imagery task; the relative accuracy of paradigms in the patient population unknown), 2017; a gap analysis of cognitive motor dissociation at its citation, Neurocritical Care 40(1):81-98, 2024, at a citing list; a magazine's report of the study's finding that more than half of the patients conscious at the bedside did not pass the scan tests. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Repeated bedside scales — improve behavioral diagnosis but depend on motor output and arousal. Functional-imaging command following — supplies strong positive evidence with transport and false-negative limits. Electroencephalographic command following — more deployable and method-sensitive. Perturbational and resting-state complexity — task-free correlates whose individual interpretation remains under validation. Multimodal assessment — raises detection while leaving discordant cases. **Narrowing recorded.** Cognitive motor dissociation names patients with neural command following despite no observable response; repeated multimodal testing narrows misclassification. **Progress.** A substantial minority of behaviorally unresponsive patients show command-following brain activity, and standardized multicenter paradigms now exist. **Own record.** A positive willful command-following signal establishes preserved cognition more strongly than a negative test establishes its absence. ## 221 Continual learning without catastrophic forgetting **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Can a learning system acquire an open sequence of tasks or data distributions while retaining prior capabilities and transferring useful knowledge without storing or replaying the complete past? In the field's own posing, incrementally learning new information from a non-stationary stream of data is a key feature of natural intelligence but a challenging problem for deep neural networks, and the many methods proposed are difficult to compare for the lack of a common framework, three scenarios with their own challenges having been described, task-incremental, domain-incremental and class-incremental learning. **The field's account of its hardness.** Parameter changes that fit new data overwrite representations used by earlier tasks; stability and plasticity conflict under bounded memory, and results depend strongly on whether task identity, boundaries and past samples are available. **Accounts.** Parameter regularization protects weights or functions important to earlier tasks. Replay restores earlier data or generated surrogates during later training. Architectural expansion or modular routing isolates task-specific capacity. Meta-learned and self-organizing representations make future updates less interfering. **Specifications.** The task, domain and class-incremental setting is declared — local. Evaluation includes every earlier task after each update — universal under continual learning. Memory, computation and task-boundary information are counted — universal under resource comparison. **Specifications withdrawn.** Unlimited access to all earlier training data, withdrawn under the bounded-memory statement while replay methods retain limited exemplars or generators. **Common requirement.** An update rule balancing new learning, retained performance and forward or backward transfer under stated resources. **Conserving relation.** Performance on each learned capability remains above the declared retention criterion as new data arrive. **Conserving reach.** The full task or distribution sequence, including long sequences, recurrences and unannounced changes. **Reach.** The statement ranges over open future sequences. Benchmarks supply finite, designed sequences with selected boundaries and shifts. The field's framework states that the three scenarios differ substantially in difficulty and in the effectiveness of the strategies, the comparison performed on designed split protocols according to each scenario: the performance measured on a designed protocol and the capability sought over open sequences are not the same, the scenario standing between them. **Unreachability.** **Persistence.** A resource-bounded learner with robust retention and transfer on long, unannounced, independently sampled sequences. **Addresses.** Statement, with the posing added there, hardness at the dependence of results on the scenario and on whether examples may be stored, approaches at the strategies compared, narrowing recorded at the three scenarios, progress at the framework and the empirical comparison, and the relation at Reach: van de Ven, Tuytelaars and Tolias, "Three types of incremental learning", Nature Machine Intelligence 4:1185–1197, 2022, doi 10.1038/s42256-022-00568-3, at its abstract as the publisher's page and a repository copy carry it verbatim (continual learning a key feature of natural intelligence but a challenging problem for deep networks; numerous methods proposed and difficult to compare for the lack of a common framework; three scenarios described, task-, domain- and class-incremental, each with its own challenges; a comprehensive empirical comparison of current strategies on the Split MNIST and Split CIFAR-100 protocols according to each scenario; substantial differences between the scenarios in difficulty and in the effectiveness of the strategies) and at a passage of its text as the repository copy carries it (class-incremental learning requiring discrimination between classes not observed together, very challenging for deep networks especially when storing examples is not allowed), the paper itself not fetched. The hardness line's clause on overwriting and on stability against plasticity, the account, the specifications and the specification withdrawn stand keyed at the next pass. Current mitigation: Zhou and Li, "Mitigating catastrophic forgetting in lifelong learning: a hybrid architecture integrating neural ordinary differential equations with memory-augmented transformers", Scientific Reports 16:2012, 2026, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a streaming-learning paper at its introduction as the arXiv copy carries it (the mitigating methods designed for settings with independent samples, known task boundaries, stored past samples and offline evaluation, impractical where continual learning is paramount; forgetting in the streaming setting more severe and largely unaddressed), arXiv:2404.00781; a 2026 preprint at its abstract (continual pre-training without replay and without task labels an unsolved architectural problem at scale; every published method retaining a buffer, requiring task identifiers, applying a penalty that scales poorly, or operating at small scale), arXiv:2605.15053; a reinforcement-learning study at its closing (the path forward synthesizing approaches to both forgetting and the loss of plasticity), arXiv:2303.07507; their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Regularization — low memory and reduced plasticity as protected structure grows. Exemplar replay — strong retention proportional to stored past samples. Generative replay — replaces stored data with model-dependent reconstructions. Dynamic architectures — isolate interference while capacity grows. Prompt and adapter routing — modularize updates with task-identification dependence. **Narrowing recorded.** Standard evaluations separate task-, domain- and class-incremental reach and report memory and task-oracle assumptions. **Progress.** Interference metrics, replay and regularization baselines, long-sequence benchmarks and bounded-memory evaluations stand. **Own record.** ## 222 Incompatible definitions of algorithmic fairness **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Can a nonperfect risk score be simultaneously calibrated within groups and equalize its positive- and negative-outcome error relations when outcome base rates differ between groups? In the field's own posing, competing notions of what it means for a probabilistic classification to be fair to different groups stand in tension, three conditions at the heart of the debates having been formalized and proved unsatisfiable together except in highly constrained special cases. **The field's account of its hardness.** Several fairness criteria formalize independently compelling relations, but under differing base rates their simultaneous satisfaction is mathematically incompatible except in degenerate cases; choosing a metric therefore cannot be replaced by optimizing all of them. **Accounts.** Calibration within groups is the required relation: equal scores carry equal outcome risk. Equalized error rates are the required relation: protected groups share false-positive and false-negative rates. Balance for positive and negative classes is the required relation: mean scores conditional on each outcome agree across groups. Fairness is a substantive institutional judgment not exhausted by any single parity condition. **Specifications.** Groups have unequal outcome base rates — local. Predictions are neither perfect nor wholly uninformative — local. Metrics are evaluated on the same population and outcome — universal under the incompatibility result. **Specifications withdrawn.** Simultaneous satisfaction of all listed parity relations as a universal technical requirement, withdrawn where the impossibility conditions hold. **Common requirement.** A declared relation governing how predictions, outcomes and protected groups may differ. **Conserving relation.** Calibration and the selected error or balance relation hold together. **Conserving reach.** Every group in a population with unequal base rates and imperfect prediction. **Reach.** The statement quantifies over simultaneous metric satisfaction on one joint distribution. Audits estimate each metric from sampled predictions and outcomes. The theorem's authors state that even satisfying the three conditions approximately requires the data to lie in an approximate version of one of the constrained special cases: the approximate satisfaction an audit measures and the simultaneous satisfaction the statement quantifies over are not the same, the special cases standing between them. **Unreachability.** With unequal base rates and imperfect prediction, calibration within groups and balance for both outcome classes cannot all hold simultaneously; equivalent tensions obtain among common calibration and error-parity criteria under their stated conditions. **Persistence.** A justified choice or reformulation of the governing fairness relation for the decision context after the incompatible conjunction is withdrawn. **Addresses.** Statement, with the posing added there, hardness at the three conditions proved unsatisfiable together except in constrained special cases and at approximate satisfaction requiring approximate special cases, unreachability at the theorem as the abstract states it, progress at the incompatibility proved and its special cases identified, Own record, and the relation at Reach: Kleinberg, Mullainathan and Raghavan, "Inherent Trade-Offs in the Fair Determination of Risk Scores", Innovations in Theoretical Computer Science, 2017, Leibniz International Proceedings in Informatics 67, paper 43, at its abstract as the arXiv page and an abstract service carry it verbatim (tension between competing notions of fairness for probabilistic classification across groups; three conditions formalized; no method satisfying the three simultaneously except in highly constrained special cases; approximate satisfaction requiring the data to lie in an approximate version of a special case; key notions of fairness incompatible, a framework for the trade-offs), the paper itself not fetched; the three conditions are not named at the abstract, so the specifications and the unreachability's base-rate clause stand keyed at the next pass. Chouldechova, "Fair prediction with disparate impact", Big Data 5:153–163, 2017, and the accounts' sources, Sahlgren, "What's Impossible about Algorithmic Fairness?", Philosophy & Technology 37, 2024, and Green, "Escaping the Impossibility of Fairness", 2021, cited here and not read this pass. Approaches: the routes and their endings at six papers returned in the search, none fetched, their authors not carried except as the copies give them: the impossibility as a 2026 preprint's introduction carries it (calibration and error-rate balance incompatible except in degenerate cases, the incompatibility of equalizing prevalence and accuracy across groups; the debate shifted from which rule is best to which tradeoff is acceptable; the impossibility no longer binding where behaviour is endogenous to classification), arXiv:2604.06378; a comparative analysis of metrics (a common framework for the trade-offs), arXiv:2001.07864; the relaxation route at two philosophy papers' abstracts (a weakening of calibration consistent with equalized odds outside trivial cases; a further weakening consistent even in non-trivial cases), Philosophy and Technology, 2024 and 2026, doi 10.1007/s13347-024-00810-3 and 10.1007/s13347-026-01043-2; an accounting identity for departures from perfect fairness as a 2026 preprint's abstract carries it (a total unfairness budget, the mean-squared error times the prevalence difference, nesting the impossibility results), arXiv:2601.20217; the institutional and causal route keyed at the next pass. Persistence, no landing: the closing condition at the papers read above (a choice of the governing metric grounded in the decision context, no one-size-fits-all; the debate shifted to which tradeoff is acceptable). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Calibration — preserves score meaning within groups and permits unequal error rates. Equalized odds — constrains error rates and generally changes calibration. Threshold adjustment — reaches selected parity at group-specific decision rules. Multiobjective optimization — traces a tradeoff surface rather than satisfying incompatible equalities. Institutional and causal analysis — specifies harms and decision processes beyond prediction metrics. **Narrowing recorded.** Each deployment narrows the reach to declared groups, outcomes, decision thresholds and one selected set of fairness conditions. **Progress.** The incompatibility is proved, its exceptional cases are known, and optimization under chosen metric sets is routine. **Own record.** The theorem's authors state that their results suggest some of the ways in which key notions of fairness are incompatible with each other, and hence provide a framework for thinking about the trade-offs between them. ## 223 The existence of one-way functions **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does there exist an efficiently computable function that every probabilistic polynomial-time algorithm fails to invert on more than a negligible fraction of inputs drawn from the prescribed distribution? In the field's own posing, the existence of one-way functions is equivalent to the existence of the central private-key primitives, and equivalent, for every polynomial time bound of the stated form, to the mild average-case hardness of time-bounded Kolmogorov complexity, the first natural computational problem characterizing the feasibility of those primitives. **The field's account of its hardness.** That the field parts five possible worlds by the status of its assumptions, and in the third of them, named Pessiland, problems in NP are hard on average and yet one-way functions do not exist, average-case hardness of NP standing not known to yield a function hard to invert on average, the existence of one-way functions implying both that P differs from NP and that NP is hard on average while neither is known to imply it back; that none of the five worlds has been ruled out in the three decades since they were named; and that the approaches proposed to rule out the middle worlds study the complexity of compression problems, the minimum circuit size problem and time-bounded Kolmogorov complexity, the hardness of approximating a variant of the latter being what would rule Pessiland out. **Accounts.** Integer, lattice, code or multivariate problems support candidate functions under explicit average-case assumptions. One-way functions exist exactly when time-bounded Kolmogorov complexity is mildly hard on average under the stated formulation. No unconditional construction is presently derivable from known worst-case separations. **Specifications.** Evaluation is polynomial-time — universal under the definition. Inversion success is negligible for every probabilistic polynomial-time adversary — universal under cryptographic security. Inputs and security parameter follow the specified ensemble — universal under average-case complexity. **Specifications withdrawn.** **Common requirement.** An unconditional average-case computational asymmetry between evaluation and inversion. **Conserving relation.** Efficient evaluation coexists with negligible efficient inversion probability as the security parameter grows. **Conserving reach.** Every probabilistic polynomial-time inverter over the specified input ensemble and all sufficiently large security parameters. **Reach.** The statement quantifies over all efficient inverters and asymptotic inputs. Security reductions and attacks treat named constructions or models. The field's theorem states that the existence of one-way functions is equivalent, for every polynomial time bound of the stated form, to the mild average-case hardness of time-bounded Kolmogorov complexity: the constructions and attacks the reductions treat and the existence the statement quantifies over are not the same, the equivalence to one named problem standing between them. **Unreachability.** **Persistence.** An unconditional construction with proof, or a proof that no such function exists. **Addresses.** Statement, with the posing added there, the approach at complexity characterizations, progress at the equivalences, and the relation at Reach: Liu and Pass, "On One-way Functions and Kolmogorov Complexity", IEEE Symposium on Foundations of Computer Science, 2020, pp. 1243–1254, doi 10.1109/FOCS46700.2020.00118, at its abstract as the arXiv page, the cryptology preprint archive and a university listing carry it verbatim (the equivalence of two fundamental problems proved; for every polynomial time bound at least a constant factor above linear, one-way functions exist, which is equivalent to secure private-key encryption, signatures, pseudorandom generators and functions and commitments, if and only if time-bounded Kolmogorov complexity is mildly hard on average, no probabilistic polynomial-time algorithm computing it on more than a polynomially bounded fraction of strings; the first natural, well-studied problem characterizing the feasibility of the central private-key primitives), the paper itself not fetched. The account, the specifications, the other approaches and the narrowing stand keyed at the next pass. Average-case view: Impagliazzo, "A Personal View of Average-Case Complexity", Structure in Complexity Theory, 1995, cited here and not read this pass. Hardness: re-formed to the field's words at the average-case paper the address cites, at its abstract as an index carries it (the state of knowledge summarized, definitions standardized), Structure in Complexity Theory 1995, pages 134-147 as a later paper's reference gives them, and at the five worlds as a lecture note and three later papers state them, returned in the search, none fetched: a 2026 preprint at its definition of the five (Pessiland, NP hard on average but no one-way functions; Minicrypt; Cryptomania), arXiv:2606.27139; a 2023 preprint on proof complexity at its plausibility section (approaches to rule out Heuristica and Pessiland by compression problems, the minimum circuit size problem and time-bounded Kolmogorov complexity), arXiv:2312.08163; a 2024 conference paper at its passage on ruling Pessiland out by the hardness of approximating a variant of time-bounded Kolmogorov complexity, at a proceedings archive; a commentary that none of the five has been rejected in almost thirty years, at its page; their authors not carried at the returns; the former line, in the file's compression (Candidate functions rest on unproved average-case assumptions, worst-case complexity separation alone does not supply the required distributional hardness, and black-box evidence and restricted-model lower bounds do not establish unconditional existence.), released with its average-case content at the first clause, its black-box and restricted-model content, released from the line, standing for the next pass. Persistence, no landing: the closing condition at the five-worlds statements read above (no world ruled out; an unconditional proof of existence not known to follow from any complexity assumption). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Factoring-, discrete-log-, lattice-, code- and multivariate-based candidates — secure under named assumptions and parameter ranges. Reductions from average-case problems — transfer hardness conditional on a source assumption. Oracle and black-box results — delimit proof techniques without deciding existence. Complexity characterizations — establish equivalence to natural average-case hardness statements. **Narrowing recorded.** Concrete candidates narrow adversaries, distributions and assumptions; cryptographic reductions specify exactly which primitive follows from a one-way function. **Progress.** One-way functions are equivalent to central private-key primitives and to a stated time-bounded Kolmogorov-complexity hardness condition. **Own record.** ## 224 Hilbert's sixteenth problem for limit cycles **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is there, for every degree n, a finite uniform upper bound H(n) on the number of isolated limit cycles of a planar polynomial vector field of degree n, and what is the bound? In the field's own posing, the maximum number of limit cycles a planar polynomial vector field of degree n can have, whenever it is finite, is realizable by structurally stable vector fields with only hyperbolic limit cycles and is a strictly increasing function of the degree. **The field's account of its hardness.** That the second part of the problem remains unsolved even for quadratic polynomials, several attempts to solve it having failed, and it is still unknown whether a bound on the number of limit cycles depending on the degree alone exists even at degree two; that the problem parts into three, the finiteness of the number of cycles of a given field, a uniform bound depending only on the degree, and an upper bound for that number, a solution to any implying a solution for the previous and only the first solved, the positive answer established by two proofs after an earlier claimed proof stood for decades before a gap was found, a published solution for the quadratic case refuted by counterexamples, quadratic examples with four cycles found, and the field's own history recording that after eighty years its knowledge of the problem was almost the same as when it was stated; and that finiteness is not obvious, smooth vector fields with infinitely many concentric limit cycles being easy to construct. **Accounts.** The conjecture as posed, that for every degree a finite uniform bound on the number of limit cycles exists: one account stands, the finiteness for each field proved and the uniform bound at degree two still unknown; no negation is defended in the field's record. **Specifications.** The vector field is planar and polynomial of fixed degree — universal under the problem. Limit cycles are isolated periodic orbits — universal under dynamical-systems terminology. Multiplicity and degenerate parameter cases are treated under a stated convention — local. **Specifications withdrawn.** **Common requirement.** A degree-dependent global bound valid independently of coefficients. **Conserving relation.** The number of isolated limit cycles does not exceed H(n). **Conserving reach.** Every real planar polynomial vector field of degree n, including degenerate and bifurcation-limit cases admitted by the convention. **Reach.** The statement quantifies over an unbounded real coefficient space and global phase portraits. Existing analyses cover selected degrees, families, neighborhoods or perturbations. The field's note states that, whenever it is finite, the maximum is realizable by structurally stable vector fields with only hyperbolic limit cycles and is strictly increasing in the degree: the bound sought over the whole coefficient space and the fields at which it is attained are not the same, the structurally stable fields sufficing. **Unreachability.** **Persistence.** Proof of finiteness with a uniform bound for each degree, followed by determination or sharp bounding of H(n). **Addresses.** Statement, with the posing added there, progress at the realization by structurally stable fields and the strict increase, and the relation at Reach: Gasull and Santana, "A note on Hilbert 16th Problem", Proceedings of the American Mathematical Society 153:669–677, 2025, at its abstract as the arXiv page carries it verbatim (the maximum number of limit cycles of a planar polynomial vector field of degree n; proved realizable by structurally stable vector fields with only hyperbolic limit cycles; proved strictly increasing whenever finite), the venue at the society's listing, the note itself not fetched. The specifications, the approaches and the other clauses of progress stand keyed at the next pass. Workshop report: Rousseau, "Mathematical developments around Hilbert's 16th problem", Banff International Research Station workshop report, 2007, cited here and not read this pass. A preprint titled as a note on a recent attempt to solve the second part of the problem, 2024, returned at its citation only, a reading for the Own record at the next pass. Hardness: re-formed to the field's words at the centennial history of the problem, at its abstract and opening as the publisher's copy, an index and two repository copies carry them verbatim (unsolved even for quadratic polynomials; several failed attempts; the three problems, finiteness, a bound by degree, an upper bound, each implying the previous, only the first solved; still unknown whether the degree-two bound exists), Bulletin of the American Mathematical Society 39(3):301-354, 2002, doi 10.1090/S0273-0979-02-00946-1, its author carried at the copies, not fetched; a 2018 survey on the problem and o-minimality at its timeline as the arXiv copy carries it (the 1923 claimed proof with the gap found in 1981; the 1955-57 solution refuted by counterexamples in 1963; quadratic fields with four cycles in 1979-80; the eighty-years remark quoted), arXiv:1804.03585, its author not carried; an encyclopedia page for the two finiteness proofs of 1991 and 1992 and the smooth fields with infinitely many cycles; the former line, in the file's compression (The global number of cycles must be controlled across all coefficient choices and bifurcations; even the quadratic case lacks a known exact maximum, while local perturbative results do not supply a global uniform bound.), released with its content at the first clause, its perturbative content, released from the line, standing for the next pass. Accounts: the conjecture as posed at the note and the centennial history read above, no negation carried at either. Persistence, no landing at the second part: the closing condition at the centennial history read above (the three problems, finiteness proved, the uniform bound by degree and its upper bound open, a solution to each implying the previous). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Local bifurcation and cyclicity analysis — bounds cycles born near specified configurations. Perturbation of integrable systems — counts zeros of displacement and Abelian integrals in restricted families. Algebraic and topological phase-plane methods — classify subsets of systems. Computer-assisted constructions — raise lower bounds by exhibiting fields with many cycles. **Narrowing recorded.** Fixed degrees, coefficient families and perturbative neighborhoods carry finite bounds and explicit cycle constructions. **Progress.** Finiteness is proved for important local settings, many lower-bound examples are known, and the linear case is settled; no global degree-uniform solution stands. **Own record.** ## 225 The quantum PCP conjecture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is approximating the ground-state energy of a local quantum Hamiltonian to a fixed constant fraction of its interaction scale complete for quantum Merlin-Arthur verification? In the field's own posing, the question is whether a quantum version of the PCP theorem holds, and the jury is still out as to whether or not such a theorem holds. **The field's account of its hardness.** That the additional difficulty of the quantum conjecture beyond that of proving the classical theorem is, in the field's account, that solutions of QMA-hard problems are expected to have high description complexity, ground states of local Hamiltonians having no classically checkable polynomial-size descriptions if NP differs from QMA, and only in one dimension is it known that low entanglement implies a classical representation from which the energy can be computed efficiently; that the apparent contradiction between the physical intuition that systems reach equilibrium at room temperature and the conjecture's implication that a QMA-hard problem must then be solved is captured by the no-low-energy-trivial-states conjecture, which isolates the robustness of entanglement from the remaining difficulties, must hold for the quantum conjecture to hold, and has since been proved by a construction based on quantum error correction; and that gap amplification, the natural proof strategy at the heart of the classical proof, is being carried to Hamiltonians, quantum de Finetti theorems having ruled out certain approaches and oracles existing relative to which quantum PCP statements are false, requiring non-relativizing techniques. **Accounts.** Constant-gap local Hamiltonian approximation remains quantum Merlin-Arthur hard. All low-energy states of broad local Hamiltonians admit enough structure for better approximation, making the conjecture false. Hardness holds only for selected interaction geometries, dimensions or local dimensions. **Specifications.** Local terms have bounded locality and norm — universal under the local-Hamiltonian problem. The promise gap is a constant fraction independent of system size — universal under the conjecture. Completeness and soundness use polynomial-size quantum witnesses — universal under quantum Merlin-Arthur verification. **Specifications withdrawn.** **Common requirement.** A complexity classification of constant-additive-error ground-energy approximation. **Conserving relation.** The verifier distinguishes low- from high-energy instances across the constant promise gap. **Conserving reach.** All Hamiltonians in the stated locality, geometry and dimension class as system size grows. **Reach.** The statement ranges over arbitrary large entangled systems in the class. Algorithms and reductions cover restricted structures or weaker gaps. The two are not the same: only in one dimension is low entanglement known to imply a classical representation from which the energy can be computed, solutions of QMA-hard problems are expected to have high description complexity, and gap amplification, the natural strategy at the heart of the classical proof, is still being carried to Hamiltonians. **Unreachability.** **Persistence.** A constant-gap hardness reduction, or a general approximation algorithm or structural theorem contradicting it. **Addresses.** Statement, with the posing added there: Aharonov, Arad and Vidick, "The Quantum PCP Conjecture", ACM SIGACT News 44:47–79, 2013, doi 10.1145/2491533.2491549, carried at the journal's page as "Guest column: the quantum PCP conjecture" and at the arXiv listing under the title above, at its abstract as an abstract service and the journal's page carry it verbatim (the classical PCP theorem arguably the most important achievement of classical complexity theory in the past quarter century; the question whether a quantum version holds; the study opening onto the foundations of quantum mechanics, the global nature of entanglement and its topological properties, quantum error correction and information theory; the jury still out; a snapshot of the status for a general theory audience), the survey itself not fetched. No Reach relation is stated at the abstract read. The account, the specifications, the approaches and progress stand keyed at the next pass. Simons Institute, "Quantum PCP, Area Laws, and Quantum Gravity", research program description, 2024, cited here and not read this pass. Hardness: re-formed to the field's words at the survey the address names, at passages of its text as a repository copy carries them verbatim (only in one dimension low entanglement known to imply a classical representation; the apparent contradiction with physical intuition captured by the no-low-energy-trivial-states conjecture, which isolates robustness of entanglement and must hold for the quantum conjecture to hold), the survey itself not fetched; and at four sources returned in the search, none fetched: a 2018 preprint at its introduction as the arXiv copy carries it (the additional difficulty that solutions of QMA-hard problems are expected to have high description complexity; no classically checkable polynomial-size descriptions if NP differs from QMA), arXiv:1802.07419, its authors carried as Nirkhe, Vazirani and Yuen; an institute's colloquium page recording proofs of the trivial-states conjecture by a construction based on quantum error correction; a 2025 preprint at its abstract (gap amplification the natural strategy at the heart of the classical proof; quantum de Finetti theorems having ruled out certain approaches), arXiv:2510.01333, its authors carried at the copy; a citing passage at a repository page on oracles relative to which quantum PCP statements are false, requiring non-relativizing techniques, its source not carried; the former line, in the file's compression (Quantum entanglement invalidates direct classical proof-checking constructions, local consistency need not extend to a global low-energy state, and known gap amplification or low-energy structure results reach restricted Hamiltonians or shrinking approximation gaps.), released with its content at the three clauses. The relation at Reach: the survey and the preprints, at the passages read for the hardness. Persistence, no landing: the closing condition at the survey and the preprints read above (a constant-gap hardness, gap amplification the natural strategy, the trivial-states obstacle since cleared). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Quantum gap amplification — established in limited constructions without a general constant-gap reduction. Hamiltonian complexity reductions — prove hardness at inverse-polynomial precision. Entanglement and area-law methods — give efficient descriptions in restricted dimensions or gapped settings. No-low-energy-trivial-states formulations — relate circuit depth to low-energy structure. **Narrowing recorded.** Hardness and algorithms are known for specified locality, geometry, commuting structure and approximation regimes. **Progress.** The local-Hamiltonian problem is complete at inverse-polynomial precision, and several equivalent or related structural conjectures and restricted results stand. **Own record.** ## 226 Deterministic polynomial identity testing **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is there a deterministic polynomial-time algorithm that decides whether an arithmetic circuit computes the identically zero polynomial? In the field's own posing, polynomial identity testing, the problem of checking whether a given arithmetic circuit is the zero circuit, ranks as one of the most important open problems at the intersection of algebra and computational complexity, with impressive progress and a complete solution that might take a while. **The field's account of its hardness.** Random evaluation gives an efficient test, but constructing a small deterministic hitting set for general circuits remains open; successful derandomization for restricted circuit classes depends on exploitable structure, and general derandomization is tied to major arithmetic-circuit lower bounds. **Accounts.** Explicit hitting sets intersect the nonzero evaluations of every circuit in the size and degree class. Hardness-versus-randomness converts sufficiently explicit circuit lower bounds into derandomization. Algebraic structure such as low depth, bounded top fan-in, sparsity or rank permits class-specific deterministic tests. **Specifications.** The circuit size, number of variables, degree and coefficient field are encoded — universal under the input model. Runtime is polynomial in the declared input parameters — universal under deterministic polynomial time. Identity means the formal polynomial is zero, not merely zero at sampled points — universal under PIT. **Specifications withdrawn.** Random bits, withdrawn in the deterministic statement while the randomized test is retained as the baseline. **Common requirement.** A polynomial-size hitting or certification procedure for nonzero general circuits. **Conserving relation.** The algorithm returns zero exactly for circuits computing the zero polynomial. **Conserving reach.** All arithmetic circuits over the stated field with polynomially encoded size and degree. **Reach.** The statement ranges over general circuits. Current deterministic algorithms cover restricted circuit families; the randomized evaluation test covers the general family with bounded error. The field's survey states that the randomized algorithms place the problem in the class BPP, conjectured equal to P, so that derandomizing is a reasonable goal: the general test with bounded error and the deterministic test sought are not the same, the conjectured equality of the two classes standing between them. **Unreachability.** **Persistence.** A deterministic polynomial-time general test, or a lower bound ruling out such a test in the stated model. **Addresses.** Statement, with the posing added there, hardness at the clause that random evaluation gives an efficient test while the deterministic test remains open, specifications withdrawn at the randomized test retained as the baseline, the approach at random evaluation, progress at the randomized test placing the problem in BPP, and the relation at Reach: Saxena, "Progress on Polynomial Identity Testing", Bulletin of the EATCS 99:49–79, 2009, at its abstract as the author's page, a repository copy and the column's copy carry it verbatim (the problem of checking whether a given arithmetic circuit is the zero circuit; one of the most important open problems at the intersection of algebra and computational complexity; impressive progress in the last few years and a complete solution that might take a while) and at passages of its text as the copies carry them (the randomized algorithms placing the problem in BPP, conjectured equal to P; derandomizing a reasonable goal), the survey itself not fetched. Saxena, "Progress on Polynomial Identity Testing-II", Perspectives in Computational Complexity, Progress in Computer Science and Applied Logic 26, 2014, pp. 131–146, doi 10.1007/978-3-319-05446-9_7, at its abstract as the publisher's page and the arXiv page carry it verbatim (a survey of algebraic complexity theory focused on the problem; the key ideas of the results of the last few years), not read beyond it. The hardness line's clauses on exploitable structure and on the tie to circuit lower bounds, the account, the specifications and the other approaches stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the founding equivalence at its abstract as the conference proceedings and a repository copy carry it (identity testing derandomizable in subexponential time provided an explicit family of polynomials has superpolynomial arithmetic circuit complexity, and the converse, derandomization implying such lower bounds), Kabanets and Impagliazzo, Computational Complexity 13(1-2):1-46, 2004; a primer at its section on hardness against randomness as the arXiv copy carries it (the problem efficiently derandomisable if and only if explicit polynomials of high circuit complexity exist), arXiv:2506.19604; a low-characteristic study at its prior work (a bootstrapping phenomenon, a barely non-trivial derandomization implying a near-complete one), arXiv:2005.10885; a citing paper's statement of the main goal, a deterministic polynomial-time algorithm, a major challenge even for restricted versions; their authors not carried except as given. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Random evaluation — polynomial time with controllable error. White-box algebraic analysis — deterministic for selected circuit structures. Black-box hitting-set construction — explicit for restricted depth, sparsity and rank. Hardness-versus-randomness — conditional general derandomization and lower-bound consequences. Variable reduction and rank concentration — extend restricted classes. **Narrowing recorded.** Deterministic polynomial-time tests stand for many bounded-depth, sparse, read-once and structured models. **Progress.** Efficient randomized PIT, equivalences to hitting sets, and strong conditional links between derandomization and circuit lower bounds stand. **Own record.** ## 227 Collective intentionality **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What makes several individuals jointly intend or act as a group rather than merely having coordinated individual intentions? In the field's own posing, the main philosophical challenge is the tension within one expression, carried verbatim in four words, "individuals as a group", a contradiction between two widely accepted claims: that collective intentionality is no simple summation, aggregate or distributive pattern of individual intentionality, and that it is had by the participating individuals, all of whose intentionality is their own. **The field's account of its hardness.** An account must preserve individual minds while explaining a genuinely joint commitment, distinguish shared agency from parallel action and coercion, and avoid defining the joint state through the very collective fact it is meant to explain. **Accounts.** Each participant has interlocking individual intentions referring to the joint activity and the others' roles. Participants enter a public normative commitment that creates mutual obligations irreducible to private intentions. Shared plans and meshing subplans organize the activity without a group mind. A group can bear attitudes as an organized agent through decision structures distinct from member attitudes. **Specifications.** Participants and the proposed joint activity are identified — local. The account distinguishes cooperation, coincident action, coercion and mass behavior — universal under collective intentionality. Individual and group attitude bearers are not conflated — universal under the ontology claimed. **Specifications withdrawn.** **Common requirement.** A constitutive condition for joint agency that explains coordination and normative relations among participants. **Conserving relation.** The condition classifies joint and merely parallel cases consistently and predicts the commitments generated. **Conserving reach.** Small informal actions, large organized groups, partial knowledge, dissent and changing membership. **Reach.** The statement ranges over what constitutes the joint attitude. Examples, linguistic reports and institutional practices supply judgments and consequences. The field's encyclopedia states that collective attitudes involve a plurality of participants such that the attitude can be ascribed to the individuals as a group, and that the challenge lies in that very expression, the two accepted claims contradicting: the ascriptions the reports return and the joint attitude sought are not the same, the expression standing between them. **Unreachability.** **Persistence.** A constitutive account resolving the difficult cases without circularity, over-demanding common knowledge or positing an unexplained group subject. **Addresses.** Statement, with the posing and the marked phrase added there, hardness at the two claims an account must hold together, and the relation at Reach: Schweikard and Schmid, "Collective Intentionality", Stanford Encyclopedia of Philosophy, at its opening as the encyclopedia's page carries it verbatim (the power of minds to be jointly directed; the modes; attitudes ascribable to individuals as a group; the main challenge the tension within that expression, spelled out as a contradiction between the Irreducibility Claim and the Individual Ownership Claim; theories proposed over the last decades pointing to different solutions), first published 2013 by a data listing, its current revision date not carried at the page read, the entry itself not fetched. "Collective Intentionality", Internet Encyclopedia of Philosophy, at its opening as the encyclopedia's page carries it verbatim (the dominant picture that intentionality is a feature of individual minds; groups without minds or brains and the question how they could have intentional states; growing attention, focused on collective intention and belief), its author not carried at the page read, the section numbers given before this pass, §§1–5, not verified. The hardness line's clauses on parallel action, coercion and circularity, the account, the specifications and the approaches stand keyed at the next pass. Content and recent dispute: Crone, "Collective intentionality: why content matters", Inquiry, 2025, cited here and not read this pass. Persistence, no landing: the closing condition at the entry the address names, at passages of its text as the encyclopedia's page and six archived editions carry them verbatim (the charge that intentions grounding collective intentionality in individuals' intentions are circular, presupposing rather than constituting the jointness; the relational alternative acknowledging genuine we-form attitudes, which does away with the problem of circularity; the nature of the first-person plural indexical an aspect whose investigation is still in the early stages; the question what sort of entity can be a subject of intentionality), the entry itself not fetched. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Interlocking-intention accounts — individualistic and vulnerable to coercion and circularity cases. Commitment accounts — explain obligation while requiring a constitutive act or convention. Shared-plan accounts — model coordination in detail with limited large-group reach. Group-agent accounts — explain institutional attitudes through aggregation structures and raise ontology questions. **Narrowing recorded.** Accounts distinguish shared intention in small action from corporate agency in organized institutions; results stand within each setting. **Progress.** Counterexample families for coincident action, coercion and insufficient mutual awareness, and formal models of shared plans and group attitudes, stand. **Own record.** ## 228 Peer disagreement **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What should a person rationally believe after learning that an epistemic peer, with the same evidence, disagrees? In the field's own posing, the Equal Weight View prescribes doxastic conciliation on discovering a peer's disagreement and meets the objection that there is disagreement about the epistemic significance of disagreement itself, so that a view calling for conciliation can have it that it calls for its own rejection. **The field's account of its hardness.** The disagreement is itself evidence about one's reliability, yet judging the other a peer and judging the first-order evidence already use the disputed reasoning; universal conciliation threatens self-defeat and steadfastness appears to ignore symmetric evidence. **Accounts.** Equal weight requires moving belief toward the peer's judgment. Total evidence permits retaining confidence when first-order reasons justify downgrading the disagreement evidence. Steadfastness is rational where the agent has justified confidence in the disputed reasoning. No single response is required because peerhood, independence and stakes vary by case. **Specifications.** The parties are peers with respect to the question — local. They share the relevant evidence — local. Their judgments are independent enough that the disagreement adds evidence — local. **Specifications withdrawn.** **Common requirement.** A norm for updating on symmetric higher-order evidence while retaining first-order reasons. **Conserving relation.** The updating norm treats evidentially symmetric positions symmetrically and remains coherent when applied to disagreement about the norm itself. **Conserving reach.** Ideal and ordinary disagreements, repeated disagreement, higher-order dispute and disagreement over peerhood. **Reach.** The statement ranges over rational response under specified epistemic relations. Cases supply contested judgments about evidence, reliability and peer status. The field's encyclopedia states the Independence principle, that reasons to discount a peer's opinion must be independent of the disagreement itself, and that a view calling for conciliation can call for its own rejection, there being disagreement about the significance of disagreement: the judgments the cases supply about reliability and peerhood and the rational response sought are not the same, the independence they must have from the disagreement standing between them. **Unreachability.** **Persistence.** A norm that handles symmetry, self-application and justified asymmetry across the canonical cases. **Addresses.** Statement, with the posing added there, hardness at the self-defeat clause and at the Independence principle, the account at the Equal Weight View, the approach at equal-weight conciliation and its self-defeat objection, progress at the self-defeat objection and the central cases, and the relation at Reach: Frances and Matheson, "Disagreement", Stanford Encyclopedia of Philosophy, first published 2018, at passages of the entry as the encyclopedia's page and its archive editions carry them verbatim (the Equal Weight View prescribing doxastic conciliation, defended by central cases, theoretical considerations and analogies, the Restaurant Check Case the central one; the objection that such views are self-defeating, there being disagreement about the epistemic significance of disagreement itself; the Independence principle, that reasons to discount a peer's opinion must be independent of the disagreement; defenses through the Uniqueness Thesis or without it), the entry itself not fetched; the substantive revision date given before this pass, 2024, and the section numbers, §§1–5, not carried at the pages read. The hardness line's clause on the disagreement as evidence of reliability, the specifications and the other approaches stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the encyclopedia's entry on social epistemology at its passage on peer disagreement as the encyclopedia's page carries it verbatim (conciliation supported by the appearance of dogmatism in refusing it and by the disagreement as higher-order evidence of error; the charge of self-refutation, conciliationism itself widely disputed), its author not carried; a paper at its abstract as the publisher's page carries it (the self-undermining objection; both families of response, solution and skeptical, argued to fail by conciliationism's own lights), Synthese, 2024, doi 10.1007/s11229-024-04590-3, its author not carried; a formal reply at its abstract (a model of conciliatory reasoning in defeasible logic answering the self-defeat objection), Review of Symbolic Logic 16(3), 2023, its author carried at the copy; a paper on the symmetry argument at its opening (the standard argument for conciliation an appeal to symmetry), Synthese, 2019. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Equal-weight conciliation — respects symmetry and risks excessive skepticism or self-defeat. Steadfast views — retain first-order warrant and risk arbitrary self-favoring. Total-evidence views — weigh both orders and require a non-question-begging reliability assessment. Permissivism — admits multiple rational responses and shifts the problem to the limits of permission. **Narrowing recorded.** Results are conditioned on shared evidence, genuine peerhood and independence; relaxing each yields distinct cases. **Progress.** The first-order and higher-order evidence distinction, self-defeat objections and canonical symmetry cases are sharply formulated. **Own record.** ## 229 Compositionality in language understanding **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How is the meaning of a novel complex expression determined from its parts and their mode of combination while idiom, context and lexical variation are retained? In the field's own posing, neural models greatly outperform grammar-based models across many tasks, raising the question whether linguistic principles such as the Principle of Compositionality still have value as modeling tools, an overly strict interpretation making broad coverage hard to achieve while compositionality remains necessary for learning the correct linguistic generalizations from limited data. **The field's account of its hardness.** Strict compositional rules support systematic generalization but lose empirical coverage on idioms and context dependence; flexible statistical systems cover usage yet can fail on novel recombinations, and benchmarks often confound memorization with composition. **Accounts.** Syntactic structure and lexical meanings determine complex meaning through fixed composition functions. Construction-level and idiomatic meanings supplement word-level composition. Context-sensitive pragmatic inference contributes variables or enrichments within a compositional architecture. Distributed learned representations approximate composition without explicit symbolic rules. **Specifications.** Parts, structure, context and target meaning are operationally represented — local. Generalization is tested on recombinations not seen in training — universal under systematicity. Coverage includes ordinary nonliteral and context-sensitive usage within the claimed language fragment — universal under the account. **Specifications withdrawn.** Strict context-free word-by-word composition as a universal coverage condition, withdrawn in hybrid accounts retaining compositional structure. **Common requirement.** A rule or learned operation yielding interpretations for novel complexes from reusable components. **Conserving relation.** Meaning predictions remain correct under new combinations of familiar parts and declared contextual changes. **Conserving reach.** Productive constructions together with idiomatic, ambiguous and context-dependent cases claimed by the system. **Reach.** The statement ranges over unbounded productive expression. Corpora and benchmarks sample finite constructions and lexical combinations. The field's review states that compositionality is still necessary for a model to learn the correct linguistic generalizations from limited data, while an overly strict interpretation makes broad coverage hard to achieve: the coverage measured on the tasks and the generalization sought over productive expression are not the same, the design space between the two standing between them. **Unreachability.** **Persistence.** A model with independently tested systematic generalization and broad empirical coverage under one explicit account of context. **Addresses.** Statement, with the posing added there, hardness at the coverage-against-generalization clauses, the account at the Principle of Compositionality as the review names it, specifications withdrawn at the overly strict interpretation, approaches at grammar-based and neural models, and the relation at Reach: Donatelli and Koller, "Compositionality in Computational Linguistics", Annual Review of Linguistics 9:463–481, 2023, doi 10.1146/annurev-linguistics-030521-044439, at its abstract as the publisher's page and a university listing carry it verbatim (neural models greatly outperforming grammar-based models across many tasks; the question whether linguistic principles such as the Principle of Compositionality still have value as modeling tools; an overly strict interpretation making broad coverage in semantic parsing hard to achieve; compositionality still necessary for learning the correct generalizations from limited data; reconciling both requiring the exploration of a novel design space; recent results reviewed) and at a passage of its text as a repository copy carries it (end-to-end networks capturing aspects of meaning without relying on the principle), the review itself not fetched. The authors and pages given before this pass, Hupkes et al., 131–151, are withdrawn, the publisher's page carrying the authors and pages above. The hardness line's clause on benchmarks confounding memorization with composition, the specifications, the other approaches and progress stand keyed at the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the meta-learning paper at its abstract as the publisher's page, a database and an archive copy carry it verbatim (the systematicity challenge persisting; evidence that networks achieve human-like systematicity when optimized for compositional skills, in a head-to-head comparison with humans on an instruction-learning paradigm; advances on the systematic-generalization benchmarks), Nature 623(7985):115-121, 2023, doi 10.1038/s41586-023-06668-3, its authors carried; a dissenting analysis at its section as the arXiv copy carries it (a severe lack of compositionality revealed in that framework; how to better test and train for it), arXiv:2506.01820; a benchmark paper at its introduction (systematic generalization remaining a significant challenge; large models often failing to combine acquired knowledge in novel scenarios), arXiv:2504.01445; a review at its abstract as the publisher's page carries it (deep networks capable of subtle grammar-dependent generalizations yet not relying on systematic compositional rules), Philosophical Transactions B, 2020; their authors not carried except as given. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Formal semantics — exact composition for specified fragments with coverage costs. Construction and type-logical extensions — cover additional phenomena through richer inventories. Neural sequence models — high corpus coverage with variable systematic transfer. Synthetic compositional benchmarks — isolate recombination but simplify natural language. Controlled natural-language evaluations — narrow the gap at smaller scale. **Narrowing recorded.** Claims are reported by construction family, split design and type of generalization rather than as one undivided compositional score. **Progress.** Multiple generalization taxonomies, adversarial data splits and hybrid symbolic-neural systems stand. **Own record.** ## 230 The causes of long-run economic development **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What causal forces explain persistent differences in productivity and income across societies over the long run? In the field's own posing, the question is carried verbatim in eight words at the chapter's opening, "Why are some countries much poorer than others?", and the institutional account makes the empirical and theoretical case that differences in economic institutions are the fundamental cause of differences in economic development. **The field's account of its hardness.** Institutions, geography, human capital, technology and culture jointly evolve; outcomes can alter their proposed causes, historical instruments and natural experiments carry exclusion assumptions, and the small number of independent long-run histories permits multiple compatible decompositions. **Accounts.** Inclusive political and economic institutions secure investment and innovation. Geography, disease environment and resource endowment shape productivity and feasible institutions. Human capital and demographic change drive sustained technological adoption. Cultural norms and social organization alter cooperation, saving and innovation. Technology and state capacity coevolve through historically contingent feedback. **Specifications.** Development and persistence are measured comparably across societies and time — universal under comparative growth. Proposed causes precede outcomes or are identified through stated exogenous variation — universal under causal inference. Historical proxies retain the construct attributed to them — local. **Specifications withdrawn.** **Common requirement.** A causal account reproducing both divergence and persistence across historical cases. **Conserving relation.** Changes in the proposed cause predict corresponding long-run development under independent comparisons and counterfactual checks. **Conserving reach.** Societies with differing geography, colonization, institutions, cultures and initial income across centuries. **Reach.** The statement ranges over historical counterfactuals and interacting causal paths. Econometric designs observe realized histories and proxy variables. The field's chapter states that the empirical importance of institutions is documented at two quasi-natural experiments in history, the division of Korea and the European colonization of much of the world, and the framework illustrated with historical examples: the realized histories the designs observe and the fundamental cause sought across societies are not the same, the quasi-natural experiments standing between them. **Unreachability.** **Persistence.** Robust causal estimates that distinguish primary drivers and feedbacks across independent historical settings. **Addresses.** Statement, with the posing and the marked question added there, hardness at the clause that outcomes alter their proposed causes, the institutional account, the approach at quasi-natural experiments, and the relation at Reach: Acemoglu, Johnson and Robinson, "Institutions as a Fundamental Cause of Long-Run Growth", Handbook of Economic Growth, volume 1A, chapter 6, pp. 385–472, 2005, at its abstract as the working-paper pages and an economics index carry it verbatim (the empirical and theoretical case that differences in economic institutions are the fundamental cause of differences in development; the empirical importance documented at two quasi-natural experiments, the division of Korea and the European colonization of much of the world; institutions changing over time because prevailing institutions affect the distribution of resources and groups with de facto power strive to change political institutions; growth-encouraging institutions emerging where political institutions allocate power to groups with interests in broad-based property rights, constrain power-holders, and leave few rents; historical examples) and at its opening as an institutional copy carries it (the question why some countries are much poorer than others; neoclassical growth models explaining differences by factor accumulation), the chapter itself not fetched; the working paper carries the title as "the Fundamental Cause", the handbook as "a Fundamental Cause". The hardness line's other clauses, the specifications, the other approaches and progress stand keyed at the next pass. Competing accounts and approaches: Spolaore and Wacziarg, "How Deep Are the Roots of Economic Development?", Journal of Economic Literature 51:325–369, 2013, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the founding handbook chapter at its abstract and aims as the working paper's page and the authors' copies carry them verbatim (the empirical and theoretical case that institutional differences are the fundamental cause, documented at two quasi-natural experiments; a large number of areas where future theoretical and empirical work would be very fruitful), Acemoglu, Johnson and Robinson, Handbook of Economic Growth, chapter 6, 2005; a data story at its passage on identification as an institute's page carries it (correlation not implying causation, reverse causality and third factors such as geography or culture; the instrumental-variable approach from the colonization natural experiment the standard answer), 2024, its author not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Cross-country growth regression — broad reach with endogeneity and measurement limits. Historical instrumental variables — isolate variation under disputed exclusion restrictions. Natural experiments and border discontinuities — strengthen local identification with limited generalization. Persistence studies — link deep history to present outcomes through multiple possible channels. Structural models — close feedbacks under stronger specifications. **Narrowing recorded.** Individual studies narrow causal claims to populations reached by an instrument, discontinuity or historical episode. **Progress.** Institutions, disease environment, human capital, culture and historical state capacity each have replicated associations and locally identified effects in selected settings. **Own record.** ## 231 Cross-national measurement equivalence **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** When do scores from a survey or test measure the same latent construct with the same scale across countries, languages or groups? In the field's own posing, measurement invariance assesses the psychometric equivalence of a construct across groups or across time, noninvariance suggesting that the construct has a different structure or meaning to different groups, so that it cannot be meaningfully tested or construed across them, and its assessment is essential before mean differences or differential relations are tested. **The field's account of its hardness.** Translation, response style, sampling and cultural interpretation can change item thresholds and loadings; exact invariance is often rejected in large comparisons, while relaxing it may restore fit without fixing which comparisons remain meaningful. **Accounts.** Configural invariance is sufficient for comparing the pattern of a construct without comparing scores. Metric invariance permits comparison of relations while scalar invariance is required for latent means. Partial or approximate invariance permits comparison when a bounded subset of parameters differs. Alignment and Bayesian models estimate group variation without requiring exact equality of every item parameter. **Specifications.** The construct and item model are specified before group comparison — local. Sampling and translation differences are represented — universal under cross-national measurement. Each substantive comparison uses the level of invariance it requires — universal under measurement theory. **Specifications withdrawn.** Exact equality of every item parameter, withdrawn in partial and approximate models while the common-construct account is retained. **Common requirement.** A transformation under which observed responses support the intended between-group comparison of one construct. **Conserving relation.** The target comparison remains unchanged by group-specific measurement artifacts admitted by the model. **Conserving reach.** All groups, languages and items included in the claimed comparison. **Reach.** The statement ranges over a common latent scale. Instruments return item responses shaped jointly by construct and group-specific response processes. The field's review states that noninvariance suggests a construct has a different structure or meaning to different groups, so that it cannot be meaningfully tested or construed across them, and that invariance is to be assessed before any comparison: the item responses the instruments return and the common latent scale the statement ranges over are not the same, the invariance test standing between them. **Unreachability.** Without at least the invariance conditions required by a target comparison, the latent scale is not identified for that comparison under the measurement model. **Persistence.** A substantively justified model showing which comparisons remain identified and replicating across instruments and populations. **Addresses.** Statement, with the posing added there, hardness at the clause on which comparisons remain meaningful, unreachability at noninvariance leaving the construct untestable across groups, approaches at the confirmatory-factor and item-response frameworks, progress at the review of practice, and the relation at Reach: Putnick and Bornstein, "Measurement Invariance Conventions and Reporting: The State of the Art and Future Directions for Psychological Research", Developmental Review 41:71–90, 2016, doi 10.1016/j.dr.2016.06.004, at its abstract as the publisher's page and an index carry it verbatim (invariance assessing the psychometric equivalence of a construct across groups or time; noninvariance suggesting a different structure or meaning, so that the construct cannot be meaningfully tested or construed across groups or time; assessing invariance essential before testing mean differences or differential relations; practices reviewed in 126 articles with 269 tests; implications and research needs) and at a passage of its text as the publisher's page carries it (invariance testable in an item-response or a structural-equation framework, the review focusing on confirmatory factor analysis), the review itself not fetched. The hardness line's other clauses, the account, the specifications, the specification withdrawn and the other approaches stand keyed at the next pass. Violations and partial interpretation: Hahn-Klimroth, Dierkes and Kleespies, "An unsupervised learning approach to evaluate questionnaire data — what one can learn from violations of measurement invariance", arXiv:2312.06309, 2023, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the editors' introduction to a special issue at its text as the publisher's page and an archive copy carry it verbatim (the requirement of invariance very hard to meet; scalar invariance often not holding, even partial scalar invariance seldom achieved across many countries or time points; approximate invariance and alignment the new approaches), Sociological Methods and Research, 2018, doi 10.1177/0049124118789708, its editors carried; the field's review at passages as two repository copies carry them (true scalar equivalence difficult to find; analysing only a subset of countries a partial and unsatisfactory solution), Annual Review of Sociology 40:55-75, 2014; a comparison of exact and approximate tests at its abstract (the exact approach proving overly strict, the approximate establishing comparability), 2015. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Multigroup confirmatory factor analysis — tests nested equality conditions with fit and sample-size sensitivity. Item-response differential-functioning analysis — localizes item differences under an anchor set. Partial invariance — frees selected parameters and conditions inference on the remaining anchors. Approximate and alignment methods — estimate small deviations at the cost of new priors or tolerances. **Narrowing recorded.** Mean, covariance and regression comparisons are licensed separately at scalar, metric and configural levels; partial models narrow the invariant item set. **Progress.** Invariance hierarchies, simulation-based error analyses and large-scale alignment methods are established. **Own record.** ## 232 The peopling of the Americas **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** When, by which routes and in how many population movements did humans first enter and spread through the Americas? In the field's own posing, current genetic and archaeological evidence allows inland, coastal or multiple pathways, the data largely consistent with either an inland route through the ice-free corridor or a Pacific coastal route, or both, and neither rejected at present. **The field's account of its hardness.** That the precise time and mode of the colonization remain, in the field's words, hotly disputed, the current genetic, archaeological and paleoecological evidence largely consistent with either an inland or a Pacific coastal route or both, neither rejected at present, systematic investigation informed by geomorphology being required to test each; that with postglacial sea-level rise inundating most paleoshorelines older than about seven thousand years, locating terminal Pleistocene sites along modern coastlines is challenging, the postulate of travel along a paleoshoreline now hidden underwater being an attractive means to posit early movement without leaving traces onshore, while geological analyses of the potential route raise questions not fully addressed and misconceptions remain about the state of knowledge, productivity and deglaciation chronology of Pleistocene coastlines; and that the search for older sites and the ability to recognize them is described as difficult but essential, accurate dating at its centre, with multiple working hypotheses warranted. **Accounts.** A population moved from Beringia southward through an interior ice-free corridor after the continental ice sheets separated. People moved along the North Pacific coast before the interior corridor became viable. More than one movement or prolonged Beringian population history contributed to the founding populations. Claims of substantially earlier entry represent either a real pre-last-glacial presence or misidentified context and artifacts. **Specifications.** Artifact manufacture and depositional context are established — local. Site ages date human activity rather than redeposition — local. Genetic models include population structure, bottlenecks and later gene flow — universal under demographic inference. **Specifications withdrawn.** The specification that the earliest widespread fluted-point complex marks the first entry, withdrawn while it remains an established later archaeological horizon. **Common requirement.** A chronology and route model consistent with securely dated sites, viable landscapes and genomic ancestry. **Conserving relation.** One population history returns the archaeological dates, paleogeography and genetic branching and diversity patterns. **Conserving reach.** Beringia, coastal and interior routes, North and South America, and populations before and after the earliest secure widespread record. **Reach.** The statement ranges over migrations across landscapes partly submerged or glaciated. The record samples preserved sites and descendant genomes. The field's review states that the current genetic, archaeological and paleoecological data are largely consistent with either an inland or a Pacific coastal route, or both, neither rejected at present, and that systematic investigation informed by geomorphology is required to test each: the sites and genomes the record samples and the routes sought are not the same, the two hypotheses both standing on one record. **Unreachability.** **Persistence.** Multiple independently dated early sites with diagnostic human traces and a demographic model fixing route and movement structure. **Addresses.** Statement, with the posing added there, the accounts at the inland and coastal routes, specifications withdrawn at the pre-Clovis expansion, progress at the expansion after sixteen thousand years ago and the earliest securely dated sites after about fifteen thousand, and the relation at Reach: Potter et al., "Current evidence allows multiple models for the peopling of the Americas", Science Advances 4:eaat5473, 2018, doi 10.1126/sciadv.aat5473, at its abstract as the publisher's page, an index and two university listings carry it verbatim (recent literature implying the problem largely resolved in favor of a Pacific coastal migration thousands of years before Clovis, the earliest widespread cultural manifestation south of the ice; speculations on maritime adaptations and typological links; a review of the genetic, archaeological and paleoecological evidence indicating ancestral Native American population expansion after sixteen thousand years ago, consistent with the earliest securely dated sites after about fifteen thousand; the data largely consistent with either an inland or a coastal route, or both, neither rejected; systematic investigation required), the paper itself not fetched. The specifications and the approaches stand keyed at the next pass. New coastal constraints: Steffen, "New age constraints for human entry into the Americas on the north Pacific coast", Scientific Reports 14:4291, 2024, cited here and not read this pass. Hardness: re-formed to the field's words at the paper the address names, at its abstract as the publisher's page, an index, two university listings and a university repository carry it verbatim (neither route rejected; systematic investigation required), and at five sources returned in the search, none fetched: a 2008 genetics paper at its abstract and opening as an archive copy carries them (the precise time and mode of the colonization hotly disputed), American Journal of Human Genetics 82(3):583-592, doi 10.1016/j.ajhg.2007.11.013, its authors carried at the copy; a 2019 paper at its abstract as a directory carries it (postglacial sea-level rise inundating most paleoshorelines older than about seven thousand years; locating terminal Pleistocene sites along modern coastlines challenging), Open Quaternary 5(1), doi 10.5334/oq.66, its authors carried; a 2020 forum at its abstract as a funding agency's copy carries it (misconceptions remaining about the state of knowledge, productivity and deglaciation chronology of Pleistocene coastlines; much remaining to be learned, multiple working hypotheses warranted), American Antiquity, its authors carried; a 2017 paper at its abstract as the publisher's page carries it (a paleoshoreline now hidden underwater an attractive means to posit early movement without traces onshore; geological analyses raising questions not fully addressed), Quaternary Research, its authors not carried; a 2014 review essay at a database listing (the difficult but essential search for and ability to recognize older sites; accurate dating); the former line, in the file's compression (The earliest sites are rare and contested, coastal evidence was submerged by sea-level rise, genetic dates depend on demographic and mutation models, and archaeological, geological and genomic clocks resolve different stages of entry and dispersal.), released with its content at the three clauses, its content on genetic dates depending on demographic and mutation models, released from the line, standing for the next pass. Persistence, no landing: the closing condition at the paper's abstract and the papers read at Reach (systematic investigation of each route informed by geomorphology; the difficult but essential search for and recognition of older sites; accurate dating). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Stratified archaeology — supplies direct behavior with context and artifact disputes at the oldest sites. Radiocarbon and luminescence dating — bound activity under sample-association requirements. Ancient and modern genomics — reconstruct branching and population size under demographic assumptions. Paleogeographic route modelling — dates corridor and coastal viability with incomplete landscape evidence. **Narrowing recorded.** Secure pre-fluted-point sites withdraw a single late-first-entry model; route viability and genomic divergence bound families of coastal and interior histories. **Progress.** A late-glacial presence predating the classic fluted-point horizon is established at multiple sites, Beringian ancestry is genomic, and both route environments are increasingly dated. **Own record.** ## 233 The origins of agriculture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why and through what processes did human populations in several regions shift from foraging to sustained cultivation and domestication? In the field's own posing, agriculture is one of the key innovations of human societies while the nature of and the reasons for its emergence are debated, the degree to which environmental and social conditions enabled its origins remaining contested among a surplus hypothesis, a necessity hypothesis and a regional-uniqueness hypothesis. **The field's account of its hardness.** Agriculture emerged independently through long, regionally different sequences; climate, population, risk, social organization and plant ecology covary, and archaeological indicators of management precede fully domesticated morphology by different intervals. **Accounts.** Postglacial climate improved plant productivity and population density until cultivation became advantageous. Resource pressure and demographic growth raised returns to intensified food production. Risk reduction, storage and seasonal scheduling favored management before population pressure. Competition, feasting, property and political organization increased demand for controllable surplus. Coevolution between human practices and responsive species produced gradual domestication without one initiating cause. **Specifications.** Cultivation, domestication and dependence are dated and distinguished — universal under agricultural origins. Environmental and demographic proxies represent local conditions before the transition — local. Independent centers are not forced into one causal sequence — universal under comparative inference. **Specifications withdrawn.** A single abrupt revolution from foraging to farming, withdrawn where protracted cultivation and domestication are retained. **Common requirement.** A causal sequence connecting environmental and social conditions to repeated management, genetic domestication and subsistence dependence. **Conserving relation.** The proposed process predicts the timing and order of cultivation, domestication traits and settlement in independent regions. **Conserving reach.** Independent centers, plant and animal taxa, mobile and settled populations, and protracted transition intervals. **Reach.** The statement ranges over motives, decisions and ecological feedback through long transitions. Archaeology observes preserved plants, animals, tools, settlements and environments. The field's model states that the degree to which environmental and social conditions enabled the origins remains contested, and tests three hypotheses by a statistical model of population density among 220 foraging societies hindcast across the world: the densities modelled and the origins sought are not the same, the hindcast standing between them. **Unreachability.** **Persistence.** Regionally tested causal histories whose comparative predictions distinguish common mechanisms from local sequences. **Addresses.** Statement, with the posing added there, hardness at the contested degree and the regional-uniqueness hypothesis, the account at the surplus hypothesis, the approach at the statistical hindcast, and the relation at Reach: Piperno, "A model of agricultural origins", Nature Human Behaviour 2:446–447, 2018, doi 10.1038/s41562-018-0390-8, a commentary, at its summary as the publisher's page carries it verbatim (agriculture one of the key innovations of human societies, the nature of and reasons for its emergence debated; a new model hindcasting past global population suggesting that an improving climate increased plant productivity and population density, facilitating domestication), and Kavanagh et al., "Hindcasting global population densities reveals forces enabling the origin of agriculture", Nature Human Behaviour 2, 2018, the paper the commentary concerns, at its abstract as the publisher's page carries it verbatim (the degree to which environmental and social conditions enabled the origins contested; three hypothesized links tested, surplus, necessity and regional uniqueness; a statistical model in which environmental, geographic and cultural variables capture 77 percent of the variation in population density among 220 foraging societies), its pages not carried at the page read; neither paper fetched. The commentary's own hardness line, the two other hypotheses beside the account, the specifications, the other approaches and progress stand keyed at the next pass. Larson et al., "Current perspectives and the future of domestication studies", Proceedings of the National Academy of Sciences 111:6139–6146, 2014, and Zeder, "The Origins of Agriculture in the Near East", Current Anthropology 52:S221–S235, 2011, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the special feature's introduction at its abstract and text as the journal's page, a database and three copies carry them verbatim (fundamental questions where, when and how many times domestication took place; controversies over the speed, intentionality and evolutionary aspects; at least eleven independent centers; multiple domestications of a species to be evaluated carefully; three key challenges for future research), Proceedings of the National Academy of Sciences 111(17):6139-6146, 2014, doi 10.1073/pnas.1323964111, twenty-five authors; a botanical review at a passage as the publisher's page carries it (debates about centers continuing so spiritedly because the question cannot be separated from the single-or-multiple, the pace, and the decoupling of domestication from agricultural economies), 2014, its author not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Archaeobotanical trait sequences — date morphological domestication after management begins. Settlement and demographic reconstruction — tracks population under preservation and proxy limits. Paleoclimate comparison — tests productivity and stress timing regionally. Optimal-foraging and cultural-evolution models — generate transitions under specified incentives and transmission. Ancient genomics — reconstructs domestication and introgression after the initial practices. **Narrowing recorded.** Cultivation, genetic domestication and agricultural dependence are separated in time; individual regions carry different sequences rather than one global date. **Progress.** Multiple independent centers, protracted domestication and recurrent gene flow between domesticates and wild relatives are established. **Own record.** ## 234 The emergence of the genus Homo **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which fossils belong to the earliest genus Homo, from which populations did they arise, and what ecological and developmental changes produced them? In the field's own posing, the traits once thought to have evolved together near the origin of the genus in response to aridity and open habitats did not arise as a single package, some arising substantially earlier and some later than thought, three lineages of early Homo evolving between about 2.5 and 1.5 million years ago in a context of habitat instability and fragmentation. **The field's account of its hardness.** The earliest record is sparse, fragmentary and variable; diagnostic traits do not appear as one package, genus boundaries depend on comparative criteria, and dates, environments and behaviors are unevenly associated with the fossils. **Accounts.** A recognizable early species marks a branching transition from an australopith ancestor. Early named species form a single highly variable evolving lineage. Several contemporaneous lineages exchanged traits or represent a bush of experiments near the genus boundary. Dietary, locomotor, developmental and technological changes accumulated mosaically rather than in one adaptive shift. **Specifications.** Genus assignment uses declared morphological and phylogenetic criteria — local. Fossils are dated and associated with environmental and archaeological contexts — local. Variation due to sex, age and geography is separated from lineage difference — universal under taxonomy. **Specifications withdrawn.** A single abrupt coappearance of large brain, humanlike body and stone technology, withdrawn while each trait remains part of later Homo evolution. **Common requirement.** A phylogenetic and functional sequence explaining the mosaic transition and observed fossil variation. **Conserving relation.** One classification and history assigns the fossil sample consistently and predicts the ordering of derived traits. **Conserving reach.** African and Eurasian fossils and archaeological contexts spanning late australopiths through early Homo. **Reach.** The statement ranges over populations and ancestry. The record consists of incomplete individuals, isolated localities and inferred environments. The field's review states that the integration of fossil, archaeological and environmental evidence has revised the timing of the traits, which did not arise as one package, and that opinions diverged on the generic status of the earlier, more fragmentary fossils: the fragments and localities the record supplies and the populations and ancestry sought are not the same, the generic criteria standing between them. **Unreachability.** **Persistence.** Fossils with associated anatomy and context sufficient to resolve genus membership, branching and the causal trait sequence. **Addresses.** Statement, with the posing added there, hardness at the traits not arising as one package and at the diverging generic status of fragmentary fossils, specifications withdrawn at the single package, the approach at paleoenvironmental reconstruction, progress at the three lineages and the revised trait timing, and the relation at Reach: Antón, Potts and Aiello, "Evolution of early Homo: An integrated biological perspective", Science 345:1236828, 2014, doi 10.1126/science.1236828, at its abstract as the publisher's page, two repositories and an index carry it verbatim (integration of evidence revising the major adaptations underlying the origin of the genus; features once thought to have evolved near the origin in response to aridity and open habitats; the traits not arising as a single package, some earlier and some later; three lineages of early Homo from about 2.5 to 1.5 million years ago in habitat instability and fragmentation), at the publisher's summary (hundreds of fossils from well-dated sites; stratigraphy, dust, cores and isotopes improving environmental reconstruction; habitat instability an important selective force) and at a passage of its text as an index carries it (opinions diverging on the generic status of the earlier, more fragmentary fossils), the review itself not fetched. The hardness line's other clauses, the account as posed, the specifications and the other approaches stand keyed at the next pass. Antón and Snodgrass, "Origins and Evolution of Genus Homo: New Perspectives", Current Anthropology 53:S479–S496, 2012, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the review the field titles for a transition that was not, at its abstract and opening as the publisher's page, a database and an index carry them verbatim (the fossil record bearing on the origin and earliest evolution of the genus virtually undocumented; the poles of the transition attached to taxa remote in time; the frustration from critical gaps in the record), Philosophical Transactions B 371(1698):20150248, 2016, doi 10.1098/rstb.2015.0248, its authors carried; a review of early members of the genus at a passage as the publisher's page carries it (the material known from before two million years too fragmentary for a meaningful assessment of species diversity), American Journal of Physical Anthropology, 2018, doi 10.1002/ajpa.23387, its author carried; a symposium's abstracts (many aspects murky and not resolvable with the available data; no clear way to define the genus let alone its species; the record notoriously poor), their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Morphometric classification — compares preserved anatomy under small, incomplete samples. Phylogenetic analysis — returns trees conditional on character coding and taxon assignment. Geochronology and paleoenvironment — order fossils and ecology with locality gaps. Archaeology — traces behavior without secure maker assignment at many sites. Developmental reconstruction — infers life history from limited tissues. **Narrowing recorded.** Mosaic traits replace a single diagnostic package; specimens and hypotheses are increasingly restricted by improved dates and quantitative shape comparison. **Progress.** Earlier and more variable Homo-like fossils, refined chronologies and evidence of overlapping lineages stand. **Own record.** ## 235 The causal production of political polarization **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Which mechanisms cause ideological, partisan and affective polarization, and under what conditions does each rise or fall? In the field's own posing, elite polarization has been discussed for decades while mass polarization received comparable attention only recently, much of the evidence used to study it presenting problems of inference that render conclusions problematic, the most direct evidence showing little or no increase in mass polarization while party sorting has clearly occurred. **The field's account of its hardness.** The term covers distinct distributions of policy positions, identities and out-group feeling; elites, media, institutions and citizens influence one another, observational trends admit selection and reverse causation, and measures can show polarization on one dimension without another. **Accounts.** Elite sorting and partisan cues align citizen attitudes and identities. Social sorting merges political identity with race, religion, geography and culture, increasing affective division. Selective exposure and media incentives amplify conflict and perceived extremity. Electoral institutions, geographic sorting and activist selection reward polarized competition. Economic and cultural change generates substantive conflict that parties organize. **Specifications.** Ideological position, partisan sorting and affective hostility are measured separately — universal under the problem. Causes precede outcomes or use stated exogenous variation — universal under causal inference. Population and political system are declared — local. **Specifications withdrawn.** One undivided polarization scale, withdrawn while its component measures are retained. **Common requirement.** A mechanism predicting change in a specified polarization dimension under intervention or independent historical variation. **Conserving relation.** The proposed cause changes the declared dimension in the predicted direction across independent settings. **Conserving reach.** Citizens, elites, parties and media across the political systems and periods claimed. **Reach.** The statement ranges over interacting causal feedbacks. Surveys, elections, media traces and experiments sample selected levels and intervals. The field's survey states that much of the evidence used to study mass polarization presents problems of inference that render conclusions problematic, the most direct evidence showing little or no increase while party sorting has clearly occurred: the trends the surveys return and the causes sought are not the same, the problems of inference standing between them. **Unreachability.** **Persistence.** Replicated causal decompositions distinguishing dimensions, directions and feedback across political systems. **Addresses.** Statement, with the posing added there, hardness at the problems of inference and at polarization on one dimension without another, specifications withdrawn at the concept parted from its measures, narrowing recorded at policy positions parted from sorting, approaches at the kinds of evidence considered, progress at party sorting established, and the relation at Reach: Fiorina and Abrams, "Political Polarization in the American Public", Annual Review of Political Science 11:563–588, 2008, doi 10.1146/annurev.polisci.11.053106.153836, at its abstract as the publisher's page, a preprint archive and an index carry it verbatim (elite polarization discussed for two decades, mass polarization studied only recently; the concept discussed, then the kinds of evidence critically considered, much of it presenting problems of inference; citizens' policy positions showing little or no increased mass polarization over two to three decades; party sorting, an increased correlation of policy views with partisan identification, clearly occurred and sometimes exaggerated), the survey itself not fetched. The hardness line's clauses on mutual influence and on selection and reverse causation, the account as posed, and the specifications stand keyed at the next pass. Affective mechanisms, accounts and progress: Iyengar et al., "The Origins and Consequences of Affective Polarization in the United States", Annual Review of Political Science 22:129–146, 2019, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the field's review at passages as the publisher's copy and a university copy carry them verbatim (research into the consequences only beginning; more research needed on how the factors play out in different political contexts; no research identified showing the affective form driving both elite and mass ideological polarization; ongoing disagreement over the extent of ideological polarization), Annual Review of Political Science 22:129-146, 2019, doi 10.1146/annurev-polisci-051117-073034, its authors carried; a nine-democracy experiment at its discussion as the publisher's page carries it (the causal relation remaining contested; causal evidence scarce outside one country; interventions changing out-party feeling reducing avoidance but not support for norms), European Journal of Political Research, 2026, its authors not carried; a modelling paper's literature review (research converging on a multicausal phenomenon; sorting against movement to the extremes), arXiv:2602.08160. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Repeated surveys — measure trends with question and scale dependence. Field and survey experiments — identify short-run cue, contact and media effects. Natural experiments — exploit institutional and media changes with local reach. Network and platform data — measure exposure and behavior under selection and access limits. Comparative politics — tests institutional variation with measurement equivalence requirements. **Narrowing recorded.** Ideological divergence, issue consistency, partisan sorting and affective hostility are entered as different outcomes. **Progress.** Elite polarization, partisan sorting and rising affective hostility are well measured in the United States; causal effects of selected cues, contacts and media exposures stand locally. **Own record.** ## 236 The core-collapse supernova explosion mechanism **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How does the stalled shock in a massive star's collapsing core revive to produce the observed range of supernova explosions and compact remnants? In the field's own posing, the neutrino-heating mechanism aided by nonradial flows drives explosions, albeit low-energy ones, of oxygen-neon-magnesium-core and some iron-core progenitors, while simulations suggest that neutrino-powered explosions might not explain the most energetic supernovae and hypernovae, which seem to demand magnetorotational driving, answers to the long-standing questions coming into reach as modeling advances from two to three dimensions. **The field's account of its hardness.** Neutrino transport, turbulence, nuclear matter, rotation and magnetic fields interact in three dimensions over extreme scales; explosion outcome varies nonmonotonically with progenitor structure, and simulations must follow both shock revival and observable nucleosynthesis and remnant properties. **Accounts.** Neutrino heating aided by multidimensional convection and shock instability revives the shock in ordinary events. Rapid rotation and strong magnetic fields power jets and the most energetic events. Acoustic, phase-transition or other additional processes contribute in restricted progenitors. The mechanism is a progenitor-dependent mixture rather than one channel for all core collapses. **Specifications.** Radiation-hydrodynamic evolution conserves energy and lepton number at stated numerical accuracy — universal under simulation. The nuclear equation of state and neutrino interactions are specified — local. The outcome reproduces explosion energy, nucleosynthesis, neutrino signal and remnant — universal under the mechanism. **Specifications withdrawn.** Spherical symmetry as sufficient for ordinary neutrino-driven explosions, withdrawn while spherical calculations remain baselines. **Common requirement.** A physically complete instability and energy-transfer sequence that turns infall into an outgoing explosion. **Conserving relation.** One simulation framework predicts explosion or failure and the associated observables from independently specified progenitors. **Conserving reach.** The massive-star progenitor distribution, ordinary and energetic explosions, failed explosions and remnant types. **Reach.** The statement ranges over three-dimensional stellar collapses. Simulations discretize finite domains and observations sample explosions, remnants and one nearby neutrino event. The field's review states that the neutrino-heating mechanism drives low-energy explosions of some progenitors in the simulations, that the most energetic events seem to demand magnetorotational driving, and that modeling was then being advanced from two to three dimensions with answers coming into reach: the explosions the simulations produce and the three-dimensional collapses the statement ranges over are not the same, the dimension of the modeling standing between them. **Unreachability.** **Persistence.** Predictive three-dimensional calculations across progenitors, validated jointly by neutrinos, gravitational waves, nucleosynthesis and remnant demographics. **Addresses.** Statement, with the posing added there, hardness at the interacting transport, microphysics and instabilities and at the dependence on progenitor, the account at neutrino heating aided by nonradial flows, specifications withdrawn at spherical symmetry, approaches at multidimensional neutrino-transport simulation and magnetorotational driving, progress at the low-energy explosions and the revised neutrino, gravitational-wave and nucleosynthesis pictures, and the relation at Reach: Janka, "Explosion Mechanisms of Core-Collapse Supernovae", Annual Review of Nuclear and Particle Science 62:407–451, 2012, doi 10.1146/annurev-nucl-102711-094901, at its abstract as the publisher's page and the arXiv page carry it verbatim (remarkable progress from sophisticated neutrino-transport tools, improved microphysics and insight into hydrodynamic instabilities; violent nonradial mass motions generic in supernova cores; neutrino heating aided by nonradial flows driving low-energy explosions of oxygen-neon-magnesium-core and some iron-core progenitors; neutrino emission revised, gravitational-wave features discovered, the notion of supernova nucleosynthesis shattered, kicks and asymmetries better understood; the most energetic events seeming to demand magnetorotational driving; modeling advancing from two to three dimensions), the review itself not fetched. The hardness line's clause on nonmonotonic outcome and on following nucleosynthesis and remnants, and the specifications, stand keyed at the next pass. Current simulation standing and progress: Burrows and Vartanyan, "Core-collapse supernova explosion theory", Nature 589:29–39, 2021, cited here and not read this pass; a later review by the first author in the same journal, volume 75, 2025, returned at its citation only, a reading for the next pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the field's review at its summary as the publisher's page carries it (the delayed neutrino-heating mechanism emerging as the key driver, many issues remaining, among them the chaos of the dynamics), Nature 589:29-39, 2021, doi 10.1038/s41586-020-03059-w, its authors carried at citing lists; a review of the mechanism at its abstract and introduction as the arXiv copy carries them (a long-standing problem; demonstrating quantitatively that shock revival can be achieved a notoriously hard problem requiring multi-physics simulations), arXiv:1702.06940, its author carried; a 2024 status report at its abstract and introduction (different groups obtaining different results; a three-dimensional code comparison missing and highly desirable; the neutrino mechanism argued short of the observed energies), arXiv:2411.08555, its author carried; a gravitational-wave paper at its introduction (neutrinos and gravitational waves the diagnostics that see into the obscured core), arXiv:0907.4762; a proceedings paper's opening (no final consensus on the mechanism operating in the obscured core), arXiv:1112.1913. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Multidimensional neutrino-radiation hydrodynamics — produces explosions for growing progenitor sets with code and input variation. Parametric heating — maps thresholds without closing first-principles transport. Magnetorotational simulation — produces jets under strong initial rotation and fields. Nucleosynthetic and remnant comparison — constrains integrated outcomes. Neutrino and gravitational-wave observation — discriminating for a future nearby event. **Narrowing recorded.** Progenitor compactness and structure narrow explodability regimes; rotation and field strength narrow magnetorotational channels. **Progress.** Self-consistent three-dimensional neutrino-driven explosions, instability mechanisms and observable multimessenger predictions stand for selected progenitors. **Own record.** ## 237 Sex bias in autoimmune disease **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What mechanisms produce the large and disease-specific differences in autoimmune incidence, severity and manifestations between the sexes? In the field's own posing, it has long been known that many autoimmune diseases are more common in females, and although much progress has been made the exact mechanisms for this sex bias are not yet fully understood, chromosomes, genes, sex hormones, immunometabolism and the skin each playing a role. **The field's account of its hardness.** Sex chromosomes, hormones, pregnancy history, immunity, microbiota and social exposures covary and change across life; female bias differs by disease and ancestry, and animal or cellular mechanisms do not automatically explain population incidence. **Accounts.** X-chromosome dosage, escape from inactivation and mosaicism alter immune regulation. Sex hormones reshape innate and adaptive immune thresholds across life stages. Pregnancy, fetal microchimerism and reproductive transitions change long-term immune states. Sex-dependent microbiota, metabolism and environmental exposure modify immune risk. Different mechanisms dominate different autoimmune diseases and clinical phases. **Specifications.** Incidence, severity, subtype and treatment response are distinguished — universal under the question. Chromosomal, hormonal and social sex-related variables are measured separately where possible — universal under causal analysis. Life stage and ancestry are represented — universal under population inference. **Specifications withdrawn.** A single female-hormone explanation for every disease, withdrawn in multifactorial accounts retaining hormonal effects. **Common requirement.** A mechanism linking sex-related biology or exposure to loss of immune tolerance and disease-specific outcome. **Conserving relation.** Manipulation or natural variation in the proposed factor predicts the immune phenotype and disease difference independently. **Conserving reach.** Female- and male-biased diseases, life stages, ancestries and clinical outcomes. **Reach.** The statement ranges over causal pathways across life. Cohorts, animal models and cells observe different factors and endpoints. The two are not the same: the risk of autoimmunity linked to female sex is substantially larger than any susceptibility gene discovered, its mechanisms remaining, in the field's words, poorly understood, and the candidate factors, the second X chromosome and its incomplete inactivation, the X-inactive transcript's complexes, sex hormones, immunometabolism and the microbiota, are each observed in different systems, mice engineered to express the transcript, patient sera and cohorts, two recent studies reaching the same transcript by different hypotheses and different proposed mechanisms. **Unreachability.** **Persistence.** Disease-specific causal decompositions that predict incidence and course and are confirmed by intervention or natural experiment. **Addresses.** Statement, with the posing added there, hardness at the several factors each playing a role, the account at the chromosomal factor among those named, specifications withdrawn at the several factors beside hormones, the approach at gender-affirming hormone therapy as a natural experiment, and progress at the proportion of patients who are women and the diseases with the strongest bias: van Drongelen, Rew and Billi, "Sex Bias in Autoimmunity: New Findings and New Opportunities", JID Innovations 5:100391, 2025, doi 10.1016/j.xjidi.2025.100391, at its abstract as the publisher's page and a repository carry it verbatim (autoimmune diseases from the immune system's inability to discriminate self from foreign; many more common in females; much progress and the exact mechanisms of the bias not yet fully understood; an update on chromosomes, genes, sex hormones including gender-affirming hormone therapy, immunometabolism and the skin; gaps identified) and at a passage of its text as the journal's copy carries it (three quarters to four fifths of autoimmune patients women; rheumatoid arthritis and systemic lupus erythematosus with particularly high female predominance), the review itself not fetched. No Reach relation is stated at the abstract read. The hardness line's other clauses, the specifications and the other approaches stand keyed at the next pass. The relation at Reach: an opinion article at its opening as an archive copy and the journal's page carry it verbatim (the risk linked to female sex substantially larger than any susceptibility gene; the mechanisms poorly understood; two studies reaching the transcript by different hypotheses, their proposed mechanisms different), Frontiers in Immunology, 2024, doi 10.3389/fimmu.2024.1399408, its author carried as Andrade; a review at its abstract and a passage as an archive copy and the publisher's page carry them (the exact mechanisms not yet fully understood; chromosomes, genes, hormones, immunometabolism and the skin; male mice expressing the transcript prone to autoimmunity), 2025, doi 10.1016/j.xjidi.2025.100391, its authors carried as van Drongelen, Rew and Billi; an earlier review at its opening (female sex a risk that dwarfs any susceptibility locus), its authors not carried; none fetched. Persistence, no landing: the closing condition in part at the reviews read at Reach (the mechanisms poorly understood, the candidate factors observed in different systems), the disease-specific decompositions keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Epidemiology across life stages — maps sex ratios with diagnostic and exposure differences. Sex-chromosome aneuploidy and mosaic studies — separate dosage from hormones in rare populations. Hormonal and reproductive natural experiments — identify selected effects with confounding. Animal models — perturb pathways with species and disease-transfer limits. Single-cell and epigenetic studies — resolve sex-linked immune states without causal closure. **Narrowing recorded.** Analyses partition chromosomal complement, gonadal hormones, life stage and disease subtype rather than treating sex as one exposure. **Progress.** X-inactivation escape, hormone-sensitive immune pathways and disease-specific age patterns are established. **Own record.** ## 238 The origins of fast radio bursts **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What sources and emission mechanisms produce repeating and apparently nonrepeating fast radio bursts? In the field's own posing, the origins of fast radio bursts are still unknown and the source class is defined only observationally, most bursts appearing one-off and perhaps cataclysmic while repeating sources clearly have a longer-lived central engine, the many proposed theories of origin awaiting the wide-field instruments to differentiate them. **The field's account of its hardness.** Millisecond bursts are often detected without counterparts, propagation through local plasma alters their spectra and polarization, activity is intermittent, and the observed population spans host environments and energetics that may require more than one channel. **Accounts.** Magnetar magnetospheres produce coherent radio emission during magnetic reconnection or crustal activity. Relativistic shocks from magnetar flares produce coherent maser emission outside the star. Compact-object interactions, accretion or mergers produce a subset of catastrophic or repeating events. Multiple source classes contribute to the observed population. **Specifications.** The mechanism produces the brightness temperature, duration, bandwidth, polarization and repetition pattern — universal under burst emission. Propagation effects are separated from source effects — universal under inference. Source rates and energetics agree with host populations — universal under population models. **Specifications withdrawn.** A single homogeneous source environment for all bursts, withdrawn in population accounts retaining a shared observable class. **Common requirement.** A source and coherent-emission process consistent with event-level and population-level observations. **Conserving relation.** The same mechanism predicts burst properties, counterparts, repetition and source rate for the population it claims. **Conserving reach.** Repeaters and one-off detections across host types, energies, activity cycles and local plasma environments. **Reach.** The statement ranges over compact sources and microscopic emission. Telescopes measure propagated radio signals and occasional multiwavelength counterparts and localizations. The field's review states that the source class is at present defined only observationally, and that propagation effects, strongly frequency dependent, must be disentangled from the intrinsic properties of the signal: the propagated signals the telescopes measure and the sources and emission sought are not the same, the propagation standing between them. **Unreachability.** **Persistence.** Source identification and mechanism-specific predictions confirmed across a defined burst population, with additional channels fixed where required. **Addresses.** Statement, with the posing added there, hardness at the propagation clause, specifications withdrawn at the one-off and repeating classes, and the relation at Reach: Petroff, Hessels and Lorimer, "Fast radio bursts", Astronomy and Astrophysics Review 27:4, 2019, doi 10.1007/s00159-019-0116-6, at its abstract as the publisher's page, the arXiv page and an index carry it verbatim (bursts discovered serendipitously in pulsar surveys; their large dispersive delays placing them far outside the Milky Way and many orders of magnitude more luminous than pulsar pulses; most appearing one-off and perhaps cataclysmic, two sources then known to repeat with a longer-lived central engine; a detectable burst roughly once a minute somewhere on the sky and fewer than a hundred found for want of field of view; wide-field instruments critical for differentiating the many proposed theories of origin) and at passages of its text as the publisher's copy carries them (the origins still unknown and the class defined only observationally; propagation effects strongly frequency dependent, to be disentangled from the intrinsic signal), the review itself not fetched. The account at the magnetar link, approaches at high time resolution, sustained monitoring and precision localization, and progress at the population counts: the same authors, "Fast radio bursts at the dawn of the 2020s", Astronomy and Astrophysics Review 30:2, 2022, at its abstract as the arXiv page carries it verbatim (over six hundred published sources, twenty-four repeaters and nineteen host galaxies; insight into repeaters, hosts, morphology, activity and progenitor models; a burst-like event from a Galactic magnetar linking bursts and magnetars), its volume and article number taken from the publisher's listing, the review itself not fetched. The hardness line's other clauses, the specifications and the other approaches stand keyed at the next pass. Current source diversity: Pearlman et al., "Multiwavelength constraints on the origin of a nearby repeating fast radio burst source in a globular cluster", Nature Astronomy 9:111–127, 2025, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the field's review of mechanisms at its summary as the publisher's page carries it (the Galactic detection suggesting magnetar engines can produce at least some, and probably all, bursts), Nature 587:45-53, 2020, doi 10.1038/s41586-020-2828-1, its author not carried; a persistent-source paper at its introduction (the origins and the immediate magneto-ionic environments remaining poorly understood despite detection and localization), arXiv:2504.01125; a statistical study at its introduction as the publisher's page carries it (the trigger mechanism still unclear; about four per cent confirmed repeaters), Monthly Notices, 2025; an irregularity study and a microshot study at their introductions (the vast majority detected only once, limiting constraints on progenitors; whether repeaters and non-repeaters share progenitors presently unclear), arXiv:2512.10249, arXiv:2308.12118; a burst-astrophysics review (the progenitor of the majority currently unknown), arXiv:2410.12105; their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Interferometric localization — identifies hosts and local environments for a growing subset. Repetition monitoring — measures activity distributions under sensitivity and cadence selection. Polarimetry and dynamic spectra — constrain magnetic and plasma geometry with propagation degeneracy. Multiwavelength coincidence — strongly constrains energetic counterparts, usually by null results. Galactic magnetar comparison — establishes that magnetars can produce burst-like radio events. **Narrowing recorded.** Host localization and repetition separate population subgroups; energy and counterpart limits restrict catastrophic channels. **Progress.** Cosmological distances, coherent emission, diverse hosts, repeaters, extreme local magnetoionic environments and a Galactic magnetar association are established. **Own record.** ## 239 Magnetospheric substorm triggering **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What physical instability triggers magnetospheric substorm onset, and in what causal order do near-Earth current disruption and magnetotail reconnection occur? In the field's own posing, it has been unclear whether a substorm is triggered by a disruption of the electrical current flowing across the near-Earth magnetotail at about ten Earth radii or by magnetic reconnection farther out at twenty to thirty, the multipoint observations reported as demonstrating that tail reconnection likely initiates substorms and a comment providing evidence that near-Earth current disruption came first. **The field's account of its hardness.** Onset evolves across widely separated regions faster than sparse spacecraft can establish timing; auroral, current, plasma-flow and reconnection signatures are observed with different cadences, and candidate processes can trigger or follow one another in simulations. **Accounts.** Reconnection begins in the mid-tail and earthward flows trigger near-Earth current disruption and auroral onset. A near-Earth plasma instability disrupts the cross-tail current and subsequently drives reconnection farther down the tail. Alfvénic coupling and ballooning or interchange instability coordinate a distributed onset without one universal first location. External solar-wind changes directly trigger some events while internally stored energy triggers others. **Specifications.** Onset time is defined independently for auroral, flow, current and reconnection observables — local. Spacecraft timing accounts for propagation and spatial separation — universal under causal ordering. The mechanism closes the stored-energy and current-system change — universal under substorm physics. **Specifications withdrawn.** **Common requirement.** A causal onset sequence connecting energy storage, instability, current reconfiguration and auroral release. **Conserving relation.** The mechanism predicts the order and propagation of onset signatures across the coupled magnetosphere-ionosphere system. **Conserving reach.** Isolated and compound substorms under differing solar-wind drivers and magnetotail configurations. **Reach.** The statement ranges over a three-dimensional rapidly evolving system. Multipoint spacecraft and ground arrays sample separated trajectories and projections. The field's exchange states that the timing turns on which signatures fix onset and reconnection, the response faulting the comment for an electrojet index in place of ground auroral and pulsation signatures and single field components in place of full vectors and particle distributions: the signatures the spacecraft and ground arrays return and the causal order sought are not the same, the choice of signature standing between them. **Unreachability.** **Persistence.** Event ensembles with sufficient multipoint coverage to establish a repeatable causal order and discriminate instability mechanisms. **Addresses.** Statement, with the posing added there, hardness at the dispute over which signatures fix onset and reconnection, the accounts at tail reconnection first and at near-Earth current disruption first, the approach at multipoint timing, progress at the observed sequence and its dispute, and the relation at Reach: Angelopoulos et al., "Tail Reconnection Triggering Substorm Onset", Science 321:931–935, 2008, doi 10.1126/science.1160495, at its abstract as the publisher's page and an index carry it verbatim (substorms explosively releasing solar-wind energy stored in the magnetotail; unclear whether triggered by near-Earth current disruption at about ten Earth radii or by reconnection at twenty to thirty; simultaneous measurements at multiple distances; reconnection observed at twenty Earth radii at least a minute and a half before auroral intensification, two minutes before expansion and about three minutes before current disruption; substorms likely initiated by tail reconnection); a comment, Science 324:1391, 2009, at its abstract as the publisher's page and an index carry it verbatim (evidence that near-Earth current disruption occurred before the conventional reconnection signatures and triggered onset; the auroral intensification and the reconnection not causally linked; the onset a continuation of earlier activity); and the authors' response at its opening as a repository copy carries it (the comment faulted for an electrojet index in place of ground auroral and pulsation signatures, single components in place of full vectors and particle distributions, and a preliminary index in place of ground and tail data), none fetched. The hardness line's other clauses, the specifications and the other approaches stand keyed at the next pass. New Alfvénic evidence: Wu et al., "Radio emissions reveal Alfvénic activity and electron acceleration prior to substorm onset", Nature Communications 16:10553, 2025, cited here and not read this pass. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: the mission's own objective as a satellite-mission page carries it (which magnetotail process is responsible for onset, a local disruption of the current or the current's interaction with the influx from lobe reconnection; correlative observations from long-baseline probe conjunctions to delineate the causal relationship); a review of flow bursts and onset at its abstract as the publisher's page carries it (causality between near-Earth and midtail processes one of the most controversial issues; outstanding issues that may serve as critical tests; the coupling to the ionosphere to be implemented in future modelling), Space Science Reviews, doi 10.1023/B:SPAC.0000042940.59358.2f; the exchange over the mission's first result at the comment and the reply as the journal's page and an archive carry them (near-Earth current disruption held to precede the reconnection signatures; the reply holding reconnection the initiator), Science, 2009, doi 10.1126/science.1167726; a radio-emissions paper at its abstract (the triggering mechanisms remaining unclear), 2025; a superstorm study at its abstract (the mechanism of directly driven substorms unresolved), Geophysical Research Letters, 2026, doi 10.1029/2025GL119629; their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Multipoint satellite timing — orders local signatures when geometry is favorable. Ground auroral imaging and magnetometers — provide broad projection with mapping uncertainty. Global kinetic and magnetohydrodynamic simulation — tests causal sequences under specified initialization. Statistical superposed epochs — recover common timing while averaging event diversity. Radio and wave observations — trace Alfvénic activity and particle acceleration. **Narrowing recorded.** Event studies constrain onset regions and propagation delays; substorms are partitioned by external driving and pre-onset tail state. **Progress.** Near-Earth reconnection, bursty flows, current disruption, auroral onset and Alfvénic coupling are all observed components; their universal causal order remains open. **Own record.** ## 240 The Unique Games conjecture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** For every positive ε, is there an alphabet size for which it is NP-hard to distinguish a unique constraint game with value at least 1 − ε from one with value at most ε? In the field's own posing, a two-prover game is unique when the answer of one prover uniquely determines the answer of the other and vice versa, the value of a game being the maximum acceptance probability of the verifier over all prover strategies, and the conjecture concerns the power of such games. **The field's account of its hardness.** That the field records a subexponential algorithm for unique games, running in time exponential in a power of the instance size that shrinks with the approximation parameter, unimproved since it was found and putting the problem among the few whose best known algorithm falls between exponential and polynomial time, which was a surprise, with the observation that were the running time improved so that its exponent shrank independently of that parameter the problem would not be NP-hard; that unique games on expanding constraint graphs are easy, no integrality gap being known for a constant number of rounds of the sum-of-squares hierarchy, while the known algorithmic attacks fall far short of disproving the conjecture; and that unlike many other conjectures the community is described as truly split on this one, the assortment of findings likened in the field to blind men touching different parts of an elephant. **Accounts.** The stated completeness-soundness gap is NP-hard for a sufficiently large alphabet. The conjecture fails because a polynomial-time algorithm distinguishes every instance at some fixed gap. Hardness holds only after changing the quantitative gap or imposing additional instance structure. **Specifications.** Each edge carries a permutation constraint with one satisfying label at one endpoint for each label at the other — universal under unique games. Completeness and soundness thresholds are 1 − ε and ε — local. Alphabet size may depend on ε but not on instance size — universal under the conjecture. **Specifications withdrawn.** **Common requirement.** A complexity classification for the promised gap version of unique constraint satisfaction. **Conserving relation.** A polynomial-time procedure either distinguishes the promised value cases, or the distinction is NP-hard under polynomial reductions. **Conserving reach.** Every promised unique game at the alphabet size supplied for each ε. **Reach.** The conjecture quantifies over all small ε and corresponding promised instances. Algorithms solve subclasses and parameter regimes; reductions establish conditional consequences. The two are not the same: the known algorithmic attacks work equally at completeness near one and at exactly one and fall far short of disproving the conjecture, unique games on expanding constraint graphs are easy, and the subexponential algorithm, unimproved since it was found, leaves the general problem's standing between exponential and polynomial. **Unreachability.** **Persistence.** A proof of the hardness reduction for every ε, or a polynomial-time distinguishing algorithm contradicting one required gap. **Addresses.** Statement, with the posing added there, the account at the conjecture as made, and progress at one conditional consequence: Khot, "On the power of unique 2-prover 1-round games", ACM Symposium on Theory of Computing, 2002, pp. 767–775, doi 10.1145/509907.510017, at its abstract as the publisher's page and a repository carry it verbatim (a two-prover game unique when each prover's answer determines the other's; the value of a game the maximum acceptance probability over prover strategies; a conjecture made regarding the power of unique games, called the Unique Games Conjecture), the paper itself not fetched, an abstract version at a complexity conference of the same year listed beside it. Progress at the consequence that vertex cover is hard to approximate within any constant better than two under the conjecture, at the abstract of the paper proving it as the publisher's page carries it, returned in the search. No Reach relation is stated at the abstract read. The specifications, the approaches and the other clauses of progress stand keyed at the next pass. Approaches and progress as posed: Klarreich, "Approximately Hard: The Unique Games Conjecture", Simons Foundation, 2011, cited here and not read this pass. Hardness: re-formed to the field's words at the exposition the address names, at its text as the foundation's page carries it (the subexponential algorithm a surprise, the problem joining a small clique between exponential and polynomial; the findings likened to the blind men and the elephant), 2011, not fetched; at the conjecture's author's exposition of the 2-to-2 games proof, at passages as a university page carries them (the algorithm's running time and the observation on its exponent; the known attacks far short of disproving the conjecture; no integrality gap known for constant rounds of the sum-of-squares hierarchy; an integrality gap on random instances among the evidence), 2019, not fetched; at a lecture note as a university page carries it (the community truly split, unlike many other conjectures; the algorithm not improved since 2010), 2021, its author not carried; and at the title of the 2008 paper that unique games on expanding constraint graphs are easy, at citing lists; the former line, in the file's compression (The conjecture implies sharp approximation thresholds for many problems, yet standard random instances are tractable and algorithmic relaxations solve broad structured classes; neither the known algorithms nor hardness reductions reach the full promised instance family.), released with its content at the second clause, its sharp-thresholds clause standing at the account. The relation at Reach: the conjecture's author's exposition and the lecture note, at the passages read for the hardness. Persistence, no landing: the closing condition at the conjecture's author's exposition read above (a hardness proof for every parameter, or an algorithm whose exponent shrinks independently of it, which would show the problem not NP-hard). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Semidefinite programming — gives strong approximations and solves high-value or structured regimes. Integrality-gap construction — matches relaxations without proving NP-hardness. Invariance and dictatorship testing — yields optimal conditional hardness for many problems. Spectral and expansion algorithms — solve broad nonrandom classes. Subexponential algorithms — constrain quantitative forms without settling polynomial-time hardness. **Narrowing recorded.** The conjecture is tested by alphabet, completeness, soundness and graph structure; many approximation thresholds stand conditional on the exact promise. **Progress.** Sharp conditional hardness consequences, matching algorithms for many target problems and substantial algorithms for special unique games are proved. **Own record.** ## 241 Odd perfect numbers **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does there exist an odd positive integer equal to the sum of its proper positive divisors? In the field's own posing, a perfect number is one whose divisors sum to twice it, the even ones characterized since antiquity, while the search for odd perfect numbers has not been as successful, the results accumulating as necessary conditions on a hypothetical number's factors. **The field's account of its hardness.** Necessary congruence, factorization and size conditions have accumulated without contradiction, but they constrain a hypothetical integer rather than produce a finite exhaustive search; multiplicative divisor identities permit many compatible local exponent patterns. **Accounts.** The conjecture as posed, that no odd perfect number exists: one account stands, the field's results accumulating as necessary conditions on a hypothetical number's factors; no existence claim is defended in the field's record. **Specifications.** The integer is positive and odd — universal under the problem. Perfection means its divisor sum is twice the integer — universal under number theory. Ordinary integer factorization and equality are used — universal under the statement. **Specifications withdrawn.** **Common requirement.** An integer satisfying the oddness and divisor-sum equation. **Conserving relation.** The divisor sum equals twice the integer. **Conserving reach.** All odd positive integers. **Reach.** The statement quantifies over an unbounded discrete set. Computation excludes finite ranges and theory excludes factorization patterns. The field's paper states that for a fixed number of distinct prime factors there are only finitely many odd perfect numbers, with an effective bound, and that each further case is settled by finding a contradiction at the count below it: the finite ranges the computations exclude and the unbounded set the statement quantifies over are not the same, the count of distinct prime factors standing between them. **Unreachability.** **Persistence.** Exhibition of one odd perfect number, or a proof that the divisor-sum equation has no odd solution. **Addresses.** Statement, with the posing added there, hardness at the accumulated necessary conditions and the finiteness at each fixed count, approaches at the factor-chain algorithm and the counting of distinct factors, progress at the nine distinct prime factors and twelve where three does not divide, and the relation at Reach: Nielsen, "Odd perfect numbers have at least nine distinct prime factors", Mathematics of Computation 76:2109–2126, 2007, doi 10.1090/S0025-5718-07-01990-4, at its abstract as the arXiv page, an abstract service and the society's copy carry it verbatim (an odd perfect number shown to have at least nine distinct prime factors, and at least twelve if three does not divide it; the proof avoiding previous computational results) and at passages of its introduction and closing as the arXiv copy carries them (the even perfect numbers characterized, the odd search less successful; lower bounds, the finiteness for each fixed number of distinct factors with an effective bound, large and small factor bounds; a factor-chain argument run by computer; the next case requiring a contradiction at nine), the paper itself not fetched; a data listing gives the last page as 2127 where the society's reference gives 2126. The hardness line's clause on local exponent patterns, the specifications and the other approaches stand keyed at the next pass; a later paper by the same author raising the count to ten, and a paper placing odd perfect numbers above ten to the fifteen-hundredth, returned at a repository copy and a reference listing, readings for progress at the next pass. Dittmer, "Spoof Odd Perfect Numbers", Mathematics of Computation 83:2575–2582, 2014, cited here and not read this pass. Accounts: the conjecture as posed at the abstract read above, no existence claim carried there. Persistence, no landing: the closing condition at the abstract read above (the search for an odd perfect number, or a proof that none exists, no existence claim carried). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Congruence and factor-form analysis — imposes a distinguished prime and exponent restrictions. Prime-factor counting — raises the minimum number of distinct factors under general and conditional cases. Lower-bound computation — excludes all candidates below enormous finite bounds. Factor-chain search — eliminates branches when forced primes or exponents contradict necessary conditions. Spoof constructions — show why divisor identities alone admit near-counterparts. **Narrowing recorded.** Any odd perfect number must have a restricted prime-power form, many distinct prime factors and size above the established computational bound. **Progress.** Strong necessary congruences, prime-factor minima, component bounds and exhaustive factor-chain exclusions stand. **Own record.** ## 242 Hadwiger's conjecture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Is every graph with no complete graph K_t as a minor colorable with at most t − 1 colors? In the field's own posing, the conjecture is that every graph with no complete minor on t vertices is colorable with t − 1 colors, for every t, and the general bounds proved run above that, from order t√(log t) to order t (log log t)^18. **The field's account of its hardness.** The conjecture links a global coloring constraint to the existence of a highly connected minor; the small cases draw on the four-color theorem, while general structure and density theorems yield bounds larger than t − 1 and do not force the exact minor. **Accounts.** The conjecture as posed, that every graph with no complete minor on t vertices is colorable with t minus one colors: one account stands, the small cases proved and the general bounds running above; no counterexample is claimed in the field's record. **Specifications.** Graphs are finite and simple — universal under the standard conjecture. A minor is obtained by vertex and edge deletion and edge contraction — universal under graph-minor theory. Proper coloring assigns different colors to adjacent vertices — universal under coloring. **Specifications withdrawn.** **Common requirement.** A complete minor whose order reaches the chromatic obstruction asserted by the statement. **Conserving relation.** Absence of a K_t minor entails a proper coloring with at most t − 1 colors. **Conserving reach.** Every finite simple graph and every positive integer t. **Reach.** The statement quantifies over all graph sizes and t. Exact proofs cover t at most six, and asymptotic theorems give weaker color bounds for general t. The field's preprint states that the general bounds proceed from the average degree of graphs with no such minor, giving colorability of order t√(log t), then t(log t)^β for every positive β, then t(log log t)^18: the asymptotic bounds the density theorems return and the exact bound the conjecture quantifies over are not the same, the gap between the two orders standing between them. **Unreachability.** **Persistence.** A proof for every t, or a graph whose chromatic number is at least t and that has no K_t minor. **Addresses.** Statement, with the posing added there, hardness at the clause that density theorems yield bounds larger than t − 1, the approach at extremal density, progress at the general upper bounds, and the relation at Reach: Postle, "Further progress towards Hadwiger's conjecture", arXiv:2006.11798, 2020, at its abstract as the arXiv page and an abstract service carry it verbatim (the conjecture that every graph with no such minor is colorable with one fewer color than the minor's size; every such graph of average degree of order t√(log t) and hence so colorable; the order improved to t(log t)^β for every β above a quarter, then for every positive β; with another paper by the author, order t(log log t)^18), the preprint recorded at the page as withdrawn by its author at the fourth version, 2022, its results stated there to be added to a self-contained standalone paper by the same author, submitted 2021 and revised 2022, whose identifier is not carried at the page read; nothing fetched. The hardness line's other clauses, the small cases, the specifications and the other approaches stand keyed at the next pass. Small cases and standing: Chudnovsky et al., "The Legacy of Robin Thomas", Notices of the American Mathematical Society 69:966–978, 2022, cited here and not read this pass. Accounts: the conjecture as posed at the abstract read above, no counterexample carried there. Persistence, no landing: the closing condition at the abstract read above (a proof for every order, or a counterexample, none claimed). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Graph-minor structure — proves deep decompositions and the small exact cases. Extremal density — forces large complete minors from average degree and yields superlinear coloring bounds. Connectivity and linkage — proves stronger results under added hypotheses. Fractional, list, odd and clustered variants — settle neighboring relations or approximations without the exact conjecture. **Narrowing recorded.** The conjecture is proved through t = 6; stronger connectivity and restricted graph classes extend exact conclusions, and general graphs carry asymptotic color bounds. **Progress.** The cases through t = 6 and increasingly sharp general upper bounds on the chromatic number of K_t-minor-free graphs stand. **Own record.** ## 243 The two-dimensional Jacobian conjecture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does every polynomial map from complex affine two-space to itself with nonzero constant Jacobian determinant have a polynomial inverse? In the field's own posing, the conjecture asks whether every polynomial map of complex n-space whose Jacobian determinant is a nonzero constant must have a polynomial inverse; refuted in dimension three, with an infinite family and a geometric explanation following within days, the counterexample sweeping the tangent lines of a plane curve, which classical duality forces to hit most points several times, the plane case remaining open. **The field's account of its hardness.** The constant determinant gives a local inverse everywhere but does not by itself control global behavior at infinity; degree reductions and special-case theorems leave the two-variable case open, while the former higher-dimensional statement has been refuted by a noninjective polynomial local isomorphism. **Accounts.** The conjecture as posed in two variables: one account stands for the plane; for the general statement the field's second account is the refutation in every dimension above two by a noninjective polynomial local isomorphism, the two-variable case standing open. **Specifications.** The map has two complex polynomial coordinates — universal under the present statement. Its Jacobian determinant is a nonzero constant — universal under a Keller map. The required inverse is polynomial and global — universal under the conjecture. **Specifications withdrawn.** The specification that the same assertion holds in every dimension, withdrawn after an explicit counterexample in dimension three while the two-dimensional statement is retained. **Common requirement.** Global injectivity and polynomial invertibility following from the nonvanishing constant determinant in two variables. **Conserving relation.** Distinct points have distinct images and the inverse coordinate functions are polynomials. **Conserving reach.** Every polynomial self-map of complex affine two-space with nonzero constant Jacobian determinant. **Reach.** The statement quantifies over all two-variable polynomial degrees and coefficient choices. Reductions and proofs cover bounded degrees, special Newton polygons and other restricted classes. The field's paper states that the tangent-sweep mechanism, generalized from plane curves to direction fields on hypersurfaces, produces counterexamples in every dimension greater than two and of arbitrarily large geometric degree, the plane case standing open in the field's records: the maps the construction reaches and the two-variable maps the statement quantifies over are not the same, the dimension standing between them. **Unreachability.** **Persistence.** A proof of global polynomial invertibility in two variables, or one two-variable constant-Jacobian map with a nontrivial fiber. **Addresses.** Statement, with the posing added there, hardness at the refutation by a map hitting most points several times, specifications withdrawn at the all-dimensions assertion, narrowing and progress at the counterexamples in every dimension above two and of arbitrarily large geometric degree, Own record, and the relation at Reach: Gao, "Counterexamples to the Jacobian conjecture in dimensions greater than two", arXiv:2608.00222, 2026, at its abstract as the arXiv page carries it verbatim (the conjecture open since 1939; refuted in dimension three by Alpöge on 19 July 2026, an infinite family by Gallagher the next day and a geometric explanation by Speyer four days later, the counterexample sweeping the tangent lines of a plane curve, a map classical duality forces to hit most points several times; the mechanism generalized to direction fields on hypersurfaces; counterexamples in every dimension greater than two and of arbitrarily large geometric degree; five new explicit maps, one three-dimensional of degree four, two four-dimensional of degrees five and ten, two five-dimensional of degrees six and twelve), the paper itself not fetched; its title page carries the author's disclosure that the main idea and framework are the author's and that a language model assisted in the proofs and the writing. The field's own record of the finding: an encyclopedia page, as the page carries it, records the three-dimensional counterexample as announced in July 2026 and credited by its announcer to a language model, on a magazine's report, with the plane case remaining open; a mathematician's weblog digesting the counterexample, returned at its listing, states the conjecture open in two dimensions and settled in one; neither fetched. The degree given at progress before this pass for the three-dimensional certificate, seven, is not carried at the pages read, the generalization paper's own three-dimensional map being of degree four; the first clause of the hardness line, the specifications and the approaches stand keyed at the next pass; the standing of the result at peer review, riding from the previous session, not carried at the pages read. Bass, Connell and Wright, "The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse", Bulletin of the American Mathematical Society 7:287–330, 1982, cited here and not read this pass. Accounts: the conjecture as posed and the refutation above the plane at the abstract read above. Persistence, no landing at the plane: the closing condition at the abstract read above (the two-variable case open, the refutation above the plane by a noninjective local isomorphism). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Degree and homogeneous reductions — reduce a general proof to structured cubic maps in higher dimension but do not settle two variables. Valuation, Newton-polygon and compactification methods — prove families and constrain behavior at infinity. Differential-operator and related conjecture programs — yield equivalences and special cases. Counterexample search — has refuted dimensions three and higher while exact searches and classifications leave two dimensions. **Narrowing recorded.** The open reach is now dimension two; many degree and support classes within it are proved, while the unrestricted two-variable degree remains. **Progress.** An explicit degree-seven map refutes the conjecture in dimension three and therefore in every higher dimension by adjoining identity coordinates. The two-dimensional case remains open. **Own record.** The field's own account states the conjecture refuted in dimension three, an infinite family of counterexamples and a geometric explanation following within days, and counterexamples then produced in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree, so that the two-variable statement is the one that remains. ## 244 The abc conjecture **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does a coprime sum a + b = c bound c by a power of the product of the primes dividing abc? In the field's own statement, carried verbatim from the society's article with its displayed product written in line: for any given ε > 0 there exists a constant κ_ε such that if a, b, and c are coprime positive integers for which a + b = c, then c ≤ κ_ε (∏_{p prime, p | abc} p)^(1+ε). **The field's account of its hardness.** The best unconditional result bounds c by the exponential of a constant times the two-thirds power of the product of the primes dividing abc, one exponential away from the conjecture, and results from these techniques, as beautiful as they are, fall short of the goal; the conjecture is equivalent to extensions of several of the most important theorems of the subject, on rational approximation, on rational points on curves, on linear forms in logarithms and on modular parametrisations, and the least desirable state of affairs would be to find it undecidable, and thus those extensions with it. **Accounts.** The conjecture as posed, its polynomial analogue proved. A claimed proof in four papers, posted and nine years later published, standing in the field's record without acceptance: the community's initial impression described as bafflement, verification ongoing for years with any consensus slow to emerge, if it has; part of the community skeptical on a published criticism identifying a specific flaw, to which the author's reply was not accepted by the other side; and a later report holding the proof incomplete where it must establish the existence of the many structures it compares and averages over. **Specifications.** The three integers are coprime and positive with a + b = c — universal under the statement. The product runs over the distinct primes dividing abc — universal under the statement. For every ε > 0 the constant depends on ε alone — universal under the statement. An explicit version assigns a value to κ_ε for each ε — local. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** c is bounded by κ_ε times the product of the primes dividing abc raised to 1 + ε, uniformly over the triples for each ε. **Conserving reach.** All coprime positive integer triples with a + b = c, for every ε > 0. **Reach.** The statement quantifies over all coprime positive triples with a + b = c and every ε > 0. Computation returns the ratio of the two sides over finite ranges of triples, never large in the ranges checked, and proof returns an unconditional bound at the exponential of the two-thirds power of the product of the primes. The article states that the unconditional bound is one exponential away from the conjecture: the bound proved and the bound conjectured are not the same, and the gap is that exponential. **Unreachability.** **Persistence.** A proof or a disproof; the article names a third outcome, undecidability, as the least desirable. **Addresses.** Statement, carried verbatim, hardness, specifications, reach, persistence, approaches, narrowing recorded and progress: Granville and Tucker, "It's As Easy As abc", Notices of the American Mathematical Society 49(10):1224–1231, 2002, read whole at the society's page, which carries the article as published (the statement displayed under the conjecture's name; the polynomial analogue and its proof; the precursor inequalities without ε and their counterexamples; the unconditional bound one exponential short; the equivalences to extensions of four theorems; the closing on proof, disproof and undecidability). Owed: the standing of the claim of proof and the field's dispute over it, at their own sources, not read this sitting. Accounts: the conjecture at the article read whole above; the claimed proof's standing at a 2022 preprint on deep disagreement in mathematics at its section on the news from Kyoto as the arXiv copy carries it (four papers of five hundred pages posted in 2012 purporting to prove the conjecture; the initial impression bafflement; a long time before any consensus began to emerge, if it has; verification ongoing five years by 2017), arXiv:2210.16488, its author not carried; at a mathematician's page on the conjecture as it carries a 2022 note (part of the community skeptical on a published criticism, the reply not accepted by the other side; the papers reviewed and published in a special volume in 2021); and at a 2025 report at its opening as the arXiv copy carries it (why the proof is incomplete: the existence of many arithmetic holomorphic structures not proved, needed where the proof compares and averages over them; the 2018 objection recognizing the problem and extrapolating, in the report's judgment, incorrectly), arXiv:2505.10568, its author carried at the copy; none fetched, the claimed proof's own papers not read. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Linear forms in logarithms — return the unconditional bound, an exponential of the two-thirds power of the product of the primes dividing abc, short of the conjecture by one exponential. Explicit versions with κ_ε assigned — would give explicit bounds on the Fermat equation and an explicit form of the finiteness theorem for rational points on curves, the conjecture standing unproved. Equivalent formulations — the conjecture is equivalent to strengthened forms of the theorems on rational approximation of algebraic numbers, on rational points on curves of higher genus, on linear forms in logarithms and on the degree of modular parametrisations. **Narrowing recorded.** The relation weakened to an exponential bound, proved unconditionally: c is at most the exponential of a constant times the two-thirds power of the product of the primes dividing abc. **Progress.** The polynomial analogue is proved: for coprime polynomials with a + b = c, the largest degree is less than the number of distinct roots of abc. The inequality without ε is refuted by 1 + 8 = 9 and further small examples, and any fixed constant multiple of the product is refuted by the family a = 1, c = 2^(p(p−1)) for large primes p. The unconditional exponential bound stands. **Own record.** ## 245 Cosmic inflation and the primordial perturbations **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What physics drove the early accelerated expansion, and are the primordial perturbations the fluctuations of that epoch? The field poses it through the reconstruction of the inflaton potential from the primordial spectra, within the framework of single-field models under the established theory of gravity, with the connection between the inflationary expansion and the subsequent thermalized era carried as an uncertainty. **The field's account of its hardness.** That while the concept resolves several fundamental puzzles in cosmology its underlying physical origin remains unknown, the field describing it after almost four decades as a very general theory with no direct link to the Standard Model of particle physics, the inflaton not observed and the number of dynamical degrees of freedom associated with the scalar potential unknown; that the reconstruction of the potential from observable quantities, the field's ultimate goal within the paradigm, meets degeneracies whereby different models give identical or near-identical predictions, the tensor spectrum corresponding uniquely to the potential in single-field slow-roll models while the scalar perturbations alone leave a one-parameter perfect degeneracy and a family of possible potentials; and that there is no universally accepted mechanism of reheating, the transition between the inflationary and the radiation eras, on whose details the number of e-folds required depends, the measurement uncertainty in the spectral index being comparable to the variation in many models' predictions over that range. **Accounts.** Single-field slow-roll inflation with a concave potential generates the perturbations. Inflation with dynamics beyond slow roll, or with features or oscillations in the primordial spectrum, generates them. Inflation with more than one field generates them, leaving isocurvature contributions. Inflation with a statistically anisotropic quadrupolar modulation generates them. **Specifications.** The primordial perturbations are adiabatic — universal under the standard account. The primordial scalar spectrum is a power law with one index — local. The reconstruction of the potential holds for a single field under the established theory of gravity — local. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The primordial spectra predicted by the inflationary dynamics return the measured temperature, polarization and lensing spectra. **Conserving reach.** All observed scales of the microwave background, with the baryon acoustic oscillation and ground-based polarization data. **Reach.** The statement quantifies over the dynamics of the inflationary epoch and its connection to the thermalized era. The instruments return the anisotropy spectra of temperature, polarization and lensing, from which the scalar spectral index, the tensor-to-scalar ratio, the curvature and the adiabaticity are inferred. The collaboration states that its reconstruction of the potential holds in the framework of single-field models under the established gravity, and that the connection to the thermalized era carries uncertainties: the potential reconstructed and the physics of the epoch are not the same without that framework. **Unreachability.** **Persistence.** A detection fixing the energy scale and dynamics of the epoch, with its connection to the thermalized era fixed, or an exclusion of the inflationary account over the stated reach. **Addresses.** Accounts at the model classes tested, specifications, reach, approaches, narrowing recorded and progress: Planck Collaboration, "Planck 2018 results. X. Constraints on inflation", Astronomy and Astrophysics 641:A10, 2020, at its abstract as the arXiv page, the abstract service and a university repository carry it verbatim (the scalar spectral index and the absence of running; the flatness with baryon acoustic oscillation data; the tensor bound, tightened with the ground-based polarization data; concave potentials favoured and no dynamics beyond slow roll found; the pure power law; no parameterized features, strengthened with bispectrum data; the adiabaticity test and the isocurvature bounds; the anisotropic modulation bound) and at its introduction as the arXiv page carries it (the quadratic potential ruled out; the large-angle and high-multipole anomalies motivating non-standard models; the uncertainties in connecting the inflationary expansion to the thermalized era), the paper itself not fetched. The hardness stands empty: the report carries constraints and a framework, not a statement of the hardness; the field's critiques and defences of the paradigm are its account, not read this sitting. The concise question at the Statement is a rendering and not the field's posing; the field's posing as a question stands keyed at the next pass. Hardness: filled at the review literature returned in the search, none fetched: a 2026 preprint on particle cosmology at its section on inflationary dynamics as the arXiv copy carries it (the underlying physical origin unknown; the inflaton not observed; different potentials, distinct predictions), arXiv:2606.31812, its authors not carried; a 2018 preprint at its introduction (after almost four decades a very general theory with no direct link to the Standard Model; no universally accepted mechanism of reheating), arXiv:1808.00835, its authors carried at the copy; a 2001 preprint at its introduction (reconstruction of the potential the ultimate goal; degeneracies; the tensor spectrum unique, the scalar alone a one-parameter perfect degeneracy), arXiv:astro-ph/0109412, its authors not carried; a 1998 preprint at its opening (the number of dynamical degrees of freedom unknown), arXiv:astro-ph/9802209, its authors not carried; the Particle Data Group's review on inflation, 2023, at its reheating passage as the group's copy carries it (the e-folds required depending on the reheating details; the spectral-index uncertainty comparable to the models' variation). Persistence, no landing: the closing condition in part at the 2001 preprint read above (the tensor spectrum corresponding uniquely to the potential, its detection fixing the reconstruction) and the 2018 preprint (no accepted mechanism of reheating, the connection to the thermalized era), the exclusion alternative keyed at the next pass. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Slow-roll constraints with Bayesian model comparison — favour concave potentials, with the uncertainties of the connection to the thermalized era carried. Potential reconstruction by two methods — finds no dynamics beyond slow roll. Non-parametric spectrum reconstruction — confirms a pure power law. Searches for parameterized features and oscillations, with bispectrum data — find no evidence. Adiabaticity and isocurvature tests — bound the non-adiabatic contribution to the temperature variance at the percent level. Tensor searches with ground-based polarization — bound the tensor-to-scalar ratio. **Narrowing recorded.** The scalar spectral index measured at 0.9649 ± 0.0042 with no running; spatial flatness to 0.4 percent with baryon acoustic oscillation data; the tensor-to-scalar ratio below 0.10, and below 0.056 with the ground-based data; non-adiabatic contributions bounded at 1.3, 1.7 and 1.7 percent for the cold dark matter, neutrino density and neutrino velocity modes. **Progress.** The quadratic slow-roll potential ruled out; concave potentials increasingly favoured; a pure power-law spectrum confirmed non-parametrically; adiabatic initial conditions tested stringently by polarization. **Own record.** ## 246 The engram **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What is a memory, and what in the brain is its substrate? In the field's own posing, at the publisher's summary of its review, the question is carried verbatim in four words, "what is a memory"; the search for the neuronal substrate that forms a memory has a long history, and the view now emerging is that ensembles of engram cells account for how memories are formed and retrieved. **The field's account of its hardness.** That memory storage is thought to be mediated by lasting physical changes in the brain, and that although many attempts have been made to localize the physical trace of a memory it remained largely elusive until recently, the field having to develop four defining criteria for identifying an engram before recent capture studies, which tag the neurons active at encoding for later manipulation, could be assessed as observing, erasing and artificially expressing one. **Accounts.** An ensemble of engram cells, changed enduringly at an experience and reactivated at retrieval, is the substrate of a memory. **Specifications.** A memory has a neuronal substrate — universal under the engram account. The substrate is an ensemble of cells rather than one cell or a diffuse change — local. The evidence is drawn particularly from rodents — local. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The identified substrate accounts for the formation, the retrieval over time and the loss of a specific memory. **Conserving reach.** All memories of the kinds studied, across formation, retrieval and loss. **Reach.** The statement quantifies over the substrate of a specific memory. The evidence for engram cells as a substrate is drawn from rodents, as the summary states, and returns the features of formation, retrieval over time and loss. The two are not the same: the field had to develop four defining criteria before capture studies, which tag the neurons active at encoding in rodents for later manipulation, could be assessed as observing, erasing and artificially expressing an engram, and a substrate shown by manipulation in rodents is read onto the specific memory the statement concerns. **Unreachability.** **Persistence.** A neuronal substrate identified for a specific memory, with its formation, retrieval over time and loss accounted for at that substrate, and how memory becomes knowledge answered at it. **Addresses.** Statement, with the posing there, the account as the emerging view, specifications, reach, persistence, approaches and progress: Josselyn and Tonegawa, "Memory engrams: Recalling the past and imagining the future", Science 367:eaaw4325, 2020, at the publisher's summary of the review as the publisher's page and its mirrors carry it verbatim (the question what a memory is; the long search for the neuronal substrate; the emerging view of engram-cell ensembles; the evidence particularly in rodents; formation, retrieval over time and loss; how memory becomes knowledge as the direction ahead), the review itself not fetched. The hardness stands empty: the summary carries the posing and the emerging view, not a statement of the hardness; the review's own statement of what remains stands keyed at the next pass. Hardness: filled at the earlier review of the field, Josselyn, Köhler and Frankland, "Finding the engram", Nature Reviews Neuroscience 16(9):521-534, 2015, doi 10.1038/nrn4000, at its abstract as the publisher's page, an index, a database and an archive carry it verbatim (memory storage thought mediated by lasting physical changes; many attempts to localize the trace, the engram largely elusive until recently; four defining criteria developed; capture studies tagging the neurons active at encoding, assessed as observing, erasing and expressing the engram), the review itself not fetched. The relation at Reach: the earlier review's abstract, at the passage read for the hardness. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Engram-cell studies in rodents — return evidence for ensembles as a substrate, with formation, retrieval over time and loss described at them. **Narrowing recorded.** **Progress.** Evidence for engram cells as a substrate of memory stands, particularly in rodents. **Own record.** ## 247 The 100,000-year glacial cycle **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why have the glacial cycles of the past million years run near 100,000 years when the insolation change at that period is small? In the field's own posing, the cycles of the past million years have a period near 100,000 years, matching the period of change in the orbit's eccentricity, yet insolation changes little at that timescale, so the amplitude of the cycles is hard to reconcile with the forcing; the problem is compounded by the unexplained change of cycle length from 41,000 to 100,000 years at the mid-Pleistocene transition. **The field's account of its hardness.** The insolation change at the eccentricity period is small against the amplitude of the cycles, and the shift of the dominant period from 41,000 to 100,000 years occurred without a discernible change in the insolation forcing. **Accounts.** The astronomical theory: orbital forcing drives the cycles, with amplification from Earth-system feedbacks. The geochemical theory: internal dynamics dominate, with orbital forcing synchronising the oscillations. Internal climate oscillations phase-locked to the 413,000-year eccentricity cycle produce the cycles. **Specifications.** The cycles of the past million years have a period near 100,000 years — universal under the record. The insolation effect of eccentricity is weak beside those of obliquity and precession — universal under the astronomical calculation. The cycles before the transition ran near 41,000 years — universal under the record. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the observed period, amplitude and timing of the cycles from the stated forcing and dynamics. **Conserving reach.** The glacial cycles of the past million years, and the transition before them. **Reach.** The statement quantifies over the mechanism producing the period. Marine and ice records return the timing and amplitude of the cycles, and astronomical calculation returns the insolation. A study of the coupling states that formal hypothesis testing shows eccentricity not significantly modulating the timing of glacial terminations: the period matched and the forcing calculated are not the same, the pacing unexplained by the forcing at that period. **Unreachability.** **Persistence.** A mechanism returning the 100,000-year period, the amplitude and the mid-Pleistocene transition from the forcing and the internal dynamics together. **Addresses.** Statement, with the posing there, hardness at the small insolation change, the phase-locking account, and approaches at the synchronization analysis: "Synchronization of the climate system to eccentricity forcing and the 100,000-year problem", Nature Geoscience, 2013, at its abstract and the journal's summary as the journal's page carries them verbatim (the cycles near 100,000 years; the insolation change small at that timescale; the problem compounded by the unexplained transition from 41,000 to 100,000 years; phase locking to the 413,000-year eccentricity cycle proposed), its authors, volume and pages not confirmed at the page read, the paper itself not fetched. The two competing theories at Accounts, and approaches at the energy-balance models: "A First Principles Approach to the 100,000-year Problem", arXiv 2604.12143, 2026, at its abstract as the arXiv page carries it verbatim (the astronomical and the geochemical theories; augmented energy-balance models failing to reproduce the period), its authors not confirmed at the page read, the paper itself not fetched. Hardness at the transition without a discernible change in forcing, and the relation at Reach: a preprint on the stochastic coupling of climate variables and ice volume over the late Pleistocene, arXiv 2603.26937, 2026, at its introduction as the arXiv page carries it (the transition without a discernible change in insolation forcing; formal hypothesis testing showing eccentricity not significantly modulating the timing of terminations, cited there to its source), the paper itself not fetched. Persistence, no landing: the closing condition at the field's statements returned in the search, none fetched: a precession-pacing paper at its introduction as the publisher's page carries it verbatim (the cause of the predominantly hundred-thousand-year cyclicity remaining unresolved; the insolation forcing over the ice sheets containing almost no power at that period; the debate whether the cycles belong to obliquity or to the precession index modulated by eccentricity; age uncertainties making the distinction difficult), Nature Geoscience 16:717-722, 2023, doi 10.1038/s41561-023-01235-x; a synchronization paper at its opening (the problem compounded by the lack of explanation for the mid-Pleistocene transition; individual discrepancies explained, a unified explanation lacking), Nature Geoscience, 2013, doi 10.1038/ngeo1756; a morphology study at its abstract (identification of the roles hindered by imprecise age control, circumvented by the morphology of deglaciation and inception; the gross morphology appearing largely deterministic), Science, 2025, doi 10.1126/science.adp3491; a stochastic-coupling paper at its introduction (the mechanisms an active area; eccentricity not significantly modulating the timing of terminations under formal testing), arXiv:2603.26937; their authors not carried. Remaining properties: keyed at the next pass. *The report.* **Approaches.** Frequency-modulation and synchronization analysis — proposes phase locking of internal oscillations to the long eccentricity cycle. Conceptual energy-balance models with augmentations — fail to reproduce the period, revealing formulation limitations. Formal hypothesis testing of eccentricity's modulation of termination timing — finds no significant modulation. **Narrowing recorded.** **Progress.** The period near 100,000 years over the past million years and the earlier period near 41,000 years stand measured; the weak insolation change at the eccentricity period stands calculated. **Own record.** ## 248 The origin of supermassive black holes **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How did supermassive black holes of about a billion solar masses come to exist within the first billion years of the Universe? In the field's own posing, their existence within the first billion years has stimulated numerous ideas for the prompt formation and rapid growth of black holes in the early Universe, the questions being how the seeds of massive black holes first assembled, how they subsequently grew as massive as a billion solar masses, and how multimessenger observations could distinguish between the different assembly scenarios. **The field's account of its hardness.** That black holes of about a billion solar masses exist within the first billion years, calling for prompt formation and rapid growth; that the ultrarare holes of that mass represent only the tip of the iceberg, early black holes likely filling a continuum from the stellar-mass to the supermassive regime that reflects a range of initial masses and growth histories; and that the stellar-mass holes likely left behind by the first generation of stars at redshifts as high as thirty typically had their initial growth stunted by the shallow potential wells of their host galaxies. **Accounts.** Seeds of stellar mass left behind by the first generation of stars, grown thereafter to the supermassive regime. A continuum of initial masses and growth histories rather than one seeding route, the observed holes the tip of the iceberg, the routes parted by the field's review and the assembly scenarios to be distinguished by multimessenger observation. **Specifications.** Supermassive black holes of about a billion solar masses exist within the first billion years — universal under the observations the review begins from. Stellar-mass black holes were likely left behind by the first generation of stars at redshifts as high as thirty — the review's conclusion. The initial growth of such seeds was typically stunted by the shallow potential wells of their host galaxies — the review's conclusion. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the observed masses at the observed epoch from the seeds and the growth histories. **Conserving reach.** The supermassive black holes of the first billion years, and the early black hole population the review places beneath them. **Reach.** The statement quantifies over the assembly of the earliest supermassive black holes. Luminous quasars at high redshift, and the multimessenger signals proposed to distinguish assembly scenarios, are what is measured. The two are not the same: the ultrarare holes observed are read by the review as the tip of a continuum whose remainder is not observed. **Unreachability.** **Persistence.** A seeding and growth history returning black holes of a billion solar masses within the first billion years, the scenarios distinguished by observation. **Addresses.** Statement, with the posing there, hardness, the accounts as the review's conclusions, the specifications and the relation at Reach: Inayoshi, Visbal and Haiman, "The Assembly of the First Massive Black Holes", Annual Review of Astronomy and Astrophysics 58:27-97, 2020, doi 10.1146/annurev-astro-120419-014455, arXiv:1911.05791, first published as a review in advance on 30 March 2020, at its abstract as the publisher's page, an abstract service, an index and the arXiv page carry it verbatim (the existence of billion-solar-mass holes within the first billion years stimulating numerous ideas for prompt formation and rapid growth; the seeds' assembly, their growth, and multimessenger discrimination among scenarios reviewed; the ultrarare holes the tip of the iceberg; early holes likely a continuum from the stellar-mass to the supermassive regime; stellar-mass holes left by the first stars at redshifts as high as thirty, their initial growth typically stunted by shallow potential wells), the review itself not fetched, its further conclusions not carried at the returns. A later review, "The origins of massive black holes", Nature Reviews Physics, 2021, arXiv:2110.10175, at its citation only, its authors not carried, a reading for the next pass. Approaches: the two routes and their endings at the review's abstract read above (the seeding routes reviewed, the continuum concluded; multimessenger discrimination proposed). Persistence, no landing: the closing condition at the review's abstract read above (the assembly scenarios to be distinguished by multimessenger observations). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Review of seeding routes and growth histories — concludes a continuum from the stellar-mass to the supermassive regime. Multimessenger observation to distinguish assembly scenarios — proposed, the discrimination not yet made. **Narrowing recorded.** **Progress.** The existence of the billion-solar-mass holes within the first billion years stands observed; the reading that early black holes fill a continuum, and that the seeds' initial growth was typically stunted, stand as the review's conclusions. **Own record.** ## 249 Weak cosmic censorship **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Are the singularities of gravitational collapse always hidden inside black holes, invisible from infinity? In the field's own posing, the conjecture asserts in essence that all singularities of gravitational collapse are hidden within black holes; as first put, the gravitational collapse of an isolated physical body, starting from generic regular initial data, cannot produce spacetime singularities that can be seen from infinity, even though observations from infinity are allowed to continue indefinitely, the question asked being whether there is a cosmic censor who forbids the appearance of naked singularities, clothing each one in an absolute event horizon. **The field's account of its hardness.** That the conjecture is completely open in its full generality, described in the field as arguably the most outstanding unsolved mathematical problem of the theory and as so open that the correct formulation of the statement is not even known; that for the model of the spherically symmetric scalar field a suitable formulation has been proved, explicit naked-singularity spacetimes being constructible but unstable and non-generic, the instability shown only in a topology of low-regularity solutions, and the question in a smoother topology remaining open even there; and that counterexamples are known in higher dimensions and, for test particles, only when backreaction is neglected, the conjecture's motivation being that the future incompleteness of null infinity would threaten the predictability of the theory. **Accounts.** The conjecture as posed: for generic initial data, naked singularities do not form, singularities being hidden inside black holes and invisible to observers who exist for all time, null infinity future complete. Naked singularities form generically: in spherically symmetric collapse models for large classes of matter fields in four dimensions, sets of initial data of non-zero measure lead to a locally naked singularity, trapped surfaces not forming early enough to shield it, with families of future outgoing non-spacelike geodesics escaping. **Specifications.** Under fairly physical situations of unstoppable collapse, spacetime singularities must occur — universal under the singularity theorems. Explicit naked-singularity spacetimes can be constructed — known. In the spherically symmetric scalar-field system, naked singularities are unstable to black hole formation and non-generic in a suitable sense — proved. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns, from generic regular initial data for physically reasonable matter, every singularity of collapse hidden behind a horizon and null infinity complete. **Conserving reach.** Gravitational collapse of isolated physical bodies in the classical theory. **Reach.** The statement quantifies over generic initial data for the collapse of physically reasonable matter. Explicit spacetimes, symmetric model systems, gedanken experiments on overspinning and overcharging, and higher-dimensional examples are what is measured. The two are not the same: the field records that the correct formulation of the statement is not even known. **Unreachability.** **Persistence.** A proof or disproof of the conjecture in full generality, in a formulation the field accepts. **Addresses.** Statement, the posing in essence and the review's standing: Wald, "Gravitational Collapse and Cosmic Censorship", arXiv:gr-qc/9710068, 1997, at its abstract and introduction as an arXiv rendering carries them verbatim (the conjecture asserting in essence that all singularities of collapse are hidden within black holes; little progress toward a general proof or disproof; notable progress in examples and special cases supporting the view that naked singularities cannot arise generically; reports of its death greatly exaggerated, the conjecture probably healthier than at any time before), the review itself not fetched, its published venue not carried at the return. The conjecture as first put and the question asked, as a 2005 preprint quotes them, arXiv:gr-qc/0503093, at its introduction (the collapse of an isolated physical body from generic regular initial data unable to produce singularities seen from infinity; the cosmic censor clothing each naked singularity in an absolute event horizon; the conjecture a hypothesis of the area theorem), its authors not carried. Hardness: a survey of open problems in mathematical physics at its section on the hypothesis as the arXiv copy carries it (so open that the correct formulation is not even known), arXiv:1710.02105; a 2026 preprint at its introduction (completely open in full generality; the spherically symmetric scalar-field formulation proved, in a low-regularity topology, the smoother topology open; null infinity's incompleteness threatening predictability), arXiv:2605.19143; a 2026 preprint on phantom fields at its introduction (unproven in full generality despite decades of effort; partial results in restricted settings; counterexamples in higher dimensions, and for test particles only neglecting backreaction), arXiv:2603.13210; a 2015 preprint on higher-dimensional examples at its introduction (limited progress toward a general proof or disproof; possibly the most important open problem in classical general relativity; trapped-surface formation without symmetry), arXiv:1509.07956. The second account at a 2015 preprint at its introduction (initial data of non-zero measure leading to a locally naked singularity in spherically symmetric collapse models), arXiv:1503.06651. The gedanken-experiment approach at a 2008 essay at its abstract (overspinning by scalar absorption; the horizon irrefutably stable when backreaction is taken into account), arXiv:0805.3873. None fetched; the authors of the preprints not carried at the returns except as the copies' bylines give them. Persistence, no landing: the closing condition at the review and the preprints read above (a general proof or disproof; the correct formulation of the statement not even known). Remaining properties: keyed at the next pass. *The report.* **Approaches.** The spherically symmetric scalar field — a suitable formulation proved, the smoother topology open. Gedanken experiments overspinning or overcharging a black hole — the horizon stable when backreaction is included. Higher-dimensional vacuum collapse — examples of naked singularity formation constructed. Trapped-surface formation without symmetry assumptions — significant progress recorded. **Narrowing recorded.** **Progress.** The model-system formulation stands proved; the trapped-surface results stand; the field's own reading that reports of the conjecture's death were greatly exaggerated stands as the review's conclusion. **Own record.** ## 250 Proton decay and grand unification **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Does the proton decay, as the unified theories predict, and at what lifetime? In the field's own posing, one of the most crucial and generic predictions of grand unification is that the proton must ultimately decay into leptonic matter such as a positron and a meson, revealing quark-lepton unity; the only known way to observe the unification phenomenon directly is a nucleon-decay measurement, the unification scale of about ten to the fifteenth or sixteenth gigaelectronvolts being unreachable by accelerators; and to date all attempts to observe such decays have failed, the experiments establishing lower limits on the lifetime instead. **The field's account of its hardness.** That the unification scale is unreachable by accelerators, its effects detectable only through rare phenomena beyond the Standard Model, the most promising such phenomenon the violation of baryon number and the most sought-after signature proton decay; that the predicted lifetime spans orders of magnitude and is highly model dependent, from ten to the thirty-first to ten to the thirty-sixth years across the theories, the lifetime in the gauge-boson mode going as the fourth power of the unification scale, some models predicting lifetimes so long that successful detection is not currently plausible while a class of well-motivated theories provides a conservative upper limit within a factor of ten of the current lower limit; and that the searches have excluded the simplest unification model and lifetimes up to ten to the thirty-fourth years, the field describing proton decay as the last great unresolved prediction of its generation, improvement requiring much bigger detectors and a better description of bound-nucleon states and of the propagation of decay products through nuclear matter. **Accounts.** The proton decays as the unified theories predict, at a lifetime beyond the present limits and within the reach of megaton-scale detectors. No grand unification inducing proton decay, the proton stable to the limits set, baryon number violated only non-perturbatively at a rate suppressed beyond observation. **Specifications.** The lower limit on the partial lifetime for decay to a positron and a neutral pion exceeds ten to the thirty-fourth years — measured. Grand unified theories predict the decay at lifetimes between ten to the thirty-first and ten to the thirty-sixth years — the theories' prediction. The simplest unification model stands ruled out by the searches — the field's record. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the proton's stability to the limits set and the meeting of the three couplings from one theory. **Conserving reach.** Baryon-number violation at the unification scale. **Reach.** The statement quantifies over the decay of the proton in any channel at any lifetime. Lower limits on partial lifetimes in particular channels from water Cherenkov exposure are what is measured, the detection efficiency dominated by absorption and charge exchange of the final-state meson in the nucleus. The two are not the same: a limit in a channel bounds that channel's partial lifetime at a confidence level and does not test decay in channels the detector does not see. **Unreachability.** **Persistence.** An observation of nucleon decay, or a limit excluding the unified theories' predicted range. **Addresses.** Statement, with the posing there, hardness, the specifications and the relation at Reach, at field statements returned in the search, none fetched, their authors not carried except as the copies' bylines give them: a proposal document for an experimental program in neutrino physics and proton decay, arXiv:hep-ex/0608023, 2006, at its nucleon-decay section as the arXiv copy carries it verbatim (the proton's ultimate stability questioned in the context of unification; one of the most crucial and generic predictions that the proton must ultimately decay into leptonic matter, revealing quark-lepton unity; a class of well-motivated theories with a conservative upper limit within a factor of ten of the lower limit; the detection efficiency dominated by final-state absorption or charge exchange); a project document, arXiv:hep-ex/0106019, 2001, at its sensitivity section (the only known way to directly observe the unification phenomenon a nucleon-decay measurement; the gauge-boson mode's lifetime possibly as short as ten to the thirty-fifth years; the supersymmetric mode highly model dependent between ten to the thirty-second and thirty-fifth); a search paper at its introduction as the arXiv copy carries it, arXiv:2001.08011, 2020, and its abstract as a repository carries it (unification at ten to the fifteenth or sixteenth gigaelectronvolts unreachable by accelerators, the effects detectable through rare phenomena, the most sought-after signature proton decay; no significant signal, lifetimes up to ten to the thirty-fourth years excluded; the simplest model ruled out; trilepton limits between nine and thirty-four times ten to the thirty-third years); a 2010 preprint at its opening (the lifetime proportional to the fourth power of the unification scale; lifetimes so long that detection is not currently plausible; the last great unresolved prediction of the generation), arXiv:1003.2570; a conference paper on nuclear effects, arXiv:0811.1892, 2008, at its abstract (a spectacular proof of unification; much bigger detectors; bound-nucleon states and propagation through nuclear matter); a 2022 preprint at its unification section (the current lower limit above sixteen times ten to the thirty-third years; the next-generation sensitivity above ten to the thirty-fifth), arXiv:2208.05257; the second account at a 2013 preprint at its lifetime section (no unification inducing proton decay assumed; non-perturbative violation suppressed), arXiv:1310.3904. Approaches: the four routes and their endings at the papers read above (the classical-mode and trilepton searches at the search paper's introduction and abstract; the liquid argon proposal at the nuclear-effects paper; the lifetime prediction at the 2010 preprint). Persistence, no landing: the closing condition at the papers read above (an observation of nucleon decay; the next-generation sensitivity above ten to the thirty-fifth years against the well-motivated theories' upper limit). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Water Cherenkov searches in the classical modes — no significant signal, limits above ten to the thirty-fourth years. Trilepton-mode searches — limits by more than an order of magnitude over earlier experiments. Liquid argon detectors proposed for the next generation — the nuclear effects to be described. Renormalization-group prediction of lifetimes from the unification parameters — model dependent, corrected in its second-order treatment. **Narrowing recorded.** **Progress.** The limits in the classical modes stand measured; the simplest unification model stands excluded; the next generation's sensitivity of ten to the thirty-fifth years stands planned. **Own record.** ## 251 The graph isomorphism problem **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Can it be decided in polynomial time whether two finite graphs are isomorphic? In the field's own posing, determining the computational complexity of the problem is regarded as a major open problem in computer science; stated as open already in the seminal paper on the NP-completeness of combinatorial problems and in the standard guide to intractability, it remains one of the few natural problems in NP that is neither known to be NP-complete nor known to be polynomial-time solvable. **The field's account of its hardness.** That the problem is one of the few natural problems in NP known neither to be NP-complete nor to be solvable in polynomial time; that after the group-theoretic methods were introduced, continually refined, analysed in the complexity of their group problems and extended to restricted classes and other structures, no real progress on the general problem was made until, in the field's words, out of the blue, a quasipolynomial-time algorithm arrived, running in time exponential in a polylogarithm of the number of vertices, its ideas spanning eighty pages in the original manuscript and combining several major, very distinct new ideas that the field finds difficult to do justice to; and that the gap between quasipolynomial and polynomial stands, isomorphism of chordal graphs being polynomial-time equivalent to the general problem. **Accounts.** The problem is solvable in polynomial time, the quasipolynomial algorithm the present bound and not the last. The problem lies between polynomial and NP-complete, an intermediate problem, the possibility the field's posing leaves open. **Specifications.** A quasipolynomial-time algorithm exists, running in time the number of vertices raised to a polylogarithm of itself — proved. For graphs of bounded maximum degree the running time is the number of vertices raised to a polylogarithm of the degree — proved. Isomorphism of chordal graphs is polynomial-time equivalent to the general problem — proved. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the problem's complexity class from an algorithm or a hardness reduction. **Conserving reach.** Isomorphism of finite graphs, and the structures polynomially equivalent to it. **Reach.** The statement quantifies over the complexity of deciding isomorphism for all finite graphs. Algorithms for restricted classes, running times parameterized by degree or treewidth, and the combinatorial refinement algorithm are what is measured. The two are not the same: a quasipolynomial upper bound and polynomial algorithms on restricted classes do not place the general problem in a class. **Unreachability.** **Persistence.** A polynomial-time algorithm, or a proof under a standard assumption that none exists. **Addresses.** Statement, with the posing there, hardness and the group-theoretic approach: Grohe and Schweitzer, "The graph isomorphism problem", Communications of the ACM, 2020, doi 10.1145/3372123, a review article, at passages of its text as the journal's page and a preprint at a university page carry them verbatim (mentioned as open in the seminal NP-completeness paper and the guide to intractability; the precise complexity regarded a major open problem; the group-theoretic methods refined and extended; no real progress on the general problem until, out of the blue, the quasipolynomial algorithm; eighty pages, several major distinct new ideas difficult to do justice to; the reduction from Cameron groups to Johnson groups), its volume, issue and pages not carried at the returns, the article itself not fetched. The standing and the parameterized bounds: Grohe and Neuen, "Recent Advances on the Graph Isomorphism Problem", arXiv:2011.01366, 2020, at its abstract and introduction as the arXiv copy carries them verbatim (a major open problem; one of the few natural problems in NP neither known NP-complete nor known polynomial-time solvable; parameterized quasi-polynomial running times; the Weisfeiler-Leman algorithm). The quasipolynomial algorithm: Babai, "Graph isomorphism in quasipolynomial time", Proceedings of the Symposium on Theory of Computing, 2016, 684-697, at a citing summary as an index carries it (chordal graphs polynomial-time equivalent to the general problem), the paper itself not fetched; the parameterization by degree at a 2019 preprint's introduction, arXiv:1911.11257, its author not carried. The third possibility, NP-completeness, and the result relating it to the polynomial hierarchy, keyed at the next pass. Persistence, no landing: the closing condition at the reviews read above (the problem's class undetermined between polynomial and NP-complete; the quasipolynomial bound the present standing). Remaining properties: keyed at the next pass. *The report.* **Approaches.** The group-theoretic method — quasipolynomial time for the general problem, and parameterized by maximum degree. The combinatorial refinement algorithm — polynomial on classes of bounded dimension. Algorithms for restricted graph classes — polynomial time for planar graphs, bounded treewidth and others. Canonical forms — quasipolynomial time. **Narrowing recorded.** **Progress.** The quasipolynomial upper bound stands proved, and canonical forms in quasipolynomial time; the parameterized bounds by maximum degree stand; the general problem's class stands undetermined. **Own record.** ## 252 The neutron lifetime discrepancy between beam and bottle measurements **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why do beam measurements of the free neutron's lifetime, counting its decay products, disagree with bottle measurements, counting the neutrons that survive, when each set of measurements is consistent within itself? In the field's own posing, there is consistency among the beam measurements and among the bottle measurements, but the two sets of experiments disagree by about eight seconds out of a lifetime of about eight hundred and eighty, a nearly four-sigma difference, the puzzle described as still unsolved. **The field's account of its hardness.** That the two methods have independent and very different systematic effects, the most likely explanation of the discrepancy in the field's judgment being an underestimation of systematic effects; that the discrepancy is large enough to be the dominant uncertainty in the prediction of the primordial helium-to-hydrogen abundance ratio; that its resolution relies to a large extent on the outcome of the beam experiments now operating, one counting protons and the other electrons, the first result of the newer of them not yet precise enough to favour either set; and that a magnetic-trap measurement of twice the earlier precision, consistent with the theoretical calculation, does not explain why neutrons in a beam last nearly ten seconds longer on average. **Accounts.** Underestimated systematic effects in one or both methods, the true lifetime a single value. A dark decay channel for the neutron, one or more dark-sector particles in the final state, the beam method counting only the visible channel. Oscillation of the ordinary neutron into a mirror neutron, the beam approach giving the true beta-decay lifetime and the bottle discrepancies indicating new physics. **Specifications.** The bottle measurements agree among themselves and the beam measurements among themselves — the field's record. The two sets differ by about eight seconds, nearly four standard deviations — measured. The most precise bottle measurement, in a magnetic trap, is consistent with the theoretical calculation — measured. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the lifetime each method reports from one decay rate and the method's counting. **Conserving reach.** Free-neutron decay as the two methods observe it. **Reach.** The statement quantifies over the free neutron's decay rate. Proton or electron counts along a beam, and the counts of neutrons surviving in material or magnetic traps, are what is measured. The two are not the same: a beam counts the products of a decay channel and a bottle counts disappearances by any channel. **Unreachability.** **Persistence.** A beam and a bottle measurement agreeing within their uncertainties, or a decay channel that parts them. **Addresses.** Statement, with the posing there, hardness at the systematics and the cosmological weight: a white paper on the determination of the free neutron lifetime, arXiv:1410.5311, 2014, at its summary as the arXiv copy carries it verbatim (consistency within each set; a disagreement of about eight seconds out of about eight hundred and eighty, nearly four sigma; underestimated systematic effects the most likely explanation; the dominant uncertainty in the primordial helium-to-hydrogen prediction; independent and very different systematic effects), its authors not carried, not fetched. The puzzle's standing and the newer beam experiment: a measurement paper with pulsed cold neutrons, arXiv:2007.11293, 2020, at its summary (the puzzle, a discrepancy of eight and a half seconds at four sigma, still unsolved; the beam method with different systematics; the first result), its authors not carried. The resolution's dependence on the operating beam experiments and the proton counter proposed: a review of neutron dark decay, arXiv:2306.11349, 2023, at its beam-and-bottle section (two beam results against multiple bottle results; the experiments counting protons and electrons complementary; the initial data not yet precise enough), its author not carried. The second account: the dark-decay proposal at its abstract, arXiv:1801.01124, 2018, its authors carried at the copy as Fornal and Grinstein. The third account: a review of the anomaly and mirror matter, Universe 9(4):180, 2023, doi 10.3390/universe9040180, at its abstract as a directory carries it, its author carried as Tan. The magnetic-trap measurement and the ten seconds: a news article as a university copy carries it, Nature, 2021 (the lifetime at 877.75 seconds with twice the precision, consistent with theory, not explaining why beam neutrons last nearly ten seconds longer). A meta-analysis by median statistics, arXiv:1812.09671, at its discrepancy section (between about four and six sigma by the set chosen), its authors not carried. None fetched. Approaches: the four routes and their endings at the sources read above (the magnetic-trap measurement at the news article; the pulsed-beam result at the measurement paper; the proton counter at the dark-decay review; the meta-analysis at its discrepancy section). Persistence, no landing: the closing condition at the dark-decay review read above (the resolution relying on the outcome of the beam experiments now operating) and the white paper (both methods improved with independent systematics). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Magnetic-trap bottle with halved error margins — consistent with theory, the discrepancy unresolved. Pulsed cold-neutron beam counting electrons — a first result not yet precise enough to favour either set. A proton counter inside a bottle — proposed. Meta-analysis by median statistics — the discrepancy between about four and six sigma by the set chosen. **Narrowing recorded.** **Progress.** The magnetic-trap lifetime at twice the earlier precision stands measured; the newer beam experiment's first result stands; the discrepancy stands. **Own record.** ## 253 The origin of viruses **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Where did viruses come from? In the field's own posing, viruses are ubiquitous parasites of cellular life and the most abundant biological entities on Earth; it is widely accepted that they are polyphyletic, but a consensus scenario for their ultimate origin is still lacking, the question described in the field as unsolved and controversial. **The field's account of its hardness.** That the three scenarios traditionally considered, descent from primordial precellular genetic elements, reductive evolution from cellular ancestors, and escape of genes from cellular hosts achieving partial replicative autonomy, give different timelines for the origin or origins of viruses and do not explain the provenance of the two key functional modules responsible, respectively, for viral genome replication and virion morphogenesis; and that viruses being polyphyletic, new groups of them are held to have emerged repeatedly at all stages of the evolution of life, often through the displacement of ancestral structural and genome-replication genes. **Accounts.** Descent from primordial, precellular genetic elements. Reductive evolution from cellular ancestors. Escape of genes from cellular hosts, achieving partial replicative autonomy and becoming parasitic genetic elements. A chimeric scenario: different types of primordial, selfish replicons gave rise to viruses by recruiting host proteins for virion formation. **Specifications.** Viruses are the most abundant biological entities on Earth — the field's record. Viruses are polyphyletic — widely accepted in the field. Viral capsid proteins have multiple origins from cellular ancestors — the field's finding. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the provenance of both the replication module and the virion-morphogenesis module of each viral lineage on one timeline. **Conserving reach.** The virosphere, capsid-encoding viruses and capsidless replicons together. **Reach.** The statement quantifies over the ultimate origin of all viruses. Shared conserved genes across the virosphere, the structural homologies of capsid proteins to cellular proteins, and the classes of replicons are what is measured. The two are not the same: a homology dates the recruitment of a module and not the origin of the lineage that recruited it. **Unreachability.** **Persistence.** A scenario returning the provenance of both modules for each viral lineage with a consistent timeline. **Addresses.** Statement, with the posing there, hardness, the four accounts, two specifications and the relation at Reach: Krupovic, Dolja and Koonin, "Origin of viruses: primordial replicators recruiting capsids from hosts", Nature Reviews Microbiology 17(7):449-458, 2019, doi 10.1038/s41579-019-0205-6, published 29 May 2019, an opinion article, at its abstract as the publisher's page, an institute's page, a database and an index carry it verbatim (ubiquitous parasites and the most abundant biological entities; polyphyletic, a consensus scenario for the ultimate origin still lacking; the three traditional scenarios; different timelines, the provenance of the two modules unexplained; the chimeric scenario; new groups emerging repeatedly through displacement of ancestral genes) and at the journal's summary (an unsolved, controversial question), the article itself not fetched. The third specification at the title of the capsid-origins paper, Proceedings of the National Academy of Sciences, 2017, doi 10.1073/pnas.1621061114, as citing pages carry it, not fetched. Approaches: the three routes and their endings at the opinion article's abstract and the capsid-origins title read above, the greater-virus-world route at a citing summary as an index carries it. Persistence, no landing: the closing condition at the opinion article's abstract read above (a consensus scenario for the ultimate origin, still lacking; the provenance of the two modules to be explained). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Comparative genomics of the shared conserved genes across the virosphere — a greater virus world as an evolutionary network of capsid-encoding viruses and capsidless replicons. Structural homology of capsid proteins to cellular proteins — multiple cellular origins of capsids. The chimeric scenario blending the primordial-replicator and escape hypotheses — proposed. **Narrowing recorded.** **Progress.** The polyphyly of viruses stands widely accepted; the multiple cellular origins of capsid proteins stand established; the consensus on the ultimate origin stands lacking. **Own record.** ## 254 The solar dynamo and the sunspot cycle **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What hydromagnetic dynamo produces the Sun's eleven-year sunspot cycle and its polarity reversals? In the field's own posing, the sunspot cycle is the magnetic cycle of the Sun produced by the dynamo process, whose central idea is that the toroidal and the poloidal magnetic fields sustain each other, the field's review of the cycle's models recording recent advances and current debates in modelling the cycle as a hydromagnetic dynamo, with its emphasis on models simple enough to be understood and detailed enough to be compared to the observed cycle. **The field's account of its hardness.** That how the toroidal field produces the concentrated flux ropes that later give rise to sunspots upon buoyant destabilization remains, in the field's review, perhaps the most severe missing link between dynamo models and solar magnetic-field observations; that the models based on poloidal-field regeneration by the decay of bipolar active regions carry, in the field's own point-and-counterpoint review, ten issues of contention, from their similarities and differences with mean-field electrodynamics, through the role of meridional circulation in setting the predicted form of the butterfly diagram, the constraints from light-element abundances and the nonlinear backreaction on the driving flows, to their use for predicting cycle amplitudes; and that global magnetohydrodynamical simulations of solar convection have only recently reached the level of generating large-scale fields undergoing more or less solar-like regular polarity reversals. **Accounts.** The flux-transport dynamo: differential rotation stretches the poloidal field into a toroidal field primarily at the bottom of the convection zone, parts of which rise by magnetic buoyancy to produce sunspots, the decay of tilted bipolar sunspot pairs regenerating the poloidal field and the meridional circulation, poleward at the surface and equatorward at the bottom, transporting the fluxes. Mean-field electrodynamics: the poloidal field regenerated by the turbulent effect of helical convection, the cycle a dynamo wave with the equatorward drift of the toroidal field its signature. Tachocline instabilities: magnetofluid instabilities in the rotational shear layer beneath the convective envelope developing a net helicity under the Coriolis force, twisting the toroidal field into a source for the poloidal component. Global magnetohydrodynamical simulation: the cycle emerging from the full turbulent convection, with solar-like reversals recently obtained. **Specifications.** The sunspot cycle is the magnetic cycle of the Sun — the field's record. Sunspots are manifestations of the toroidal field — the field's record. The tachocline is a rotational shear layer immediately beneath the convective envelope, uncovered by helioseismology — measured. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the cycle's period, the equatorward drift of the butterfly diagram and the polarity reversals from the measured flows and the field's induction. **Conserving reach.** The Sun's large-scale magnetic field over its cycle. **Reach.** The statement quantifies over the mechanism regenerating the Sun's large-scale field. Sunspot numbers and positions, the surface poloidal field, and the flows helioseismology returns are what is measured. The two are not the same: the toroidal field at the bottom of the convection zone, where the models place its generation, is not observed. **Unreachability.** **Persistence.** A dynamo model returning the observed cycle, its polarity reversals and its variability from measured flows. **Addresses.** Statement, with the posing there, hardness at the missing link and the simulations, the third account and the third specification: Charbonneau, "Dynamo models of the solar cycle", Living Reviews in Solar Physics 17:4, 2020, doi 10.1007/s41116-020-00025-6, at its abstract and passages of its text as the publisher's page and an index carry them (recent advances and current debates; emphasis on relatively simple models comparable to the observed cycle; the tachocline uncovered by helioseismology; instabilities within it developing a net helicity), at the society's notice of its update as the society's page carries it (major developments in the models based on the decay of bipolar active regions; global simulations generating solar-like reversals), and at a passage of its earlier edition as a repository copy carries it (the most severe missing link), the review itself not fetched. The central idea, the first account and the first two specifications: a review of the flux-transport model, Reviews of Modern Plasma Physics, 2023, doi 10.1007/s41614-023-00120-9, at its abstract as the publisher's page carries it verbatim, its author not carried. The ten issues: a point-and-counterpoint review of the same models, Advances in Space Research, 2007, at its abstract as the publisher's page carries it verbatim, its authors not carried. The surface field's role at the title of a 2015 paper in Science, as a citing list carries it. None fetched. Approaches: the four routes and their endings at the living review's abstract, the society's update notice and the flux-transport review's abstract read above; the prediction schemes at the citing list of a prediction review, their endings keyed at the next pass. Persistence, no landing: the closing condition at the living review's abstract read above (models detailed enough to be compared to the observed cycle; the missing link between the toroidal field and the sunspots). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Flux-transport models with the decay of bipolar regions regenerating the poloidal field — major developments, used for cycle prediction. Mean-field models — dynamo waves. Global magnetohydrodynamical simulations — solar-like polarity reversals recently reached. Cycle-prediction schemes with a dynamo basis — precursor methods. **Narrowing recorded.** **Progress.** The tachocline stands uncovered by helioseismology; global simulations stand at solar-like reversals; the crucial role of the surface field stands established in the field's record. **Own record.** ## 255 The etiology of major depression **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** What causes major depressive disorder? In the field's own posing, the aetiology of the disorder is multifactorial, its heritability estimated at approximately thirty-five per cent, with environmental factors such as sexual, physical or emotional abuse during childhood strongly associated with the risk of developing it; and no established mechanism can explain all aspects of the disease. **The field's account of its hardness.** That no established mechanism can explain all aspects of the disease, its pathogenesis described in the field as poorly understood; that the disorder is highly polygenic, no single gene sufficient to cause it and each susceptibility gene contributing a small fraction of the total genetic risk, a heritability of thirty to forty per cent meaning that genetics alone does not account for most of the risk; and that the disorder results from a complex interplay of vulnerability genes and environmental factors acting cumulatively throughout the individual's lifetime, stressful experiences early in life suggested to exert a crucial impact on brain development, the interaction of polygenic risk with stressful life events and childhood trauma tested, and the biological impact of the environment held to be mediated by epigenetic modifications. **Accounts.** A polygenic liability interacting with environmental adversity, childhood abuse and recent life stress, the interaction cumulative over the lifetime. Disturbance of the main neurobiological stress-responsive systems, the hypothalamic-pituitary-adrenal axis and the immune system, with alterations in regional brain volumes, particularly the hippocampus, and functional changes in the cognitive-control and affective-salience circuits. Epigenetic mediation of environmental stress, its modifications altering neuroendocrine responses, neuroplasticity, neurotransmission and neuroglial function. **Specifications.** The disorder occurs about twice as often in women as in men and affects one in six adults in their lifetime — measured. Heritability is estimated at approximately thirty-five per cent, with a two-to-threefold lifetime risk among first-degree relatives — measured. The disorder is associated with alterations in regional brain volumes, particularly the hippocampus, and with functional changes in the cognitive-control and affective-salience networks — measured. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the disorder's heritability, its environmental associations and its neurobiological alterations from one causal structure. **Conserving reach.** Major depressive disorder as diagnosed, across its episodes. **Reach.** The statement quantifies over the causes of the disorder in every case. Twin and family heritability estimates, polygenic scores, reported adversity, imaging of regional volumes and circuits, and measures of the stress-hormone and immune systems are what is measured. The two are not the same: an association at the population level does not return the cause in a case, and the diagnosis gathers heterogeneous presentations under one name. **Unreachability.** **Persistence.** A mechanism explaining all aspects of the disease, or a parting of the diagnosis into disorders each with its own. **Addresses.** Statement, with the posing there, hardness at the absence of a mechanism, the second account and the three specifications: the field's primer on the disorder, Nature Reviews Disease Primers 2:16065, 2016, doi 10.1038/nrdp.2016.65, published 15 September 2016, at its abstract as the publisher's page and a university index carry it verbatim (multifactorial aetiology, heritability approximately thirty-five per cent; childhood abuse strongly associated; no established mechanism explaining all aspects; regional volumes and circuits; the stress-responsive systems; twice as often in women, one in six adults), its authors not carried at the returns, the primer itself not fetched. Hardness at the polygenic architecture and the interplay, and the first account: a review of gene-environment interaction, Frontiers in Psychiatry, 2015, doi 10.3389/fpsyt.2015.00068, at its introduction as the journal's page carries it verbatim (the etiopathogenesis not fully understood; a two-to-threefold risk in first-degree relatives; heritability around 0.37; no single gene sufficient; a complex interplay acting cumulatively; early stress and brain development), its authors not carried; a polygenic-interaction study, Psychological Medicine 46(4):759-770, 2015, doi 10.1017/S0033291715002172, at its abstract as an archive copy and a university repository carry it (highly polygenic; interaction with stressful life events and childhood trauma tested), its first author carried as Mullins with twenty-one others. The third account and the pathogenesis poorly understood: a review of epigenetic modifications, Signal Transduction and Targeted Therapy, 2023, doi 10.1038/s41392-023-01519-z, at its abstract as the publisher's page carries it verbatim (heritability thirty to forty per cent, genetics alone not most of the risk; environmental stressors; epigenetic mediation altering neuroendocrine responses, neuroplasticity, neurotransmission and neuroglia), its authors not carried. None fetched. Approaches: the four routes and their endings at the primer's abstract, the gene-environment review's introduction, the polygenic-interaction study's abstract and the epigenetics review's abstract read above. Persistence, no landing: the closing condition at the primer's abstract read above (a mechanism explaining all aspects of the disease, none established). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Polygenic risk scores tested for interaction with life events and childhood trauma — in case-control samples. Neuroimaging of regional volumes and circuits — alterations associated. Epigenetic studies of stress mediation — marks associated with diagnosis and treatment. Candidate-gene interaction studies — the earlier approach, the disorder known to be highly polygenic. **Narrowing recorded.** **Progress.** The heritability stands measured at approximately thirty-five per cent; the polygenic architecture stands established; the hippocampal and circuit alterations stand associated; no established mechanism stands. **Own record.** ## 256 The causes of rising obesity **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** Why has the prevalence of obesity risen across the world in recent decades? In the field's own posing, the increased prevalence of obesity in recent decades is a topic of great scientific and medical interest, but despite many advances the causes of this increase have not been adequately identified, and the pandemic continues unabated despite a persistent public-health campaign to decrease energy intake and increase energy expenditure. **The field's account of its hardness.** That the causes of the increase have not been adequately identified, two conflicting models being vigorously debated by distinct cohorts of experts and differing fundamentally, in a neutral assessment, in what they seek to explain; that the model attributing the rise to changes in the food environment does not identify how those influences promote obesity in some but not all members of the population, its own proponents stating that its further development requires elucidating the factors most responsible for instigating obesity, the mechanisms by which they alter the brain circuits controlling food intake, and why some individuals are more susceptible; and that the campaign's failure admits two explanations, that the approach has not been adequately embraced by the public or that it rests on an erroneous paradigm, a constructive paradigm clash being called for to resolve long-standing controversies. **Accounts.** The energy-balance model: overeating, intake exceeding expenditure, is the primary cause, under complex endocrine, metabolic and nervous-system signals that control food intake below the conscious level, the rising prevalence primarily due to changes in the food environment, the increased availability and marketing of inexpensive, convenient, energy-dense, ultra-processed foods high in portion size, fat and sugar and low in protein and fiber. The carbohydrate-insulin model: hormonal responses to highly processed carbohydrates shift energy partitioning toward deposition in adipose tissue, leaving fewer calories available for the body's metabolic needs, and increasing adiposity causes overeating to compensate for the sequestered calories, a reversal of the causal direction. **Specifications.** Body-mass index has risen across nations since the nineteen-eighties, in a systematic analysis of over nine million participants — measured. The obesity pandemic continues unabated despite the campaign to eat less and move more — the field's record. Both models target processed carbohydrates as major drivers — the field's record. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the population rise and the individual susceptibility from one causal structure. **Conserving reach.** The prevalence of obesity across populations over recent decades. **Reach.** The statement quantifies over the causes of the rise across populations. National surveys of body-mass index, measured and reported energy intake, food-supply availability and controlled feeding trials are what is measured. The two are not the same: energy intake and energy availability are both measured with error, and a model of the population rise must also say who rises and who does not. **Unreachability.** **Persistence.** A model nominating specific and quantifiable environmental influences and defining how each promotes obesity despite the homeostatic system, and why in some and not others. **Addresses.** Statement, with the posing there, hardness and the neutral assessment: a perspective moderating the debate between the two models, Cell Metabolism, 2023, at its summary and passages as the journal's page and the publisher's page carry them verbatim (the causes of the increase not adequately identified; two conflicting models vigorously debated by distinct cohorts; the models differing fundamentally in what they seek to explain; the environment model not identifying how its influences promote obesity in some and not all; its proponents' statement of what its further development requires), its author not carried at the returns, not fetched. The posing's second sentence, the two accounts, the second and third specifications and the paradigm clash: a paper on the competing paradigms, European Journal of Clinical Nutrition, 2022, doi 10.1038/s41430-022-01179-2, at its abstract as the publisher's page, an archive copy and an index carry it verbatim (the pandemic unabated despite the campaign; two explanations of the failure; the energy-balance model's overeating and its food-environment attribution; the carbohydrate-insulin model's partitioning and reversal of causation; a constructive paradigm clash needed; both models targeting processed carbohydrates), its authors not carried at the returns, not fetched. The relation at Reach: a perspective on obesity as an unexplained epidemic, American Journal of Clinical Nutrition, 2022, at a passage as the journal's page carries it (energy intake and availability both measured with error; the core hypothesis of rising intake relatively unquestioned), its author not carried. The first specification at the citation of a systematic analysis of body-mass-index trends across nine million participants, The Lancet, as citing pages carry it. None fetched. Approaches: the three routes at the competing-paradigms paper's abstract (the paradigm clash), the unexplained-epidemic perspective's passage (the intake hypothesis questioned) and a citing title (the genetic loci) read above, the third's ending keyed at the next pass. Persistence, no landing: the closing condition at the moderating perspective read above, the environment model's own proponents' statement of what its development requires (the factors most responsible, the mechanisms altering the brain circuits, why some individuals are more susceptible). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Controlled feeding trials testing the two models' contrasting predictions — proposed as the paradigm clash. Analysis of national intake and availability trends — the core hypothesis of rising intake questioned. Genetic studies of body-mass index — polygenic loci found, at a citing title. **Narrowing recorded.** **Progress.** The rise stands measured since the nineteen-eighties; the two models stand articulated with their contrasts; the causes stand, in the field's words, not adequately identified. **Own record.** ## 257 The origin of the state **Conditions prior.** The statement holds across readings. The terms have determinate reference. The accounts hold their truth value while tested. A procedure repeated returns the same outcome independently of what precedes it. Nothing prior enters. **Statement.** How did the first states arise from autonomous villages, and why where they did? In the field's own posing, the state was not a unique event but a recurring phenomenon, arising independently in different places and at different times where the appropriate conditions existed, and the serious theories of its origin are of two general types, voluntaristic, on which peoples spontaneously, rationally and voluntarily gave up their individual sovereignties to unite with other communities in a larger political unit, and coercive, on which force and not enlightened self-interest is the mechanism by which political evolution led step by step from autonomous villages to the state. **The field's account of its hardness.** That the factual evidence stood, in an earlier statement of the field, buried deep in the unrecorded past, a statement no one in the field would now utter after half a century of work; that the founding coercive theory, holding that no political unit ever gave up its sovereignty of its own accord and pointing to tight environmental circumscription, has been amplified to allow resource concentration to produce political complexity even without circumscription, and remains the central point of contention of a field a quarter of whose recent literature concerns the origin of the state and the early state; that recent measurement finds both cooperative and coercive mechanisms likely at play, early states showing a more rapid rise in trade and public goods and also signs of conflict sooner, with persistent inequality a consequence rather than a precursor of state formation; and that the early-state school reads the first rulers as combining consensus and coercion under a sacred legitimacy. **Accounts.** The circumscription theory: in areas of circumscribed agricultural land, population pressure led to warfare in which the vanquished could not disperse, and so to subjugation, tribute and the state, the theory explaining why states arose where they did and failed to arise elsewhere, the state a predictable response to specific cultural, demographic and ecological conditions. Voluntaristic theories: peoples spontaneously, rationally and voluntarily gave up their individual sovereignties to form a larger political unit. The early-state reading: the leader combining consensus and coercion, legitimate as a descendant of gods, spirits or forefathers, and gathering an inner circle of clients, priests and servants. Cooperative and coercive mechanisms together: circumscription fostering trade, public goods and urbanization as well as conflict. **Specifications.** Pristine states arose independently in several regions — the field's record. Persistent inequality is found a consequence rather than a precursor of state formation — measured in the recent analysis. Environmental circumscription is recorded as fitting the regions most likely on archaeological evidence to have been the centers of pristine state formation — the field's assessment of the founding theory. **Specifications withdrawn.** **Common requirement.** **Conserving relation.** The account returns the rise of states where they arose and their absence elsewhere from the stated conditions. **Conserving reach.** The pristine states, and the secondary states formed in contact with them. **Reach.** The statement quantifies over the formation of every pristine state. Archaeological records of settlement, walls and conflict, the geography of arable land and its periphery, and the trajectories of trade and public goods are what is measured. The two are not the same: the record shows the conditions and the outcomes, and the mechanism between them is read. **Unreachability.** **Persistence.** A theory returning the pristine states where they arose and their absence elsewhere, with the mechanism tested against the record. **Addresses.** Statement, with the posing there, the first account and the third specification: Carneiro, "A Theory of the Origin of the State", Science 169(3947):733-738, 1970, doi 10.1126/science.169.3947.733, at its summary as the publisher's page carries it verbatim (why states arose where they did and failed elsewhere; a predictable response to specific cultural, demographic and ecological conditions; the most important single step in political evolution) and at its opening as a summary page and a commentary carry it (a recurring phenomenon; two general types of theory; the voluntaristic and the coercive defined), the paper itself not fetched; the theory's fit with the archaeological evidence as a later theoretician's assessment, quoted at the author's own clarification. Hardness at the founding claim and its amplification: the author's clarification, amplification and reformulation of the theory, Social Evolution and History, 2012, at its opening as the journal's page carries it verbatim (no political unit ever gave up its sovereignty of its own accord; a coercive theory required; resource concentration producing complexity even without circumscription), not fetched. Hardness at the earlier statement and the field's proportion: a review of twenty years of that journal, Social Evolution and History 22(1):179-196, 2023, doi 10.30884/seh/2023.01.06, at its abstract and opening as an archive copy carries them (the origin of the state and the early state twenty-four per cent of articles; the circumscription theory the central point; the earlier statement that the evidence is buried in the unrecorded past, which no one would now utter), its author not carried, not fetched. The third account: a paper on the emergence of pristine states at its passage on consensus and coercion as the journal's page carries it, its author not carried at the return. The fourth account, the second specification and the hardness at the recent measurement: a paper on environmental circumscription and the origins of the state at its summary as the author's page carries it (both cooperative and coercive mechanisms likely at play; trade and public goods rising faster; conflict sooner; inequality a consequence), its author carried at the page's name only, not fetched. Approaches: the three routes and their endings at the founding paper's summary, the journal's twenty-year review, the early-state paper's passage and the measurement paper's summary read above. Persistence, no landing: the closing condition at the founding paper's summary and the twenty-year review read above (a theory explaining why states arose where they did and failed elsewhere, tested against the archaeological record). Remaining properties: keyed at the next pass. *The report.* **Approaches.** Comparative ethnography and archaeology of the regions of pristine states — the circumscription theory, judged by one review to have passed the test. The early-state school's comparative study — consensus and coercion combined under legitimacy. Quantitative measurement of early state formation and circumscription — both mechanisms found, inequality a consequence. **Narrowing recorded.** **Progress.** The two-type classification of theories stands; the circumscription theory stands as the field's central point of contention; the finding that inequality follows state formation stands. **Own record.** ## The collection The entries stand here, one to a problem, **and their count is recomputed at them** — a count carried beside them ages the moment one arrives. --- # 6 Checks **C1.** A property without an address. **C2.** An approach in the published literature, unlisted, or listed with an ending other than where the literature has it. **C3.** A statement other than the canonical one, or a variant omitted. **C4.** A specification the statement carries, unlisted. **C5.** A condition prior stated other than as the five stand. **C6.** A property stating more than its address carries. **C7.** An approach's ending stated more strongly or more weakly than the record supports. **C8.** An address keyed from no property. **C9.** A property keyed to an address not carrying it. **C10.** A date or a person functioning as attribution in a property. **C11.** An unreachability stated where the result reports an attempt rather than a failure of the relation over the reach. **C12.** A conserving relation or conserving reach stated other than as the problem states it. **C13.** An account stating the condition of the effort rather than proposing an account. **C14.** A specification recorded as withdrawn, not carried at the property for it. **C15.** A narrowing in the record, not carried at the property for it. **C16.** A property carrying a comparison with another entry, or a relation between accounts. **C17.** A field's statement of its own problem's hardness, standing in its literature, not carried at the property for it. **C18.** A hardness line stating the compendium's reading rather than the field's words. **C19.** An approach count matching the count at neighbouring entries where the fields' records differ. Corrections are applied against the literature. --- # 7 The aggregate Each figure is a count over filled properties. The count of entries. The count of entries by the name their specifications are universal under. The distribution of specifications carried per entry. The count of entries carrying an unreachability result. The count carrying a narrowing recorded, by narrowing of reach and by weakening of relation. The count carrying a specification withdrawn. The distribution of approaches per entry and of approach standings. The count whose settling result is specified as one result, as either of two, or as proved unavailable. The count of properties standing empty, by property. Each figure is recomputable from the entries.