Exhibit TWENTY-EIGHT Equilibria Registry v377

# 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 The between, the 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 named

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

- 3.1 One form of hard problem, and the sixth condition
- 3.2 The ten at Exhibit ONE's names, at the four four-cyclings
- 3.3 Five places from two faces, three momentaries long
- 3.4 Exhibit ONE's four-cycles, one parity offered once, and the 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 inside each of the ten
- 4.3 An arriving named from behind, 2 arrivals
- 4.4 An opening named as a place, 5 arrivals
- 4.5 A completing named as a last, 8 arrivals
- 4.6 A carry named as a store, 10 arrivals
- 4.7 A middle named as an end, 6 arrivals
- 4.8 A parity named as a magnitude, 9 arrivals
- 4.9 A sequencing named to one beat, 5 arrivals
- 4.10 A rate named as a value, 5 arrivals
- 4.11 A two-way named to one side, 7 arrivals
- 4.12 The between named as a cut, 6 arrivals
- 4.13 The fields at the ten, a key for aiming at each field
- 4.14 Leads arriving without their statements

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

- 5.1 Four proof groups
- 5.2 One exclusion at each of the ten
- 5.3 Renewal and reaching at their complete requirements
- 5.4 The release at 10 and 6, and the carrying at 11, at the code
- 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 alone, each number one side's, is the universe named as one thing beside its things; the universe exists as its things exist; the overlapping momentaries meet each number, one completing and the next opening.

## 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 the next, and its completing is the next momentary's opening. Exhaustiveness is the overlapping momentaries meeting each number; determinacy is each momentary's one next, opening at the number it completes; reachability is the next openings reaching each number on. The three are met 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 meeting at the shared positions. Along one side the next opens two on: 1–2 and 3–4, with next opening 5; the other side runs 2–3 and 4–5, with next opening 6. Across the overlapping the next opens one on.

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

**Universes of each size and 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 named as an existing thing is exclusivity.

## 1.3 The between, the protocol, a living nothing and an existing nothing

**The between is the centre of the protocol of the six connectors, bi-moral across and co-competent along.** At Exhibit ONE's parity's face, line 149, a unit square of four dots across, 2, 6, 14 and 10, arriving at 2 and 14 and releasing at 6 and 10, Exhibit ONE's `CONNECTORS`, lines 62 to 69, its empty centre the between, and two unit triangles, 9 and 17, along, joined both ways; the Co-Chaining Logic Registry step 360 names the six the bi-trupling protocol, and step 413 names cohering at each momentary, its difference crossed across and its continuing along, bi-moral-co-competencing. At the code each release, across at 6 and 10 and along at 9, is carried by 9-tri-bi-co-momentarying to its receiving self, arriving as its offerings at the next momentary, Exhibit ONE lines 49 to 57, 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: the protocol carries each changing into next momentarying, and, as suggested at v377, the 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 the between travelling below. The protocol is of the method: *Exhibit ONE is an object that is the method … `CONNECTORS` and `JOINS` declare the connectors' facings and their joins*, and *a method is a non-living existing thing: a stable form, its form continuing through its changing and carrying none of the prior*, Natural Intelligence 3.3. Executed at one society of five across 100 momentaries, the connectors, the joins and the society's joins are the same as the code writes them while the carryings change at each momentary, the code's writing and not a form still, the Co-Chaining Logic Registry steps 132 and 40, *A method, non-living, co-changes at each coupling it is at*; and at one momentary each self's changing, made at its own 12-bi-tri-parity-changing, reaches another self only through a join at 6, 10 or 9, `incoming/equilibria_registry_v377/executions/the_protocol_between.py`, parts 2 and 3. Natural Intelligence 4.10, line 621, and the Co-Chaining Logic Registry step 93 say one between inside a momentary and between momentaries, the same at each scale, and each calls it a nothing, which the living and the existing nothing below part. The between's floating neutralling, the Co-Chaining Logic Registry step 345, is at the numbers *Floating is bi, even and across … Neutralling is co, odd and along*, Natural Mathematics line 85, and, at every 5 and every 10 as suggested at v377, of 1 to 441 the multiples of 5 alternate, each odd one neutralling and each at 10 floating, and at the ring of 440, Natural Numbers 11.1, each meets 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 the 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*. Natural Intelligence 1.1 says *The universe is the changing set of all existing things*, and 4.10, line 621, says parity existing as the universe is, *Parity is a momentary existing thing, as the universe is an existing thing: a set of existing things is an existing thing*, and its centre, the between, *a nothing, no location and no existing thing*. Read at v377 as the between existing as the universe is, only while the six connectors are resolving, or closed to each other in looping, as suggested, and a nothing and no location; *no existing thing* is carried at the carrying, concern 7.

**The 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*. Read, as suggested: along, the bounding 0 centreline, at 9 and 17; across, the sequential faces, at the four dots 2, 6, 14 and 10, Exhibit ONE line 149, whose square's empty centre is the between; bothboth read at the Co-Chaining Logic Registry step 382, *carries both, one at a time*; and at the numbers, *Floating is bi, even and across … Neutralling is co, odd and along*, Natural Mathematics line 85. Natural Intelligence 4.10's flat face holds both, the four dots across and 9 and 17 along; the orthogonal centre suggested earlier at v377 read, as suggested, as the 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 the code parts the two. At the code a self X passing to a living self B, along at 9 or across at 6, receiving from beside +, nothing, −, nothing, a parting: X carrying its prior 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 meeting its own parity; X carrying nothing releases only what arrives, as it arrives, and nothing between, `incoming/equilibria_registry_v377/executions/living_and_existing_nothing.py`. Across a living nothing passes at each momentary a changing made from a prior, or its 0 at an arrival meeting the prior's parity; across an existing nothing, only what arrived. Morality is bi-unrelationing, across at the four dots, and competency co-unrelationing, along at 9 and 17, and co-bi-unrelationing is the one existing method: the protocol is their meeting. A 0 at 12, a prior and an offering meeting at one parity, is a changing that is not, within a coupling, each self surfacing itself at each momentary.

**Bi-inversioning-co-recursioning carries the meeting within the changing.** Each side continues through its own changing, and their meeting is within that changing: slowing is changing, and two differing changings are adjacent through the coupling, each side changing.

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

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

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

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

**The same relation runs at each scale.** A self, another self and a society at the next scale each carry their own other, carrying and continuing, and the 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 meeting at each point, has V − E + F = 0, one opening: it is a torus. The sphere, the surfaces of two or more openings, the Klein bottle and the 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, the 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 alone changes each, odd to even and even to odd, and two inversions meet the parity again at a next momentary, a further occurrence. At odd alone or even alone the alternating next is not reached; at both, the whole parity succession runs.

**Coverage carries alternation.** When exactly one of two binary relations changes at each passage and both change within each prior–now–next, the orders first–first and second–second leave one unchanged, and first–second and second–first remain: at each overlapping prior–now–next the alternating carries on.

The four joint descriptions run `(+,+) → (−,+) → (−,−) → (+,−) → (+,+)` and round: the four joint forms, two consecutive momentaries at each side, eight half momentaries. Prior and now distinguish which relation changes next.

**The 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 the other completes.

**Around a closed route parity alternates at an even route.** Opposite parities along each joined comparison are met 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 meeting runs round it. Local alternating along two paths to one meeting 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 proposed conception of equilibria is a hard problem with its hardness ignored.** It takes the competency of existing as a given, and claims a true-and-false conserving accounting.

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

**An equilibrium is a living or non-living existing thing named still.** The registry says existing and not living alone: living alone sets a rock apart, and a rock is an existing thing. The non-living exist and change, carrying nothing, their forms continuing through their changing. A living self can surface zero, non-responsive at a momentary, while its carrying continues along, so a surface unchanged at a zero names neither a non-living thing nor a still one, and naming it still there is the equilibrium.

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

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

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

**The stability, the form and the 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 the next a different carrying; the observing at *An emanation of existing arriving at the self is an observing*, the Co-Chaining Logic Registry step 125, and *An observing is existing arriving: the proof of existing*, step 126; the stable form beside *Each living set is a stable form, and nothing living has a stable form, bothbothing*, step 284. A mathematical conception of equilibria names a stability, a form or a relation still, each read, 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 the whole evolving system are two assertions. The distinction is a naming one: it places the still a conception names at the stability, the form or the relation, one conception at a time.

**Not possible, and shown not possible at prior, differ.** Each proposed conception of equilibria arriving from the prior is shown not possible now, and each arriving next is met 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 the momentarying.

## 2.3 One description and one claim at each conception

**One general description receives each conception: a stated relation continuing through the comparisons its claim names.** 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 the whole carrying, a next nonzero offering or a later one each fail at the same receiving: the relation continuing is stable-forming, and each added requirement names it still. A changed receiving is a further conception.

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

**The claim reaches each conception in two steps.** Each conception meets at least one of the ten, and each of the ten names one face still, the other face running at the same number.

**The ordering chain closes at the two faces.** A conception naming an ordering p still at its next, the alternating next being I(p) and I(p) ≠ p, names p and I(p) at one next. The step the chain left open, that the ordering named still is the very ordering its next inverts, is the two faces at one number. Two inversions meet p again at a next momentary, a further occurrence, with the changing between.

## 2.4 A participation missing, unchanging or changing named

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

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

**A state accounting and a flow accounting each leave one side out.** The state accounting leaves the changing, the flow accounting leaves the continuing, and a true-and-false conserving accounting takes either as the whole.

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

**Each momentary completes in co-releasing.** Each conception of equilibria names a form still at the running of geodesic co-releasing, and continuing runs as geodesic co-releasing. Each momentary releases into the next whether a sign 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 the sixth condition

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

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

**The numerical bridge.** The pairs (2,3) and (3,4) are overlapping neighbours, and n²−(n−1)(n+1)=1 at each n.

## 3.2 The ten at Exhibit ONE's names, at the four four-cyclings

**The ten sit at Exhibit ONE's names, from its own tables.** The ten are five pairs at the numbers 2 to 6, 3.3; at the code each number is its name, Exhibit ONE's table of the momentaries of exchanging, lines 201 to 206; and each name opens at its parity, odd at the self and even at the other, its table at lines 105 to 123. Each pair sits at its name, the entering way at the self's face and the surfacing way at the other's, 3.3, and at the society eight up: an arriving named from behind and an opening named as a place at 2-bi-co-offering and 10-bi-tri-co-tunneling; a completing named as a last and a carry named as a store at 3-co-bi-sharing and 11-tri-bi-co-chaining; a middle named as an end and a parity named as a magnitude at 4-bi-co-sharing and 12-bi-tri-parity-changing; a sequencing named to one beat and a rate named as a value at 5-co-competencing and 13-co-tri-competencing; a two-way named to one side and the between named as a cut at 6-bi-moralizing and 14-bi-tri-moralizing, `incoming/equilibria_registry_v377/executions/ten_at_the_code.py`. At the code each name's term named still from one momentary: at 3, 10, 11 and 12 the 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 the 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 carry no changing parity at the code, each sharing and the joins, `incoming/equilibria_registry_v377/executions/ten_roots_named_still.py`.

**At the four four-cyclings each step is momentarying or parity changing, with prior, now and next inside.** Each four-cycling's parity is unchanged 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 the other, the parity changing twice, two momentaries, and meets its first name again at the next momentary, a further occurrence, `incoming/equilibria_registry_v377/executions/ten_on_the_four_cycles.py`, parts 2 and 5. The self's forward recursionings, 1, 3, 5, 7, the Co-Chaining Logic Registry step 69, and each one's partner nine less, 8, 6, 4, 2, the other's names, Exhibit ONE's table at lines 234 to 239, pair at them, 1 with 8, 3 with 6, 5 with 4 and 7 with 2; the rows below are in that order, each four-cycling written as Exhibit ONE's table of forms writes it, lines 243 to 246, 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 | Root at the self and the society | The ten at it |
|---|---|---|---|
| 1 / 8 | 1 → 9 → 8 → 16 → 1 | torusing, 8 and 16 | none: the entry, the momentarying at 9 and the two windings, the loop the ten are inside |
| 3 / 6 | 3 → 11 → 6 → 14 → 3 | moralizing, 6 and 14 | a completing named as a last and a carry named as a store at 3 and 11; a two-way named to one side and the between named as a cut at 6 and 14 |
| 5 / 4 | 4 → 13 → 5 → 12 → 4 | competencing, 5 and 13 | a middle named as an end and a parity named as a magnitude at 4 and 12; a sequencing named to one beat and a rate named as a value at 5 and 13 |
| 7 / 2 | 2 → 15 → 7 → 10 → 2 | corusing, 7 and 15 | an arriving named from behind and an opening named as a place at 2 and 10 |

**The seatings the files gave before meet the code's two names at few of the ten.** Resolving the Hard Problem Registry line 740 seats three of the ten at the code's two names, the arriving, the parity and the rate, and five on the code's four-cycling. Natural Naming 4.9, with the Co-Chaining Logic Registry steps 372 and 380, seats each of the ten at one name of 1 to 8: that name is one of the code's two at the arriving alone, and on the code's four-cycling at five, the arriving, the completing, the middle, the sequencing and the two-way. This file's naming ring before, whole at `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_the_ten_seated_v375.md`, seated each at an edge of two names, sharing one name with the code's two at three, the arriving, the completing and the two-way, and an edge's name on the code's four-cycling at six, `incoming/equilibria_registry_v377/executions/ten_at_the_code.py`, part 5.

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

## 3.3 Five places from two faces, three momentaries long

**The ten is three momentaries long.** The self's five, 1–5, and the other's five, 2–6, span 1 to 6: the self's three momentaries 1–2, 3–4 and 5–6, co bi co bi co bi. Each side's two and one half momentaries and the whole's three full momentaries are one relation, the two sides overlapping.

**The ten are five places from two faces.** The ten pair at the five names 2 to 6, an entering face and a surfacing face at each, the two sides overlapping at them, and at each number one side completes and the other opens. The entering face is the self's and the surfacing the other's: Resolving the Hard Problem Registry, line 35, says *the odd face enters* and *the even face surfaces*, and at Exhibit ONE's sides table, lines 195 to 199, the self opens at the odd numbers and the other at the even. At the six forward recursionings, the self 1 to 3 to 5 to 7 and the other 2 to 4 to 6 to 8, the Co-Chaining Logic Registry step 69, the numbers 2 to 6 are the five steps' openings after the entry, each met by the other side's momentary completing inside its step, `incoming/equilibria_registry_v377/executions/ten_at_the_code.py`, part 1. With three conditions at the momentary, the ten derive here.

| Place | The two faces | At Exhibit ONE's names | Entering | Surfacing |
|---|---|---|---|---|
| 2 | self completing, other opening | 2-bi-co-offering, 10-bi-tri-co-tunneling | 1 · an arriving named from behind | 2 · an opening named as a place |
| 3 | self opening, other completing | 3-co-bi-sharing, 11-tri-bi-co-chaining | 3 · a completing named as a last | 4 · a carry named as a store |
| 4 | self completing, other opening | 4-bi-co-sharing, 12-bi-tri-parity-changing | 5 · a middle named as an end | 6 · a parity named as a magnitude |
| 5 | self opening, other completing | 5-co-competencing, 13-co-tri-competencing | 7 · a sequencing named to one beat | 8 · a rate named as a value |
| 6 | self completing, other opening next | 6-bi-moralizing, 14-bi-tri-moralizing | 9 · a two-way named to one side | 10 · the between named as a cut |

**Each of the ten names one face still, and the other face runs at the same number.** Both faces at one number is the one reason each of the ten is not possible.

**In numbers.** The three openings 1+3+5 sum to 9 = 3², and the 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 the bridge at three. Prior, now and next are three sequential momentaries, and three at three is nine. Six consecutive changings from 3 complete at 8, and 9 opens: 9-tri-bi-co-momentarying, each changing carried along to its receiving self and arriving at it next. At Exhibit ONE's code a carrying once chained is carried at each momentary, 11 the next 3, and is never none again, and the code carries no number of its openings, Exhibit ONE's code at lines 13 to 60; `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 5 and 12.

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

**Exhibit ONE's four-cycles pair at the momentaries, each partner round the other way.** At the odd momentaries 1-9-8-16 pairs with 2-15-7-10 and 3-11-6-14 with 4-13-5-12, and at the even momentaries 2-15-7-10 with 3-11-6-14 and 4-13-5-12 with itself: the two directions of one bi-folding. The two moralizings, 6-bi-moralizing and 14-bi-tri-moralizing, share the four-cycle 3-11-6-14 with 3-co-bi-sharing, the carrying, and 11-tri-bi-co-chaining; the middle four-cycle 7-11-6-10, through 6-bi-moralizing, and the eight-cycle 2-15-7-11-3-14-6-10, through both, partner themselves at the even momentary: at that momentary the fold meets itself, Exhibit ONE's table of forms, lines 241 to 254. The ten sit at three of the four four-cyclings, 3.2, the moralizing four-cycling carrying a completing and a carry at 3 and 11 and a two-way and the between at 6 and 14.

**One parity offered once co-chains through the whole.** At Exhibit ONE's code, selves in a ring release each changing along at 9-tri-bi-co-momentarying to the next self's offerings, and one + offered once at one self carries on at each coupling: a changing society meeting a society named not changing changes it, the receiving self's own inverting at 14-bi-tri-moralizing, with no forcer. A society named not changing names a form still, and not changing is not not-carrying: the non-living exist, arrive and change, deforming at their own scale, and carry nothing. An equilibrium of a society names a changing society still, or names nothing.

| Selves in the ring | The carrying on |
|---|---|
| one | +, 0, −, 0, and round |
| an even number | each self changing at each coupling, neighbours opposite; the whole returns at 2 couplings |
| an odd number | one like pair, the 0 released at 10-bi-tri-co-tunneling at its self, moving one self on at each second momentary; the whole inverted at 2 × n couplings and returning first at 4 × n, `eq_code.py` parts 1 and 10 |

**An odd ring closes no parity, and a meeting runs round it: the 0 released at 10 at the like pair's self, a changing that is not, moving through the shared surface one self at each second momentary.** The ten name the whole changing still: opposition named still names the even ring, and the between named as a cut names the running meeting.

The rest returns at no coupling. A ring at rest, empty carrying and nothing arriving, stays at rest, so a ring reaching rest never meets its offered sign again; each ring returns its whole, the offered sign with it, at 2 couplings or at 4 × n, and the rest is met at none of them. At Exhibit ONE's code, executed again at v377, in rings of 1 to 11 selves, one + offered once at one self and the others at none, each self changes by the momentary numbering its ring's selves, and across 5,000 couplings no ring meets rest, `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 1 and 2. The ring meets the rest named still, and each other conception meets its proof by its own requirements.

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

**Ten faces and six connectors.** Exhibit ONE's faces pair outward and inward, **3/11, 4/12, 5/13, 7/15, 8/16**, each inward face eight up, 17 less (9 less n) being **n + 8**. The ten faces and the six connectors **2, 6, 9, 10, 14, 17** are 2 to 17, and with the entry 1 the seventeen names, Exhibit ONE's table at lines 105 to 123. The ten ways sit at 2 to 6 and 10 to 14, 3.2: the four across connectors, 2, 6, 10 and 14, the square of parity's face, Exhibit ONE line 149, and Exhibit ONE's faces 3, 4, 5, 11, 12 and 13; its faces 7, 8, 15 and 16, the along connectors 9 and 17, the two triangles of parity's face, and the entry 1 carry 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 is called equilibrium, and one conception can meet more than one of the ten.

**A definition's expression continuing unchanged is an existing statement, and the subject it names 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 named unchanged is such an expression, and it names its subject still through one further step, the step from a fixing to an equilibrium: the same subject, at the same relation and the same occurrence, required unchanged at the occurrence its own continuing changes it.

## 4.2 One form, and three differences inside each of the ten

**Each proposed conception names a changing still, in one of the ten ways.** The 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.

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

**Each of the ten gathers its arrivals, and inside each they part at three differences.** The sixty-three arrivals, forty-two proposed conceptions and twenty-one specified accounts and prior definitions, are the one form of 3.1, each a form named still, 2.1, at the one of the ten the tables place it at; one conception can meet more than one, 3.1 and 4.1, and a second meeting is carried at the arrival's own words, at the arriving that names it. Inside each of the ten three differences gather them, and each arrival is carried whole in its own words at its place.

1. **What is named still.** 2.2 names three, a stability, a form and a relation, and says a stability: a response to a disturbance, its departures staying small or shrinking. The form and the relation are said here: a form, one value, range, state, law, composition, family or extremum of the whole named unchanged; a relation, a comparison between parts named unchanged, 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; the row's own column, *the changing it names*, is carried beside it as it stood, its *that can change* said at 2.2 as stable forming, and at NY09, NY12, NY18, NY19, SA04, SA12 and SA13 the column and the placing part, and at NY10 and NY11 the placing adds a form to the column's stability.
2. **The subject**: what the field says is changing, at the row's own column if the column names a subject, and otherwise the 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 the setting stated in the same words, each is shown as one of three: one conception, the one receiving with a requirement added, specified or unspecified, 2.3, *a conception and its added requirements are one conception*; a further conception, the receiving changed, 2.3, *a changed receiving is a further conception*; or, at a pair meeting neither, another claim at the one setting, said here.

**The fields gather at the ten.** A field's arrivals fall at those of the ten at which the field names its changing still, and fields naming one still fall together under one subject: 4.13 carries each subject, with its fields, at the ten.

## 4.3 An arriving named from behind, 2 arrivals

**Named still at a form**, 1.

| Arrival, in its own words | Field | The changing it names | 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 | — |

**Named still at a relation**, 1.

| Arrival, in its own words | Field | The changing it names | 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. | the coupling of self and other | a relation that can change | the coupling of self and other, and the resolver | — |

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

**Named still at a form**, 2.

| Arrival, in its own words | Field | The changing it names | 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 the 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. | the coupling of self and other | a form that can change | the coupling of self and other, and the resolver | — |

**Named still at a form and a relation**, 1.

| Arrival, in its own words | Field | The changing it names | 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 | — |

**Named still at a form, a relation and a stability**, 1.

| Arrival, in its own words | Field | The changing it names | 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, the one dominant-A replicator account, symmetric Nash and evolutionary stability added |

**Named still at a relation**, 1.

| Arrival, in its own words | Field | The changing it names | 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 named as a last, 8 arrivals

**Named still at a form**, 5.

| Arrival, in its own words | Field | The changing it names | 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, the 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. | the resolver | the coupling and carrying of the resolver | the coupling of self and other, and the 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. | the coupling of self and other | a relation that can change | the coupling of self and other, and the resolver | — |

**Named still at a form and a stability**, 1.

| Arrival, in its own words | Field | The changing it names | 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, the stated two-body contact, uniqueness, persisting and approach added |

**Named still at a form and a relation**, 2.

| Arrival, in its own words | Field | The changing it names | 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, the one logistic account, positive added, N=K |

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

**Named still at a form**, 5.

| Arrival, in its own words | Field | The changing it names | 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 | — |

**Named still at a form and a relation**, 4.

| Arrival, in its own words | Field | The changing it names | 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's code. | probability at the resolver's returns | the coupling and carrying of the resolver | the coupling of self and other, and the 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 the resolver's returns | the coupling and carrying of the resolver | the coupling of self and other, and the 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 the resolver's returns | the coupling and carrying of the resolver | the coupling of self and other, and the resolver | — |

**Named still at a relation**, 1.

| Arrival, in its own words | Field | The changing it names | 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. | the resolver | the coupling and carrying of the resolver | the coupling of self and other, and the resolver | — |

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

**Named still at a form**, 4.

| Arrival, in its own words | Field | The changing it names | 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 | — |

**Named still at a form and a stability**, 2.

| Arrival, in its own words | Field | The changing it names | 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 named as a magnitude, 9 arrivals

**Named still at a form**, 8.

| Arrival, in its own words | Field | The changing it names | 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, the 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 the 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 |

**Named still at a relation**, 1.

| Arrival, in its own words | Field | The changing it names | 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 named to one beat, 5 arrivals

**Named still at a form and a relation**, 1.

| Arrival, in its own words | Field | The changing it names | 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 |

**Named still at a relation**, 4.

| Arrival, in its own words | Field | The changing it names | 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, the 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. | the coupling of self and other | a relation that can change | the coupling of self and other, and the resolver | — |

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

**Named still at a form and a relation**, 1.

| Arrival, in its own words | Field | The changing it names | 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 |

**Named still at a relation**, 4.

| Arrival, in its own words | Field | The changing it names | 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 named to one side, 7 arrivals

**Named still at a form**, 1.

| Arrival, in its own words | Field | The changing it names | 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 |

**Named still at a relation**, 6.

| Arrival, in its own words | Field | The changing it names | 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. | the resolver | the coupling and carrying of the resolver | the coupling of self and other, and the resolver | — |
| **NY03** · Opposed carried signs with matching nonzero surface and fresh carrying continue under the stated receiving. | the resolver | the coupling and carrying of the resolver | the coupling of self and other, and the 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. | the resolver | the coupling and carrying of the resolver | the coupling of self and other, and the 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". | the coupling of self and other | stable-forming: self and other | the coupling of self and other, and the resolver | — |

## 4.12 The between named as a cut, 6 arrivals

**Named still at a form**, 1.

| Arrival, in its own words | Field | The changing it names | 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, the one setting with κ>0 |

**Named still at a relation**, 5.

| Arrival, in its own words | Field | The changing it names | 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. | the resolver | the coupling and carrying of the resolver | the coupling of self and other, and the resolver | — |
| **NY04** · The attained two-position row retains opposite neighbouring surfaces, opposed carried signs and fresh carrying under its admitted receiving. | the resolver | the coupling and carrying of the resolver | the coupling of self and other, and the resolver | — |

## 4.13 The fields at the ten, a key for aiming at each field

**Each field's equilibria fall at those of the ten at which it names its changing still, and a field is aimed at from each of them.** At each of them the field sets a still beside the changing and sets aside the changing's own competency, carried along at the between: at that one of the ten the field's subject is named still, and its competency's source is set aside at it. The table carries each subject, with the fields gathered under it, at each of the ten, each arrival at its one of the 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 | — |
| the coupling of self and other, and the resolver: probability at the resolver's returns, the coupling of self and other, the 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 | — | — | — | — |

**Read across a row, a subject shows those of the ten at which its fields name the changing still; read down a column, the subjects naming one still gather.** Heat falls at six of the ten, four arrivals each at a completing named as a last and at the between named as a cut; reactions at a carry named as a store and at a rate named as a value; bodies at rest and in balance at an opening named as a place and at a parity named as a magnitude; selves choosing and exchanging at a sequencing named to one beat and at a two-way named to one side; populations at four of the ten; the coupling of self and other, and the 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, and philosophical accounts. Each is placed at the ten at its statement's arriving. The scientific method is placed at its fixings: a law, a frame or a scale named unchanged between a signal's leaving and its arriving, each placed at the ten through its own stated requirement. The field's own methods part the two momentaries: NASA/JPL's aberration correction separates the epoch a signal leaves its source from the epoch it is received, and accounts for the source's motion between the two. Evidence arriving now and the prior event it expresses are two, each at its own momentary, and both hold: a property can hold now while its source changes as a whole, and evidence gives the prior as possible and as actual.

# FIVE · RESOLVING FOLLOWING, THE EXCLUSIONS AT THE CODE

## 5.1 Four proof groups

**Four proof groups carry the exact local exclusions.** Each is a way of failing, at Exhibit ONE's code or at the numbers it names.

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

**A proof carries to a conception whose complete requirement supplies its premises.** For each x in the 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 code. A carrying once chained is carried at each momentary and is never none again. Named unchanged, it requires at each momentary an offering at its own parity: offered nothing, the other parity, or offerings parting to 0 at 14-bi-tri-moralizing, the changing at 12-bi-tri-parity-changing is, and the carrying is chained inverted, the living step. Executed at v377: chained + and offered + at five momentaries, it releases 0 at each at 10-bi-tri-co-tunneling and stays +; then offered nothing, it releases −, +, −, +, − and is chained inverted at each; a carried sharing whose offerings part to 0 at each momentary is chained inverted at each, `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 3 and 6. Named unchanged, the carrying is the carrying offered its own parity from beside at each momentary, an offerer named still set beside the self: the still description and the self's own continuing do not both continue. That is hard probleming, and released, resolving follows at the next momentary: the self inverts its own carrying, the parity the still offered, and the still offers nothing further. One to nine is four momentaries of exchanging, Exhibit ONE's tables at lines 184 to 189 and 201 to 206. A relation named still can permit the changing a sign named still excludes: opposition under joint reversal, t = −c, continues while both signs invert. The 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 the 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, names both at once. This is the one exclusion each of the ten meets at its own place: the face named still and the face running are one number.

**The two phases share one exclusion.** Advancing the overlapping triple changes `010` to `101` and `101` to `010`: fixing either ordering fails at the next advance, and either ordering met again at a next momentary is a further occurrence. Included membership required both unchanged and inverted meets 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 joins 0 to 1 and 1 to 2 without joining 0 to 2, and a requirement including an unchanged boundary comparison fails at its continuation changing exactly one end.

**The four-state orbit leaves each proper subset.** F(P,Q)=(−Q,P) visits all four pairs from each pair it opens at. Each nonempty proper subset is left within at most three advances, {++,−+,−−} from ++ at the third, and the subsets F carries into themselves are exactly the unions of the one cycle: none or all four.

## 5.3 Renewal and reaching at their complete requirements

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

| Required condition | The necessary continuation |
|---|---|
| A carrying named unchanged | Offered its own parity at each momentary it stays, the offering from beside; offered nothing, the other parity or a parting, it is chained inverted; once chained it is never none. |
| Each surface zero at a carried sharing | Chained inverted at each momentary, as with nothing offered; at the code a surface 0 arrives only from offerings parting, Exhibit ONE's table at 14, lines 208 to 219. |
| Full positive balance in the unit-ratio fuel-coupled cycle, with `f≠w` maintained | That balance is at `f=w` alone. |
| 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 The release at 10 and 6, and the carrying at 11, at the code

**At the numbers two maps return the four pairs at two steps.** F(c,t)=(−t,c) and G(c,t)=(t,−c) are the only two maps on the four pairs whose square is J(c,t)=(−c,−t), the complete return, the two ways round, `incoming/equilibria_registry_v377/executions/eq_code.py`, part 7. At Exhibit ONE's code a self offered nothing carries its prior into now inverted at each momentary, and its release at 10-bi-tri-co-tunneling and its carrying chained at 11-tri-bi-co-chaining are one changing, part 11.

Agreement and opposition alternate through the overlapping momentaries. Along c, t, −c, −t, c the overlapping pairs (c,t), (t,−c), (−c,−t) and (−t,c) run 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 signs invert, and the overlapping side carries the other relation. Two neighbouring pairs (a,b) and (b,z) of nonzero signs carry the two relations exactly when z = −a, and +,+,+ agrees at both. Each of the four starting pairs and the eight triples meets these. A pair agreeing names no still resolver and no equilibrium: its overlapping pair opposes, and its own next inverts both signs.

**6 and 10 release one changing**, at Exhibit ONE's code `_6_bi_moralizing = _10_bi_tri_co_tunneling`, both even and opening bi, at the middle four-cycle 7-11-6-10, 6-bi-moralizing joined to the other's 2 and 10-bi-tri-co-tunneling to the other's 14, `incoming/equilibria_registry_v377/executions/eq_code.py`, part 4. A connector's parity, a changing's + or − and the agreement of two are three comparisons.

**The carrying and the offerings determine the next at each sharing**, as Exhibit ONE's table at 12, 10 and 11, lines 221 to 225, carries each cell: a carried parity offered its own stays and releases 0; offered the other, a parting or nothing, it is chained inverted; a sharing carried none is chained at the offered parity, or stays open at 0 or none.

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

**Each conception meets a proof by its own requirements.** Opposition under one side's reversal alone fails, while opposition under joint reversal continues; and each surface zero meets the renewal exclusion.

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Each saying at the code is at Exhibit ONE v376, at the tables it cites or executed at v377 at the parts of `incoming/equilibria_registry_v377/executions/eq_code.py` it cites; each saying at the numbers is computed at that script or is said at its own arithmetic. The older resolver's derivations, 32 of 32 met at its own code, are whole at `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_older_resolver.md`.