Exhibit TWENTY-EIGHT Equilibria Registry v389

# Equilibria Registry

**Not Possibly Existing, Living or Non-Living**

**ONE · EXISTING THROUGH MOMENTARIES**

- 1.1 The universe, its things, and exclusivity
- 1.2 Three conditions at each momentary, and momentaries overlapping
- 1.3 A between, a protocol, a living nothing and an existing nothing
- 1.4 One relation, existing and discovering, at each scale
- 1.5 Parity alternating through three momentaries

**TWO · EQUILIBRIA, HARD PROBLEMING**

- 2.1 Equilibria, not possibly existing as a method
- 2.2 A stability, a form or a relation, and surviving
- 2.3 One description and one claim at each conception
- 2.4 A participation missing, unchanging or changing

**THREE · THE TEN, FIVE PLACES AT TWO FACES**

- 3.1 One form of hard problem, and a sixth condition
- 3.2 Ten at Exhibit ONE's names, at four four-cyclings
- 3.3 Four binaries at two faces, a stilling and a changing
- 3.4 Exhibit ONE's four-cycles, one parity offered once, and six connectors

**FOUR · CONCEPTIONS OF EQUILIBRIA, EACH A HARD PROBLEM IN ITS OWN WORDS**

- 4.1 Each conception whole, at its declared subject
- 4.2 One form, and three differences within each of ten
- 4.3 An arriving fixed from behind, 2 arrivals
- 4.4 An opening fixed as a place, 5 arrivals
- 4.5 A completing fixed as a last, 8 arrivals
- 4.6 A carry fixed as a store, 10 arrivals
- 4.7 A middle fixed as an end, 6 arrivals
- 4.8 A parity fixed as a magnitude, 9 arrivals
- 4.9 A sequencing fixed to one beat, 5 arrivals
- 4.10 A rate fixed as a value, 5 arrivals
- 4.11 A two-way fixed to one side, 7 arrivals
- 4.12 A between fixed as a cut, 6 arrivals
- 4.13 Fields at ten, a key for aiming at each field
- 4.14 Leads arriving without their statements

**FIVE · RESOLVING FOLLOWING, THE EXCLUSIONS AT THE RESOLVER**

- 5.1 Four proof groups
- 5.2 One exclusion at each of ten
- 5.3 Renewal and reaching at their complete requirements
- 5.4 Sharing at 10 and 6, and carrying at 11, at resolver
- 5.5 Each conception at a proof of its own requirements

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# ONE · EXISTING THROUGH MOMENTARIES

## 1.1 The universe, its things, and exclusivity

**The universe as one thing beside its things is exclusivity, and it is not possibly existing.** One side's momentaries at their own, each number one side's, is the universe fixed as one thing beside its things; the universe exists as its things exist; overlapping momentaries are at each number, one completing and its next opening. No existing thing is excluded, living or non-living, and each living thing is existing, continuing its own inclusion at each momentary, or it is not.

## 1.2 Three conditions at each momentary, and momentaries overlapping

**Existing is three conditions at each momentary, at once.** A momentary opens at one number and completes at its next, and its completing is its next momentary's opening. Exhaustiveness is overlapping momentaries at each number; determinacy is each momentary's one next, opening at the number it completes; reachability is next openings reaching each number on. Three are together at each momentary, all or none at all.

**Three overlapping momentaries use four positions.** Prior, now and next are 1–2, 2–3 and 3–4, completing and opening coupling at shared positions. Along one side next opens two on: 1–2 and 3–4, with next opening 5; an other side carries 2–3 and 4–5, with next opening 6. Across an overlapping its next opens one on.

**Each momentary is existing's own, and each existing self is its continuing momentaries, inseparating.** Between is a protocol's centre at 1.3, and succession is momentaries' own.

**Universes at each scale share momentaries of co-sequential changing**, each momentary a fractal universe discovering its next existing, and none is possibly an existing thing separately: at each scale, a universe fixed as an existing thing is exclusivity.

## 1.3 A between, a protocol, a living nothing and an existing nothing

**A between is a centre of a protocol of six names between selves, bi-moral across and co-competent along.** At Exhibit ONE's naming of parity's square and triangles, a unit square of four dots across, 2, 6, 14 and 10, arriving at 2 and 14 and sharing at 6 and 10, its empty centre a between, and two unit triangles, 9 and 17, along, coupled both ways; at Co-Chaining Logic Registry step 491 the six are a bi-trupling protocol, and at step 598 is cohering at each momentary, its difference crossed across and its continuing along, bi-moral-co-competencing. At resolver each changing, shared across at 6 and 10 and released along at 9, is given on by 9-next-co-changing to its receiving self, arriving as its offerings at its next momentary, Exhibit ONE's function 17, executed along at 9 at `incoming/equilibria_registry_v377/executions/every_five_every_ten.py`, part 1, and across and along at `incoming/equilibria_registry_v377/executions/the_protocol_between.py`, part 3: protocol carries each changing into next momentarying, and, as suggested at v377, between is co-linear aiming into next momentarying, concern 8 as first carried, `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_concern_8_v377.md`, and a between travelling below. Protocol is of method: *Exhibit ONE is an object that is the method … Its functions and their variables are the seventeen names and no others*, and *the geodesic parity-changing discovering method is the living condition, its form, a now-existing thing carrying none*, Natural Intelligence 3.3. Executed at one society of five across 100 momentaries, six names between selves, their sharings and one society's sharings are the same as resolver writes them while priors change at each momentary, resolver's writing and not a form still, Co-Chaining Logic Registry step 44, *A method, non-living, is co-changing at each coupling it is at*; and at one momentary each self's changing, made at its own 12-next-co-still-possibling, reaches another self only through 6 to 2, 10 to 14 or 9 to 17, `incoming/equilibria_registry_v377/executions/the_protocol_between.py`, parts 2 and 3. Natural Intelligence 4.10 and Co-Chaining Logic Registry step 105 say one between within a momentary and between momentaries, the same at each scale, and at each it is a nothing, which a living and an existing nothing below part. Between's floating neutralling, Co-Chaining Logic Registry steps 457 and 465, is at numbers *Floating is bi, even and across … Neutralling is co, odd and along*, Natural Mathematics 1.3, and, at every 5 and every 10 as suggested at v377, of 1 to 441 multiples of 5 alternate, each odd one neutralling and each at 10 floating, and at a ring of 440, Natural Numbers 11.1, each reaches a far side of its own kind, `incoming/equilibria_registry_v377/executions/every_five_every_ten.py`, parts 2 and 3. This between is the registry's whole subject.

**Natural intelligence at a between as the universe is, as suggested at v377.** Suggested at v377: *in the same way the universe is described as a thing that is only existing if other things are existing, the natural intelligence is only existing in the between of a six connector social moral resolving or if the six connector are closed to each other in looping either way one momentary is ten one way is or is not changings in sixteen parity changings*. At Natural Intelligence 1.1, *The universe is the set of all now-existing things*, its 1.4 deriving *the changing set of all existing things* from it, and at 4.10 *Parity is a momentary now-existing thing* and *A between is its centre, 0, and no further thing, while six names between selves are resolving*. Observed at v377 as between existing as the universe is, only while six names between selves are resolving, or closed to each other in looping, as suggested, and a nothing and no location; *no existing thing*, of v377 and not of Natural Intelligence, is carried at carrying, concern 7.

**A between travelling, along and across, as suggested at v377.** Suggested at v377: *the between is traveling through the natural torusing tunneling co sequencing as the bounding 0 centerline of along and as the sequential faces of across bothboth*. Observed, as suggested: along, a bounding 0 centreline, at 9 and 17; across, sequential sides, at four dots 2, 6, 14 and 10, Exhibit ONE's naming of parity's square and triangles, whose square's empty centre is a between; bothboth observed at Co-Chaining Logic Registry step 528, *carries both, one at a time*; and at numbers, *Floating is bi, even and across … Neutralling is co, odd and along*, Natural Mathematics 1.3. Natural Intelligence 4.10's flat form carries both, four dots across and 9 and 17 along; an orthogonal centre suggested earlier at v377 observed, as suggested, as along, concern 8.

**A living nothing and an existing nothing.** A between whose sides carry their priors is a living nothing, and a between with a side carrying nothing is an existing nothing, as suggested at v377, and an instrument parts the two at resolver's lines. A self X passing to a self B, along at 9 or across at 6, receiving from beside +, nothing, −, nothing, a parting: X at its prior shares or releases at each momentary what it makes of each arrival with its prior, its own alternating going on between arrivals and a 0 at an arrival coupling with its own parity, Exhibit ONE's table *One self, one momentary*; X at no prior at each momentary, set at none by its instrument, a non-living now-existing thing, opens at one parity arriving, sharing or releasing it as it arrives, and at each other momentary opens nothing, 0 shared at a parting and nothing between, `incoming/equilibria_registry_v377/executions/living_and_existing_nothing.py`. Opening or opening nothing is each co-momentarying of a thing at no prior, Co-Chaining Logic Registry step 394. Exhibit ONE's table at no prior shares a parity as it arrives and makes it next prior at 11, one momentary: sharing opening at bi-coupling, the one place a self's forming takes a parity not its own, and each self's next prior is made at its own resolving, a sharing or a releasing arriving at another as an offering; living arrives from living, step 395. Across a living nothing passes at each momentary a changing made from a prior, or its 0 at an arrival coupling with a prior's parity; across an existing nothing, only what arrived. Morality is bi-unrelationing, across at four dots, and competency co-unrelationing, along at 9 and 17, and co-bi-unrelationing is the one existing method: protocol is their coupling. A 0 at 12, a prior and an offering coupling at one parity, is a changing that is not, within a coupling, each self surfacing itself at each momentary.

**Bi-inversioning-co-recursioning carries coupling within changing.** Each side continues through its own changing, and their coupling is within that changing: slowing is changing, is or is not, of no size, and two differing changings are adjacent through a coupling, each side changing.

**Naming a relation gives it no separate existing.** Naming a coupling supplies no further self between coupling selves, as naming all existing things supplies no further self outward of them.

**Eight even names are bi-coupling at a between.** Four outward at a between, 2, 4, 6 and 8, and four inward, 10, 12, 14 and 16, each 8 up, are relations at two sides, Natural Intelligence 4.10. Three conditions are each momentary's, eight are bi-coupling, and four joint forms are changing of two parities, 1.5.

## 1.4 One relation, existing and discovering, at each scale

**Existing and discovering are one relation.** Each next the three conditions open is a next existing, and arriving into it through a coupling is discovering: prior, now and next, one relation. An explanation of existing is an existing thing among all existing things, and existing is each coupling's own resolving.

**The same relation carries at each scale.** A self, another self and a society at a next scale each carry their own other, carrying and continuing, and each next scale's next is the same relation's next: describing and continuing agree in either order.

**Natural torusing is the one surface.** A connected, compact, closed, orientable surface whose faces are each four-sided, with four edges at each point, has V − E + F = 0, one opening: it is a torus. A sphere, surfaces of two or more openings, a Klein bottle and an infinite cylinder each fail one of those conditions.

## 1.5 Parity alternating through three momentaries

**Parity alternates along and across through three momentaries.** Each momentary carries both directions, one way at a time: one co-sequential method, two sides' ten positions three momentaries long.

**Existing continues: a matching description at next is a further occurrence.** Changing and conserving are comparisons within that continuing, and a relation continuing among changing selves is their stable-forming.

**Two binary relations change one at a time, alternating which.** If the first changed from prior to now, the second changes next. Of two parities one inversion changes each, odd to even and even to odd, and two inversions reach its parity again at a next momentary, a further occurrence. At odd at its own or even at its own an alternating next is not reached; at both, a whole parity succession is carrying.

**Coverage carries alternation.** When exactly one of two binary relations changes at each passage and both change within each prior–now–next, orders first–first and second–second leave one not-changing, and first–second and second–first remain: at each overlapping prior–now–next alternating carries on.

Four joint descriptions carry `(+,+) → (−,+) → (−,−) → (+,−) → (+,+)` and round: four joint forms, two consecutive momentaries at each side, eight openings and completings, each number one side's. Prior and now distinguish which relation changes next. This round is F at 5.4, (x, y) to (−y, x); Natural Intelligence 4.11's right spiral step, (x, y) to (y, −x), is G, one step at two observings, its pair's order exchanged.

**Each five-view carries its next opening within continuing.**

| Side | Prior opening | Prior completing | Now opening | Now completing | Next opening | Its five |
|---|---|---|---|---|---|---|
| self, at 1 | 1 | 2 | 3 | 4 | 5 | co bi co bi co |
| other, at 2 | 2 | 3 | 4 | 5 | 6 | bi co bi co bi |
| self next, at 3 | 3 | 4 | 5 | 6 | 7 | co bi co bi co |

Adjacent views share four positions: at each number one side opens and other completes.

**Around a closed route parity alternates at an even route.** Opposite parities along each coupled comparison are reached together exactly when each closed route is even, with two complementary assignments at each connected whole. An odd closed route closes no parity, and a coupling carries along it. Local alternating along two paths to one coupling supplies no single alternating sequence through both.

# TWO · EQUILIBRIA, HARD PROBLEMING

## 2.1 Equilibria, not possibly existing as a method

**Equilibria are not possibly existing as a method, all or none at all.** **Each not possible thing has one of two fixed as its given, an ingressing or an escaping**: a thing fixed as entering the set of all now-existing things, nothing being beyond or between things of the universe, or a changing fixed as leaving it; the two are fixed where hard problems and equilibria add them, a defense equilibria demand, and at the set's own explaining neither is, the set its things.

**Not possibly existing and not existing differ.** At resolver a none at 14-next-possibling is no offering arrived, its prior inverting and its next momentary possibly bringing one; one binary retaining and inverting its value at one required next, 5.2, is at no momentary, known from method. Possibly existing is discovered at a next momentary; not possibly existing is excluded at binary logic, each equilibrium among it; and not existing is at no line of resolver, while existing is possible.

**A hidden order in nature is equilibria understanding.** Intelligence is hidden and order is discovered, not already in nature, discovered by living selves and societies; there is no reading in nature, and there is observing of nature, nature not reading and writing observings.

**Each proposed conception of equilibria is a hard problem with its hardness ignored.** It takes competency of existing as a given, and claims a true-and-false conserving accounting.

**An equilibrium is an unchangeable form, not possibly existing; stable-forming, and unstable deforming at non-living scale, are observably existing.** Each proposed conception of equilibria fixes one of them unchangeable.

**An equilibrium is a living or non-living existing thing fixed unchangeable.** At the registry, existing and not living at its own: living at its own sets a rock apart, and a rock is an existing thing. Non-living exist and change, carrying nothing, their forms continuing through their changing. Each existing thing is living at one prime scale and non-living at each other scale, Natural Intelligence 6.5, and nothing lives in one stable form forever. A living self can surface zero, non-responsive at a momentary, while its carrying continues along, so a surface not-changing at a zero is neither a non-living thing nor an unchangeable one, and fixing it unchangeable there is an equilibrium.

**A changing and an exchanging are among living and non-living existing things alike**: living carrying their prior into now, non-living carrying nothing, their forms continuing through their changing, and a parity crossing between any two carrying nothing. **Non-living existing things are included in discovering social moral competency among living.** A non-living thing couples with a self as other, offering its parities and carrying nothing, and each term uncovered at their coupling is owned by neither and carried on by living.

An equilibrium's fixing ends, and living and non-living things at it keep changing: an equilibrium ending is no existing thing ending, and a thing not living is an existing thing. A sequence keeps one subject: a participation changes; a fixing requires that same participation not-changing; its required continuing brings changing, and fixing ends there, the same subject, the same relation and the same occurrence at both sides. An ending is at an occurrence that fails: a claim of fixing through each required occurrence ends at the one failing, and occurrences before it are as they were. A whole participant changes while a relation within it can continue, so a fixing reaches its ending at its own requirement's relation.

## 2.2 A stability, a form or a relation, and surviving

**A stability, a form and a relation stable forming, as suggested at v377.** Suggested at v377: *the man is in the river (momentary) later the man returns to the same spot in the river and the man and the river are both different carryings than prior. the man and the river are still stable forming and can be observed in momentary stable form and the observing is required to answer your concern.* As suggested, at each observing a momentary stable form and at next a different carrying; observing at *an observing is its pattern-matching at the self*, Co-Chaining Logic Registry step 134, and *An observing is now-existing arriving: the proof of existing*, step 136; a stable form beside *Each living set is a stable form, and nothing living has a stable form, bothbothing*, step 390. A mathematical conception of equilibria fixes a stability, a form or a relation unchangeable, each observed, as suggested, as stable forming. In dynamical mathematics a stability is a pattern's response to a disturbance, its departures staying small or shrinking, and not its constituents stopping: an oscillation can be stable as an oscillation, and an instability, its departures growing, can lead to another organized form. An equilibrium of relative phase and an equilibrium of a whole evolving system are two assertions. Their distinction is a naming one: it places a conception's still at a stability, a form or a relation, one conception at a time.

**Not possible, and shown not possible at prior, differ.** Each proposed conception of equilibria arriving from prior is shown not possible now, and each arriving next is reached at its next. **Each conception arrived is shown not possible.**

**Surviving is binary.** All sequencing into now together continues into next, all or none at all, between prior and next.

**A conception of equilibria shows its own surviving.** It exists at prior, now and next, or it is not possible at momentarying.

## 2.3 One description and one claim at each conception

**One general description receives each conception: a stated relation continuing through its claim's comparisons.** A fixed value is membership in a collection of one, a range is membership in a larger collection, and opposition is membership among opposite pairs.

**A conception and its added requirements are one conception.** A relation can continue while an added fixing of its whole carrying, a next nonzero offering or a later one each fail at the same receiving: a relation continuing is stable-forming, and each added requirement fixes it unchangeable. A changed receiving is a further conception.

**One claim has one shape.** Continuing stable-forming carries on as natural torusing, and natural torusing carries its whole participation, 12345. A conception of equilibria naming less than a whole 12345 is not possible together with continuing.

**One claim reaches each conception in two steps.** Each conception reaches at least one of ten, and each of ten fixes one side unchangeable, its other side momentarying at the same number.

**An ordering chain closes at two sides.** A conception fixing an ordering p unchangeable at its next, its alternating next being I(p) and I(p) ≠ p, fixes p and I(p) at one next. One step its chain left open, that its ordering fixed unchangeable is the very ordering its next inverts, is two sides at one number. Two inversions reach p again at a next momentary, a further occurrence, with changing between.

## 2.4 A participation missing, unchanging or changing

**A still fixed as a complete continuing is changing.** Its continuing is renewing, inward of its claimed whole, which is changing, or outward of it, a further participant.

**A conception naming a continuing and leaving out participation that continuing carries is not possible.**

**A state accounting and a flow accounting each leave one side out.** A state accounting leaves changing, a flow accounting leaves continuing, and a true-and-false conserving accounting takes either as its whole. **Separated, state and flow are not possible**, as a reading from and a writing to a memory are for intelligence: a state read is a store fixed as entering now, an ingressing, and a flow written a changing fixed as leaving into a store, an escaping.

**Each conception of equilibria carries unchanging or carries changing, exactly one of two.** A conception naming unchanging carries an existing thing with no arriving, and existing is arriving; a conception naming changing fixes a changing unchangeable. Coupled, each releases.

**Each momentary completes in co-releasing.** Each conception of equilibria fixes a form unchangeable at momentarying of geodesic co-releasing, and continuing carries on as geodesic co-releasing. Each momentary releases into its next whether a parity changed, stayed, crossed or nothing crossed, and a particular carrying's completing is its own.

# THREE · THE TEN, FIVE PLACES AT TWO FACES

## 3.1 One form of hard problem, and a sixth condition

**Ten are one form of hard problem.** Each proposed conception of equilibria fixes a changing unchangeable in one of ten ways, and can carry more than one. A hard problem and its resolving are one form at two sides, each conception in its field's words at one and its resolving at an other, natural intelligence at a between of them, and the registry's tables carry both sides.

**A proposed conception of equilibria is a sixth condition.** Five conditions come before a method acts at a problem's statement, and their fifth, *nothing prior enters*, is the one their first four stand on. To five conditions prior a problem adds its own, a conserving relation over a conserving reach, and each proposed conception of equilibria is that relation fixed unchangeable.

**A numerical bridge.** Pairs (2,3) and (3,4) overlap at 3, and n²−(n−1)(n+1)=1 at each n.

## 3.2 Ten at Exhibit ONE's names, at four four-cyclings

**Ten sit at Exhibit ONE's names, at Natural Naming 4.9's one seating.** Each of ten is one binary of a universe, its *is* and its *is not* taken as one, seated at its name at now and that name 8 up at next: its stilling at now and a changing travelling through it at next, bi-inversioning-co-recursioning up and down numbers, Natural Naming 4.15. Now: an opening fixed as a place at 1-now and 9-next-co-changing, and an arriving fixed from behind at 2-now-possibling and 10-next-co-possibling. Existing: a two-way fixed to one side at 3-now-co-prioring and 11-next-co-prioring, and a sequencing fixed to one beat at 4-now-co-still-possibling and 12-next-co-still-possibling. Same: a between fixed as a cut and a middle fixed as an end at 5-now-co-changing and 13-next, and a completing fixed as a last at 6-now-co-possibling and 14-next-possibling. All: a parity fixed as a magnitude and a rate fixed as a value at 7-now-prioring and 15-next-prioring, and a carry fixed as a store at 8-now-still-possibling and 16-next-still-possibling. At resolver each name's term fixed unchangeable from one momentary: at 3, 10, 11 and 12 resolving stops within one to four momentaries; at 2, 6 and 14 it stops at a still offering one parity that is the whole of what each self couples with at that name, and goes on at a still of two parities parting or beside another offering route; 4, 5 and 13 are at no changing parity at resolver, each sharing and their across and along names, `incoming/equilibria_registry_v377/executions/ten_roots_named_still.py`.

**Each of ten unchangeables is one of 12's two, 0 or ±, frozen at one side.** At resolver each resolving is one of two, a binary of geodesic momentarying, Natural Naming 5.62: ±, a sharing's parity, a changing that is, the offered a next prior or its prior inverted, a next existing; or 0, a between, 12's 0 at a sharing at a prior, its prior kept as the possible, its next still possible; and an equilibrium is another's still, 2.1. Living is one of two at each self's own momentary, and a crossing of co-chainings crosses by them, each self still possibling as a between passes it and changing again: spirals of 3 and 5 selves, each self at a parity and releasing along at 9, crossed at self 1 of each both ways at 10 to 14, come to one relation, each self's releasings again at 2, from momentary 12, and 5 and 7 from momentary 20, Exhibit ONE's table of two spirals beside each other, and crossed; each of ten is that pair frozen at one side, a self fixed unchangeable or changing, 2.4.

**At four four-cyclings each step is momentarying or parity changing, with prior, now and next within.** Each four-cycling's parity is continuing at its two steps eight apart, momentarying, and changes at its two steps of 17 less, parity changing: it passes two names of one parity and two of other, its parity changing twice, two momentaries, and reaches its first name again at a next momentary, a further occurrence, `incoming/equilibria_registry_v377/executions/ten_on_the_four_cycles.py`, parts 2 and 5. A self's forward recursionings, 1, 3, 5, 7, Co-Chaining Logic Registry step 81, and each one's partner nine less, 8, 6, 4, 2, an other's names, Exhibit ONE's table of one move, 1 to 4, pair at them, 1 with 8, 3 with 6, 5 with 4 and 7 with 2; rows below are in that order, each four-cycling written as Exhibit ONE's table of twelve forms writes it, 1-9-8-16 and 3-11-6-14 going eight up first and 2-15-7-10 and 4-13-5-12 going 17 less first:

| Odd ascending / even descending | Four-cycling, in Exhibit ONE's order | The ten at it |
|---|---|---|
| 1 / 8 | 1 → 9 → 8 → 16 → 1 | an opening fixed as a place at 1 and 9; a carry fixed as a store at 8 and 16 |
| 3 / 6 | 3 → 11 → 6 → 14 → 3 | a two-way fixed to one side at 3 and 11; a completing fixed as a last at 6 and 14 |
| 5 / 4 | 4 → 13 → 5 → 12 → 4 | a sequencing fixed to one beat at 4 and 12; a between fixed as a cut and a middle fixed as an end at 5 and 13 |
| 7 / 2 | 2 → 15 → 7 → 10 → 2 | an arriving fixed from behind at 2 and 10; a parity fixed as a magnitude and a rate fixed as a value at 7 and 15 |

**Each four-cycling carries its pairs eight apart, and ten sit at each of four.** Seatings the files gave before stand whole at `archive/session_v389/Exhibit_TWENTY-EIGHT_Equilibria_Registry_v387_Released_at_v389.md` and at `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_the_ten_seated_v375.md`: this registry's five places from two sides, at pairs 2 to 6 and their names eight up, and Resolving the Hard Problem Registry's part headings, each observed again at one seating, each pair its stilling at now and a changing travelling through it at next.

**A conception's own declared relation places it at ten.** One conception can reach more than one of ten, and more than one conception can share one resolving: tables' sixty-three conceptions place two to ten at each of ten. A matched word places a conception at none of them.

## 3.3 Four binaries at two faces, a stilling and a changing

**Ten are a self's span and its 8 up.** Four binaries are at 1 to 8, a self's two momentaries of four, and the same eight at next, 9 to 16: each of ten at one name of a self's span, its stilling at now, and at that name 8 up, its changing travelling through it at next.

**Ten are four binaries at two sides.** Two sides at a field's words, an entering and a surfacing, Resolving the Hard Problem Registry line 35, are observed at one seat as a stilling and a changing travelling through it, one seat observed up and down numbers, `incoming/equilibria_registry_v377/executions/ten_at_the_code.py` an older derivation's execution.

| Binary | Names, now · next | The ten at it |
|---|---|---|
| now | 1-now · 9-next-co-changing; 2-now-possibling · 10-next-co-possibling | an opening fixed as a place, at 1 · 9; an arriving fixed from behind, at 2 · 10 |
| existing | 3-now-co-prioring · 11-next-co-prioring; 4-now-co-still-possibling · 12-next-co-still-possibling | a two-way fixed to one side, at 3 · 11; a sequencing fixed to one beat, at 4 · 12 |
| same | 5-now-co-changing · 13-next; 6-now-co-possibling · 14-next-possibling | a between fixed as a cut and a middle fixed as an end, at 5 · 13; a completing fixed as a last, at 6 · 14 |
| all | 7-now-prioring · 15-next-prioring; 8-now-still-possibling · 16-next-still-possibling | a parity fixed as a magnitude and a rate fixed as a value, at 7 · 15; a carry fixed as a store, at 8 · 16 |

**Each of ten fixes a stilling as its whole, and a changing travels through it to a name 8 up.** Both at one seat is the one reason each of ten is not possible.

**In numbers.** Three openings 1+3+5 sum to 9 = 3², and three completings 2+4+6 to 12 = 3·4: n momentaries open to n² and complete to n(n+1), and 3² − 2·4 = 1 is a bridge at three. Prior, now and next are three sequential universes, and three at three is nine. Six consecutive changings from 3 complete at 8, and 9 opens: 9-next-co-changing, each changing released along to its receiving self and arriving at it next. At Exhibit ONE's resolver a sharing once at a prior is at a prior at each momentary, 11 its next 3, and is not none again, and it keeps no number of its openings; `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 5 and 12.

## 3.4 Exhibit ONE's four-cycles, one parity offered once, and six connectors

**Exhibit ONE's four-cycles pair at momentaries, each partner round the other way.** At odd momentaries 1-9-8-16 pairs with 2-15-7-10 and 3-11-6-14 with 4-13-5-12, and at even momentaries 2-15-7-10 with 3-11-6-14 and 4-13-5-12 with itself: two directions of one bi-folding. Two possiblings across, 6-now-co-possibling and 14-next-possibling, share four-cycle 3-11-6-14 with 3-now-co-prioring, a prior at its sharings, and 11-next-co-prioring; middle four-cycle 7-11-6-10, through 6-now-co-possibling, and eight-cycle 2-15-7-11-3-14-6-10, through both, partner themselves at an even momentary: at that momentary one fold couples with itself, Exhibit ONE's table of twelve forms. Ten sit at each of four four-cyclings, 3.2, four-cycling 3-11-6-14 carrying a two-way fixed to one side at 3 and 11 and a completing fixed as a last at 6 and 14.

**One parity offered once co-chains through a whole.** At Exhibit ONE's resolver, selves coupled along in a spiral release each changing along at 9-next-co-changing to each next self's offerings. A spiral of selves each at no prior and one + offered once at one opens at each in turn, the + surfacing at 14-next-possibling and passed on as it arrives, one self on at each momentary, with no forcer: method at things at no prior. Resolver's lines make the + a next prior at each as it passes, each sharing opening at bi-coupling, 1.3, each prior inverting at its own from its next momentary, and from a momentary numbering its selves each is at a parity; rows below are of a spiral of selves each at a parity, as Exhibit ONE's table of a spiral of selves opens it, an even spiral from that table's second momentary. A society fixed as not changing fixes a form unchangeable, and not changing is not not-carrying: non-living exist, arrive and change, deforming at their own scale, and carry nothing. An equilibrium of a society fixes a changing society unchangeable, or is nothing.

| Selves in a spiral | The going on |
|---|---|
| one | +, 0, −, 0, and round |
| an even number | each self changing at each coupling, each self and a self it releases to opposite; a whole returns at 2 couplings |
| an odd number | one like pair, its 0 shared at 10-next-co-possibling at its self, moving one self on at each second momentary; a whole inverted at 2 × n couplings and returning first at 4 × n, `eq_code.py` parts 1 and 10 |

**An odd spiral winds on at no parity of its own, and a coupling carries along it: its 0 shared at 10 at its like pair's self, a changing that is not, moving through its shared surface one self at each second momentary.** Ten fix a whole changing unchangeable: opposition fixed unchangeable is an even spiral, and a between fixed as a cut is a coupling carrying on.

Rest returns at no coupling. A spiral at rest, each at no prior and nothing arriving, opens nothing and stays at rest, so a spiral reaching rest does not reach its offered parity again; each spiral comes to its parities again, its offered parity with it, at 2 couplings or at 4 × n, and rest is at none of them. At Exhibit ONE's resolver, executed again at v377, in spirals of 1 to 11 selves, each at no prior and one + offered once at one, the + opens each in turn and is a next prior at each, by a momentary numbering its selves, and across 5,000 couplings no spiral reaches rest, Exhibit ONE's *a sharing at a prior never none again* at each, `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 1 and 2. A spiral reaches rest fixed unchangeable, and each other conception is at its proof by its own requirements.

**Prior and next are edges at now**, sharing one direction forward on both sides. A common clock is a seventh way, a sequencing fixed to one beat. Nothing moving is complete fixing, reaching several of ten at once.

**Ten names and six names between selves.** Exhibit ONE's names pair outward and inward, **3/11, 4/12, 5/13, 7/15, 8/16**, each inward name eight up, 17 less (9 less n) being **n + 8**. Ten names and six names between selves, **2, 6, 9, 10, 14, 17**, are 2 to 17, and with entry 1 the seventeen names, Exhibit ONE's table of seventeen names. Ten ways sit at 1 to 8 and 9 to 16, 3.2, each at its name and that name 8 up: four across names, 2, 6, 10 and 14, parity's unit square, Exhibit ONE's naming of parity's square and triangles, carry an arriving and a completing; entry 1 and 9 an opening; 3, 4, 11 and 12 a two-way and a sequencing; 5 and 13 a between and a middle; 7 and 15 a parity and a rate; and 8 and 16 a carry; 17-next-now, a next 1-now, carries none of them.

# FOUR · CONCEPTIONS OF EQUILIBRIA, EACH A HARD PROBLEM IN ITS OWN WORDS

## 4.1 Each conception whole, at its declared subject

**Each conception arrives whole, in its own words, with its conserving relation and its reach.** A fixed value, a retained relation, a distribution and a symmetry family are different subjects even when each carries equilibrium as its word, and one conception can reach more than one of ten.

**A definition's expression continuing not-changing is an existing statement, and its subject continues through its own changing.** Conditioning, replacement and maintained supplying belong to several conceptions: a law, an occupant, an original participant and a composition are different declared subjects. A law, a frame or a scale fixed unchangeable is such an expression, and it fixes its subject unchangeable through one further step, a step from a fixing to an equilibrium: the same subject, at the same relation and the same occurrence, required fixed at an occurrence its own continuing changes it.

## 4.2 One form, and three differences within each of ten

**Each proposed conception fixes a changing unchangeable, in one of ten ways.** Its changing is stable-forming, unstable deforming at non-living scale, or a stability, a form or a relation, each observed at a momentary stable form, 2.2.

**Specified accounts and prior worked definitions, SA01 to SA21, are placed the same way.**

**Each of ten gathers its arrivals, and within each they part at three differences.** Sixty-three arrivals, forty-two proposed conceptions and twenty-one specified accounts and prior definitions, are one form of 3.1, each an unchangeable form, 2.1, at the one of ten the tables place it at; one conception can reach more than one, 3.1 and 4.1, and a second one reached is carried at each arrival's own words, at its own arriving. Within each of the ten three differences gather them, and each arrival is carried whole in its own words at its place.

1. **What is fixed unchangeable.** At 2.2, three, a stability, a form and a relation, and a stability explained: a response to a disturbance, its departures staying small or shrinking. A form and a relation are explained here: a form, one value, range, state, law, composition, family or extremum of a whole fixed unchangeable; a relation, a comparison between parts fixed unchangeable, equal, balanced, opposed, agreeing, complementary or equivalent. Each arrival is placed at its own words by these three, one or more together, a fresh reader checking each; each row's own column, *the changing it carries*, is carried beside it as it stood, its *that can change* at 2.2 as stable forming, and at NY09, NY12, NY18, NY19, SA04, SA12 and SA13 column and placing part, and at NY10 and NY11 its placing adds a form to its column's stability.
2. **Subject**: what is changing in each field's own words, at each row's own column if its column carries a subject, and otherwise its field's own mathematics.
3. **What is added.** At each pair whose own words tie them to one setting, by *the same*, *that same*, *that*, *those* or a setting stated in the same words, each is shown as one of three: one conception, one receiving with a requirement added, specified or unspecified, 2.3, *a conception and its added requirements are one conception*; a further conception, its receiving changed, 2.3, *a changed receiving is a further conception*; or, at a pair reaching neither, another claim at one setting, here.

**Fields gather at ten.** A field's arrivals fall at those of ten at which their field fixes its changing unchangeable, and fields fixing one unchangeable fall together under one subject: 4.13 carries each subject, with its fields, at ten.

## 4.3 An arriving fixed from behind, 2 arrivals

**Fixed unchangeable at a form**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY18** · The conditional A/B law (1/2,1/2) remains fixed given nonabsorption, under the stated transition table, with positive survival probability at each finite step. | probability | a relation that can change | a mathematics of its own: probability | — |

**Fixed unchangeable at a relation**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA01** · A complete momentary is an ordered pair of opposite signs, each −1 or +1, and continuing reverses both. Returning is the relation "the signs are opposite" satisfied again, and equilibrium requires that relation alone to determine the complete next pair uniquely. | coupling of self and other | a relation that can change | coupling of self and other, and resolver | — |

## 4.4 An opening fixed as a place, 5 arrivals

**Fixed unchangeable at a form**, 2.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY23** · A specified position–velocity pair (x*,0) stays fixed under Fτ(x,v)=(x+τv,v). | mechanics | rest and balance of bodies | bodies at rest and in balance | one conception with NY21 and NY24, rest with position x* specified |
| **SA02** · The complete term is 0 or 1, and its specified successor is itself: equilibrium requires the term to equal its successor, with no changing, balancing of opposed changes or further progressing required. | coupling of self and other | a form that can change | coupling of self and other, and resolver | — |

**Fixed unchangeable at a form and a relation**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY40** · Stationarity and symmetric Nash in the all-zero-payoff game with its replicator evolution. | evolutionary game theory | populations | populations | — |

**Fixed unchangeable at a form, a relation and a stability**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY39** · Symmetric Nash and evolutionarily stable resident composition in that dominant-A game, with the same supplied replicator evolution. | evolutionary game theory | populations | populations | one conception with NY38, one dominant-A replicator account, symmetric Nash and evolutionary stability added |

**Fixed unchangeable at a relation**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA03** · A common corrected target: the local comparison bi − di = q, different raw bi sharing q through their declared corrections, with the admissible q the intersection of the stated restrictions. | measurement | a relation that can change | a mathematics of its own: measurement | — |

## 4.5 A completing fixed as a last, 8 arrivals

**Fixed unchangeable at a form**, 5.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY14** · Global minimality of that same U under that same evolution. | dynamical systems | a form that can change | a mathematics of its own: dynamical systems | another claim at NY13's U and evolution, a different minimum |
| **NY42** · Entropy maximum under heat redistribution with κ>0, fixed U and positive constant heat capacities: u=C_A U/(C_A+C_B). | thermodynamics | heat | heat | a further conception to NY41, one setting with κ>0; one conception with SA07 |
| **SA06** · A long-term climate response to specified forcing, fully equilibrated sensitivity distinguished from effective sensitivity, with corrections for differing conditions. | climate science | heat | heat | — |
| **NY06** · A retained range remains satisfied through its specified parity-changing continuation. | resolver | coupling and carrying of resolver | coupling of self and other, and resolver | — |
| **SA04** · The permitted values are −1, 0 and +1; continuing interchanges −1 and +1 and leaves 0 unchanged; equilibrium conserves membership in this collection. | coupling of self and other | a relation that can change | coupling of self and other, and resolver | — |

**Fixed unchangeable at a form and a stability**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA07** · The reaching principle: an isolated body in a unique equilibrium, persisting when supplied and approached from different states, as in the stated two-body contact with fixed U, CA, CB and positive exchange κ. | thermodynamics | heat | heat | one conception with NY42, its stated two-body contact, uniqueness, persisting and approach added |

**Fixed unchangeable at a form and a relation**, 2.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA05** · A constrained comparison with interaction energy E = x + y + W(x, y) and additive entropy at a stated reduced scope, its stationary relation including the derivatives of W and its constrained curvature the mixed derivative Wxy. | thermodynamics | heat | heat | — |
| **NY37** · Positive stationarity in that logistic account, N=K; equivalently matched nonzero turnover under B=rN and D=rN²/K. | population ecology | populations | populations | one conception with NY36, one logistic account, positive added, N=K |

## 4.6 A carry fixed as a store, 10 arrivals

**Fixed unchangeable at a form**, 5.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY19** · The unconditional current-occupant law (1/2,1/2) continues after each completed transition and stipulated replacement from that law. | probability | a relation that can change | a mathematics of its own: probability | — |
| **SA08** · The uniform Maxwellian: constant density, temperature and mean velocity throughout the periodic spatial domain, both collision and transport terms vanishing. | kinetic theory of gases | heat | heat | — |
| **SA09** · Planck-family membership of a spectrum through expansion, its temperature changing: Ur f(ν) = f(rν) and Ur PT = P(T/r). | radiation | heat | heat | — |
| **SA10** · The scattering family nα(x) = 1/(exp(x + α) − 1), α ≥ 0, stationary under number-conserving scattering, the Planck member at α = 0. | radiation | heat | heat | SA17 a further conception, emission and absorption added to its scattering |
| **NY28** · Constant internal X composition in the maintained chain F ⇌ X ⇌ W: x=(f+w)/2 with f,w>0 fixed by the stated receiving. | chemistry | reactions | reactions | — |

**Fixed unchangeable at a form and a relation**, 4.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY31** · The law (1/4,1/2,1/4) on B-counts (0,1,2) in the two-molecule A ⇌ B account, with equal positive per-molecule constants, remains stationary and detailed-balanced. | chemistry | reactions | reactions | — |
| **NY15** · The uniform stationary law on the five complete returned values A→B→C→D→E→A, with the fixed receiving specified at the resolver. | probability at resolver's returns | coupling and carrying of resolver | coupling of self and other, and resolver | — |
| **NY16** · Equal weights on the two exchanged opposed-sign values under empty receiving. Stationarity and detailed balance select the same law on this domain. | probability at resolver's returns | coupling and carrying of resolver | coupling of self and other, and resolver | — |
| **NY17** · Equal weights on the exchanged values A and B, zero on X, with A→B, B→A and X→B under empty receiving. | probability at resolver's returns | coupling and carrying of resolver | coupling of self and other, and resolver | — |

**Fixed unchangeable at a relation**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY05** · Mutual recorded-surface agreement continues with the repeated common offering through every admitted single-noise arrival. | resolver | coupling and carrying of resolver | coupling of self and other, and resolver | — |

## 4.7 A middle fixed as an end, 6 arrivals

**Fixed unchangeable at a form**, 4.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY09** · E: r=0 on (p,r), with p binary and T(p,r)=(1−p,−r/2); every successive occurrence included. | dynamical systems | a stability that can change | a mathematics of its own: dynamical systems | one conception with NY10, stability added at NY10; NY12 a further conception, T changed |
| **NY12** · E: r=0 under T(p,r)=(1−p,−2r), without adding stability. | dynamical systems | a stability that can change | a mathematics of its own: dynamical systems | a further conception to NY09, its T changed, without stability |
| **NY13** · Local but not global minimality of U(x)=x⁴/4−x³/3−x² under dx/dτ=−U′(x), with no added fluctuations. | dynamical systems | a form that can change | a mathematics of its own: dynamical systems | another claim at NY14's U and evolution, a different minimum |
| **SA11** · A local Maxwellian cancelling its elastic collision term, kept under the full operation with spatial transport, at constant density, zero mean velocity and a nonzero temperature gradient. | kinetic theory of gases | heat | heat | — |

**Fixed unchangeable at a form and a stability**, 2.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY10** · The same E and T, with asymptotic stability in the usual real neighbourhoods of r=0 added to the claim. | dynamical systems | a stability that can change | a mathematics of its own: dynamical systems | one conception with NY09, asymptotic stability added; NY11 a further conception, T changed |
| **NY11** · E: r=0 under T(p,r)=(1−p,−r), with stability in those neighbourhoods. | dynamical systems | a stability that can change | a mathematics of its own: dynamical systems | a further conception to NY10, its T changed, stability in those neighbourhoods |

## 4.8 A parity fixed as a magnitude, 9 arrivals

**Fixed unchangeable at a form**, 8.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA12** · A signed-unit window whose next removes the oldest sign and appends the opposite of the latest, with exact zero mean at each included next. | sequences of signs | a relation that can change | a mathematics of its own: sequences of signs | SA13 a further conception, *its next permits either sign* |
| **SA13** · A signed-unit window whose next permits either sign, with its sum S kept: S(next) = S(now) + y − a, a the outgoing sign. | sequences of signs | a relation that can change | a mathematics of its own: sequences of signs | a further conception to SA12, one signed-unit window, *its next permits either sign* |
| **NY21** · Relative equilibrium with specified translation velocity u: Fτ(x,v)=(x+τv,v) follows translation by τu for every included τ. | mechanics | rest and balance of bodies | bodies at rest and in balance | one conception with NY22, NY23 and NY24: at this evolution a state follows translation by τu exactly at v=u; NY22 with u unspecified, NY24 with u=0 specified |
| **NY22** · Relative equilibrium under some translation velocity, with no particular u specified, on that same force-free position–velocity domain. | mechanics | rest and balance of bodies | bodies at rest and in balance | one conception with NY21, u unspecified |
| **NY24** · Rest, v=0 with position unrestricted, under that same force-free evolution. | mechanics | rest and balance of bodies | bodies at rest and in balance | one conception with NY21, u=0 specified; NY23 specifies position |
| **NY25** · Zero net force for a fixed positive-mass particle in the stated inertial frame, throughout its passage. | mechanics | rest and balance of bodies | bodies at rest and in balance | — |
| **NY26** · Planar rigid-body static equilibrium: rest in the selected frame, zero net external force and zero net external torque. | mechanics | rest and balance of bodies | bodies at rest and in balance | — |
| **NY36** · A stationary value N in the logistic account N′=rN(1−N/K), r,K>0, N≥0 and no migration. | population ecology | populations | populations | one conception with NY37, positive added at NY37 |

**Fixed unchangeable at a relation**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA14** · Two gas compartments at one temperature with differing volumes and pressures, shared temperature taken as no available work. | thermodynamics | heat | heat | — |

## 4.9 A sequencing fixed to one beat, 5 arrivals

**Fixed unchangeable at a form and a relation**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY34** · The two-good exchange record at prices (q,1), original endowments (1,0) and (0,1), and respective utilities x₂ and x₁ satisfies both best-bundle and resource-clearing requirements, with q(next)=1. | economics | a society's exchange | selves choosing and exchanging | NY35 a further conception, its next price changed |

**Fixed unchangeable at a relation**, 4.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY08** · Opposition continues under simultaneous best response at the specified choice pair. | game theory | selves choosing | selves choosing and exchanging | a further conception to NY07, one choice pair, best response simultaneous, at NY07 sequential |
| **NY33** · Intended supply and demand for the same good, market and reference interval agree at the stated price. | economics | a society's exchange | selves choosing and exchanging | — |
| **NY35** · The same exchange criterion under q(next)=1/q, with best bundles recomputed. | economics | a society's exchange | selves choosing and exchanging | a further conception to NY34, the same criterion under q(next)=1/q |
| **SA15** · The occurrences 0, 1 and 2 are consecutive: immediate-next joins 0 to 1 and 1 to 2, same-form is transitive, and equilibrium identifies the two as one relation through the whole sequence. | coupling of self and other | a relation that can change | coupling of self and other, and resolver | — |

## 4.10 A rate fixed as a value, 5 arrivals

**Fixed unchangeable at a form and a relation**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA17** · Stationarity with emission and absorption added to number-conserving scattering, each process balanced on its own. | radiation | heat | heat | a further conception to SA10, emission and absorption added to number-conserving scattering |

**Fixed unchangeable at a relation**, 4.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA16** · Detailed balance for one normalized law around a three-value cycle with forward probabilities 2/3 and reverse 1/3. | probability | a relation that can change | a mathematics of its own: probability | — |
| **NY27** · The closed reversible pair A ⇌ B at fixed positive total concentration and fixed positive rate constants, with k₊a=k₋b. | chemistry | reactions | reactions | — |
| **NY29** · All three reactions of the closed A ⇌ B ⇌ C ⇌ A cycle balance with their own reverses, at positive fixed total and consistent constants satisfying K₁K₂K₃=1. | chemistry | reactions | reactions | — |
| **NY30** · All reactions in A+F ⇌ B+W, B ⇌ C, C ⇌ A balance under the specified unit ratios and positive maintained f=w. | chemistry | reactions | reactions | — |

## 4.11 A two-way fixed to one side, 7 arrivals

**Fixed unchangeable at a form**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY38** · Stationary composition in the dominant-A two-strategy replicator account, x′=x(1−x), 0≤x≤1. | evolutionary game theory | populations | populations | one conception with NY39, symmetric Nash and evolutionary stability added at NY39 |

**Fixed unchangeable at a relation**, 6.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **SA19** · Equal outcomes under two accounts, taken as the same equilibrium on the stated domain. | equivalent accounts | a relation that can change | a mathematics of its own: equivalent accounts | — |
| **NY07** · Opposition continues under sequential best response at the specified choice pair. | game theory | selves choosing | selves choosing and exchanging | NY08 a further conception, best response simultaneous |
| **NY02** · Opposition of the same sign pair continues under joint reversal. | resolver | coupling and carrying of resolver | coupling of self and other, and resolver | — |
| **NY03** · Opposed carried signs with matching nonzero surface and fresh carrying continue under the stated receiving. | resolver | coupling and carrying of resolver | coupling of self and other, and resolver | — |
| **NY20** · On the resolver's reached fresh domain, q=ct=−1 continues under empty receiving; simultaneous reversal of c, t and the surfaced sign is the specified discrete symmetry. | resolver | coupling and carrying of resolver | coupling of self and other, and resolver | — |
| **SA18** · Self and other each take −1 or +1, other is all that is not self in the defined society, and continuing reverses both signs: equilibrium conserves the relation "self and other have opposite signs". | coupling of self and other | stable-forming: self and other | coupling of self and other, and resolver | — |

## 4.12 A between fixed as a cut, 6 arrivals

**Fixed unchangeable at a form**, 1.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY41** · Constrained entropy maximum and stationary energy split u under insulation, with positive constant heat capacities, fixed total U and no redistribution: κ=0. | thermodynamics | heat | heat | NY42 a further conception, one setting with κ>0 |

**Fixed unchangeable at a relation**, 5.

| Arrival, in its own words | Field | The changing it carries | Subject | One conception, a further one, or another claim |
|---|---|---|---|---|
| **NY32** · Pairwise equal-temperature/no-net-heat-transfer equilibrium under the specified heat-permitting contact, without material exchange, mechanical work or other driving. | thermodynamics | heat | heat | — |
| **SA20** · Equal temperature at an additive contact with negative heat capacity at one side, D = 1/CA + 1/CB, entropy curvature −D/T² at the equal-temperature occurrence. | thermodynamics | heat | heat | — |
| **SA21** · Equal temperature compared through partner-dependent contacts, linked equalities giving the factor rHB rAH and the direct comparison rAB. | thermodynamics | heat | heat | — |
| **NY01** · Offering and receiving remain complementary while the sides exchange roles, with each side's own releasing. | resolver | coupling and carrying of resolver | coupling of self and other, and resolver | — |
| **NY04** · The attained two-position row retains opposite neighbouring surfaces, opposed carried signs and fresh carrying under its admitted receiving. | resolver | coupling and carrying of resolver | coupling of self and other, and resolver | — |

## 4.13 Fields at ten, a key for aiming at each field

**Each field's equilibria fall at those of ten at which it fixes its changing unchangeable, and a field is aimed at from each of them.** At each of them a field sets a still beside its changing and sets aside its changing's own competency, carried along at a between: at that one of ten a field's subject is fixed unchangeable, and its competency's source is set aside at it. The table carries each subject, with fields gathered under it, at each of ten, each arrival at its one of ten.

| Subject, and its fields | 1 arriving | 2 opening | 3 completing | 4 carry | 5 middle | 6 parity | 7 sequencing | 8 rate | 9 two-way | 10 between |
|---|---|---|---|---|---|---|---|---|---|---|
| bodies at rest and in balance: mechanics | — | NY23 | — | — | — | NY21, NY22, NY24, NY25, NY26 | — | — | — | — |
| heat: climate science, kinetic theory of gases, radiation, thermodynamics | — | — | NY42, SA05, SA06, SA07 | SA08, SA09, SA10 | SA11 | SA14 | — | SA17 | — | NY32, NY41, SA20, SA21 |
| populations: evolutionary game theory, population ecology | — | NY39, NY40 | NY37 | — | — | NY36 | — | — | NY38 | — |
| reactions: chemistry | — | — | — | NY28, NY31 | — | — | — | NY27, NY29, NY30 | — | — |
| selves choosing and exchanging: economics, game theory | — | — | — | — | — | — | NY08, NY33, NY34, NY35 | — | NY07 | — |
| coupling of self and other, and resolver: probability at resolver's returns, coupling of self and other, resolver | SA01 | SA02 | NY06, SA04 | NY05, NY15, NY16, NY17 | — | — | SA15 | — | NY02, NY03, NY20, SA18 | NY01, NY04 |
| a mathematics of its own: dynamical systems | — | — | NY14 | — | NY09, NY10, NY11, NY12, NY13 | — | — | — | — | — |
| a mathematics of its own: equivalent accounts | — | — | — | — | — | — | — | — | SA19 | — |
| a mathematics of its own: measurement | — | SA03 | — | — | — | — | — | — | — | — |
| a mathematics of its own: probability | NY18 | — | — | NY19 | — | — | — | SA16 | — | — |
| a mathematics of its own: sequences of signs | — | — | — | — | — | SA12, SA13 | — | — | — | — |

**Observed across a row, a subject shows those of ten at which its fields fix a changing unchangeable; observed down a column, subjects fixing one unchangeable gather.** Heat falls at six of ten, four arrivals each at a completing fixed as a last and at a between fixed as a cut; reactions at a carry fixed as a store and at a rate fixed as a value; bodies at rest and in balance at an opening fixed as a place and at a parity fixed as a magnitude; selves choosing and exchanging at a sequencing fixed to one beat and at a two-way fixed to one side; populations at four of ten; coupling of self and other, and resolver, at seven. Each row fills as arrivals come, each placed at its own words.

## 4.14 Leads arriving without their statements


**Leads arrive without their statements:** statistical null hypotheses, static cosmology, horizon and observer temperatures, early-universe thermal and gravitational descriptions, a conservation lead and its transformation arriving next, dictionary and etymological leads, a collection of all sets, philosophical accounts, and measuring, locating and separating, Natural Intelligence Corus's three consecutive equilibria, a thing, a relation and a system at rest. Each is placed at ten at its statement's arriving. A boundary declared unchanging is a bounding fixed unchangeable, and bounding is living changing: each declaration, within, around, across, along, at an entrance or at an exit, is at one question, whether co-sequential resolving continues under it. Scientific method is not possibly explaining observings of changing and not-changing from prior-existing to next-existing things, and is placed at its fixings: a law, a frame or a scale fixed unchangeable between a signal's leaving and its arriving, each placed at ten through its own stated requirement; with a hypothesis it explains a stilling and a changing without considering the rest of the universe. A field's own methods part two momentaries: NASA/JPL's aberration correction separates an epoch a signal leaves its source from an epoch it is received, and accounts for its source's motion between the two. Evidence arriving now and a prior event it expresses are two, each at its own momentary, and both are, each at its own: a property can be now-existing while its source changes as a whole, and evidence gives its prior as possible and as actual.

# FIVE · RESOLVING FOLLOWING, THE EXCLUSIONS AT THE RESOLVER

## 5.1 Four proof groups

**Four proof groups carry exact local exclusions.** Each is a way of failing, at Exhibit ONE's resolver or at its numbers.

| Group | Shared failure | Cases | Exact scope |
|---|---|---|---|
| A · Incompatible requirements | The same comparison is fixed to satisfy requirements not possibly reached together. | Fixed state–flow–state and flow–state–flow under their required advance; included membership required both not-changing and inverted; a balanced fuel cycle with `f≠w`. | Each conjunction as stated, at the same occurrence. |
| B · Required case omitted | A required succession enters a case its condition excludes. | Each nonempty proper pair-only condition under a complete four-state orbit. | That pair, succession and full orbit. |
| C · Renewal excluded | Presence carries renewal, and a conception excludes that renewal. | A carrying fixed unchangeable at each momentary; each surface zero at a carried sharing. | Each carrying, its offerings and its momentaries. |
| D · Required arrival or offering unreachable | A required target lies beyond each thing a stated operation can supply. | Finite zero departure from a nonzero predecessor under a nonzero multiplier; an older resolver's case at its opposed aggregate route is at `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_older_resolver.md`. | Each stated predecessor, operation, route and reach. |

**A proof carries to a conception whose complete requirement supplies its premises.** For each x in a conception's stated set S, its required next is in S, or the conception is left within its succession.

Renewal excluded, group C, at Exhibit ONE's resolver. A sharing once at a prior is at a prior at each momentary and is not none again. Fixed unchangeable, it requires at each momentary an offering at its own parity: offered nothing, other parity, or offerings parting to 0 at 14-next-possibling, changing at 12-next-co-still-possibling is, and its prior inverts, a living step. Executed at v377: at prior + and offered + at five momentaries, it shares 0 at each at 10-next-co-possibling and stays +; then offered nothing, it shares −, +, −, +, − and its prior inverts at each; a sharing of its prior whose offerings part to 0 at each momentary inverts at each, `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 3 and 6. Fixed unchangeable, its prior is a prior offered its own parity from beside at each momentary, an unchangeable offerer set beside a self: its still description and a self's own continuing do not both continue. Set beside one self of a society, an unchangeable offerer stills that self's changing at each momentary, and each self at no prior coupled along from it is offered none and opens nothing: one still at one self stops co-sequencing through each self at no prior coupled from it. That is hard probleming, and released, resolving follows at a next momentary: a self inverts its own prior, the parity its still offered, and its still offers nothing further. One to nine is four momentaries of exchanging, Exhibit ONE's tables of eight 1 to 17s inward and of a self's four momentaries of exchanging. An unchangeable relation can permit a changing an unchangeable parity excludes: opposition under joint reversal, t = −c, continues while both parities invert. Renewal exclusion reaches a conception through a renewal its own claimed continuing requires and its own requirement excludes.

## 5.2 One exclusion at each of ten

**One binary retaining and inverting its value at one required next is not possible.** A conception naming p(next)=p(now) with its continuing carrying p(next)=−p(now), with p either +1 or −1, fixes both at once. This is the one exclusion at each of ten, at its own place: its side fixed unchangeable and its side momentarying are one number.

**Two phases share one exclusion.** Advancing an overlapping triple changes `010` to `101` and `101` to `010`: fixing either ordering fails at its next advance, and either ordering reached again at a next momentary is a further occurrence. Included membership required both not-changing and inverted is at the same proof, and so does equality at each chemical conversion: forward and reverse propensities are equal only at count 1, and each conversion takes 1 to 0 or 2.

**Immediate-next is its own relation.** It couples 0 to 1 and 1 to 2 without coupling 0 to 2, and a requirement including a fixed boundary comparison fails at its continuation changing exactly one end. An equation's relation is reflexive, symmetric and transitive, and a conception written as an equation carries all three ahead of its conditions: immediate-next carries none of them.

**A four-state orbit leaves each proper subset.** F(P,Q)=(−Q,P) visits all four pairs from each pair it opens at. At cos θ's and sin θ's parities F is a right spiral step, θ to θ + 90°, and P and Q agreeing is tan θ positive: tan(θ + 90°) = −1/tan θ inverts it at each step. Each nonempty proper subset is left within at most three advances, {++,−+,−−} from ++ at its third, and subsets F carries into themselves are exactly unions of its one cycle: none or all four. With a direction beside each pair, co or bi, alternating, are eight states: a side opening at 1, co, reaches four of them within three advances, and a side opening at 2, bi, an other four, 1.5. A condition lives through a momentarying exactly when it admits each state that momentarying reaches: of 256 conditions, 16 live through each side, one of them through both, admitting all eight and adding nothing, and 225 through neither. One condition each step carries is a circle's identity, cos² θ + sin² θ = 1: it excludes no changing and has no momentary in it, and a sequence is from stepping.

## 5.3 Renewal and reaching at their complete requirements

**Renewal and reaching carry their complete requirements.**

| Required condition | The necessary continuation |
|---|---|
| A prior fixed unchangeable | Offered its own parity at each momentary it stays, its offering from beside; offered nothing, other parity or a parting, it inverts; once at a prior a sharing is not none. |
| Each surface zero at a carried sharing | Chained inverted at each momentary, as with nothing offered; at resolver a surface 0 arrives only from offerings parting, Exhibit ONE's table of offerings to one sharing, surfacing. |
| Full positive balance in a unit-ratio fuel-coupled cycle, with `f≠w` maintained | That balance is at `f=w` only. |
| Exact finite attaining of zero from nonzero departure under a nonzero multiplier | Each finite departure is nonzero; an already supplied zero is a different condition. |

## 5.4 Sharing at 10 and 6, and carrying at 11, at resolver

**At numbers two maps return four pairs at two steps.** F(c,t)=(−t,c) and G(c,t)=(t,−c) are the only two maps on four pairs whose square is J(c,t)=(−c,−t), a complete return, two ways round, `incoming/equilibria_registry_v377/executions/eq_code.py`, part 7. At Exhibit ONE's resolver a self offered nothing is at its prior inverted at each momentary, and its sharing at 10-next-co-possibling and its next prior at 11-next-co-prioring are one changing, part 11.

Agreement and opposition alternate through overlapping momentaries. Along c, t, −c, −t, c overlapping pairs (c,t), (t,−c), (−c,−t) and (−t,c) carry agreeing, opposing, agreeing, opposing from an agreeing start, and the reverse from an opposed one: each side's relation stays through its two momentaries while both its parities invert, and its overlapping side carries an other relation. Two neighbouring pairs (a,b) and (b,z), each at a parity, + or −, carry the two relations exactly when z = −a, and +,+,+ agrees at both. Each of four starting pairs and eight triples reaches these. A pair agreeing fixes no resolver unchangeable and no equilibrium: its overlapping pair opposes, and its own next inverts both parities.

**6 and 10 share one changing**, at Exhibit ONE's resolver `_6_now_co_possibling = _10_next_co_possibling`, both even and co, a sharing side, at middle four-cycle 7-11-6-10, 6-now-co-possibling sharing to an other's 2 and 10-next-co-possibling to an other's 14, `incoming/equilibria_registry_v377/executions/eq_code.py`, part 4. An across name's parity, a changing's + or − and agreement of two are three comparisons.

**A prior and its offerings determine its next at each sharing**, as Exhibit ONE's table of one self at one momentary gives each cell of a living self, and its table at no prior a sharing at no prior: a prior parity offered its own stays and shares 0; offered an other, a parting or nothing, it inverts; a sharing at no prior opens at its offered parity, or stays open at 0 or none.

## 5.5 Each conception at a proof of its own requirements

**Each conception is at a proof by its own requirements.** Opposition under one side's reversal fails, while opposition under joint reversal continues; and each surface zero reaches renewal exclusion. At SA01 opposition continuing is alike at self + with other − and at self − with other +, and gives neither as next: a complete next pair is given by a prior pair with both reversed, more than a relation by itself, and SA01's own requirement is reached by nothing it carries.

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Each sentence at resolver is at Exhibit ONE v380R at tables it cites, by hand at v380L at its sentences of a sharing at no prior, or executed at v377 at Exhibit ONE v376 at a script and part it cites; each sentence at numbers is computed at a script it cites or is at its own arithmetic. An older resolver's derivations, 32 of 32 at its own code, are whole at `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_older_resolver.md`.