Exhibit TWENTY-ONE Hard Problem Registry v393L

 

# Hard Problem Registry

 

**An Open Registry of Hard Problems in Science**

 

**Each observing is here whole with fields' own words, its incoming side: a hard problem as scientific method poses it.** Same observings are resolving with its momentarying side, living and non-living existing things co-sequencing, prior, now and next, each resolving with its field's own next. **One learning with two sides.**

 

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# The conditions prior

Where science has a method for problems, five conditions stand prior to its acting.

The statement holds across readings.

The terms have determinate reference.

The accounts hold their truth value while under test.

A procedure repeating returns the same outcome independently of what precedes it.

Nothing prior enters: what comes to the statement brings nothing of what it held before.

Replication requires the first, measurement the second, inference the third, replication the fourth. The fifth is a condition on what reaches a statement, and the first four stand on it: a statement stands across readings where nothing reaching it brings a prior reading.

Five fix what enters and how it executes. A problem adds a sixth, its own, fixing its outcome: its outcome preserves a relation over a range, each as the problem states it. This condition is a problem's and not a method's, and it stands with Conserving relation and Conserving reach of every entry.

Everything below follows from these and from nothing else.

---

# 1 What stands open, and what stands firm

This is an open registry: a set of entries over a set of problems it states, each with one same set of properties.

**Problems are open.** No entry holds a settling answer to its own statement, except where its record settles one and its entry holds that settling.

**Intake is open.** This registry turns nothing away. It enters each observing, and its collection grows as observings come.

**Properties and checks are open.** Anyone with fields' literature executes any property or check against that literature.

**Each line is open with its observing.** A property with a filling carries an observing or its problem as its field poses it, said as itself, and never its source; a claim not observed directly is an anticipated observing. A line carrying neither is a claim nothing reaches, and it is check C1's case.

Against these, one thing stands firm: properties themselves. Every entry has one same set of properties, with one same order, whatever its field. Uniform elements are what make two entries comparable, and comparability is what a registry is for. Its schema closes once, one place only, so that nothing else has to.

---

# 2 Hard problems in science

**Science's problems** are hypotheses. A hypothesis is an account with consequences following from it, under test together with specifications it has. Where a consequence fails, a test refutes its conjunction of account and specifications and identifies no member of it. Where a consequence holds true, nothing follows about its account, a rival account entailing that same consequence.

**Hard** is where testing goes on and returns no refutation: either no consequence separating accounts is derivable, or accounts entail one same set of consequences. Both are properties of a hypothesis together with what it has, and neither is a property of what a hypothesis is about.

**Progress** is what stands: an establishing, a measure, a proof, or an exclusion. It accumulates whether or not a problem settles, every approach that ends returning a result about itself.

**Persistence** is a remainder between what stands and a settling result.

Each entry is one problem's statement as its field poses it, with progress and persistence and a remainder between them visible. Each entry has one same set of properties, each filling with an observing or with its problem, never with its source.

---

# 3 Derivation

## 3.1 What a refutation identifies

A refutation falls on a conjunction of account and specifications. It identifies no member of it. A test therefore returns no indication of which specification falls, where a settling result requires one to fall.

## 3.2 Specification

A hypothesis has every specification a consequence derives from. A hypothesis with fewer entails no consequence.

A specification is universal under a name where it holds across everything a statement says under that name, and local where it holds to one statement.

Nothing identifies a specification as one to withdraw, and specifications stand as many as a field brings.

## 3.3 Repetition

A result replicates where one same procedure returns one same outcome independently of what precedes it. A test with its own history is not replicable, and no test reads what a prior test produces.

Accumulation stands as specifications. A result that stands becomes a specification of tests that follow. Inference reads specifications as a test opens and writes to none of them.

Where a problem is durable, specifications stand firm across tests running against them. Testing's measure is its count of hypotheses under test.

## 3.4 The two responses

Where a test returns no refutation, refinement or revision follows.

Refinement sharpens a statement, its accounts, or its instruments, within specifications standing.

Revision withdraws a specification, retains its accounts, and tests again. Inference licenses neither, refuting a conjunction and identifying no member.

Both operate on what specifications permit. Neither response reaches a settling result specifications do not permit.

---

# 4 What an entry has

An entry has two parts. Its properties are of a problem, and stand whether or not any work goes on with it. Its report is of work on it, and stands empty where no record has any.

Each carries its observing and its problem, never its source, or is empty.

## 4.1 The properties

**Conditions prior.** Five, standing within every entry. Every unreachability result derives from these together with specifications below.

**Statement.** A problem's form as its field poses it, with its own words. Variant formulations stand, with no ranking.

**The field's account of its hardness.** Where a field states, with its own words, what is hard about its own problem — before anyone proposes an account. A field's own words, or empty. No inference from its record's difficulty.

**Accounts.** What each account proposes to exist, on its own content, as strongly as its record has it. Their number is as the field poses them. No relation between accounts stands here. A statement about an effort's condition proposes no account.

**Specifications.** Each specification a statement has besides five, with a name it is universal under, or local. Their number stands against a count of consequences derivable.

**Specifications withdrawn.** Each specification a record shows a field giving up while keeping its accounts. No reason stands, a refutation identifying no member.

**Common requirement.** What accounts require alike, where a field states it.

**Conserving relation.** What an outcome has to preserve, as its 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.** A range over which its problem requires a relation, as the problem states it: all profiles, all programs, all statements of a language, all cases.

**Reach.** What a statement quantifies over. What its instruments return values over. Where a field states a relation between these two — an equivalence, a limitation with a proof, an account of what a method does not deliver — that statement stands.

**Unreachability.** A result establishing that a conserving relation does not hold over its conserving reach. A result is about its conditions and stands whether or not anyone makes an attempt. What anyone attempts stands with Approaches.

**Persistence.** A settling result, as its field specifies it.

## 4.2 The report

**Approaches.** Each direction a field takes, with its present reach by the measures it uses, and its standing: a refutation; a limitation with a proof blocking it; equivalent by construction to a rival; answering an adjacent question by a method's own account; instruments disagreeing on their object; narrowing; open.

**Narrowing recorded.** Where a record has its reach narrowing, or its relation weakening, with results following. Each stands as its record states it.

**Progress.** What stands, with its measures.

**Own record.** Where a field makes a statement about a problem's own standing, in the field's own phrasing.

## 4.3 Register

Properties have 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, no source and no date. Eponyms that are technical vocabulary stand where that vocabulary puts them.

## 4.4 Statement rule

A precise statement has nothing its original statement together with its own specifications does not have.

Short verbatim phrases stand where an exact formulation is itself an entry's object — a thesis by name, a canonical question — with a mark as such.

## 4.5 What an entry leaves out

Priority and sequence. In contest wherever a record has them, and no standing rests on it. A source, a citation, a finder's name and a publishing year: an entry carries its observing and its problem, never its source.

Importance, difficulty, promise. No statement has these.

An account no one proposes.

Any property of a problem by comparison with 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.**

*The report.*

**Approaches.**

**Narrowing recorded.**

**Progress.**

**Own record.**

Five stand in every entry as they stand at its front, a problem being what a field fixes in order for a method to act on it.

**A property standing empty is an observing and not a gap.** Nothing on record at that property, and its entry has that; a record having it and its entry not, and check C1 through C19 finds it. **An emptiness is on record at its entry and not at intake.**

**The entries number from one, in the order of their entering.** Numbers 155 and 172 are at no entry: 255 entries, numbering to 257. **A number is at an observing's entering and not what it is**, and the aggregate is over properties with a filling 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 under test. A procedure repeating 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.

Higher-order vagueness is repudiated as an illusion on one side; on the other, borderline behaves like a vague predicate and can be embedded in a sorites argument.

*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 under test. A procedure repeating 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.

*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 under test. A procedure repeating 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.

The ambiguity in the term's use is problematic, leading to confusion and hampering research.

*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 under test. A procedure repeating 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 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.

An explaining: no account of the constitution of the standards of correctness is possible without presupposing a commitment to them already in place.

*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 under test. A procedure repeating 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 does not go, 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.

*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 under test. A procedure repeating 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.

Ranking functions are a sound and complete approach to termination for non-probabilistic programs, transition invariants unifying termination metrics and invariants, and a tool built on them verifies device drivers.

*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 under test. A procedure repeating 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.

Most probes indicate a crossover rather than a transition at the pseudogap's onset temperature, and a direct probe of the pairs tests the fluctuation view.

*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 under test. A procedure repeating 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 with 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.

Non-interferometric experiments are the most effective tests of the collapse models, cold atoms, optomechanics, X-rays, bulk heating and cosmological comparisons bounding them.

*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 with its own internal disputes on record, 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 under test. A procedure repeating 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.

*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 has 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 under test. A procedure repeating 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 holds 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, standing 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 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 holds the split as it stands: 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.

*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, facing 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 under test. A procedure repeating 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.

*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 under test. A procedure repeating 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 went past 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.

*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 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 under test. A procedure repeating 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 holds the split over what significance requires.

*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 under test. A procedure repeating 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 holds the split over whether a principled criterion is owed or the practice suffices.

*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 under test. A procedure repeating 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.

*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 under test. A procedure repeating 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 holds the split over what would count as settling.

*The report.*

**Approaches.** Essentialism — has 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 under test. A procedure repeating 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.

An anticipated observing: unsupervised clustering returns satellites as distinct from planets, and dynamical dominance as a natural organizing principle.

*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 under test. A procedure repeating 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 holds the split over whether one criterion is owed.

*The report.*

**Approaches.** Evolutionary accounts — hold the paradigm cases, the symbiont cases standing against them. Physiological and immunological accounts — hold 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 under test. A procedure repeating 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 holds 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 holds the split over whether any empirical result could distinguish accounting schemes, the causal-versus-bookkeeping statements standing on both sides of the reviews.

*The report.*

**Approaches.** Kin-selection formalism — holds the social-evolution cases. Multilevel formalism — holds 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 under test. A procedure repeating 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 fixed 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.

*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 under test. A procedure repeating 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 have 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.

Quantum mechanics explains the periodic table without a full and exact reduction. An anticipated observing: relativistic quantum chemistry gives fifteen elements in the f-block.

*The report.*

**Approaches.** Measuring the configurations — both candidates have 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 under test. A procedure repeating 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 holds the split between stability and systematic correspondence as the criterion of good naming.

*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 under test. A procedure repeating 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 resting on clinical utility and on 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 have 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 holds 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.

*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 with 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 under test. A procedure repeating 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 standing 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.

*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 under test. A procedure repeating 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 holds 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 has 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 holds the split over whether a linguistic criterion is owed or the sociopolitical account suffices.

Intelligibility is often not symmetrical, exposure being set by historical and sociological factors that do not bear on a linguistic classification, and in-between situations are a transition zone.

*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 under test. A procedure repeating 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.

No direct-detection experiment has confirmed a dark matter event, and a discovery is an excess over the predicted neutrino events. The neutrino floor is a boundary below which discovery is extremely challenging without further constraints on the backgrounds, its location evolving; the backgrounds impose no hard limit, the effect gradual: a fog rather than a floor.

*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 under test. A procedure repeating 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.

Heritability is key to the response to selection and to the prediction of disease risk.

*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 under test. A procedure repeating 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.

*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 under test. A procedure repeating 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, in the survey's words, 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.

A free action is not determined: needing no cause, caused with no necessity, or caused by its agent as a substance.

**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 holds the split over whether the ability in dispute is the one the compatibilist analysis defines.

*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 under test. A procedure repeating 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 not 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 not 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 occurring after them — 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 occurring after them, 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.

*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 under test. A procedure repeating 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 enacted included.

**Reach.** The statement quantifies over behaviour under policies not 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.

*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 under test. A procedure repeating 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 run, and that the second bears on the first.

**Conserving relation.** A finding holding at studies not run, the holding standing independently of the setting sampled.

**Conserving reach.** All studies, those not 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.

Replications return weaker evidence despite the original authors' materials and advance review, and the strength of the initial evidence is more predictive of replication success than the replicating team. Replication is an important, uncommon and misunderstood practice, framed as a personal attack rather than a healthy practice.

*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 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 under test. A procedure repeating 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.

An anticipated observing: a protein chain folds during its making, co-translationally.

**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, has its answer 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 at no property here and stands with no work 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 alone.** 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 goes with every approach above and with 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 under test. A procedure repeating 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.

Advances in the genetics of lifespan are not matched by advances in understanding the causes and mechanisms of aging; the hallmarks framework is largely descriptive, not answering why aging occurs, and insinuates rather than states damage as the primary cause. An explaining: the error-based and programmatic models are not exclusive where regulatory networks modulate the rate.

*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 under test. A procedure repeating 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.

*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 under test. A procedure repeating 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, in the field's own words and undisputed there, 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.

*The report.*

**Approaches.** Testing an anomaly — any rejection being a rejection of the joint hypothesis, in the field's own words and undisputed there. 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 taking hold 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 under test. A procedure repeating 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 of 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 proposes 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 with one.

**Persistence.** A determination of the outcome from the payoff structure holding across arrangements, or agreement that arrangements are what determines it.

*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 under test. A procedure repeating 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.

*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 under test. A procedure repeating 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.

*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 under test. A procedure repeating 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 ongoing 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.

A cause raises the probability of its effect in one population and lowers it in another, the populations having different sets of interactive background factors.

*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 under test. A procedure repeating 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 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. 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.

*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, estimates sensitive to elicitation format by amounts the field measures.

**Narrowing recorded.** Tradable permits narrow the reach: pricing stands for a bounded pollutant with measurable emissions. Pigouvian taxation narrows the reach: 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 under test. A procedure repeating 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 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.

*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 under test. A procedure repeating returns the same outcome independently of what precedes it. Nothing prior enters.

**Statement.** Neurons signal with discrete spikes. Is 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 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 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 instruments differ. The identification is not reached over systems by decoding from spike trains.

**Persistence.** The used variable identified downstream rather than information available upstream, generalised beyond systems where timing manipulation has behavioural effects.

*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 specific systems manipulated. Downstream biophysics narrows the reach: usability stands for the auditory brainstem.

**Progress.** The measured information in rate and in timing. 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 under test. A procedure repeating 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 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. Designs' assumptions hold in 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 designs' assumptions.

**Conserving reach.** All traits 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 intermediate estimates are consistent with both accounts. The agreement is not preserved across traits by designs with different assumptions.

**Persistence.** The gap closed by variants recovered, or the twin estimate shown inflated by a measured amount sufficient to account for it.

The diallel panel partitioning additive from non-additive contribution is of 3025 yeast hybrids.

*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 under test. A procedure repeating 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 is, in its own words, 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 premises contain. The point is logical rather than substantive, holds whatever 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 meanings of terms: institutional and functional facts have 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 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 at each exhibited derivation an evaluative premise is located 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.

*The report.*

**Approaches.** Exhibiting a derivation — the promising argument, its critics locating an evaluative premise in the institutional description. Naturalising 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 with 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 has 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 under test. A procedure repeating 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 passes for 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 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 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.

*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 under test. A procedure repeating returns the same outcome independently of what precedes it. Nothing prior enters.

**Statement.** Do smooth solutions of equations of fluid motion in three dimensions remain smooth for all time, or can they develop a singularity? And the turbulence problem, also at the title, posed by the field on its own: whether a predictive theory of turbulent flow follows from equations, averaged equations for the mean flow with 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 equations describe fluids accurately across the range where they are tested.

Not established, and blow-up is plausible: scaling of the nonlinear term against dissipation is supercritical in three dimensions, which is exactly the regime where 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 equations go on describing 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 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 equations rather than supplied by a model.

*The report.*

**Approaches.** Energy methods — give global weak solutions, uniqueness and smoothness of weak solutions standing open, which is the same question. Partial regularity — singular set of parabolic one-dimensional measure zero, containing no curve through time, 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 reaching 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 reaching 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 under test. A procedure repeating 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's answer is 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 correlations between interior and exterior.

**Reach.** The statement quantifies over the interior, and 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, premises shown incompatible being resolved by giving one up.

An explaining: gravity is always attractive, by the principle of equivalence and the positivity of energy, leading to singularities in any reasonable theory of gravitation.

*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 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 under test. A procedure repeating returns the same outcome independently of what precedes it. Nothing prior enters.

**Statement.** Two particles prepared together are measured far apart. 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 has 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 correlations at a different cost, and the field has no agreed criterion for choosing among them.

**Specifications.** Each system has 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 holds no agreement on which is given up.

**Common requirement.** That the correlation is produced by something specifiable at one side or transmitted between sides.

**Conserving relation.** An account of what produces the correlation, the account standing on the coincidence counts.

**Conserving reach.** All accounts preserving 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 correlations at a different cost, and the field has 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.

*The report.*

**Approaches.** Closing loopholes — detection, locality, and freedom-of-choice loopholes closed, the last using cosmic photons and human random choices, the correlation standing and 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 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 under test. A procedure repeating 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 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 not 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 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 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 interpretations are empirically equivalent by construction. The fact is not reached over arrangements by preparation settings and detection events.

**Persistence.** A determination of the entity's nature between preparation and detection, which the empirical equivalence of interpretations excludes.

*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, interpretation disputed. Quantum eraser — interference recoverable by correlation with the marker, widely misreported as retrocausal, the field stating 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 under test. A procedure repeating 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 reach completion.

The programme survives in modified form: 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 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 has its own unprovable sentence, by construction at every stage, and consistency is unprovable within the system. The capture is not reached over arithmetical truths by derivation in any system strong enough for arithmetic.

**Persistence.** Within conditions, nothing: results are proved. Outside them, which axioms suffice for which theorems, settled case by case.

*The report.*

**Approaches.** Strengthening the system — each strengthening with 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, a proof-theoretic ordinal established at each system analysed, each needing a stronger metatheory. Set-theoretic realism — holds truth outruns provability and 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 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 under test. A procedure repeating returns the same outcome independently of what precedes it. Nothing prior enters.

**Statement.** Is there a set larger than integers and smaller than 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 standard axioms and so is its negation. It is independent, and independence is the answer.

Independence from one axiom set is not the end: 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. 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 axioms.

**Conserving reach.** All models of standard axioms.

**Reach.** The statement quantifies over whether a further axiom is true of 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 models by standard axioms or by large-cardinal extensions.

**Persistence.** A criterion for truth of a further axiom, beyond its consistency.

*The report.*

**Approaches.** Proving it from 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 models.

**Progress.** The two independence results. The proof that large-cardinal axioms do not settle it. 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 under test. A procedure repeating 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 sorting.

**Conserving relation.** An analysis reproducing sorting, the reproduction standing over cases not 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 reached 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.

*The report.*

**Approaches.** Adding a condition — each reached 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: 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 under test. A procedure repeating returns the same outcome independently of what precedes it. Nothing prior enters.

**Statement.** Any claim can be asked for its grounds. 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 trilemma's founding treatise 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 extends 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 goes on, and that these three exhaust 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 has 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.

*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 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 under test. A procedure repeating 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 reached by a revenge sentence, which its own authors record. The removal is not preserved over 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.

*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 reasoning and facing their own revenge cases. Denying truth-bearing — each form reached 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 reaches 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 under test. A procedure repeating 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 analyses agree on, applied to the construction.

*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 under test. A procedure repeating 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: 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. Contents are fixed before the choice — local. Both principles — maximise expected utility given evidence, and dominance — are correct — local.

**Specifications withdrawn.**

**Common requirement.** That 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 principles are applied to.

**Reach.** The statement quantifies over which decision principle governs when they conflict. Cases, judged by 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. 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.

*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. Constructed variants and the cases each reaches.

**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 not 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 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. 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 supertasks by arithmetic of the sequence.

**Persistence.** A determination of the final state from the sequence for supertasks generally, or agreement that arithmetic is what the question asks.

*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 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 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 arithmetic of the sequence rather than for the completion of 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 motion of three bodies under their mutual gravitation admits a general solution. In the classical sense — an expression in known functions giving 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 check's computational periodicity.

**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. 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.

The periodic orbits of three equal masses in a plane with zero angular momentum classify topologically into families falling into four classes, the three families found in three centuries in one of them.

*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 absence of evidence is what rarity looks like.

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.

An explaining: the Great Silence is a formidable challenge for any framework assuming a naturalistic origin of life and intelligence, equivalent to the absence of cosmic miracles.

*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 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 models.

**Conserving reach.** All 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 two accounts, which is 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 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.

*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 reached by 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 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 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 tissues.

**Conserving reach.** All tissues the question is asked of.

**Reach.** The statement quantifies over what determines placement. Concentrations, expression domains, and 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 tissues by measurements taken where both processes run together.

**Persistence.** A measurement in tissue separating two processes, or agreement that both operate and the question is their proportion.

*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 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.

*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. Measured excesses from polarised light and the identified astronomical sources. 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: 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. 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 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.

*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.

Strong CP conservation is a natural symmetry where the full Lagrangian has a chiral U(1) invariance, a natural result for any theory in which at least one flavour of fermion acquires its mass through a Yukawa coupling.

*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 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 scales the reconciliation concerns.

**Reach.** The statement quantifies over the structure of a theory at scales not accessible. 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. Covering is not reached over 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.

*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 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, 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 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.

An anticipated observing: chronic pain affects more than thirty per cent of people worldwide. No clear threshold parts acute from chronic pain, and pain persisting beyond the expected healing period is generally accepted as chronic. An explaining: chronic pain, unlike acute pain, which carries survival value, is a disease unto itself; the nociceptive, neuropathic and nociplastic descriptors overlap considerably, the classification a continuum.

*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 imaging, which the field states. Clinical assessment of sensitization features — quantitative sensory testing and symptom-based instruments, assessing 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 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 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. 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 sets.

**Conserving relation.** A determination of whether the axiom holds of 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 models by consequences of chosen systems.

**Persistence.** A criterion for truth of the axiom beyond its consistency, or agreement on whether non-measurable sets are acceptable.

*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, 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: 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 absence of a bijection, and the required bijection exists outside the model but not inside it, countability being relative to a model.

Relativity is the difficulty, not its removal: if "uncountable" holds relative to a model, then either there is an absolute notion models approximate, or set-theoretic language does not determine its own subject matter.

**Specifications.** First-order axioms characterise structures they are about — local. Cardinality is a property of a set — local.

**Specifications withdrawn.**

**Common requirement.** That axioms are about something, and the question is whether they determine it.

**Conserving relation.** A determination of the subject matter by axioms, standing across models.

**Conserving reach.** All models satisfying axioms.

**Reach.** The statement quantifies over whether axioms fix their subject matter. Models satisfying 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 models by models themselves.

**Persistence.** A characterisation of the subject matter not presupposing it, or agreement on what follows from relativity.

*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 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 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 predicates by observations describable in either vocabulary.

**Persistence.** A criterion of projectibility not presupposing the vocabulary it selects.

*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. 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 relative sizes of classes, and the appearance of absurdity is a failure of intuition about small quantities.

The calculation depends on assumptions about 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. 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.

*The report.*

**Approaches.** Bayesian quantification — yields tiny confirmation under standard assumptions, 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, 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. 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 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 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.

*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. 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 standard examples found the alleged superior alternatives not superior on 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 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.

*The report.*

**Approaches.** Formal modelling — produces lock-in under stated conditions, whether conditions obtain being the empirical question. Historical case studies — 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. 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 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 interpretation is of, against which readings can be better or worse.

**Conserving relation.** A criterion for correctness of a reading, standing outside readings it adjudicates.

**Conserving reach.** All readings the criterion is asked of.

**Reach.** The statement quantifies over a criterion for 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 readings by readings taken from inside.

**Persistence.** A criterion standing outside readings it adjudicates, or agreement that no independent criterion is owed.

*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 words meant at the time, with real results, underdetermining readings in dispute, which are usually about significance rather than lexis. Rejecting the whole-part framing — reframes, dependence reappearing at whatever units are chosen.

**Narrowing recorded.** Philological method narrows the reach: the criterion stands for questions of lexis rather than for 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 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 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 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 materials determine the case.

**Conserving relation.** A determination of whether materials fix the outcome, standing separately from the outcome reached.

**Conserving reach.** All hard cases.

**Reach.** The statement quantifies over whether 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 materials fixed the outcome from the fact that an outcome was reached.

*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, 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, 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. Sorting is not preserved across the cases by legal practice.

**Persistence.** A case sorting differently under two accounts and agreed to have been sorted correctly by one.

*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. Dependence is not established over systems by measurements describable in either description.

**Persistence.** A distinguishing prediction between two readings, which no informational ontology has produced.

*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, 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 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, 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 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, 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 primitives, which is the objection; and the nativist argument stands unrefuted rather than accepted. The account is not reached over acquisitions by learners already equipped.

**Persistence.** An account of primitive acquisition not presupposing primitives, or a refutation of the argument that none is possible.

*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 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 primitives.

**Progress.** The dissolution of the classical version. 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 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.

The equity premium observed is six per cent, in data over ninety years. An anticipated observing: for the class of general equilibrium economies, the average real annual yield on equity is at most four-tenths of a per cent above short-term debt, and the largest premium obtainable in the model is thirty-five hundredths of a per cent. The violation of the models' restrictions by the data is robust to specification and measurement. An explaining: an equilibrium model that is not an Arrow-Debreu economy is the one most likely to rationalize both the equity and the Treasury-bill returns.

*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 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 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. 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 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 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.

*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 — 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. 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 axioms wrong, or 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 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 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 axioms are asked of.

**Reach.** The statement quantifies over whether 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 patterns by observed choices or by post-reflection reports.

**Persistence.** A standard the competent reflective agents who reject axioms would endorse, or agreement on what settles a normative question empirically.

*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 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 reversals directly and is documented across humans and animals, 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 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 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.

*The report.*

**Approaches.** Eliciting at individual level — reduces the aggregation artefact, heterogeneity within individuals across domains remaining. Varying elicitation format — changes 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, 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 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 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, 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. 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 accounts do not price.

**Conserving reach.** Output including what accounts do not price.

**Reach.** The statement quantifies over output including what 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.

*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: returns stand at firm level where organisational change accompanies the investment. Intangible capital measurement narrows the reach: the correction stands for 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 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 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 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 equalities are definitional.

*The report.*

**Approaches.** The corrected procedure — aggregate equalities proved not all simultaneously preservable in the standard framework. Single-system interpretations — preserve equalities by redefining the value of inputs, their critics holding the redefinition makes 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: equalities stand under a redefinition of input value, which their critics hold makes them definitional.

**Progress.** The proof that aggregate equalities cannot all hold. The physical determination result. 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 diminishing of illusion under salient inflation is predicted by both accounts. The identification is not reached over decisions by choices under nominal presentation.

**Persistence.** A decision setting in which two accounts predict differently, or a measurement of the decision weight rather than the registration.

*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: 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 maintenance of sex is accounted for by a benefit exceeding the cost.

**Conserving relation.** A benefit measured against the two-fold cost, measurement standing across taxa.

**Conserving reach.** All multicellular taxa where sex is maintained.

**Reach.** Its statement quantifies over the advantage, measured against a stated two-fold cost. Particular systems where 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 recombination benefit relative to the two-fold cost is contested. Measurement is not preserved across taxa by particular systems.

**Persistence.** A benefit measured against the two-fold cost across taxa, or independently measured weights for combined mechanisms.

*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, 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 categories. Theoretical pluralism — most current accounts combining mechanisms, fitting, the combination's weights not independently measured.

**Narrowing recorded.** The parasite account narrows reach: the advantage stands measured in systems where parasite pressure is measured. Experimental evolution narrows reach: the benefit stands under stress in 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 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, comparison standing on standardised incidence.

**Conserving reach.** All species comparison is asked of.

**Reach.** Its 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; sampling bias is acknowledged in studies themselves. 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.

*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 studies themselves. Multistage modelling — reproduces patterns under fitted parameters, 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 framing and not resolving the cross-species case.

**Narrowing recorded.** Within-species scaling narrows reach: relation stands within a species rather than across species. Cell-level experiments narrow reach: 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. 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 poses the problem as 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, sorting standing across instruments.

**Conserving reach.** All sequence the question is asked of.

**Reach.** Its 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 its field debated exactly whether activity is function in print. Sorting is not preserved across instruments by any of the three criteria.

**Persistence.** A criterion of function the three instruments agree on, applied to the same sequence.

*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 reach: the functional fraction stands under a conservation criterion. Deletion experiments narrow reach: dispensability stands under laboratory conditions.

**Progress.** The measured activity fraction. The conserved fraction. 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.** In competition theory, 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 on its own being insufficient, while continuing to be cited as the explanation.

**Accounts.** Equilibrium does not arrive: environmental fluctuation, mixing, and predation reset competition before exclusion completes, and exclusion principle applies to an equilibrium the system does not reach.

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.** Its 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 coexistence framework accommodates both accounts without measuring their weights in natural communities. The count is not reached over 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.

*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. 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 its field in specific systems. Relative nonlinearity — species specialising on resource variance rather than mean level, variation generated externally by seasonal turnover or internally by consumer-resource dynamics, supporting a number growing quadratically with 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 reach: sufficiency stands established in models. Microcosm experiments narrow reach: 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 pattern-matching it has been identified: hierarchical convergence onto neurons responsive to conjunctions, plus attention, accounts for 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, identification standing as cause rather than correlate.

**Conserving reach.** All cases unity is asked of.

**Reach.** Its statement quantifies over what makes 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 its field's analyses disagree, with disputes in print on both sides; and no mechanism pattern-matching synchrony has been identified. Identification is not reached over cases by neural activity and behavioural report.

**Persistence.** A mechanism pattern-matching synchrony identified, or a manipulation separating cause from consequence across cases.

*The report.*

**Approaches.** Gamma synchrony studies — documented across species and tasks, whether causal or a consequence of shared drive disputed, 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, phenomena remaining to be accounted.

**Narrowing recorded.** Optogenetic manipulation narrows reach: the causal result stands for specific paradigms manipulated. Conjunction cells narrow reach: the account stands for conjunctions convergence covers.

**Progress.** The documented gamma coherence across species and tasks. The measured binding failures under attentional load. The identified conjunction cells. 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, identification standing separately from the state that carries it.

**Conserving reach.** All animals with nervous systems.

**Reach.** Its 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 instruments in dispute. Identification is not reached over animals by correlates and deprivation effects.

**Persistence.** A measurement separating the two functions, or agreement that both operate and the question is their proportion.

*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 reach: the informational role stands established in structures where replay is documented. Clearance measurement narrows 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 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 behaviour.

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 concepts — local.

**Specifications withdrawn.**

**Common requirement.** That there is something in others whose presence inference is about.

**Conserving relation.** A justification of inference, standing on a check independent of the single case.

**Conserving reach.** All others inference is made about.

**Reach.** Its statement quantifies over experience in another. Behaviour and physiology are what is measured. The two are not the same.

**Unreachability.** 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 inference and are contested as changing the question. Justification is not reached over the others by behaviour and physiology.

**Persistence.** A check on inference independent of the single case, or agreement that no justification is owed.

*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 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 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, in the paper's words 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, fixing standing against alternative assignments.

**Conserving reach.** All symbols content relation is asked of.

**Reach.** Its statement quantifies over 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. Fixing is not reached over symbols by behaviour and causal history.

**Persistence.** A relation fixing content against alternative assignments, or agreement that content is not fixed by such a relation.

*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, 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 reach: content stands fixed by selection history where a function was selected. Robotic grounding narrows reach: categories stand acquired through interaction in systems built.

**Progress.** Acquirable categories in built systems. The documented disjunction and misrepresentation cases. 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 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 its 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.** Its 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 recovers, by experiment, information a system does not use, a measured 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.

A property not varying in training, unusable by its task, is still recovered by probing at 72 to 83 per cent against 50 by chance.

*The report.*

**Approaches.** Feature visualisation and probing — produces plausible descriptions, probes recovering information the system does not use, which its 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 reach: the guarantee stands for small networks and specific properties. Causal ablation narrows reach: results stand component by component. Sparse dictionary methods narrow reach: coverage stands measured and partial.

**Progress.** Identified circuits and localised features. The measured coverage of sparse decompositions. The established limitation on probing. 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 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.

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, bearing standing on validation independent of resemblance.

**Conserving reach.** All systems the question is asked of.

**Reach.** Its 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. Bearing is not reached over systems by behaviour, architecture, and physiology.

**Persistence.** A marker validated independently of resemblance to the case we know.

*The report.*

**Approaches.** Marker frameworks — applied to invertebrates with legislative consequence in several jurisdictions, establishing a graded evidential practice, the validation problem stated within frameworks themselves. Comparative neuroanatomy — finds candidate structures in distant lineages, establishing analogous organisation, 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 reach: graded evidence stands for taxa 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 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 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, identification standing without a model or reward specification supplied.

**Conserving reach.** All sequences the assignment is asked of.

**Reach.** Its 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. Identification is not reached over sequences by outcomes and within-model rollouts.

**Persistence.** An identification of contribution not resting on a supplied model or reward specification.

*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-execute, 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 reach: convergence stands proved for stated conditions. Counterfactual estimation narrows reach: identification stands where a model or re-executing 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. In classical theory, 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, proposed measures fail to rank models correctly in large-scale empirical comparisons, and 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.** Its 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 its 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 settings in use.

Networks easily fit a random labeling of the training data, unaffected by explicit regularization.

*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, facing 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 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 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 its field states.

**Narrowing recorded.** Implicit bias analysis narrows reach: the result stands proved for specific architectures and losses. PAC-Bayes bounds narrow reach: non-vacuous bounds stand in specific settings. Scaling laws weaken 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. 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 and an overly narrow circuit-based reading cautioned against in the current reviews.

**Accounts.** Convergent network effects: 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 concentrations where responsiveness is lost, and reversing in the same order as responsiveness returns.

Distinct agents produce distinct signatures: agents differ in their spectral and connectivity effects, treating them as one phenomenon obscures 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 agents produce.

**Conserving relation.** One state identified as what agents produce, identification standing across agents.

**Conserving reach.** All agents that abolish responsiveness.

**Reach.** Its 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 disagreement. Reports of experience without responsiveness after emergence show the index is not responsiveness. Identification is not preserved across agents by responsiveness and neural correlates.

**Persistence.** A measure of the state independent of responsiveness, applied across agents.

A relatively small number of important molecular targets of general anaesthetics has emerged. An explaining: the thalamus and the networks regulating it are the key to the loss of consciousness.

*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 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 reach: the discrimination stands across conditions it is calibrated on. Arousal circuit work narrows 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. Its 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, determination standing across imaging paradigms.

**Conserving reach.** All amputees the question is asked of.

**Reach.** Its 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 its field has not reconciled them. Determination is not preserved across paradigms by imaging.

**Persistence.** A reconciliation of the two imaging paradigms, or a measurement of the map's state independent of paradigm.

*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, its 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 reach: the contribution stands established for patients they help.

**Progress.** 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 selecting instances as 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, fixing standing without the other assumed.

**Conserving reach.** Relation between criterion and extension.

**Reach.** Its statement quantifies over 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 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. Fixing is not reached over 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.

*The report.*

**Approaches.** Particularism — starting from clear instances, its critics noting that selecting 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 relation: the criterion stands as a fact about reliability rather than something the knower must fix. Reflective equilibrium weakens 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.

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 carries 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 carrying from experience to its cause, standing without presupposing standards its statement questions.

**Conserving reach.** All rivals accounting for experience.

**Reach.** Its statement quantifies over the cause of experience. Experience is what is measured. The two are not the same.

**Unreachability.** Skeptical scenarios are empirically equivalent by construction, and transcendental arguments establish at most how one must think, by the standing objection. Justification is not reached over rivals by experience.

**Persistence.** A justification not presupposing standards in question, or agreement that no justification is owed.

*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 reach: justification stands within ordinary contexts rather than against skeptical rivals. Transcendental arguments weaken 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 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 only — 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 properties do not.

**Conserving reach.** All objects the question is asked of.

**Reach.** Its statement quantifies over whether a bearer does work 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 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. Determination is not reached over objects by measured properties.

**Persistence.** A case where the two accounts give different verdicts and one verdict is agreed correct.

*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 relation: properties stand as particulars, which removes the indiscernibles case and leaves the bundling question. Spatiotemporal responses narrow reach: the account stands for objects distinguishable by location.

**Progress.** The indiscernibles case and 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.** Its statement is malformed: explanation relates states of affairs, and there is no prior state from which existence of anything could be accounted for. Nothing is not a possible alternative that failed to obtain.

Its statement is intelligible and unanswered: physical accounts of a universe arising from a quantum vacuum account for 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 its 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.** Its 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 its statement is malformed accepted by both accounts.

*The report.*

**Approaches.** Necessary being — a being that could not fail to exist, the modal status being what is in question, 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 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. 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 executing 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 execute many high-fidelity ancestor simulations, simulated observers vastly outnumber unsimulated ones, so one of three disjuncts holds and the hypothesis is live.

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.** Its 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. Determination is not reached over the level by observations within it.

**Persistence.** A differential prediction distinguishing levels, or agreement on the indifference principle over observers.

*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 observed physics, 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 apart from empirical inquiry, its proponents accepting the disjunctive form is not a prediction.

**Narrowing recorded.** Signature tests narrow 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 axioms of the simplest, strongest true systematisation of facts. Nothing governs, regularity is all there is, and this avoids positing a modal relation nobody has accounted for.

Prescriptive: regularities on their own 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, determination standing on regularities.

**Conserving reach.** All regularities the distinction is asked of.

**Reach.** Its statement quantifies over the modal status of a regularity. 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 identification problem — what makes the necessitation relation necessitate — is unanswered to general satisfaction. Determination is not reached over regularities by regularities.

**Persistence.** A criterion distinguishing law from accident not resting on a primitive, or agreement that a primitive is acceptable.

*The report.*

**Approaches.** Best-system analysis — yields the distinction from simplicity and strength, measures being language-relative, which is the standing objection, addressed by appeal to natural properties, which is a primitive. Necessitation between universals — identification problem, what makes 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 relation: models that fit stand in place of a modal status determined. Primitivism weakens relation: governing stands basic rather than accounted for.

**Progress.** The best-system construction and its language-relativity. 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 now, 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.** Its 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. Determination is not reached over 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.

*The report.*

**Approaches.** The regress argument against objective tense — widely regarded as showing a difficulty, reached by taking tense as primitive, the exchange old and unresolved. Relativity — excludes a frame-independent present, reached by 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, reached by 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 reach: a present stands by stipulation rather than by detection. Growing block narrows reach: existence stands for past and present rather than all times.

**Progress.** 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, branches that apply were often built for application, and successes are salient while the vast unapplied remainder is not counted.

Selection account does not cover 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 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 needing explanation — local.

**Specifications withdrawn.**

**Common requirement.** That applicability is a phenomenon with an explanation.

**Conserving relation.** An account of why relation obtains, standing across cases including striking ones.

**Conserving reach.** All instances of application.

**Reach.** Its statement quantifies over why 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 applicability and inherits the access problem. The account is not preserved across cases by instances and the historical record.

**Persistence.** An account covering striking cases that both accounts accept, or agreement on what the historical record shows.

*The report.*

**Approaches.** Selection effect accounting — accounts for many cases, both accounts disputing whether it covers 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 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 striking cases are specific and documented.

**Narrowing recorded.** Structural realism weakens relation: applicability stands constitutive rather than accounted for. Mathematical naturalism narrows reach: the account stands for branches with physical origin.

**Progress.** Documented cases of application without prior physical motivation. Selection accounting for 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 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 permutations, quantum mechanics supplies this, and the discontinuity is real because identity is not a matter of degree.

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 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 entropy change on mixing.

**Conserving relation.** Entropy determined as a property of the system, determination standing independently of operations available.

**Conserving reach.** All mixings, including those interpolating between the two cases.

**Reach.** Its 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. Determination is not reached over mixings by extractable work.

**Persistence.** A measurement of entropy independent of operations available, or a direct probe of the interpolating case.

*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 entropy relative to available operations, reproducing every measurement, disagreement being over whether entropy is thereby subjective. Isotope experiments — separable isotopes giving mixing entropy, 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 reach: entropy stands derived relative to a stated operation set. The information-theoretic treatment weakens 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. 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 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 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.** Its 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 reached by a reformulation restoring the puzzle, documented over seventy years. The location is not preserved across formulations by any reconstruction.

**Persistence.** A location of the error standing against reformulations, or agreement on which assumption reformulations must drop.

*The report.*

**Approaches.** Epistemic blocking — the announcement not knowable in the relevant sense, formalised, reformulations restoring the puzzle with weaker assumptions. Temporal retention analysis — knowledge not retained through 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 reach: the account stands for a strategic formulation its authors distinguish from the original. Epistemic blocking narrows reach: the solution stands for formulations with the stronger knowledge assumption.

**Progress.** The seventy-year record of solutions and reformulations reaching each. 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 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.** Its 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 deterministic interpretations are empirically equivalent by construction. Determination is not reached over outcomes by observed frequencies.

**Persistence.** A measurement distinguishing fixed from chancy outcomes, which the empirical equivalence excludes.

*The report.*

**Approaches.** Statistical tests for randomness — confirm 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 reach: local determination stands excluded, nonlocal determination surviving. Algorithmic randomness narrows reach: randomness stands defined for sequences rather than for what produced them.

**Progress.** 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? Its 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 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 physical facts either fix experiential ones or do not.

**Conserving relation.** A determination of whether physical facts fix experiential ones, standing on what is examinable.

**Conserving reach.** The modal relation between the two sets of facts.

**Reach.** Its 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 concepts physically or it concedes the point. 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.

*The report.*

**Approaches.** Two-dimensional semantics — a framework separating 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 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 relation: the conceivability stands accounted for by the character of 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, 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 masses the question concerns.

**Conserving reach.** All masses, those not reached included.

**Reach.** Its statement quantifies over behaviour at masses not 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. Determination is not reached over 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.

An anticipated observing: molecules beyond 25 kDa are in quantum superposition.

*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, 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 performed.

**Narrowing recorded.** Collapse model constraints narrow reach: exclusion stands over the parameter ranges bounded rather than over the class.

**Progress.** The interference demonstrated to tens of thousands of atomic mass units. 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 its field where 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.** 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.** Its 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 unspecified settings by results on specified problems.

**Persistence.** An optimum defined where the option set is not enumerated in advance.

*The report.*

**Approaches.** The index policy — optimal for discounted independent-arm problems, 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 reach: optimality stands for discounted independent-arm problems under stated conditions. Non-stationary extensions narrow reach: bounds stand under a supplied variation budget. Regret minimisation narrows reach: the measure stands against a presupposed option set.

**Progress.** The proved index optimality under its conditions. The matched regret bounds. 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 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 anomalous cases.

**Conserving reach.** All responses, the microbiota, fetal, and tumour cases included.

**Reach.** Its 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 cases by responses consistent with more than one.

**Persistence.** A criterion accounting for anomalous cases without being defined by what it accounts for.

*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 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, 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 reach: active tolerance stands established for the foreign organisms studied. Pattern recognition receptors narrow reach: foreignness detection stands for 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 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, identification standing on measurement in established tumours.

**Conserving reach.** All tumours the question is asked of.

**Reach.** Its 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 systems tested, and sequencing of normal tissue finds driver mutations at high density without cancer, which constrains the dominant account from within and is accepted in its field. Identification is not reached over 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.

*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 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 its field. Field cancerisation and immune surveillance — accommodated by both accounts.

**Narrowing recorded.** Tissue recombination narrows reach: the result stands for systems tested. Normalisation experiments narrow reach: context dependence stands in 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, 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 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.** Behaviour has a locus — neural pathology or choice under constraint — local. Identifying it determines the response — local.

**Specifications withdrawn.**

**Common requirement.** That behaviour is produced by an identifiable mechanism at one level.

**Conserving relation.** The governing level identified, identification standing on measures altered in affected people.

**Conserving reach.** All affected people the question is asked of.

**Reach.** Its statement quantifies over which account governs 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 its field states, and remission rates and incentive responsiveness are read oppositely by the two accounts. 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.

*The report.*

**Approaches.** Neuroimaging — documents circuit changes reliably, cause and consequence not separated by imaging, which its 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 reach: neuroadaptation stands established in models, with enrichment effects measured there. Contingency management narrows reach: the responsiveness stands demonstrated under incentive schedules trialled.

**Progress.** The imaged dose-dependent circuit changes. The measured abstinence effects under incentives. The measured remission rates without treatment. 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 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 observed patterns bear on the mechanism.

**Conserving relation.** The mechanism producing community structure identified, identification standing on abundance and distribution data.

**Conserving reach.** All communities the question is asked of.

**Reach.** Its 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. Identification is not reached over communities by abundance and distribution data.

**Persistence.** Measured weights for stabilising and equalising mechanisms in natural communities, or a prediction on which the two models differ and a test discriminating them.

*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, weights not independently measured.

**Narrowing recorded.** Invasion experiments narrow reach: stabilising differences stand demonstrated for the species pairs tested.

**Progress.** 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-term 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 variables measured.

**Conserving reach.** All the stability variables the question is asked of.

**Reach.** Its statement quantifies over which stability the question is about, 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 variables by measurements.

**Persistence.** A stability concept both accounts accept, applied across variables.

*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 reach: an answer stands per stability concept rather than across them. Portfolio effects narrow 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 as it was. The hypothesis is carried by its field separately as an account of 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 its 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.** Its 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, identification standing on fossil ranges.

**Conserving reach.** All taxa the pattern is measured over.

**Reach.** Its statement quantifies over what produces constant extinction risk. Fossil ranges, with sampling and taxonomic definition affecting curves, are what is measured. The two are not the same.

**Unreachability.** Several mechanisms fit the same survivorship curves, genus definitions affect curves, and comparative analyses find environmental drivers dominating some intervals with weights unmeasured. Identification is not reached over taxa by fossil ranges.

**Persistence.** Measured weights for coevolutionary and environmental drivers, or a survivorship pattern only one mechanism fits.

*The report.*

**Approaches.** Survivorship analysis — its 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 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 curves, constraining the original inference.

**Narrowing recorded.** Coevolution experiments narrow reach: the mechanism stands demonstrated at short timescales in microbial systems.

**Progress.** The corrected survivorship curves and the reversal of its founding pattern. The demonstrated arms-race dynamics. The measured rapid evolution at host-parasite interfaces. The quantified effect of taxonomic definition on 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, mechanisms are traceable to immune regulation, and 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, identification standing among co-varying exposures.

**Conserving reach.** The rise across industrialised populations.

**Reach.** Its 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. 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 co-varying exposures.

*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 reach: causality stands established for one exposure and one outcome. Diagnostic-change analyses narrow 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, determination standing on measurement.

**Conserving reach.** Doses near or below detection.

**Reach.** Its 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 relevant magnitudes, which is measured. Determination is not reached over the low doses by measurements at detectable ones.

**Persistence.** A measurement at doses of regulatory interest, or agreement on a criterion for extrapolating below detection.

*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, doses of regulatory interest remaining below the reachable range for many agents. Epidemiology at low exposure — confounding dominating at 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 reach: adaptive responses stand established in specific systems. Large animal studies narrow reach: the reachable dose range stands extended without reaching regulatory magnitudes.

**Progress.** The assembled curve collections with their stated heterogeneity. Mechanisms established in specific systems. The measured confounding at low exposure. 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 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, separation standing across conditions.

**Conserving reach.** All conditions comparison is made over.

**Reach.** Its 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. Separation is not reached over conditions by comparisons of whole treatments.

**Persistence.** A design separating technique from context, or agreement on the level at which comparison should be made.

*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 reach: superiority stands established for disorders compared. Dismantling studies narrow reach: component contributions stand where 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 function, 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 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, identification standing separately from sleep state.

**Conserving reach.** All sleep in which dreaming occurs.

**Reach.** Its 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. Identification is not reached over sleep by stages and waking reports.

**Persistence.** A manipulation of dreaming separate from sleep state, or a content-specific effect established.

*The report.*

**Approaches.** Selective deprivation — confounded by arousal and rebound, which its field has documented extensively. Lesion cases abolishing dream report — rare, 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 unclear now.

**Narrowing recorded.** Targeted memory reactivation narrows reach: the effect stands established for sleep rather than for dreaming. Lucid dreaming narrows reach: the route stands open for 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 timing of divergence.

**Conserving relation.** Origination times determined, determination standing across the two records.

**Conserving reach.** The interval the concentration is measured over.

**Reach.** Its 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 recalibrated analyses the divergence has narrowed to the Ediacaran and 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. Determination is not reached over the interval by first appearances and clocks.

**Persistence.** Agreement between the two records on origination times, or a preservation model accounting for the remaining difference.

*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 morphological events.

**Narrowing recorded.** Preservation modelling narrows reach: part of the concentration stands accounted for as bias. Trace fossil work narrows reach: the record of activity stands extended earlier for 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 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, indexing standing across the sampled record.

**Conserving reach.** The interval between anatomical and behavioural appearance.

**Reach.** Its 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. 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.

*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 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 relation: modernity stands as a set of separately appearing traits rather than a package indexing capacity. Demographic modelling narrows reach: the pattern stands accounted for by population size and connectedness where those are estimated.

**Progress.** The repeatedly extended record of symbolic behaviour. 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 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.** 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.** Its 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 comparison its statement requires; and gains are concentrated on the least g-loaded subtests, measured consistently. The sameness is not preserved across cohorts by test scores.

**Persistence.** A measure demonstrated invariant across cohorts compared.

*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 reach: reversal stands established within families for the population analysed. Subtest decomposition narrows reach: gains stand located by subtest rather than in the construct.

**Progress.** Established failures of measurement invariance. The measured subtest concentration of gains. The within-family reversal. 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.** Its 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.

*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 reach: the host lineage stands established without the order. Culturing narrows reach: the cell biology stands observable for 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.** Its 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. Determination is not reached over variables by measured trajectories.

**Persistence.** A trajectory the two descriptions predict differently, or agreement on what a moving setpoint concedes.

*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 reach: setpoint machinery stands established for temperature and osmolality. Moving setpoint models weaken relation: the target stands variable within the feedback framework.

**Progress.** 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, determination standing for regimes not tried.

**Conserving reach.** All regimes, those not tried included.

**Reach.** Its statement quantifies over the outcome under regimes not tried, 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 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. Determination is not reached over 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.

*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 reach: the reduction stands in settings where the programme continues, and reverses when relaxed. Mixing strategies narrow reach: the advantage stands in 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 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 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.** Its 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.

*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 observe 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 reach: results stand in constructed settings where evaluation remains available. Red-teaming narrows reach: presence of failure stands established for cases constructed.

**Progress.** The measured behavioural improvements from feedback training. The documented gameability of reward models. 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.** Benchmark samples a capability — local. 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, measurement standing after benchmark is published.

**Conserving reach.** Capability on tasks not in benchmark.

**Reach.** Statement quantifies over capability on tasks not in benchmark. Benchmark items, and constructed held-out sets that become public in turn, are what is measured. 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. 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.

*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, 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: measurement stands where the set is unreleased, and is not independently verifiable there. Aggregate suites narrow the reach: single-benchmark artefacts stand reduced across suite.

**Progress.** Measured correlations with downstream performance where both are available. Documented contamination and its partial detection. 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.** Counterfactual — value under an unadopted alternative or no standard — is estimable — local.

**Specifications withdrawn.**

**Common requirement.** That the value under unadopted arrangement is comparable.

**Conserving relation.** A comparison of adopted and unadopted arrangements, standing at scale the adopted one reached.

**Conserving reach.** All cases the comparison is asked of.

**Reach.** Statement quantifies over a counterfactual arrangement at scale. Adopted arrangement, and pre-standard periods that differ in many other respects, are what is measured. Two are not the same.

**Unreachability.** Case studies on both sides are contested, formal models produce both outcomes under stated conditions with conditions being the empirical question, and competing-standard episodes are confounded by firm strategy and timing. Comparison is not reached over cases by adopted arrangement and pre-standard periods.

**Persistence.** A comparison at scale between adopted and unadopted arrangements, or agreement on conditions under which each outcome follows.

*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, 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: comparison stands for period and cases in which two standards ran together.

**Progress.** Measured gains where standards were adopted. Documented lock-in costs in specific cases. 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. Work's partitionability is fixed — local.

**Specifications withdrawn.**

**Common requirement.** That coordination cost and output are measurable against team size.

**Conserving relation.** Effect of adding people isolated, isolation standing with all else held equal.

**Conserving reach.** All projects the effect is asked of.

**Reach.** Statement quantifies over effect of adding people, all else equal. Projects where adding people co-occurs with schedule pressure, scope change, and architecture choices are what is measured. 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 executed at scales far below cases in question. Isolation is not reached over projects by observations where additions co-occur with other changes.

**Persistence.** A measurement of effect with architecture and scope held fixed, at scale of cases in question.

*The report.*

**Approaches.** Case histories — support law in 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 conditionality. Software estimation models — incorporate diseconomies of scale as fitted parameters, fitting, parameters estimated from the same data. Controlled experiments — small-team laboratory studies finding coordination overhead, at scales far below cases in question. Architecture studies — modularity reducing coupling measurably, establishing mechanism's dependence on structure.

**Narrowing recorded.** Open-source studies narrow the reach: positive returns stand in modular projects. Controlled experiments narrow the reach: overhead stands measured at small-team scale.

**Progress.** Measured returns to contributors in modular projects. Measured coordination overhead at small scale. 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 not 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 measure is estimable.

**Conserving relation.** Total harm compared against counterfactual regime, comparison standing with concurrent changes accounted for.

**Conserving reach.** All settings the measure is adopted in.

**Reach.** 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. Two are not the same.

**Unreachability.** Before-after series are confounded by concurrent trends, which literature addresses with varying success, and risk transfer to other users is measured in some settings and concealed by aggregate series. Comparison is not reached over settings by outcomes after adoption.

**Persistence.** A comparison against counterfactual regime with concurrent changes accounted for, or measured magnitudes for mechanical and behavioural components across settings.

*The report.*

**Approaches.** Before-after mortality series — show large reductions, confounded by concurrent trends, which literature addresses with varying success. Behavioural measurement — documents adaptation directly in specific behaviours, establishing mechanism, magnitude relative to 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 aggregate series can conceal. Full-compensation theory — predicts complete offset, its critics holding mortality series refute it, its defenders disputing counterfactual.

**Narrowing recorded.** Staggered-adoption designs narrow the reach: net reduction stands for jurisdictions and measures analysed. Risk transfer studies narrow the reach: transfer stands measured in settings studied.

**Progress.** Measured reductions in death and serious injury. Documented behavioural adaptation. Staggered-adoption results with partial offsets. 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 requested.

**Specifications.** There is a present cost incurred by a past choice — local. Cost is attributable to identifiable code properties — local.

**Specifications withdrawn.**

**Common requirement.** That the later cost is attributable to earlier choice.

**Conserving relation.** Later cost attributed to earlier choice, attribution standing for changes not requested.

**Conserving reach.** All changes, those not requested included.

**Reach.** Statement quantifies over cost of changes not requested. Code properties now, and change history so far, are what is measured. Two are not the same.

**Unreachability.** Counterfactual code carrying disputed cost is unavailable, and change-cost measurement covers changes made while unmade changes carry disputed cost. Attribution is not reached over unrequested changes by code properties and change history.

**Persistence.** A measurement of cost of changes not requested, or agreement on a proxy that tracks it.

*The report.*

**Approaches.** Metric-defect correlation — established for several metrics, attribution to earlier decision being the gap. Change-cost measurement — measures cost of changes made, unmade changes carrying disputed cost. Refactoring experiments — show improvements in specific measures, cost-benefit over 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: improvement stands in measures taken. Change-cost measurement narrows the reach: cost stands measured for changes made.

**Progress.** Established metric-defect correlations. Measured cost of changes made. Measured improvements from refactoring. 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.** Channel's content decomposes into wanted and unwanted components — local. 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.** Statement quantifies over which component is which, independent of purpose. Total variation, and a model supplied by analyst, are what is measured. Two are not the same.

**Unreachability.** Every decomposition is exact given its model, and model is supplied; and added noise improving detection in nonlinear systems, measured across physical and biological settings, establishes labels are purpose-relative. Decomposition is not reached over 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.

*The report.*

**Approaches.** Modelling source — makes decomposition exact given model, 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 labels are purpose-relative. Matched filtering — optimal given expected signal shape, which is supplied. Information-theoretic bounds — exact given channel and source distributions.

**Narrowing recorded.** Blind source separation narrows the reach: recovery stands under stated independence or sparsity assumptions. Matched filtering narrows the reach: optimality stands for a supplied signal shape.

**Progress.** Exact decompositions under stated models. Measured stochastic resonance across settings. 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 device.

**Conserving relation.** A determination of which output is the product, standing on energy flows.

**Conserving reach.** All devices the question is asked of.

**Reach.** Statement quantifies over which output is the product. Energy flows, all measurable, none labelled, are what is measured. 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. Determination is not reached over devices by energy flows.

**Persistence.** A measure of product not resting on a specified reference or convention, or agreement that the labels are set by purpose.

*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 reversible ideal, requiring specification of which output the ideal produces. Cogeneration accounting — standards allocating between outputs by convention, conventions differing between jurisdictions, which is documented. Life-cycle accounting — extends boundary, boundary being drawn.

**Narrowing recorded.** Exergy analysis narrows the reach: measure stands relative to a specified reference environment. Cogeneration standards narrow the reach: allocation stands by convention within a jurisdiction.

**Progress.** Exact thermodynamic accounting and its invariance under relabelling. Exergy measure with its stated reference. 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 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 rate, standing on parameters estimated independently of rate.

**Conserving reach.** All taxa the rate is measured across.

**Reach.** Statement quantifies over what sets rate. Measured rates across taxa, and population-genetic parameters estimated from the same genomes, are what is measured. 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 trait. Account is not reached over taxa by rates and parameters estimated together.

**Persistence.** An estimate of effective population size independent of genomes the rates are measured in.

An explaining: the drift barrier accounts for error-prone polymerases and parts from the view that selection fine-tunes gene-specific mutation rates.

*The report.*

**Approaches.** Mutation accumulation experiments — measure rates directly across many species, establishing 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 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 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.** Directly measured rates across species. Measured inverse correlation with effective population size. Measured proofreading costs. 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, comparison standing at scale the adopted regime reached.

**Conserving reach.** All sectors and regimes the question is asked of.

**Reach.** 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. 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. Comparison is not reached over sectors and regimes by proxies under regimes adopted.

**Persistence.** A comparison against an unadopted regime at scale, or agreement on conditions under which each effect follows.

*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 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 domains where prizes and open collaboration operate. Sector comparison narrows the reach: account stands per sector by imitation cost rather than generally.

**Progress.** Measured cross-country associations. Historical episode results. Measured litigation and blocking costs. 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. Determinant is either external backing or expectations — local.

**Specifications withdrawn.**

**Common requirement.** That there is a determinant of value.

**Conserving relation.** A determinant of value identified, identification standing for regimes with the same fundamentals.

**Conserving reach.** All regimes the question is asked of, fiat regimes included.

**Reach.** Statement quantifies over why acceptance persists. Prices and holdings under regimes that persisted, and episodes where they did not, are what is measured. 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 episodes studied. Identification is not reached over regimes by prices and holdings.

**Persistence.** A selection among multiple equilibria, or a determinant predicting onset in episodes where acceptance failed.

*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 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: account stands for commodity and convertible regimes. Tax-driven demand narrows the reach: acceptance stands accounted for within jurisdiction imposing obligation.

**Progress.** Historical record of backing and issuance against loss of value. Proved equilibrium multiplicity. 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.** Agent's action affects an observable outcome — local. A contract on observables can approximate one on action — local.

**Specifications withdrawn.**

**Common requirement.** That the divergence is representable as a contracting problem.

**Conserving relation.** A contract aligning parties, alignment standing on observables where action is unobserved.

**Conserving reach.** All settings the contracting problem is posed in.

**Reach.** Statement quantifies over agent's unobserved action. Outcomes and observables, from which action is inferred under assumptions, are what is measured. 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. Alignment is not reached over settings by contracts on observables.

**Persistence.** A contract aligning parties where tasks are multiple or production is joint, or agreement that alignment is partial by construction.

*The report.*

**Approaches.** Optimal contracting theory — solves stated problem, 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 phenomenon is not uniform. Relational contracts and reputation — achieve alignment where repeated interaction is available, conditions being the empirical question.

**Narrowing recorded.** Optimal contracting narrows the reach: solution stands under stated informational conditions. Relational contracts narrow the reach: alignment stands where repeated interaction is available.

**Progress.** Characterised optimal contracts under stated conditions. Two proved limitations. 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. Operative regime is identifiable — local.

**Specifications withdrawn.**

**Common requirement.** That the regime determining which relation holds is identifiable in advance.

**Conserving relation.** Operative regime identified in advance, identification standing on aggregate series.

**Conserving reach.** All periods the question is asked of.

**Reach.** Statement quantifies over which regime obtains. Aggregate series in which regime and outcome are jointly determined are what is measured. 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 literature documents. Identification is not reached over periods by series in which regime and outcome are jointly determined.

**Persistence.** A regime identified independently of outcome, or agreement on an estimation method for multiplier.

*The report.*

**Approaches.** Model comparison — both results derivable in their own models, accepted by both accounts. Zero lower bound episodes — providing conditions where 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 dispute, estimates varying widely by method and by regime, which literature documents. Microfoundation — reproduces result in models with nominal rigidity, which is the disputed assumption.

**Narrowing recorded.** Zero lower bound analysis narrows the reach: mechanism stands supported in episodes where constraint binds.

**Progress.** Derivations in both model classes. Empirical work at zero lower bound. 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 process, identification standing on reproductive outcomes.

**Conserving reach.** All cases the accounting is applied to.

**Reach.** Statement quantifies over which accounting describes process. Reproductive outcomes, decomposable either way, are what is measured. Two are not the same.

**Unreachability.** Two accountings are proved to give the same predictions under general conditions, so no reproductive outcome separates them. Identification is not reached over cases by outcomes decomposable either way.

**Persistence.** Conditions under which two accountings predict differently, or agreement that the rivalry is between accountings rather than hypotheses.

*The report.*

**Approaches.** Kin selection tests — relatedness predicting cooperation across many taxa, robustly, establishing pattern under either accounting. Experimental evolution — produces cooperation and cheating dynamics as predicted, under either accounting. Formal equivalence results — two accountings agreeing under general conditions, the strongest item, dissolving rivalry rather than settling it. Public goods experiments in humans — establish conditional cooperation and punishment, mechanism-level findings. Eusociality dispute — a large public disagreement with a rebuttal signed by many, mathematical points not resolved by exchange.

**Narrowing recorded.** Mechanism-level tests narrow the reach: each mechanism stands confirmed in its own domain rather than across cases.

**Progress.** Measured relatedness-cooperation association across taxa. Quantified threshold. Formal equivalence result. 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 relation between two tasks — local. Task similarity is measurable independently of transfer — local.

**Specifications withdrawn.**

**Common requirement.** That similarity is specifiable before transfer is measured.

**Conserving relation.** Similarity specified before transfer is measured, specification standing for tasks not chosen.

**Conserving reach.** All task pairs, those not specified included.

**Reach.** Statement quantifies over application to a task not specified. Pairs of tasks selected by experimenter, whose similarity is judged, are what is measured. Two are not the same.

**Unreachability.** Similarity is characterised after tasks are chosen, and far-transfer null record is now the field's own summary while expertise is established as domain-bound. 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.

*The report.*

**Approaches.** Identical-elements analysis — predicts near transfer well, similarity characterised after 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 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.** Century-long near-transfer record. Consistently null far-transfer trials. Established domain-boundedness of expertise. 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, identification standing on a cost specified independently of movements.

**Conserving reach.** All movements the rule is asked of.

**Reach.** Statement quantifies over selection rule. Recorded kinematics and muscle activity, fit by several models, are what is measured. 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. Identification is not reached over movements by kinematics and muscle activity.

**Persistence.** A cost specified independently of movements it accounts for, or a measurement distinguishing candidate costs.

*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 cost limited and known to be so. Muscle synergy analysis — finds low-dimensional structure, whether synergies are neural or an artefact of decomposition disputed, 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 task-relevant variables specified.

**Progress.** Trajectory fits under several cost functions. Stated identifiability limits. Measured low-dimensional structure. 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, independence standing in tasks employed.

**Conserving reach.** All tasks the question is asked of.

**Reach.** Statement quantifies over thought as distinct from language use. Task performance, in participants who have language available, is what is measured. Two are not the same.

**Unreachability.** Verbal interference removes many effects, establishing they are partly strategic rather than absent, and remaining effects are read oppositely; spatial frame differences are confounded by environment and practice, which the field notes. Independence is not reached over tasks by performance in participants who have language available.

**Persistence.** A task measuring thought with language unavailable, or agreement on what interference results establish.

*The report.*

**Approaches.** Verbal interference designs — reduce or remove many effects, the strongest route, establishing 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 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: residual effects stand for tasks where language is unavailable as a strategy. Infant studies narrow the reach: capacities stand established before language.

**Progress.** Replicated lateralised colour effects. Measured reduction under verbal interference. Cross-community spatial differences. Prelinguistic capacities. 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.** Representation has a determinate sparseness — local. Recording samples population without bias — local.

**Specifications withdrawn.**

**Common requirement.** That the sampled neurons represent population.

**Conserving relation.** Sparseness determined across a population, determination standing on an unbiased sample.

**Conserving reach.** The whole population the concept is represented in.

**Reach.** 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. Two are not the same.

**Unreachability.** Human recordings and animal population recordings give opposite pictures with sampling procedures differing; and both codes are proved efficient under different constraints, which does not select. Determination is not reached over population by a screened sample.

**Persistence.** An unbiased sample of population in areas where selective cells are found.

*The report.*

**Approaches.** Single-unit recording in humans — finds highly selective cells reliably, sampling driven by a screening procedure, which authors state. Large-scale population recording — finds broad, weak tuning dominating in animal cortex, establishing distributed picture in accessible areas. Decoding comparisons — population decoding generally outperforming sparse, depending on 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 areas imaged. Population recording narrows the reach: distributed picture stands established in accessible areas.

**Progress.** Invariant selective responses across format. Measured broad tuning in animal cortex. Decoding comparisons. 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 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, reconstruction standing across proxy families.

**Conserving reach.** Early state, which occurred once.

**Reach.** 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. Two are not the same.

**Unreachability.** Proxy bounds and greenhouse requirements conflict, and proxy families are contested in calibration against each other, with newer aqueous-chemistry technique returning bounds raised by an order against classic studies. Reconstruction is not preserved across proxy families.

**Persistence.** Proxy families agreeing on early composition, or a constraint independent of model-dependent calibration.

*The report.*

**Approaches.** Carbon dioxide greenhouse — requires concentrations the classic proxy studies bounded below, newer aqueous-chemistry paleosol technique returning bounds raised by an order and reopening route, 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, proxy estimates having converged low, with vesicle, imprint, and isotope instruments independently bounding early pressure at or below modern, closing thick-atmosphere version of this route. Revising solar model — a more massive early sun losing mass, constrained by other observations and not favoured.

**Narrowing recorded.** Nitrogen route stands closed in its thick-atmosphere version by three independent instruments converging low. Aqueous-chemistry technique narrows the reach: raised bound stands under that technique's calibration.

**Progress.** Quantified weathering feedback. Classic and revised carbon dioxide bounds. Photochemical limits on methane. 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.** Mapping was fixed by a process — local. Code's present structure carries evidence of it — local.

**Specifications withdrawn.**

**Common requirement.** That present structure constrains origin.

**Conserving relation.** Origin of mapping identified, identification standing on code's present structure.

**Conserving reach.** Fixing of mapping, which occurred once.

**Reach.** Statement quantifies over origin of mapping, constrained by its present structure. Code's structure, extant variants, and laboratory affinity measurements are what is measured. Two are not the same.

**Unreachability.** Error-minimisation, stereochemical, and coevolutionary accounts each fit code's structure, which is consistent with several origins, and accounts are not separable by that structure. Identification is not reached over fixing by present structure.

**Persistence.** A feature of code that one account predicts and the others do not, or a constraint on origin independent of present structure.

*The report.*

**Approaches.** Error-minimisation statistics — 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 code. Coevolution accounts — block structure accounted for by biosynthetic relatedness, with good fit, also consistent with other origins. Code variants — natural and engineered variants showing 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 pairs tested. Coevolution accounts narrow the reach: block structure stands accounted for rather than the whole mapping.

**Progress.** Measured ranking against random alternatives on every metric tried. Measured codon-related affinities. Block-structure fit. 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. Capacity's components are identifiable — local.

**Specifications withdrawn.**

**Common requirement.** That the capacity's components are identifiable.

**Conserving relation.** Emergence of syntactic capacity determined as gradual or discrete, determination standing on anatomy, artefact, and comparative data.

**Conserving reach.** Emergence, which occurred once.

**Reach.** Statement quantifies over syntactic capacity, indexed by anatomy and artefact. Comparative animal data, anatomy, ancient DNA, and material culture are what is measured. 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. Determination is not reached over emergence by comparative data, anatomy, and material culture.

**Persistence.** An index of syntactic capacity in record, or an animal analogue of recursive combination.

*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 capacity. Computational modelling narrows the reach: sufficiency stands established in simulation.

**Progress.** Established precursors for each component but recursion. Anatomical indicators with their stated inferential limits. Documented revision of candidate-gene account. 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 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.** Statement quantifies over all simply connected closed three-manifolds. Flow, its singularities, and surgery construction are what is worked with. Two are the same: method reaches domain the statement quantifies over.

**Unreachability.**

**Persistence.** Settled. Proof stands, verified by several independent groups over years, and is accepted without dissent.

*The report.*

**Approaches.** Geometrisation programme — frames conjecture within a decomposition of three-manifolds. Ricci flow — founds method. Surgery construction — completes it, continuing flow through singularities. Verification — carried out by several independent groups over the following years.

**Narrowing recorded.**

**Progress.** Proof. Geometrisation theorem it establishes. Verification record.

**Own record.** Difficulty stood in constructing proof, not in 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 laws, in a boundary condition, or in observer's position — local. These are exhaustive — local.

**Specifications withdrawn.**

**Common requirement.** That the asymmetry has a locus.

**Conserving relation.** Locus of asymmetry determined, determination standing without boundary condition stipulated.

**Conserving reach.** Status of boundary condition.

**Reach.** Statement quantifies over status of boundary condition. Consequences of whatever condition obtained are what is measured. Two are not the same.

**Unreachability.** Statistical derivation is complete given past hypothesis, and each cosmological proposal contains a stipulation at the same place. Determination is not reached over boundary condition by its consequences.

**Persistence.** A derivation of boundary condition not stipulating it, or agreement that a stipulation is acceptable.

*The report.*

**Approaches.** Statistical derivation — complete given past hypothesis. Cosmological derivation of 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 entropy gradient, consistent and not deriving 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 gradient.

**Progress.** Statistical derivation given hypothesis. Memory-asymmetry account of temporal experience. 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.** Functional equation, the Hodge decomposition, the L-function's analytic continuation, and gauge symmetry are each proved — universal under respective field. Conjecture is not proved — local.

**Specifications withdrawn.**

**Common requirement.**

**Conserving relation.** In each, conjecture holding over everything it quantifies over: every zero on line, every Hodge class algebraic, 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 theory.

**Reach.** Statements quantify over every non-trivial zero, every Hodge class on every such variety, every elliptic curve over rationals, and every compact simple gauge group. Verified instances and partial results are what is measured. Two are not the same.

**Unreachability.**

**Persistence.** In each case, a proof of conjecture, or a counterexample: a zero off 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.

An anticipated observing: planar maximally supersymmetric Yang–Mills amplitudes are computed to eight loops by antipodal duality from a form factor, and to nine loops with an independent validation and a concurrent result.

*The report.*

**Approaches.** In each, the field's own record of verified instances and partial results, standing in that field's literature: zeros verified on critical line and 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; mass gap seen in lattice computation and constructed in no continuum theory; perturbative amplitudes of planar maximally supersymmetric Yang–Mills computed to nine loops at six particles, an expansion at many colours and no existence or mass gap at a compact simple gauge group, nothing changed at Persistence. Each open.

**Narrowing recorded.**

**Progress.** Proved symmetries. 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 regime in question — local. 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 epochs the question concerns.

**Conserving reach.** Conditions at and before the earliest accessible epoch.

**Reach.** Statement quantifies over conditions at and before the earliest accessible epoch. Microwave background and later are what is measured, with earlier epochs reached only through model-dependent inference. Two are not the same.

**Unreachability.** Singularity theorems are proved under energy conditions quantum fields violate, which is established; and distinguishing signatures between accounts are proposed and unobserved, while dark energy equation of state is measured with improving precision and not to accuracy that separates fate scenarios. Determination is not reached over epochs by observation of accessible ones.

**Persistence.** A distinguishing signature observed in primordial gravitational waves, or a dark energy equation of state measured to accuracy separating scenarios.

*The report.*

**Approaches.** Singularity theorems — proved under energy conditions, conditions violated by quantum fields, which is established. Past-incompleteness arguments — apply to expanding spacetimes on average, their reach for quantum regimes disputed by 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 accuracy that separates scenarios.

**Narrowing recorded.** Singularity theorems narrow the reach: past-incompleteness stands proved under stated energy conditions. Observational cosmology narrows the reach: constraint stands for the last thirteen billion years.

**Progress.** Proved theorems under their stated conditions. Precision constraints on the last thirteen billion years. Established violation of energy conditions by quantum fields. 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. Causes are located either in properties of persons or in properties of arrangements — local. 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.** Explanatory level identified, identification standing where two levels vary independently.

**Conserving reach.** All outcomes the question is asked of.

**Reach.** Statement quantifies over which level does explanatory work. Outcomes in which both levels are present, and cannot be varied independently, are what is measured. Two are not the same.

**Unreachability.** Structuration makes two inseparable, which its critics hold removes the question; multilevel modelling returns a partition conditional on levels specified; and natural experiments where one level changes and the other does not are rare and confounded by selection into change. Identification is not reached over outcomes by measurements in which both levels are present.

**Persistence.** A comparison in which two levels vary independently, or agreement on a level at which explanation is owed.

*The report.*

**Approaches.** Structuration — structure as both medium and outcome of action, widely adopted and criticised as redescribing relation rather than analysing it, making 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 aggregation problem, the same regularity following from many individual-level specifications, and emergence objection. Multilevel statistical modelling — partitions variance between individual and contextual levels, returning a partition conditional on levels specified. Natural experiments in institutional change — comparison available where arrangements change and personnel do not, or reverse, rare and confounded by selection into change.

**Narrowing recorded.** Critical realism narrows the reach: separation stands over a stipulated temporal sequence. Multilevel modelling narrows the reach: partition stands conditional on levels specified.

**Progress.** Measured persistence of regularities across changes of personnel. Variance partitions under specified levels. 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 conventional phonon-mediated mechanism established — universal under superconductivity, and given against which statement is posed. Metals are described by Fermi-liquid theory — universal under condensed matter, and normal-state given against which strange metal stands. Physics of 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 pairing — expressible within framework.

**Conserving relation.** A mechanism established for real materials, establishment standing from minimal model to materials.

**Conserving reach.** Real materials' pairing and normal state.

**Reach.** Statement quantifies over real materials' pairing and normal state. The field's best numerical methods disagree on minimal model's ground state at relevant parameters, by the field's own multi-method benchmark; at strong coupling in relevant regime, stripe order is established as model's low-energy state; and spectroscopic probes reach different regimes of phase diagram, with their reaches stated in review literature. Two are not the same.

**Unreachability.** The best methods disagree on minimal model, by the field's own benchmark, and model-versus-material gap stands stated in the field's own record; probes disagree on pseudogap's onset, a standing characterization dispute recorded in reviews. Establishment is not reached over materials by results within minimal model or by probes.

**Persistence.** Mechanism established to the field's agreement; a theory with predictive capacity for critical temperatures from composition.

An explaining: spin-fluctuation-mediated pairing is the common thread 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 their observed properties.

*The report.*

**Approaches.** Materials search — critical temperatures measured across copper-oxide families, hydride superconductivity standing at higher temperatures under pressure with mechanism there recorded as conventional, nickelate superconductivity established, extending unconventional family. Numerical attack on minimal model — multi-method benchmark standing, stripe order established as model's relevant low-energy state, pairing glue identified within model at intermediate coupling, model-versus-material gap stated in the field's own record. Spectroscopic mapping — pairing symmetry established, phase diagram mapped, pseudogap characterized by multiple probes which disagree on its onset. Quantum simulation — pseudogap observed in a cold-atom realization of model, by thermodynamic and spectroscopic measurement at newly attainable temperatures, a new instrument class entering record. Phenomenology of intertwined orders — orders characterized and their relations stated, answering adjacent questions by its own account where mechanism is concerned.

**Narrowing recorded.** Numerical work narrows the reach: pairing glue stands identified within minimal model at intermediate coupling rather than for materials. Quantum simulation narrows the reach: pseudogap stands realized in a model system.

**Progress.** Established pairing symmetry. Mapped phase diagram across material families. Stripe order established as minimal model's relevant low-energy state. Superconducting state qualitatively understood, in the field's own summary. Cold-atom realization. A multilayer study: one relation measured within an unresolved subject, its mechanism no nearer established.

**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.** Probability calculus as shared formalism, standing axiomatized — universal under probability theory. Applications — statistical mechanics, quantum theory, statistical inference — each presupposing a reading — local.

**Specifications withdrawn.**

**Common requirement.** A fact about truthmakers of probability statements.

**Conserving relation.** A reading of what probability is, standing on formalism and its applications.

**Conserving reach.** All probability statements the readings are applied to.

**Reach.** Statement quantifies over what probability is. Frequencies, and calculus's theorems, are what is measured — recorded in literature as fitting every interpretation by construction. Two are not the same.

**Unreachability.** Calculus and frequencies fit every reading by construction, and interpretations stand empirically equivalent over shared formalism. A reading is not reached over statements by frequencies and theorems.

**Persistence.** A consequence on which readings differ, which interpretation-neutrality of calculus excludes over shared formalism.

*The report.*

**Approaches.** Frequentism — carries statistical practice, single-case and reference-class objections standing recorded. Subjectivism — coherence and convergence results standing, priors dependence recorded. Propensity accounts — standing defended, inversion objection recorded. Best-system chances — standing as recorded middle, standards of the best system disputed.

**Narrowing recorded.** Best-system chances weaken relation: chance stands extracted from a pattern under disputed standards rather than a reading established.

**Progress.** Axiomatization. Representation and convergence theorems. Interpretation-neutrality of 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 updating rule — universal under probability theory. Rule's founding scope — evidence as propositions learned — standing silent on centered information — local.

**Specifications withdrawn.**

**Common requirement.** An extension of updating rule to self-locating evidence.

**Conserving relation.** An extension of updating rule to centered information, standing across the case family.

**Conserving reach.** All cases in family the extensions are applied to.

**Reach.** Statement quantifies over rule's extension. Theorems of each proposed extension are what is measured — each consistent, returning different verdicts on the case. Two are not the same.

**Unreachability.** Consistent extensions stand on both sides and case's verdict varies with extension chosen; betting setups encode answer in their stakes, which the field records. Extension is not reached over the case family by theorems of extensions.

**Persistence.** A criterion selecting between consistent extensions, or a case on which they agree and one is shown correct.

*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 answer in their stakes. Applications — cosmological and many-worlds applications standing recorded, importing dispute into physics' own reasoning.

**Narrowing recorded.**

**Progress.** 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. Adequacy conditions the field states — avoiding repugnant and sadistic conclusions, respecting dominance and transitivity — local.

**Specifications withdrawn.**

**Common requirement.** A ranking satisfying stated conditions.

**Conserving relation.** A ranking of outcomes satisfying stated adequacy conditions jointly.

**Conserving reach.** All outcomes differing in number of people.

**Reach.** Statement quantifies over ranking. Impossibility theorems are what is established: stated adequacy conditions are jointly unsatisfiable, standing proved. Two are not the same — instruments prove conditions cannot all be satisfied, and do not select which to give up.

**Unreachability.** No ranking satisfies stated conditions jointly, standing proved. Relation is not preserved over outcomes by any principle meeting conditions.

**Persistence.** Agreement on which adequacy condition is given up.

*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. Impossibility programme — ending in proved limitation, no theory satisfying conditions jointly, theorems standing as the field's settled meta-result. Condition-rejection — each condition's rejection standing defended somewhere in record.

**Narrowing recorded.** Each principle weakens relation: a ranking stands with one stated condition given up, violated condition proved in each case.

**Progress.** Impossibility theorems. Mapped violation for each principle. 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. 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 dilemma, standing on practice.

**Conserving reach.** All mathematical practice the readings are applied to.

**Reach.** Statement quantifies over existence of abstract objects. Proofs and applications are what is measured — recorded as invariant under ontological readings. Two are not the same.

**Unreachability.** Practice is invariant under readings, and dilemma's two horns stand as posed. Account is not reached over practice by proofs and applications.

**Persistence.** An account meeting both horns, or agreement on which horn is given up.

*The report.*

**Approaches.** Indispensability — standing defended, nominalist reconstructions standing against it, with their scope recorded. Reconstruction programmes — standing partially executed, their completeness disputed. Fictionalism — standing defended, applicability of fiction standing as its recorded question. Structuralism — standing as recorded middle, ontology of structures inheriting dispute.

**Narrowing recorded.** Reconstruction programmes narrow the reach: reformulation stands for portion executed. Fictionalism weakens relation: usefulness stands in place of truth.

**Progress.** Stated dilemma. Executed portion of reconstructions with its scope recorded. Recorded invariance of practice under 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.** Sciences' causal practice — experiment, inference, explanation — standing as phenomenon — universal under scientific practice. Fundamental physics' time-symmetric laws standing as constraint — universal under physics.

**Specifications withdrawn.**

**Common requirement.** An account of relation between practice and fundamental description.

**Conserving relation.** Standing of causal structure determined, determination reaching from practice to fundamental description.

**Conserving reach.** Both special sciences' practice and fundamental description.

**Reach.** Statement quantifies over standing of causal structure. Correlations and intervention outcomes are what is measured, discovery formalisms recovering causal structure from them under stated assumptions, recorded as assumptions. Two are not the same.

**Unreachability.** Formalisms carry practice under assumptions, and interventionist account's circularity for foundations stands recorded by its own literature; grounding of asymmetry in thermodynamic gradient stands disputed in its reach. Determination is not reached over both levels by correlations and intervention outcomes.

**Persistence.** An account of relation between practice and fundamental description that both accounts accept.

*The report.*

**Approaches.** Regularity and counterfactual analyses — standing, with their counterexample records. Interventionist formalism — carries special sciences' practice, its circularity for foundations standing recorded by its own literature. Physics-based accounts — ground asymmetry in thermodynamic gradient, grounding's reach disputed. Eliminativism at base — standing defended from founding critique onward.

**Narrowing recorded.** Interventionist formalism narrows the reach: account stands for special sciences' practice under stated assumptions. Physics-based accounts narrow the reach: asymmetry stands grounded where thermodynamic gradient obtains.

**Progress.** Discovery formalisms and their stated assumptions. Counterexample records for each analysis. Recorded circularity of 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. Concordance measurement of accelerating expansion — universal under cosmology.

**Specifications withdrawn.**

**Common requirement.** An account of measured value.

**Conserving relation.** An account of measured value, standing on a measurement distinguishing candidates.

**Conserving reach.** All accounts of value.

**Reach.** Statement quantifies over value's explanation. Value is measured, estimate computed, and discrepancy stands; explanation candidates stand recorded as returning no differing measurements within reach. Two are not the same.

**Unreachability.** Candidates return no differing measurements within reach, natural symmetry candidates stand recorded as excluded or incomplete, and no-go results stand against simple dynamical adjustment. Account is not reached over candidates by measured value.

**Persistence.** A measurement on which candidate accounts differ, or agreement on whether environmental selection explains.

An anticipated observing: the vacuum contributions, determined independently with no reason to cancel, part from observation by roughly a hundred and twenty orders of magnitude.

*The report.*

**Approaches.** Symmetry mechanisms — natural candidates recorded as excluded or incomplete in the field's reviews. Dynamical adjustment — no-go results against simple adjustment standing. Anthropic route — standing defended, with the field's recorded dispute over whether environmental selection explains. Modified-gravity routes — standing constrained by multimessenger bounds.

**Narrowing recorded.** Anthropic route weakens relation: value stands selected environmentally rather than accounted for dynamically.

**Progress.** Measured value and computed estimate. Exclusions of natural symmetry candidates. No-go results against simple adjustment. 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. Collider's null results at 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.** Statement quantifies over criterion of explanatory debt in the field's method. Particle spectrum to measured energies is what is measured; criterion is not measured by it. Two are not the same.

**Unreachability.** Measurements bound mechanisms and criterion dispute stands over measurements; criterion is not among things the instruments return. Criterion is not reached over parameter values by particle spectrum.

**Persistence.** A criterion of explanatory debt the field agrees on, or a mechanism observed at a reachable scale.

*The report.*

**Approaches.** Stabilizing mechanisms — canonical candidates standing unobserved at reached scales, parameter spaces narrowing. Anthropic and landscape readings — standing defended, importing neighbouring dispute. Criterion critique — naturalness-as-heuristic literature standing published and growing in the field's record.

**Narrowing recorded.** Criterion critique weakens relation: no explanatory debt stands owed, in place of a criterion established.

**Progress.** Measured particle spectrum to reached energies. Narrowed parameter spaces for 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.** Two programs measure the same quantity — universal under cosmology. Concordance model relates early inference to 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, agreement standing under strengthened instruments.

**Conserving reach.** Both measurement programs.

**Reach.** Statement quantifies over expansion rate. Local ladder measures present rate directly, at one percent from a community-built distance network across calibrators, with crowding systematics excluded at high confidence. Early-universe determination is an inference calibrated on sound horizon under concordance model and carried forward to a present-day value, with model-dependent nature of that inference stated in the field's own literature. The sharpest form of tension stands between sound-horizon-calibrated determinations under concordance model and determinations calibrated otherwise. Two are the same only under model relating them, and that model is among accounts under test: early reach is frame-conditioned, and a discriminating programme is definable and existing.

**Unreachability.**

**Persistence.** Convergence of ladders under identified systematics, or an extension fitting both legs — the field's stated closure conditions, in-domain.

*The report.*

**Approaches.** Local-ladder cross-checks — space-telescope cross-checks verified local ladder rather than locating a systematic, deepening discrepancy, with one-percent community distance network standing across calibrators. Early-universe measurement — ground-based microwave programme's final data reinforced that side. Alternative calibrations — time-delay cosmography aligned with local values without a ladder; tip-of-the-red-giant-branch calibrations returning intermediate values, local record's own spread; standard sirens entering as an independent method. Late-time model modifications — difficulty results standing published, narrowing model route to early-universe physics, where early-dark-energy fits stand as live candidates, extended-model fits converging to low value wherever sound-horizon calibration is held, and remaining viable routes standing recorded as correlated modifications across epochs or as physics at homogeneity transition. An independent third measurement programme — primary results standing expected.

**Narrowing recorded.** Late-time difficulty results narrow the reach: model route stands narrowed to early-universe physics.

**Progress.** Two programmes' internal consistency. Persistence of gap across method variations and strengthened instruments. 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.** 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 test can be executed.

**Reach.** 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. Two are the same: instrument reaches the case the statement concerns.

**Unreachability.**

**Persistence.** A no at first sight, measured; open at its question as first posed. Immediate cross-modal matching fails, and transfer develops within days, measured.

*The report.*

**Approaches.** Philosophical treatment — three centuries of argument, with negative answer defended from founding correspondence onward. Late-childhood cataract surgery — permitted experiment when surgical instrument arrived, measurement returning immediate failure and rapid acquisition.

**Narrowing recorded.**

**Progress.** Measured initial failure and rapid acquisition of cross-modal matching.

**Own record.** Philosophical literature's continuing dispute over what measurement answers — 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 science — universal under biology. Natural selection standing as naturalizing resource — universal under evolutionary biology.

**Specifications withdrawn.**

**Common requirement.** An account fitting science's functional attributions.

**Conserving relation.** One account of function ascription, account standing across science's attributions.

**Conserving reach.** All functional attributions the science makes.

**Reach.** Statement quantifies over content of function ascription. Selection histories where reconstructible, and current causal roles, are what is measured — each account's extension measured, and attributions sorting differently at recorded cases. Two are not the same.

**Unreachability.** Cases sort differently under two accounts and science's usage divides by subfield, both accounts standing. Account is not preserved across attributions by selection histories and causal roles.

**Persistence.** An account fitting attributions across subfields, or agreement that usage is divided and both stand.

*The report.*

**Approaches.** Etiological account — carries malfunction and normativity, epiphenotype and novel-trait cases standing recorded against it. Causal-role account — carries physiology's practice, normativity of function standing recorded as its cost. Pluralism — standing published, science's usage recorded as divided by subfield.

**Narrowing recorded.** Pluralism narrows the reach: each account stands for subfield whose usage it fits.

**Progress.** Measured extensions of each account. Recorded cases sorting differently. 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.** Mathematical thesis standing as founding agreement on computable — universal under computability theory. 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.** Statement quantifies over nature's computational bound. Realized computations are what is measured — all within model — and exceeding constructions stand as descriptions whose physical realizability is disputed. Two are not the same.

**Unreachability.** All realized computation stands within model, and exceeding constructions are descriptions whose realizability is disputed, so no realized computation bears on modal question. Bound is not reached over realizable processes by realized computations.

**Persistence.** A physically realized process exceeding model, or a demonstration that the exceeding constructions are unrealizable.

*The report.*

**Approaches.** Quantum computation — preserves computability bound, alters complexity, standing as record's main extension. Hypercomputation constructions — standing described, relativistic and analog, their realizability disputed in record. Physical formulations — thesis' formalizations against physical principles standing published, principles' status disputed.

**Narrowing recorded.** Quantum computation narrows the reach: complexity claim stands altered while computability bound stands preserved.

**Progress.** Preserved computability bound under quantum computation. Described exceeding constructions. 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 founding micro theory — universal under microeconomics. Equilibrium as aggregate framework — universal under general equilibrium theory.

**Specifications withdrawn.**

**Common requirement.** An account of aggregate's relation to individuals.

**Conserving relation.** Structure inherited from individuals to aggregate, inheritance standing under aggregation.

**Conserving reach.** All aggregates the individuals compose.

**Reach.** Statement quantifies over inheritance of structure under aggregation. Theorems, and empirical aggregates the practice fits, are what is measured. Two are not the same.

**Unreachability.** Rationality's restrictions wash out in aggregation, standing proved. Inheritance is not preserved over aggregates by any restriction on individual behaviour.

**Persistence.** Conditions under which structure is inherited, or agreement on practice's standing given theorems.

*The report.*

**Approaches.** Aggregation theorems — ending in proved limitation, rationality's restrictions washing out in aggregation. Representative-agent practice — standing as the field's working method, with theorems recorded against its foundations in the field's own methodological literature. Heterogeneous-agent programmes — standing as recorded response, rebuilding aggregates from distributions.

**Narrowing recorded.** Heterogeneous-agent programmes narrow the reach: aggregate stands rebuilt from a specified distribution rather than inherited.

**Progress.** Aggregation theorems. Practice's empirical fits. 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 shared instrument — universal across host fields. Each host field's own dynamics standing as its candidate mechanism — local.

**Specifications withdrawn.**

**Common requirement.** An account of ubiquity — one mechanism, or a classification of many.

**Conserving relation.** An account of shared spectrum, standing across systems it is measured in.

**Conserving reach.** All systems the spectrum is measured in.

**Reach.** Statement quantifies over origin of shared spectrum. Spectra are measured everywhere, mechanisms are established in particular systems, and no cross-system mechanism stands established by record. Two are not the same.

**Unreachability.** Mechanisms established are local, and broad classes of processes are shown to yield spectrum, which supports many-roads reading and does not select. Account is not preserved across systems by spectra measured in each.

**Persistence.** A mechanism established across systems sharing no dynamics, or a classification of roads accepted as complete.

An explaining: the noise is represented as a superposition of many relaxation processes.

*The report.*

**Approaches.** Criticality programmes — standing proposed as universal, their scope disputed in record. System-specific mechanisms — standing established case by case. Mathematical genericity results — standing recorded, broad classes of processes yielding spectrum, supporting many-roads reading.

**Narrowing recorded.** System-specific mechanisms narrow the reach: an account stands established per system rather than across them.

**Progress.** Spectra measured across fields. Mechanisms established in particular systems. 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.** 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 physical theories standing beyond axioms' reach. Two are not the same.

**Unreachability.**

**Persistence.** Stated per part in each programme's own literature; no whole-physics criterion stands recorded.

*The report.*

**Approaches.** Probability — axiomatized. Quantum mechanics — carrying its axiomatic reconstructions. Field-theoretic programmes — founding axiom systems standing, constructive results standing in lower dimensions, physical theories standing beyond axioms' reach, standing gap being the four-dimensional interacting construction.

**Narrowing recorded.** Constructive results narrow the reach: construction stands in lower dimensions.

**Progress.** Probability axiomatization. Reconstruction theorems. 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 framework the state defies — universal under condensed matter. Transport measurements standing as record — local.

**Specifications withdrawn.**

**Common requirement.** An account of linearity and its cross-material reach.

**Conserving relation.** An account of state, standing across material families.

**Conserving reach.** All materials the linearity is measured in.

**Reach.** Statement quantifies over nature of state. Transport record across families, scale analyses, and proposed bound's saturation — measured in cases and disputed in scope — are what is measured. Two are not the same.

**Unreachability.** Linearity stands measured across families and material-by-material accounts stand established in cases, while proposed bound's universality stands disputed in scope. Account is not preserved across families by transport record.

**Persistence.** An account of state holding across material families, or a demonstration that the linearity has separate origins in each.

*The report.*

**Approaches.** Dissipation-bound reading — standing proposed and tested, its universality disputed in record. Exactly solvable models — correlated models yielding linearity standing as theoretical anchors, their relation to materials open. Material-by-material accounts — standing established in cases.

**Narrowing recorded.** Material-by-material accounts narrow the reach: linearity stands accounted for per material. Solvable models narrow the reach: linearity stands derived in models whose relation to materials is open.

**Progress.** Transport record across families. Measured saturations of proposed bound. Solvable models yielding linearity. A shot-noise measurement and its critique at geometry and material: one relation within transport record, and no proof that carriers are other than electrons.

**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 not taken up by the school that posed it, its criteria become a single criterion, 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 theorem.

**Reach.** Statement quantifies over the simplest proof. Formalized measures of proof size and depth, with lower bounds proved in restricted systems, are what is measured. Two are not the same.

**Unreachability.**

**Persistence.** A canonical formulation maintained by the field, with defended positions upon it.

*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: results stand for restricted proof systems.

**Progress.** Formalized measures. Lower bounds proved in restricted systems.

**Own record.** Statement stands recorded from its archive; no canonical formulation stands maintained upon it. None stands on 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 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.** Additive and multiplicative structures of 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.** Statement quantifies over all even numbers. Verification to a bound, and asymptotic methods with stated limits, are what is measured. Two are not the same.

**Unreachability.** Circle method proves ternary case and binary case lies beyond its reach for reasons the method itself makes precise; and parity obstruction in sieve theory is a known and stated block to reaching two primes, the last step not being a matter of refinement. Proof is not reached over even numbers by computation, circle method, or sieve methods.

**Persistence.** A proof, or a counterexample.

*The report.*

**Approaches.** Computation — verified past 4 × 10¹⁸, returning more of the same domain. Circle method — proves ternary case by estimating relevant exponential sums, 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 quintessential exhibition of parity barrier, sieve methods counting numbers with few prime factors more readily than numbers with exactly one. Bilinear form estimates — ternary resolution introduced idea that estimating certain bilinear forms can break barrier, with further instances recorded since; binary case is not among them. Parity-sensitive sieves — developed from late 1990s and recorded as making obstruction less of an obstacle, without removing it here. Bounded-error variants — an argument recorded that the obstruction rules out modifications of bounded-gaps method for binary case, with stated escapes being a failure of 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 binary case.

**Narrowing recorded.** Ternary result narrows the reach: sum-of-three statement stands proved for sufficiently large odd numbers. Sieve results narrow relation: a prime plus a near-prime stands proved in place of two primes.

**Progress.** Verification to 4 × 10¹⁸. Unconditional ternary theorem. Sieve result and 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 integers.

**Conserving relation.** A proof of infinitude for gap of two.

**Conserving reach.** All primes.

**Reach.** Statement quantifies over all primes. Sieve methods with a proved obstruction, and computation to a bound, are what is measured. Two are not the same.

**Unreachability.** Parity obstruction is proved: sieve methods cannot distinguish numbers with an even number of prime factors from an odd number, so reduction from bounded gap to two is blocked by parity rather than by effort. Proof is not reached over primes by sieve methods or computation.

**Persistence.** A proof of infinitude for gap of two, or a proof that the pairs are finite.

*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 exhibition of 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 generalised distributional conjecture and even using sieves other than the one at that analysis. Parity-sensitive sieves — developed from late 1990s, recorded as making obstruction less of an obstacle, without reaching two. Related-multiplicative-function routes — 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 multiplicative-correlation conjectures formulated, with averaged and logarithmically averaged versions proved. Conditional routes through irregular zeros — a recorded result that a strong violation of underlying randomness principle would imply conjecture. Convergence of the sum of reciprocals of twin primes — consistent with finitely or infinitely many. Random models of primes — predict density accurately and are models.

**Narrowing recorded.** Bounded-gap results narrow relation: infinitude stands proved for some gap at most 246 in place of gap of two.

**Progress.** Bounded-gap results and their stated floor. Proved parity obstruction. Convergence result. Measured agreement of 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, dispute concerning where its boundary falls.

**Conserving relation.** Two structures assigned to the same fold, assignment standing independently of which classification makes it.

**Conserving reach.** All pairs of structures, boundary cases included.

**Reach.** Statement quantifies over fold assignment of an arbitrary structure pair. Assignments made by particular classifications under particular criteria are what is measured. Two are not the same.

**Unreachability.** Structural alignment scores are continuous and assignment is a cut on them, cut declared and not measured; classifications built on different criteria differ at boundary cases and agree at centres; no procedure settles a case where they differ without adopting one classification's criterion as standard.

**Persistence.** Fold assignment standing for an arbitrary structure pair, boundary cases included.

*The report.*

**Approaches.** Evolutionary classification — assigns by relationship inferred from structure, inference disputed at boundary. Geometric classification — assigns by alignment score against a threshold, 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: assignment stands under that classification's criterion.

**Progress.** Recurrence, undisputed. Count of clusters at any chosen threshold. 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 folds to one native structure determined by its sequence. What is 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.** Native state is one structure, and ensembles are exception: most proteins fold to a defined structure, and disordered ones are a distinct class.

Native state is an ensemble, and one structure is exception: every protein occupies a distribution of conformations, and a single reported structure is distribution's most populated member or its average.

**Specifications.** A protein has a native state which is global minimum of its free energy — universal under 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, dispute concerning whether it is one conformation or a distribution.

**Conserving relation.** Native state determined for a sequence, determination standing independently of measurement that reports it.

**Conserving reach.** All sequences, disordered and ensemble-carrying sequences included.

**Reach.** Statement quantifies over 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.** Native state determined for an arbitrary sequence, ensemble-carrying sequences included.

The twenty datasets are at 1.1 to 3.1 angstroms, ensemble refinement reducing the cross-validated free R by 0.3 to 4.9 per cent.

*The report.*

**Approaches.** Crystallography — returns one model fitted to averaged data, averaging stated in method. Ensemble refinement — returns a distribution fitting the same data, and distribution is not unique. NMR — returns a set of models consistent with restraints, set reported rather than resolved. Single-particle cryo-EM — classifies particles into states, classification count declared. Learned prediction — returns one structure per sequence, trained on single deposited structures.

**Narrowing recorded.** Ensemble refinement narrows relation: a distribution consistent with data stands, and which distribution does not.

**Progress.** Structures reported across observed domain. Measured degeneracy of models against one dataset. 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 character they are accompanied by. Is gap in explanation, or in what 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.** Gap is epistemic: facts are physical and an account of them will close gap, or gap follows from how the physical is represented and not from what obtains.

Gap is ontological: an explanatory residue standing where identities elsewhere leave none is evidence that facts are not exhausted by the physical.

Gap is permanent and neither: 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. Residue is a property of explanation and not of 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 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.** 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 observed at an explanation is observed by a self carrying concepts explanation is given in, and no procedure separates a residue in account from a residue in observing. Determination is not reached over all accounts by explanations given.

**Persistence.** An account of a physical process from which experiential character follows in the way character of water follows from its account, agreed by both.

*The report.*

**Approaches.** Closing by further physical account — gap held to narrow as functions are explained, with residue reported continuing at each closing. Phenomenal concept strategy — residue accounted for by character of concepts an explanation is observed in, its critics arguing that strategy either fails to account for concepts physically or concedes the point. Denying residue — further question held not to stand once functions are explained, defended and a minority position. Holding gap permanent — residue held to follow from concepts available and to license no conclusion about what exists.

**Narrowing recorded.** Phenomenal concept strategy weakens relation: residue stands accounted for at observing rather than determined at account.

**Progress.** Residue reported continuing across four decades of further physical account. Parting of epistemic from ontological claim, stated at term's own introduction.

**Own record.** Gap is stated as a gap in explanation, and step from it to a claim about what exists is a separate step 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, in the field's own posing: 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. Universal set retained, with comprehension or logic weakened to hold it. Totality standing as a proper class, gatherable and a member of nothing. Concept indefinitely extensible: every proposed totality of sets is exceeded by a set 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 accounts retained, refutation identifying no member of 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, division standing without exception.

**Conserving reach.** All sets, gathering itself included.

**Reach.** Statement quantifies over absolutely all of kind. Theories' quantifiers range over domains each of which is again a totality next quantification exceeds. The two are not the same, and accounts state parting themselves.

**Unreachability.** Comprehension instance for non-self-membership yields a member exactly when it is not one. Power of any totality strictly exceeds it, and no greatest totality stands. Division is not preserved over full reach by any account retaining both: each repair on record narrows reach, weakens division, or places totality beyond 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.

Type theory refers to all objects for which a condition holds only where they are all of the same type.

*The report.*

**Approaches.** Separation and foundation — reach narrowed to subsets of given sets, standard resolution and standard theory. Type stratification — division kept by ranking reach. Universal set retained under stratified comprehension. Proper classes — totality moved out of membership relation's range. Indefinite extensibility — reach denied completion by concept's own extending. Each approach is placed by field against four broad response types available to any paradox of this form.

**Narrowing recorded.** Separation narrows reach to what given sets already contain. Class theory moves totality beyond relation. Extensibility denies reach a last.

**Progress.** Failure carries at theorem grade, accounts themselves stating it. Every consistent repair on record is recorded as reach-narrowing or division-relocation, and recurrence of totality question at each repair — class of all classes, domain of all domains — is stated in a field's own surveys.

**Own record.** Argument contains no material of any kind: sets are incidental, and the same argument carries at any totality gathered under its own kind — catalogue of all catalogues, word for words not true of 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 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 gravitational field — universal under quantum theory. 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.** Account returns one probability assignment for every physical process in its domain and assignment reduces to 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.** Statement ranges over a joint quantum-gravitational domain. Present experiments test low-energy limits and indirect consequences of candidate frameworks. The two are not the same: characteristic scale is so far out of range of experiment that direct tests have long seemed impossible, 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 quantum-gravitational regime while numerous models give different subtle modifications to low-energy physics, and recovering 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 two established limiting theories recovered.

The quantised spin-two theory is non-renormalisable, and the canonical approaches meet a problem of time. No experiment provides evidence that gravity requires a quantum description.

*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 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 reach to regimes below 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 primordial lithium-7 abundance inferred from old metal-poor stars lower than abundance predicted by standard big-bang nucleosynthesis using 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.** Stellar atmospheres do not preserve primordial abundance because lithium is depleted or transported.

Nuclear reaction network or its measured rates omit an effect that lowers predicted abundance.

Physics beyond standard cosmological model alters nucleosynthesis or destroys lithium afterward.

Abundance inference carries an unrecognized systematic offset.

**Specifications.** Baryon density inferred from cosmic microwave background applies during nucleosynthesis — universal under standard cosmology. Stellar plateau preserves or permits reconstruction of primordial abundance — local. Standard nuclear network contains material production and destruction channels — local.

**Specifications withdrawn.**

**Common requirement.** One primordial lithium abundance consistent with nucleosynthesis, baryon density and 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 independently measured baryon density.

**Reach.** Statement ranges over primordial lithium. Direct record consists of present stellar spectra and cosmological and nuclear inputs from which primordial value is inferred. The two are not the same: primordial abundance is derived from spectroscopy of metal-poor halo stars, and controversy concerns whether observed plateau reflects true primordial value or has been depleted from an initially higher abundance, plateau's flatness and small scatter main difficulty for depletion, while stars identical to plateau stars with no detectable lithium and plateau's meltdown at the lowest metallicities show that some depletion mechanism exists.

**Unreachability.**

**Persistence.** An account that closes lithium discrepancy while retaining successful light-element and cosmological predictions.

The plateau abundance is a factor of three to four below the prediction, a mismatch of four to five sigma.

*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 required uniform correction unsettled, plateau carried as a surface abundance and no untouched primordial sample among stars measured. 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 main discrepancy remaining.

**Narrowing recorded.** Nuclear measurements exclude many formerly allowed reaction-rate solutions; stellar models restrict depletion accounts by small observed scatter.

**Progress.** Baryon density, deuterium agreement, lithium plateau and leading reaction rates are measured more precisely than when 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 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.** Kinetic decomposition assigns gauge-invariant total angular momenta to quarks and gluons and separates quark spin contribution.

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.** Four contributions sum to one-half — universal under 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 partonic quantities used — local.

**Specifications withdrawn.**

**Common requirement.** A complete decomposition whose terms are defined, calculable and connected to observables.

**Conserving relation.** Contributions sum to measured proton spin under one stated decomposition and scale.

**Conserving reach.** All quark flavors, gluons, spin terms and orbital terms across full momentum-fraction domain.

**Reach.** Statement ranges over a complete decomposition. Scattering observables and lattice matrix elements constrain selected combinations and ranges. A 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 spin sum rule closed within uncertainties.

An anticipated observing: quark helicity is near three tenths and well constrained, the gluon contribution significant with large uncertainties at small momentum fraction; lattice calculation gives the quark spin from the axial-vector current at three to four tenths, smaller than constituent models give, the disconnected insertion negative, the glue spin and angular momentum large and the quark orbital contribution notable. An anticipated observing: the lattice resolves the contentious interpretation of the spin crisis, and orbital angular momentum is reached from generalized and transverse-momentum-dependent parton distributions.

*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 proton spin, gluon helicity is constrained over a growing range, polarized direct-photon production at Brookhaven's collider among its constraints, and several gauge-invariant sum rules stand; angular momentum is conserved at each measurement, and decomposition's terms are no more unique for it.

**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.** Photospheric abundance inference contains residual line-formation or atomic-data systematics.

Radiative opacities in solar interior are underestimated.

Standard solar model omits mixing, accretion, mass loss, magnetic effects or another transport process.

Helioseismic or compositional data require a combination of smaller revisions.

**Specifications.** Present photospheric abundances represent bulk composition supplied to model — local. Opacity and equation-of-state inputs apply throughout 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.** Photosphere and solar interior across radiative and convective zones.

**Reach.** Statement ranges over the Sun's composition and structure. Spectroscopy measures line formation at surface, while helioseismology and neutrinos constrain different interior quantities. The two are not the same: problem is conflict between observed data from helioseismology, sound-speed profile, depth of convection zone and envelope helium abundance, and predictions of interior models built with composition that three-dimensional, non-equilibrium models of line formation return, a large error in photospheric estimate ruled out, heavy elements entering interior only through opacity, and a field's own summary that better is worse.

**Unreachability.**

**Persistence.** A common composition and solar model reproducing photospheric, helioseismic and neutrino record.

An anticipated observing: solar models with accretion bring either the convective-zone depth or the surface helium abundance into agreement with helioseismology, not both.

*The report.*

**Approaches.** Higher-abundance mixtures — improve helioseismic agreement but conflict with modern spectroscopic scale. Opacity increases — targeted by calculations and experiments, with required profile not established. Revised atmosphere modelling — reduces several systematics while retaining low abundances. Nonstandard mixing, accretion and mass loss — can alter selected observables, without one model fitting full set.

**Narrowing recorded.** Helioseismic inversions fix locations and size of 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.** Measured oscillation probabilities arise from propagation of distinct mass eigenstates — universal under oscillation account. Mechanism reproduces observed mixings and mass-squared differences — universal under neutrino phenomenology. Completed theory respects all established electroweak constraints — universal under the Standard Model limit.

**Specifications withdrawn.** Massless neutrinos in minimal Standard Model, withdrawn while its established interactions are retained.

**Common requirement.** An interaction producing observed neutrino mass matrix and mixings.

**Conserving relation.** Generated mass matrix returns observed oscillation probabilities and all established non-oscillation constraints.

**Conserving reach.** All neutrino flavors, mass eigenstates, energies and baselines in established record.

**Reach.** Statement ranges over mass-generating physics. Oscillations measure differences and mixing combinations rather than generating interaction. 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: method does not deliver what statement ranges over.

**Unreachability.**

**Persistence.** Identification of mass-generating mechanism through consequences that exclude the other viable mechanisms.

*The report.*

**Approaches.** Oscillation measurements — determine mixing parameters and mass-squared differences, not generating operator. Endpoint kinematics — constrains a mass combination independently of 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 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 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 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.** Spectrum has a lightest state well below measured splittings.

Spectrum is quasi-degenerate, with a common scale larger than measured splittings.

Cosmological mass effects follow standard thermal history and structure-growth model.

**Specifications.** Oscillation mass differences apply to the same three mass eigenstates measured by other methods — universal under 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.** Statement ranges over individual eigenvalues. Each instrument returns a different aggregate: an effective electron-neutrino mass, a mass sum, or a decay-weighted combination. A 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 scale, and each direct method returns its own aggregate.

**Unreachability.**

**Persistence.** A measured absolute mass observable sufficient, together with oscillation data, to determine individual masses without an unresolved model dependence.

The kinematics of weak decays is the most model-independent approach to the absolute scale, and time of flight is a route to it beside oscillations, neutrinoless double-beta decay, cosmology and direct laboratory measurement. An anticipated observing: a direct measurement reaches a sensitivity of two tenths of an electronvolt, and further measurement reaches below fifty millielectronvolts, probing the full mass range the oscillation bounds allow.

*The report.*

**Approaches.** Beta-decay endpoint measurement — direct and model-light, presently an upper bound. Cosmological inference — tighter under standard cosmological model, with model dependence. Oscillation — precise differences, exactly insensitive to 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 allowed lightest-mass interval; oscillations impose lower bounds on mass sum for each ordering.

**Progress.** Two independent mass-squared splittings are measured and 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 mass term is Dirac.

Lepton number is not exact, neutral mass eigenstates are self-conjugate, and mass term is Majorana.

**Specifications.** Neutrinos have nonzero mass — universal under 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.** 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.** Statement ranges over a 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 two natures is proportional to square of neutrino mass, suppressed by mass over energy, two Majorana helicity states becoming equivalent to the Dirac particle and antiparticle states as mass goes to zero, and neutrinoless double-beta decay stands as unique test.

**Unreachability.**

**Persistence.** Observation of an unambiguous lepton-number-violating neutrino process, or another measurement fixing a field's self-conjugacy.

*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 neutrino mass-to-energy ratio. Cosmological probes — constrain masses but do not 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 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.

Parameters are environmental or statistically selected from a broad landscape.

Strong dynamics or compositeness generates observed masses and mixings.

**Specifications.** Measured masses and mixing matrices are low-energy parameters to be recovered — universal under the Standard Model. Renormalization carries high-scale structures to 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 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.** Statement ranges over origin of parameter pattern. Experiments measure low-energy parameters and rare processes, not mechanism that selects them. The two are not the same: hierarchies in masses and mixings are, in a field's words, technically natural within the Standard Model and remain unexplained, model providing no explanation of parameters it measures, measured pattern pointing to a deeper organizing principle without selecting one.

**Unreachability.**

**Persistence.** A mechanism with discriminating consequences that generates complete observed flavor structure.

Most charged-fermion flavor parameters are small and hierarchical. Routes to the puzzle include flavor symmetries, the Froggatt-Nielsen and clockwork constructions, supersymmetry and composite Higgs.

*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 construction. Extra-dimensional localization — converts positions into couplings, with geometry unselected. Statistical and anthropic models — return distributions rather than observed realization uniquely.

**Narrowing recorded.** Precision flavor and neutrino measurements eliminate textures and restrict symmetry-breaking patterns.

**Progress.** 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 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.** Proposed new sector satisfies every established flavor and charge-parity bound — universal under extension. Its suppression mechanism remains stable under quantum corrections — universal under quantum field theory. Sector retains phenomena for which it was proposed — local.

**Specifications withdrawn.**

**Common requirement.** A suppression structure making new interactions consistent with observed rarity of flavor change.

**Conserving relation.** 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 new sector contributes.

**Reach.** 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 teraelectronvolt scale with a generic flavor structure would contribute to flavor-changing neutral currents orders of magnitude above observed rates, and absence of deviations constrains such physics to scales well beyond reach of direct searches, at least petaelectronvolt scale by generic arguments, each process constraining a combination of operators and couplings rather than sector itself.

**Unreachability.**

**Persistence.** Discovery of a new sector together with principle that fixes its flavor structure, or exclusion of accessible generic alternatives over stated reach.

*The report.*

**Approaches.** Minimal flavor violation — imposes Standard Model spurions and sharply correlates effects, with 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 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 particles.

Starburst galaxies and their compact objects or winds supply particles.

Transient explosions and mergers supply particles.

More than one source population contributes with energy- and composition-dependent weights.

**Specifications.** Candidate sources meet 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 spectrum, shower composition, anisotropy and secondary messengers together.

**Conserving relation.** One source-and-propagation account returns joint observed distributions within their uncertainties.

**Conserving reach.** Full ultrahigh-energy range, sky, nuclear composition and associated neutrino and photon channels.

**Reach.** Statement ranges over accelerators and injected nuclei. Observatories measure arrival directions, energies and air showers after propagation. A 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: arrival directions measured and sources sought are not the same, deflection between them unknown.

**Unreachability.**

**Persistence.** Source identification with a composition and propagation account that predicts joint cosmic-ray and multimessenger record.

A departure from isotropy is at the ninety-five per cent level, with no clear correlation with the extragalactic matter distribution, and composition above ten exaelectronvolts is uncertain. An anticipated observing: the anisotropy is of extragalactic origin, with anisotropies at intermediate scales.

*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 comparatively local universe; photon and neutrino limits exclude classes of top-down and highly opaque accounts.

**Progress.** 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 solar surface into corona, dissipated as heat, and converted into 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 corona and accelerates wind.

Magnetic reconnection and impulsive nanoflares supply dominant heat.

Interchange reconnection releases closed-field plasma into slow wind.

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 energy budget. Candidate mechanisms reproduce species temperatures, charge states and velocity distributions — universal under solar-wind physics. Magnetic topology conditions source region — local.

**Specifications withdrawn.**

**Common requirement.** A measured energy pathway connecting photospheric motions, coronal heating and terminal wind.

**Conserving relation.** Inferred energy input, partition and outflow close one energy budget and reproduce plasma observables.

**Conserving reach.** Open and closed magnetic structures, fast and slow wind, and scales from photosphere to supersonic flow.

**Reach.** Statement ranges over connected energy pathway. Remote images, spectra and spacecraft samples measure different locations and plasma populations. A 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: regions measured and region where conversion occurs are not the same.

**Unreachability.**

**Persistence.** A quantitatively closed, predictive account of heating and acceleration in each wind source regime.

The region where the energy is channeled into the wind requires in situ observation from a few solar radii out to about twenty solar radii. How the turbulent flux increases toward the Sun, whether it suffices to power the heating and acceleration, and how it is driven are not known. No candidate process significantly accounts for the corona's million-degree temperature. A definitive solution begins with an identification of the energy source, passes through realistic modelling of the energy release and of the plasma's response, and ends with a prediction of observable quantities compared directly with observations.

*The report.*

**Approaches.** Wave and turbulence measurements — show energy transport and dissipation signatures, with dominant channel varying by region. Reconnection and nanoflare searches — detect impulsive events, with 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 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 are facing 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 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 acceleration.

A dynamical field with negative effective pressure supplies acceleration.

Gravity differs from general relativity on cosmological scales and produces apparent dark energy.

Large-scale inhomogeneity or observational systematics produce part of inferred acceleration.

**Specifications.** Account reproduces 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 background expansion and growth of structure.

**Conserving relation.** The same account returns consistent expansion, growth and lensing observables.

**Conserving reach.** Observed cosmic history from recombination through accelerating epoch and all measured linear scales.

**Reach.** Statement ranges over physical cause. Surveys return distances, redshifts, clustering and lensing from which dynamics are inferred. A 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: histories measured and cause are not the same, two families of account standing on one record.

**Unreachability.**

**Persistence.** A discriminating observation or derivation fixing source of acceleration and reproducing expansion and structure jointly.

Supernovae, baryon acoustic oscillations, weak lensing and cluster abundance are the four established methods probing the acceleration.

*The report.*

**Approaches.** Standard-candle and standard-ruler distances — establish acceleration under 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 constant-vacuum account fits 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 seed magnetic fields of galaxies and larger cosmic structures generated, and how were they amplified to 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 early universe before structure formation.

Plasma batteries in forming stars, galaxies and shocks generated seeds.

Outflows from compact objects and galaxies magnetized 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. Account respects cosmological bounds and reproduces present strengths and coherence scales — universal under cosmic magnetism. Dynamo calculations include relevant turbulence and transport — local.

**Specifications withdrawn.**

**Common requirement.** A seed-and-amplification history producing measured field strengths, spectra and coherence lengths.

**Conserving relation.** One magnetic history returns joint field observables without violating cosmological and plasma constraints.

**Conserving reach.** Voids, filaments, clusters, galaxies and early-universe epochs sampled by present constraints.

**Reach.** 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. A 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: fields observed and seed sought are not the same, dynamo standing between them.

**Unreachability.**

**Persistence.** Identification of seed mechanism and amplification history through a spectrum, coherence scale or redshift dependence that excludes viable alternatives.

Fields of micro-Gauss strength in galaxies and clusters are coherent up to ten kiloparsecs. An anticipated observing: a weak field fills the voids, and primordial fields are generated during inflation and phase transitions.

*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 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.** Thermodynamic and long-time limits define phase — universal under equilibrium phase classification. Model's disorder distribution and dimensionality are fixed before extrapolation — local. Diagnostics distinguish asymptotic behavior from crossover — universal under question.

**Specifications withdrawn.**

**Common requirement.** An asymptotic classification of dynamics and eigenstates under fixed model specifications.

**Conserving relation.** Phase diagnosis agrees across spectral, transport, entanglement and memory observables in stated limits.

**Conserving reach.** Arbitrarily large systems and arbitrarily long evolution for each stated dimension and disorder class.

**Reach.** Statement quantifies over two asymptotic limits. Exact diagonalization, tensor methods and experiments return bounded sizes and times. A field's review parts localized regime, found numerically and experimentally at finite sizes and times where one or both limits cannot be taken, from localized phase question asks about: regime measured and phase sought are not the same, two limits between them.

**Unreachability.**

**Persistence.** A controlled limit or theorem establishing stability or instability and locating transition wherever it exists.

An anticipated observing: larger simulations show the transition point drifting to stronger disorder. The problem has no established theoretical framework, resting mainly on numerical simulation on classical computers. An anticipated observing: avalanches, spreading by strong many-body resonances between rare near-resonant eigenstates, end localization above one dimension, the same in question in one dimension, and a phase diagram resolving the asymptotic stability holds for one model.

*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 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 anomalous response functions.

An apparent singularity is avoided by crystallization or by a continuous but noncritical structural crossover.

Two interconverting local structures generate anomalies without requiring an accessible macroscopic phase separation.

**Specifications.** States sampled before crystallization are liquid — local. Observed structural crossover and heat-capacity increase are equilibrium or controlled metastable properties — local. A critical point has thermodynamic and fluctuation signatures required by critical phenomena — universal under phase-transition theory.

**Specifications withdrawn.** Specification that only indirect experimental signs are available, withdrawn by ultrafast x-ray measurements while 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 pressures and temperatures surrounding proposed critical region, before crystallization.

**Reach.** Statement ranges over metastable equilibrium phase diagram. Experiments access brief liquid states prepared from amorphous ice, and simulations access model-specific phase diagrams. 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: transient states probed and metastable diagram statement ranges over are not the same, vicinity probed and the point inferred.

**Unreachability.**

**Persistence.** Independent reproduction and mapping of critical region with concordant thermodynamic and structural observables, or evidence excluding a critical point across stated region.

An explaining: the critical point rationalizes water's density maximum and compressibility minimum, both more pronounced when supercooled. An anticipated observing: a glassy arrest stands behind the apparent second liquid.

*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.** Ultrafast experiment narrows 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.** 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; 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 response toward upper assessed range.

Weaker cloud feedback and compensating processes place response toward lower assessed range.

State-dependent feedbacks make 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 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 conventional doubled-carbon-dioxide definition.

**Reach.** Statement ranges over an equilibrium counterfactual. Observations sample transient and past climates, and models compute counterfactual under specified feedbacks. The two are not the same: a field's assessment characterizes equilibrium sensitivity by an effective sensitivity based on behaviour within a century or two of applying forcing, more relevant to time scales of greatest interest, and reads it from three lines of evidence, feedback process understanding, historical record and paleoclimate record, each limited in traditional forcing-feedback paradigm for interpreting past changes and combined by a Bayesian calculation whose range depends on lines being judged largely independent.

**Unreachability.**

**Persistence.** A narrower, stable interval supported jointly by independent evidence and a resolved account of dominant feedback uncertainty.

Assessed very likely range is 2 to 5 degrees Celsius. An anticipated observing: lines of evidence combined by an objective Bayesian method give a median of 2.16 degrees Celsius with a 5 to 95 percent range of 1.55 to 3.2, the result sensitive to the assumptions employed.

*The report.*

**Approaches.** Instrumental energy-budget inference — direct to 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.** Assessed likely range is 2.5 to 4 degrees Celsius, with a best estimate of 3 degrees Celsius under 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 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.

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 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.** Rule maintains stated sensitivity and false-alarm performance beyond data used to construct it.

**Conserving reach.** All qualifying events and non-events in declared region, magnitude range and evaluation interval.

**Reach.** 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 location, time and magnitude of an impending event, and error rates and large areal extent of predictions from candidate precursors do not provide that diagnostic capability now, 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.

*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 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 lithosphere until it founders.

Delamination and tearing propagate into a self-sustaining subduction system.

**Specifications.** Process creates a downgoing plate and a persistent plate boundary — universal under subduction initiation. 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 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.** Statement ranges over full initiation sequence. Observations provide present snapshots and altered geological remnants. A field's review parts present-day initiation, where sinking slabs can supply far-field stresses, from first initiation, raising a paradox of first initiation, existing subduction appearing to be required before first subduction began: initiations observed on present-day Earth and first initiation sought are not the same, paradox standing between them.

**Unreachability.**

**Persistence.** A directly constrained sequence and force budget that establishes how initiation becomes self-sustaining across known cases.

In the Cenozoic record, a third of active subduction zones formed in the last sixty-five million years. An anticipated observing: the plume–lithosphere-interaction and mantle-convection models initiate subduction with no existing plate tectonics, mantle drag is a missing driving force, and the driving mechanism is a composite.

*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 inverse problem.

**Specifications withdrawn.**

**Common requirement.** A force budget reproducing plate velocities, boundary stresses and mantle flow.

**Conserving relation.** 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 present global system.

**Reach.** Statement ranges over separate force contributions. Geodesy measures net motion, while tomography and stress indicators constrain inputs indirectly. A 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: motions observed and force contributions sought are not the same, a prediction fitted to observed motion standing between them.

**Unreachability.**

**Persistence.** A globally consistent force and torque decomposition whose terms are independently constrained and predict plate-specific motion.

An anticipated observing: with upper-mantle slabs attached, slab pull is about half the driving force on the plates, and lower-mantle slabs, supported by viscous forces, drive the plates through suction. An explaining: slab pull and ridge push drive the plates, opposed by basal drag and collisional resistance.

*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 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 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 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 volcanic, geochemical, topographic and seismic record together.

**Conserving relation.** One source-and-motion account returns concordant ages, locations, compositions and mantle structure.

**Conserving reach.** Lifetime of each candidate province from deep source through surface volcanism.

**Reach.** Statement ranges over past three-dimensional mantle flow. Observations supply present tomography and discontinuous surface and rock records. A 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: anomalies imaged and upwellings sought are not the same, resolution of images standing between them.

**Unreachability.**

**Persistence.** Source depths and histories established for candidate provinces by mutually consistent, discriminating observations.

An anticipated observing: plumes are complex and broad in shape. Hotspot geochemistry carries deeply recycled subducted components and primordial domains. Resolving deep plume structures asks for seismic tomography at scales smaller than a hundred kilometres, with concerted multi-proxy geochemical and dating efforts on the spatiotemporal and chemical evolution of long-lived plumes. An anticipated observing: the Iceland mantle plume collapses and resurges.

*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 geodynamo to reverse polarity, and what controls 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 dynamo between polarity states stochastically.

Changes in heat flux at core-mantle boundary modulate reversal frequency and geometry.

Inner-core growth and long-timescale core evolution change dynamo's stability.

Reversals arise from low-dimensional interactions among dipole and multipole magnetic modes.

**Specifications.** Account is generated by electrically conducting fluid motion in core — universal under dynamo theory. It reproduces stable chrons, excursions, full reversals and secular variation — universal under geomagnetic record. Paleomagnetic directions and intensities preserve relevant field history — local.

**Specifications withdrawn.**

**Common requirement.** A dynamo mechanism yielding observed polarity statistics and transition morphology.

**Conserving relation.** One dynamo account returns field behavior consistent across secular variation, excursions and reversals.

**Conserving reach.** Geological reversal record and range of core states compatible with present observations.

**Reach.** Statement ranges over unobserved core dynamics through geological time. Measurements sample present field and discontinuous remanent magnetization. A statistical model of 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: sequence recorded and 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.

The record has a few hundred stochastic polarity switches, and cycles of variable frequency are found in the reversal rate. An explaining: short-term polarity switching is a chaotic dynamical system, and the transition to superchrons is a second-order phase transition driven by heat-flux variations at the core–mantle boundary.

*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 beyond 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 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 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 geological and climatic constraints for each glacial interval.

**Conserving reach.** Tropical and global oceans throughout the Sturtian and Marinoan glaciations.

**Reach.** Statement ranges over global, time-varying ice cover. Record samples preserved basins and proxy-forming environments. A 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: low-latitude ice recorded and 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.

An anticipated observing: in general circulation model simulations, snow cover and the nature of the frozen surface are the key factors, initiation is possible at a plausible reduction of carbon dioxide, and deglaciation requires elevated carbon dioxide and tropical dust with some cloud warming. The Neoproterozoic is a transition between the microbial world and the Phanerozoic, with extreme carbon-isotope fluctuations and glaciation resuming, the cause of the biogeochemical turbulence and its connection to the glaciations obscure. An anticipated observing: snowball self-termination, falsifiable geologically, gives long-lived cryochrons, their onsets synchronous at low latitudes and their terminations synchronous everywhere.

*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 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. Reconstructed history satisfies sulfur, chromium, iron, carbon and trace-metal evidence together — universal under 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 onset, overshoot and aftermath of event.

**Reach.** Statement ranges over continuous global oxygen abundance. Proxies sample local chemical thresholds in discontinuous sedimentary archives. A 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: threshold crossings recorded and abundance sought are not the same, correlations standing between them.

**Unreachability.**

**Persistence.** A proxy-consistent quantitative trajectory with bounded peak, floor, spatial variation and duration.

The first Paleoproterozoic global glaciation and the onset of the Great Oxidation Event are between about 2,460 and 2,426 million years ago, the Ongeluk magmatism dated at 2,426 ± 3 million years. An anticipated observing: the oscillations of oxygen continue over the following two hundred million years. The first significant buildup of atmospheric oxygen begins in the early Paleoproterozoic with global glaciations and continues to the end of the Lomagundi excursion, free oxygen accumulating about 2.45 billion years ago. Whether the initial rise is monotonic or oscillatory is unclear, the record poorly constraining oxygen after it, and vanadium isotopes capture a unidirectional transition in surface-ocean oxygenation.

*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.** Event's principal onset interval and irreversible loss of 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 oxygen sink.

Organic-carbon burial and changes in weathering shifted global redox balance.

Hydrogen escape and long-term mantle-crust evolution progressively oxidized surface system.

Threshold feedbacks among methane, climate and ozone converted a gradual flux change into persistent oxygenation.

**Specifications.** Global oxygen source exceeds combined reductant sinks over 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 account.

**Specifications withdrawn.** 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 delayed onset and permanent crossing of oxygen threshold.

**Conserving relation.** One redox budget reproduces timing and persistence of oxygenation without violating geological proxy record.

**Conserving reach.** Biological, atmospheric, oceanic, crustal and mantle reservoirs across pre-event interval and transition.

**Reach.** Statement ranges over global source and sink fluxes through deep time. Record contains local proxies and model-derived flux bounds. A 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: scenarios modelled and fluxes that ran are not the same, model's determinants standing between them.

**Unreachability.**

**Persistence.** A quantitatively closed redox budget and causal sequence distinguishing trigger from permissive background change.

The Great Oxidation Event is 2.4 billion years ago. An anticipated observing: in a biogeochemical model of the Earth system, an early origin of oxygenic photosynthesis lengthens the delay to permanent oxygenation. An anticipated observing: the Great Oxidation Event triggers when the difference between the influxes of reductants and of phosphate falls below a critical value, the transition gradual and reversible or sudden and irreversible by the sources and sinks.

*The report.*

**Approaches.** Molecular and geochemical dating of photosynthesis — moves 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 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 thermodynamic impact state — universal under geochemistry. Account reproduces present lunar mass, core fraction and orbit — universal under lunar formation.

**Specifications withdrawn.** A strictly fixed post-impact angular momentum, withdrawn in accounts retaining resonant transfer to 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.** Statement ranges over an unobserved formation sequence. Samples and dynamics constrain its final products, and simulations map specified initial conditions to outcomes. A 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: constraints matched and collision sought are not the same, parameter space standing between them.

**Unreachability.**

**Persistence.** A formation history independently fixed by discriminating sample, geophysical or dynamical evidence and reproducing complete constraint set.

The traditional isotopic proxy's constraining power is limited, insufficient on its own to part the partial-mixing from the high-energy models, and the close isotopic likeness of Earth and Moon stands as an isotopic crisis for the canonical model.

*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 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 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 inner Solar System's early bombardment a short late spike, a monotonic decline, or a more complex sequence, and what supplied 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.

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.** Inferred flux returns surviving chronological, spatial and compositional record across planetary bodies.

**Conserving reach.** Inner planets and Moon from accretion through end of basin-scale bombardment.

**Reach.** Statement ranges over a continuous multi-body flux. Samples and craters preserve a filtered, reset and spatially incomplete record. A field's review defines bombardment as impacts after planetary lithospheres stabilized such that they could be preserved as craters and basins, melt-rock and shock ages pointing toward a discrete episode: flux recorded and flux sought are not the same, stabilization standing between them.

**Unreachability.**

**Persistence.** A flux history and source mixture distinguished by samples with known context and cross-planet chronology.

An anticipated observing: lunar melt rocks and meteorite shock ages point to 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 billion years ago, and Precambrian spherule layers to a long-lived tail to about 2.0–2.5 billion years ago; a hybrid model of discrete early and later populations is the most parsimonious; and the impactors of the bombardment and of late accretion are the same bodies, from the terrestrial-planet region, the bombardment the tail end of accretion.

*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 sharp classical spike is no longer 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 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 transition.

**Specifications.** Account is consistent with 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.** 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.** Stem lineage from unicellular holozoans to last common ancestor of living animals.

**Reach.** Statement ranges over a singular ancient transition. Comparative genomes, living relatives and fossils preserve different fragments of it. A 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: ancestor reconstructed and transition sought are not the same, few data standing between them.

**Unreachability.**

**Persistence.** An ordered, experimentally supported reconstruction connecting ancestral states to integrated animal multicellularity.

The choanoblastaea and synzoospore theories carry flaws. Whether multicellularity evolved independently in the three clades remains to be tested, and the molecular mechanisms first supporting it are not studiable in animals alone.

*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 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 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 observed connected pathway, consistent with enzyme and organismal histories.

**Conserving relation.** Each historical step preserves organismal viability and yields reactions and regulation of extant pathway.

**Conserving reach.** Complete pathways, their enzymes, intermediates and network context across lineages carrying them.

**Reach.** Statement ranges over ancient sequences of functional intermediates. Comparative data observe surviving enzymes and networks after duplication, loss and transfer. A 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: pathways observed in their current state and sequence of their emergence are not the same, current state observed backward standing between them.

**Unreachability.**

**Persistence.** A supported historical sequence for representative pathways that explains intermediate utility, enzyme origin and network integration.

An explaining: metabolism's complexity unlocks where survival of the fittest applies to metabolites as well as to genes and enzymes, metabolites and enzymes coevolving. Whether reconstructions replicate an ancestral pre-genetic metabolism is debated.

*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 phenotypes sequence can produce.

**Specifications.** Genotype, phenotype, environment and developmental stage are operationally fixed — local. Map predicts interventions rather than association — universal under causal prediction. Predictions transfer across genetic backgrounds within claimed reach — universal under 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 observed phenotype for each admissible genotype-environment-development combination.

**Conserving reach.** Sequence, background, environment and developmental ranges claimed by account.

**Reach.** Statement ranges over combinatorial genotype and environment space. Experiments sample selected variants, backgrounds and conditions. A 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: molecular maps measured and organismal map sought are not the same, simpler maps standing between them.

**Unreachability.**

**Persistence.** Prospective prediction of phenotypic effects for unmeasured combinations across a declared biological reach.

Empirical evidence for phenotypic bias mounts. The networked organization of the maps conditions evolution and adaptation, and several topological properties of genotype spaces are universal.

*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 evolutionary target itself dynamic.

**Specifications.** Ancestral state, environment and prediction target are declared — local. Replicates are independent under experimental or comparative design — local. Predictability is scored against a probabilistic baseline at a stated biological level — universal under 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.** Genotypic, phenotypic, ecological and temporal range claimed by prediction.

**Reach.** Statement ranges over possible evolutionary histories. Experiments and comparative records sample finite lineages and environments. A 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: recurrence measured in experiments and predictability sought over histories are not the same, extrapolation standing between them.

**Unreachability.**

**Persistence.** Prospectively calibrated predictions whose success and failure boundaries are established across levels and timescales.

Mutation bias is a source of parallel evolution, and repeated evolution attributed to selection with mutation biases unconsidered is misattributed. The contributions of determinism and stochasticity depend strongly on the conditions, the selective pressure and the number of available beneficial mutations. Whether convergence and predictability are common is unclear, the evidence mixed, and parallel and analytic replay experiments are its instruments.

*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 rate shift — universal under causal history.

**Specifications withdrawn.**

**Common requirement.** A cause of joint lineage multiplication and ecological or phenotypic divergence.

**Conserving relation.** 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.** Statement ranges over counterfactual lineage outcomes. Phylogenies and trait records contain surviving, sampled histories. A 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: bursts recorded at macroevolutionary scale and mechanisms sought at population scale are not the same, gap between scales standing between them.

**Unreachability.**

**Persistence.** Prospective or replicated causal evidence separating opportunity, innovation and lineage predisposition across comparable clades.

Whether and how magic traits, phenotype matching and physical linkage promote rapid bursts is unclear. An explaining: beyond ecological opportunity, ancient adaptive alleles and the transporter hypothesis, sexual signal complexity, fitness landscape connectivity, diversity begetting diversity, and plasticity first are mechanisms of rapid radiation. Ecological opportunity is inferred only from its effects, and whether it carries predictive value or serves only post hoc explanation is unclear. Support for the key-innovation hypothesis is scarce even in textbook examples.

*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.** 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 not change 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 lineages that persist.

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 question.

**Specifications withdrawn.**

**Common requirement.** A causal rate effect of genome duplication on lineage origination and loss.

**Conserving relation.** 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.** Statement ranges over all duplication outcomes including extinct lineages. Comparative data preferentially retain surviving and detectable events. A 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: abundance recorded and rates sought are not the same, formation rate standing between them; forming is observed within a century and within fifty days.

**Unreachability.**

**Persistence.** Bias-controlled estimates fixing parity, timing and conditions of duplication's effect on speciation and extinction.

An anticipated observing: likelihood-based analyses give recently formed polyploids lower speciation and higher extinction rates than diploids. Sister-clade comparisons do not part speciation from extinction, and time-scaled rates meet the objection from clade age and clade size.

*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; 132 ancient duplications across 470 flowering plants clustering at end-Cretaceous extinction, at global cooling and at Paleocene–Eocene Thermal Maximum, and a reply finding no evidence that polyploids arise at environmental upheaval, five conceptual and methodological issues undermining those associations: instruments disagreeing on their object. 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. Forming is observed directly and recurring: two goatsbeard species, Tragopogon, arose in eastern Washington within a century of their parents' arrival from Europe, at twice their parents' chromosome number, losing and silencing gene copies from a first generation in a repeatable and predicted order; diploid snowflake yeast selected for larger size became tetraploid within fifty days and stayed so across nearly five thousand generations, conserved aneuploidy following; a New Zealand mud snail, Potamopyrgus antipodarum, carries a whole-genome duplication under a million years old, 3,040 genes at two copies where its relatives carry one and some sixty percent of doubled genes already lost; in yeast, extra copies of each gene hinder adapting to nystatin.

**Own record.** Its field's own posing of what persists: that it is not known, for snails or any species, why this happens so reliably; that most polyploids really are dead ends.

## 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 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 proximate drivers of degeneration.

Immune and glial dysregulation initiates or amplifies pathological process.

Vascular, metabolic and barrier dysfunction drives or conditions common disease.

Several interacting pathologies reach a common clinical syndrome through heterogeneous routes.

**Specifications.** 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 proposed driver changes predicted downstream pathology and clinical course.

**Conserving reach.** Preclinical and symptomatic course of sporadic late-onset cases, including common mixed pathologies.

**Reach.** Statement ranges over decades-long individual causal histories. Biomarkers, autopsy, genetics and trials sample different stages and populations. A 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: clearance and clinical change measured in trials and causal sequence sought are not the same, three disputes standing between them.

**Unreachability.**

**Persistence.** Replicated interventions and longitudinal evidence fixing initiating and progression-driving processes across biological subtypes.

The role of oligomeric amyloid species in cognitive impairment and neuronal death stands open. The trial outcomes stand by the amyloid hypothesis while a series of trial failures remains a question. Whether infection is a primary trigger, and whether it applies to all sporadic cases, is undetermined.

*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 jointly decisive now.

**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.** Entry is restricted to 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.** Account explains lesion initiation, persistence and anatomical distribution — universal under pathogenesis. Pain and fertility outcomes are separately represented — universal under 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 heterogeneous clinical manifestations.

**Conserving relation.** Intervention on proposed process changes predicted lesion and symptom outcomes in populations to which account applies.

**Conserving reach.** Superficial, ovarian and deep disease, symptomatic and asymptomatic lesions, pain and infertility outcomes.

**Reach.** Statement ranges over initiation before clinical detection and later symptom production. Surgery and imaging sample selected lesions after variable disease duration and treatment. A 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: disease at a delayed diagnosis and initiation sought are not the same, delay standing between them.

**Unreachability.**

**Persistence.** A prospectively validated causal taxonomy that predicts lesion course, pain, fertility and treatment response.

Retrograde menstruation is a physiologic process in nine tenths of women. No cure is identified, and the disease carries pelvic organ dysfunction and secondary mental issues beside severe pelvic pain and infertility.

*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.

**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 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.** Recognized symptom clusters, severities, acute histories and durations admitted by case definition.

**Reach.** 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, illness estimated in at least a tenth of infections with count likely much higher for undocumented cases, onset of symptoms variable, and a field's own recommendations are for research that would identify mechanisms and part overlap with other viral-onset illnesses.

**Unreachability.**

**Persistence.** Replicated causal biomarkers and interventions that assign patients to mechanisms and predict course and treatment response.

An anticipated observing: at least sixty-five million individuals worldwide have Long COVID, a conservative estimate.

*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 a complete discriminator now.

**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 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 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 declared outcome or intervention response in independent populations.

**Conserving reach.** Infection sources, host backgrounds, care settings, disease stages and therapies covered by clinical syndromes.

**Reach.** Statement ranges over latent biology and counterfactual treatment response. Studies measure time-specific biomarkers under ongoing treatment. A 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: subgroups identified in data and treatment response sought are not the same, translation standing between them.

**Unreachability.**

**Persistence.** Prospective phenotype assignment with replicated biological meaning and differential benefit from a targeted intervention.

The biological processes driving each phenotype are poorly understood. A platform trial phenotypes at enrolment by point-of-care biomarkers. An anticipated observing: without enrichment, large trials keep yielding neutral results.

*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 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 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 clinical reach. Intervention targets connect to 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 distinct symptom phases.

**Conserving relation.** 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.** Statement ranges over unobserved onset before and during spontaneous attacks. Imaging, electrophysiology and blood measures sample selected phases and locations. The two are not the same: field parts its questions into why, how and when, and when, emergence of an attack from a cycling disorder of sensory processing, is examined mostly through triggered attacks and serial scanning of cycle, difficulties in examining spontaneous attacks sending studies there, 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.

Many of the pathways are monoaminergic and peptidergic, calcitonin gene-related peptide and pituitary adenylate cyclase-activating polypeptide among them.

*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 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; 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 present clinical syndrome.

**Specifications.** 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 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.** Symptom, developmental and outcome heterogeneity admitted by diagnosis and related psychosis spectrum.

**Reach.** Statement ranges over individual developmental causes. Genetic, imaging, molecular and clinical studies measure population-level associations at selected stages. A 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: risk factors measured across populations and developmental causes sought in an individual are not the same, absent marker standing between them.

**Unreachability.**

**Persistence.** Validated mechanistic subtypes or a general causal model that prospectively predicts onset, course and response.

An anticipated observing: schizophrenia's lifetime prevalence is 0.4 percent. It manifests in late adolescence or early adulthood after a long prodromal phase, and treatment resistance is frequent.

*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 apart from 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 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 at its own 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 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: sensitivity a decoding returns and 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.

Correlation, causal role and teleology are each implied by representation's explanatory role, evidence on all three is rarely presented in a single study, and a causal effect on behaviour is harder to establish than a correlation.

*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 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 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 relevant impairments.

Task-based imaging and electroencephalography together detect more sensitively than either alone, and the standard imagery task has high false-negative rates. An anticipated observing: more than half of the patients conscious at the bedside do not pass the scan tests. Standardized behavioral evaluation is the reference standard.

*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 is patients' 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.

Class-incremental learning asks for discrimination between classes not observed together, very challenging for deep networks where storing examples is not allowed. The scenario differences are measured on the Split MNIST and Split CIFAR-100 protocols. The mitigating methods are designed for independent samples, known task boundaries, stored past samples and offline evaluation, and in the streaming setting forgetting is more severe and largely unaddressed. Continual pre-training without replay and without task labels is unsolved at scale, every published method retaining a buffer, requiring task identifiers, applying a penalty that scales poorly, or operating at small scale.

*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 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.

An anticipated observing: a weakening of calibration is consistent with equalized odds outside trivial cases, and a further weakening is consistent even in non-trivial cases; the impossibility is no longer binding with behaviour endogenous to classification; and a total unfairness budget, the mean-squared error times the prevalence difference, nests the impossibility results.

*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 particular 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: constructions and attacks the reductions treat and existence the statement quantifies over are not the same, the equivalence to one particular problem standing between them.

**Unreachability.**

**Persistence.** An unconditional construction with proof, or a proof that no such function exists.

The polynomial time bound is at least a constant factor above linear. The central private-key primitives are secure private-key encryption, signatures, pseudorandom generators and functions, and commitments. Mild average-case hardness is that no probabilistic polynomial-time algorithm computes time-bounded Kolmogorov complexity on more than a polynomially bounded fraction of strings.

*The report.*

**Approaches.** Factoring-, discrete-log-, lattice-, code- and multivariate-based candidates — secure under particular 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 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).

*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.

*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 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.

An anticipated observing: polynomial identity testing is efficiently derandomisable if and only if explicit polynomials of high circuit complexity exist, and a barely non-trivial derandomization implies a near-complete one.

*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 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: 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.

An explaining: a relational account acknowledging genuine we-form attitudes does away with the problem of circularity. The nature of the first-person plural indexical is investigated only at its early stages.

*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 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: judgments the cases supply about reliability and peerhood and the rational response sought are not the same, the independence they must have from disagreement standing between them.

**Unreachability.**

**Persistence.** A norm that handles symmetry, self-application and justified asymmetry across canonical cases.

The Restaurant Check Case is the central case for the Equal Weight View, defended through the Uniqueness Thesis or without it, refusing conciliation appearing dogmatic. On the self-defeat objection, a model of conciliatory reasoning in defeasible logic answers it; on the other side, both families of response to it, solution and skeptical, fail by conciliationism's own lights.

*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 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 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.

Networks optimized for compositional skills achieve human-like systematicity in a head-to-head comparison with humans on an instruction-learning paradigm. In the same framework a severe lack of compositionality is found.

*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: 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.

An explaining: institutions change because prevailing institutions affect the distribution of resources and groups with de facto power strive to change political institutions; growth-encouraging institutions emerge where political institutions allocate power to groups with interests in broad-based property rights, constrain power-holders and leave few rents. Neoclassical growth models explain the differences by factor accumulation.

*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 unchangeable 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.

Even partial scalar invariance is seldom achieved across many countries or time points, and analysing only a subset of countries is a partial and unsatisfactory solution. The exact approach proves overly strict where the approximate establishes comparability.

*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 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 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: sites and genomes the record samples and routes sought are not the same, 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.

The earliest securely dated sites are after about fifteen thousand years ago. An anticipated observing: ancestral Native American population expansion is after sixteen thousand years ago.

*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 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: densities modelled and 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.

Agriculture has at least eleven independent centers. An anticipated observing: in a statistical model, environmental, geographic and cultural variables capture 77 percent of the variation in population density among 220 foraging societies.

*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 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: fragments and localities the record supplies and 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.

The material known from before two million years is too fragmentary for a meaningful assessment of species diversity, and the record bearing on the genus's origin and earliest evolution is virtually undocumented, the poles of the transition attached to taxa remote in time. Hundreds of fossils come from well-dated sites, stratigraphy, dust, cores and isotopes reconstructing their environment. An explaining: habitat instability is an important selective force.

*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 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: trends the surveys return and causes sought are not the same, problems of inference standing between them.

**Unreachability.**

**Persistence.** Replicated causal decompositions distinguishing dimensions, directions and feedback across political systems.

Interventions changing out-party feeling reduce avoidance but not support for norms. Citizens' policy positions show little or no increased mass polarization over two to three decades, while party sorting, clearly occurring, is sometimes exaggerated. No observing shows affective polarization driving both elite and mass ideological polarization, and causal evidence is scarce outside one country.

*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 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: explosions the simulations produce and 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.

Violent nonradial mass motions are generic in supernova cores, and gravitational-wave features are found in them. Different groups obtain different results, and a three-dimensional code comparison is missing. The chaos of the dynamics is among the issues remaining. An anticipated observing: the neutrino mechanism falls short of the observed energies.

*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.

Three quarters to four fifths of autoimmune patients are women, with rheumatoid arthritis and systemic lupus erythematosus particularly female-predominant. Male mice expressing the X-inactive transcript are prone to autoimmunity. An explaining: autoimmune diseases come from the immune system's inability to discriminate self from foreign.

*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: propagated signals the telescopes measure and sources and emission sought are not the same, propagation standing between them.

**Unreachability.**

**Persistence.** Source identification and mechanism-specific predictions confirmed across a defined burst population, with additional channels fixed where required.

Bursts are many orders of magnitude more luminous than pulsar pulses. Over six hundred sources are published, twenty-four of them repeaters, with nineteen host galaxies; about four per cent are confirmed repeaters, and the vast majority are detected only once. An anticipated observing: a detectable burst occurs roughly once a minute somewhere on the sky. An anticipated observing: magnetar engines produce at least some, and probably all, bursts.

*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: 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.

Reconnection is 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. Parting from it: near-Earth current disruption occurs before the conventional reconnection signatures and triggers onset, the auroral intensification and the reconnection not causally linked and the onset a continuation of earlier activity, that timing resting on a preliminary index in place of ground and tail data.

*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.

Under the conjecture, vertex cover is hard to approximate within any constant better than two. An integrality gap on random instances is among the evidence for the conjecture. An anticipated observing: the 2-to-2 games conjecture is proved.

*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.

It is proved that an odd perfect number has at least nine distinct prime factors, and at least twelve if three does not divide it. An anticipated observing: an odd perfect number has at least ten distinct prime factors and is above ten to the fifteen-hundredth.

*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 at their own 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: asymptotic bounds the density theorems return and the exact bound the conjecture quantifies over are not the same, the gap between 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.

*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: maps the construction reaches and 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.

Five explicit maps refute the conjecture: one three-dimensional of degree four, two four-dimensional of degrees five and ten, and two five-dimensional of degrees six and twelve. In one dimension the conjecture is settled.

*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 only on ε — 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 primes dividing abc raised to 1 + ε, uniformly over 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 two sides over finite ranges of triples, not large in ranges checked, and proof returns an unconditional bound at the exponential of the two-thirds power of the product of 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; at the article a third outcome, undecidability, as the least desirable.

The structures whose existence the claimed proof must establish are arithmetic holomorphic structures; the published criticism recognizes that problem and extrapolates from it incorrectly.

*The report.*

**Approaches.** Linear forms in logarithms — return the unconditional bound, an exponential of the two-thirds power of the product of 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 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 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. 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.

Large-angle and high-multipole anomalies motivate non-standard models.

*The report.*

**Approaches.** Slow-roll constraints with Bayesian model comparison — favour concave potentials, with 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. Evidence is drawn particularly from rodents — local.

**Specifications withdrawn.**

**Common requirement.**

**Conserving relation.** The identified substrate accounts for formation, the retrieval over time and the loss of a specific memory.

**Conserving reach.** All memories of kinds studied, across formation, retrieval and loss.

**Reach.** The statement quantifies over the substrate of a specific memory. Evidence for engram cells as a substrate is drawn from rodents, as the summary states, and returns 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.

*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. 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 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 timing and amplitude of cycles, and astronomical calculation returns 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.

Whether the cycles are paced by obliquity or by the precession index modulated by eccentricity is debated, age uncertainties making the distinction difficult. The morphology of deglaciation and inception circumvents imprecise age control. An anticipated observing: the gross morphology of the cycles is largely deterministic.

*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 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 observed masses at the observed epoch from seeds and 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, scenarios distinguished by observation.

*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 made.

**Narrowing recorded.**

**Progress.** The existence of 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.

Examples and special cases support the view that naked singularities do not arise generically. The conjecture is a hypothesis of the area theorem.

*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; 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 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 searches — the field's record.

**Specifications withdrawn.**

**Common requirement.**

**Conserving relation.** The account returns the proton's stability to limits set and the meeting of three couplings from one theory.

**Conserving reach.** Baryon-number violation at the unification scale.

**Reach.** The statement quantifies over 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.

The lower limit on the lifetime is above sixteen times ten to the thirty-third years, and trilepton-mode limits are between nine and thirty-four times ten to the thirty-third years. An anticipated observing: the lifetime in the gauge-boson mode is as short as ten to the thirty-fifth years, and in the supersymmetric mode, highly model dependent, between ten to the thirty-second and ten to the thirty-fifth years.

*The report.*

**Approaches.** Water Cherenkov searches in 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 unification parameters — model dependent, corrected in its second-order treatment.

**Narrowing recorded.**

**Progress.** The limits in 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.

The quasipolynomial algorithm carries a reduction from Cameron groups to Johnson groups.

*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; 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. 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 two methods observe it.

**Reach.** The statement quantifies over the free neutron's decay rate. Proton or electron counts along a beam, and 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.

The neutron lifetime measured in a magnetic trap is 877.75 seconds, and the beam–bottle discrepancy is also given as eight and a half seconds at four sigma.

*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 precise enough now to favour either set, and a later result compatible with bottle measurements within its uncertainties, beam against bottle no longer parting by method by itself; earlier beam results and discrepancy stand. A proton counter in 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 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, structural homologies of capsid proteins to cellular proteins, and 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.

*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 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.

*The report.*

**Approaches.** Flux-transport models with 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 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 diagnosis gathers heterogeneous presentations under one name.

**Unreachability.**

**Persistence.** A mechanism explaining all aspects of the disease, or a parting of diagnosis into disorders each with its own.

An anticipated observing: heritability is around 0.37.

*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 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.

*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; two models stand articulated with their contrasts; 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 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 formation of every pristine state. Archaeological records of settlement, walls and conflict, the geography of arable land and its periphery, and trajectories of trade and public goods are what is measured. The two are not the same: the record shows conditions and 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.

An explaining: the state is the most important single step in political evolution.

*The report.*

**Approaches.** Comparative ethnography and archaeology of 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

Entries stand here, one to a problem, **and their number is recomputed at them** — a number carried beside them is prior the moment one arrives.

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# 6 Checks

**C1.** A property with a filling carrying neither an observing nor its problem as its field poses it.

**C2.** An approach of published literature, unlisted, or listed with an ending other than where literature has it.

**C3.** A statement other than its canonical one, or a variant omitted.

**C4.** A specification its statement carries, unlisted.

**C5.** A condition prior stated other than as its five stand.

**C6.** A property stating more than its observing carries.

**C7.** An approach's ending stated more strongly or more weakly than its record supports.

**C8.** A source, a citation or a version within an entry.

**C9.** A claim not observed directly said as an observing, and not as an anticipated observing.

**C10.** A date or a person functioning as attribution within a property.

**C11.** An unreachability stated where its result reports an attempt rather than a failure of its relation over its reach.

**C12.** A conserving relation or conserving reach stated other than as its problem states it.

**C13.** An account stating an effort's condition rather than proposing an account.

**C14.** A specification recorded as withdrawn, not carried with its own property.

**C15.** A narrowing of a record, not carried with its own property.

**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 within its literature, not carried with its own property.

**C18.** A hardness line stating this compendium's observing rather than its field's words.

**C19.** A number of approaches matching neighbouring entries' number where fields' records differ.

Corrections are applied against literature.

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# 7 The aggregate

Each figure is a number over filled properties.

A count of entries. A count of entries by each name their specifications are universal under. A count of entries by each number of specifications carried, every entry counted. A count of entries carrying an unreachability result. A count carrying a narrowing recorded, by narrowing of reach and by weakening of relation. A count carrying a specification withdrawn. A count of entries by each number of approaches, and of approaches by each standing, every entry counted. A count whose settling result is specified as one result, as either of two, or as proved unavailable. A count of properties standing empty, by property.

Each figure is recomputable from entries.