Exhibit TWENTY-EIGHT v371

# Equilibria Registry

**Forms Named Still, Not Possibly Existing**

**1 · Existing through momentaries**

- The universe as an existing thing is exclusivity, and it is not possibly existing
- Existing is three conditions at each momentary, at once
- Three overlapping momentaries use four positions
- Each momentary is existing's own, and each existing self is its continuing momentaries, inseparating
- The membrane is no separable thing
- Bi-inversioning-co-recursioning carries the meeting within the changing
- Naming a relation gives it no separate existing
- The eight even names are bi-coupling at the membrane
- Existing and discovering are one relation
- The same relation runs at each scale
- Natural torusing is the one surface
- Parity alternates along and across through three momentaries
- Existing continues: a matching description at next is a further occurrence
- Two binary relations change one at a time, alternating which
- Coverage carries alternation
- The five-view carries its next opening within continuing
- Around a closed route parity alternates at an even route

**2 · The binary claim**

- 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
- An equilibrium is a form named still, not possibly existing; stable-forming, and unstable deforming at non-living scale, are observably existing
- The stability can change, the form can change, the relation can change
- Not possible, and shown not possible at prior, differ
- Surviving is binary
- A conception of equilibria shows its own surviving
- One general description receives each conception: a stated relation continuing through the comparisons its claim names
- A conception and its added requirements are one conception
- The claim has one shape
- The claim reaches each conception in two steps
- The ordering chain closes at the two faces
- A still named as the complete continuing is changing
- 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
- Each conception of equilibria names unchanging or names changing, exactly one of the two
- Each momentary completes in co-releasing

**3 · Five pairs, the ten**

- The ten are one form of hard problem
- A proposed conception of equilibria is the sixth condition
- The numerical bridge
- The ten have addresses at the resolver's names
- A conception's own declared relation places it at the ten
- The ten is three momentaries long
- The ten are five places from two faces
- Each of the ten names one face still, and the other face runs at the same number
- In numbers
- The resolver's pairs are the momentaries, each round the other way
- One sign offered once co-chains through the whole
- An odd ring closes no parity, and a meeting runs round it: the non-existing between, moving through the shared surface one momentary at a time
- Prior and next are the edges at now
- Ten internal names and six connectors

**4 · Proposed conceptions of equilibria**

- Each conception arrives whole, in its own words, with its conserving relation and its reach
- A definition's expression continuing unchanged is an existing statement, and the subject it names continues through its own changing
- Each proposed conception names a changing still, in one of the ten ways
- The specified accounts and the prior worked definitions, SA01 to SA21, are placed the same way
- Leads arrive without their statements

**5 · Shared exclusions, run at the code**

- Four proof groups carry the exact local exclusions
- A proof carries to a conception whose complete requirement supplies its premises
- One binary retaining and inverting its value at one required next is not possible
- The two phases share one exclusion
- Immediate-next is its own relation
- The four-state orbit leaves each proper subset
- Renewal and reaching carry their complete requirements
- At the resolver's code the joint forms run one way
- The release pair and the fresh carrying are complementary
- 6 and 10 release at one parity
- The full carrying determines the next
- Each conception meets a proof by its own requirements

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## Existing through momentaries

**The universe as an existing thing is exclusivity, and it is not possibly existing.** One side's momentaries alone, each number one side's, is the universe as an existing thing; the overlapping momentaries meet each number, one completing and the next opening.

**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 membrane, 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.

**The membrane is no separable thing.** The between of two existing things co-bi-coupling is a membrane, not possibly an existing thing, as the universe is not possibly an existing thing, and this between is the registry's whole subject. Morality is bi-unrelationing and competency is co-unrelationing, and co-bi-unrelationing is the one existing method: the membrane is their meeting. A 0 at two signs meeting is a nothing within a coupling, existing due to each self surfacing itself 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 membrane.** Four at the membrane, 2, 4, 6 and 8, and four within, 10, 12, 14 and 16, each 8 on, are relations at two faces. The three conditions are the momentary's, the eight are bi-coupling, and the four joint forms are the changing of two signs.

**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.

**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.

## The binary claim

**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.

**The stability can change, the form can change, the relation can change.** A mathematical conception of equilibria names a stability, a form or a relation still, and each can change. 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.

**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.

**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.

## Five pairs, the ten

**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 membrane between 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.

**The ten have addresses at the resolver's names.** The ring runs **3→2→4→1→14→12→6→10→11→16→next 3**, a naming ring and not an order of running, and the ten at it are **1,10,7,9,8,4,6,5,3,2**.

**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.

**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 through opening, ageing, middling, rating and co-offering, an entering face and a surfacing face at each, and the five pairs are at the numbers 2 to 6, the two sides overlapping at them, and at each number one side completes and the other opens. With three conditions at the momentary, the ten derive here.

| Place | The two faces | Pair | Entering | Surfacing | Addresses at the resolver's names |
|---|---|---|---|---|---|
| 2 | self completing, other opening | Opening | 1 · an arriving named from behind | 2 · an opening named as a place | 3→2 / 16→3 |
| 3 | self opening, other completing | Ageing | 3 · a bound named as a last | 4 · a carry named as a store | 11→16 / 12→6 |
| 4 | self completing, other opening | Middling | 5 · a middle named as an end | 6 · a sign named as a magnitude | 10→11 / 6→10 |
| 5 | self opening, other completing | Rating | 7 · a sequencing named to one beat | 8 · a rate named as a value | 4→1 / 14→12 |
| 6 | self completing, other opening next | Co-offering | 9 · a two-way named to one side | 10 · a membrane named as a cut | 1→14 / 2→4 |

**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-other-releasing, co-releasing, the self, the other and the social releasing, and its released sign arriving next downstream. At the resolver's code a retained carrying continues through its openings 1, 2 and 3, and to a fourth at a positive second sign: three momentaries, and a fourth at positive competency.

**The resolver's pairs are the momentaries, each round the other way.** Its four row cycles pair at the odd momentaries 1 with 2 and 3 with 4, and at the even momentaries 2 with 3 and 4 with itself, each partner running round the other way: the two directions of one bi-folding. The two crossings, 6-other-crossing and 14-social-crossing, share row 3 with carrying and chaining, and the forms through both crossings, 7-11-6-10 and 2-15-7-11-3-14-6-10, partner themselves at the even momentary: at the crossing the fold meets itself. The ten's ring passes both crossings, 1→14→12→6→10.

**One sign offered once co-chains through the whole.** At the resolver's code, selves in a ring pass each surfacing through 9-other-releasing to the next self's offering, and one +1 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 inversion at 14-social-crossing, 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 meeting, the 0 at signs meeting, running one self each momentary; the whole returns at 4 × n couplings |

**An odd ring closes no parity, and a meeting runs round it: the non-existing between, moving through the shared surface one momentary at a time.** The ten name the whole changing still: opposition named still names the even ring, and a membrane 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 the code, in rings of 1 to 11 each self changes within 40 couplings, and across 5,000 couplings no ring meets rest. 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 internal names and six connectors.** The resolver's internal pairs are **3/11, 4/12, 5/13, 7/15, 8/16**, and **17−(9−n)=n+8** joins its two inversion faces to the advance by eight. The ten internal names and the six connectors **2,6,9,10,14,17** exhaust positions 2–17, and with the entry 1, all seventeen named positions. Their correspondence with the ten ways is the next discovering.

## Proposed conceptions of equilibria

**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.

**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 that can change.

| The ten | Proposed conception, in its own words | Field | The changing it names |
|---|---|---|---|
| 1 · an arriving named from behind | **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 |
| 2 · an opening named as a place | **NY23** · A specified position–velocity pair (x*,0) stays fixed under Fτ(x,v)=(x+τv,v). | mechanics | rest and balance of bodies |
| 2 · an opening named as a place | **NY39** · Symmetric Nash and evolutionarily stable resident composition in that dominant-A game, with the same supplied replicator evolution. | evolutionary game theory | populations |
| 2 · an opening named as a place | **NY40** · Stationarity and symmetric Nash in the all-zero-payoff game with its replicator evolution. | evolutionary game theory | populations |
| 3 · a bound named as a last | **NY06** · A retained range remains satisfied through its specified parity-changing continuation. | the resolver | the coupling and carrying of the resolver |
| 3 · a bound named as a last | **NY14** · Global minimality of that same U under that same evolution. | dynamical systems | a form that can change |
| 3 · a bound named as a last | **NY37** · Positive stationarity in that logistic account, N=K; equivalently matched nonzero turnover under B=rN and D=rN²/K. | population ecology | populations |
| 3 · a bound named as a last | **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 |
| 4 · a carry named as a store | **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 |
| 4 · a carry named as a store | **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 |
| 4 · a carry named as a store | **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 |
| 4 · a carry named as a store | **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 |
| 4 · a carry named as a store | **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 |
| 4 · a carry named as a store | **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 |
| 4 · a carry named as a store | **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 |
| 5 · a middle named as an end | **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 |
| 5 · a middle named as an end | **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 |
| 5 · a middle named as an end | **NY11** · E: r=0 under T(p,r)=(1−p,−r), with stability in those neighbourhoods. | dynamical systems | a stability that can change |
| 5 · a middle named as an end | **NY12** · E: r=0 under T(p,r)=(1−p,−2r), without adding stability. | dynamical systems | a stability that can change |
| 5 · a middle named as an end | **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 |
| 6 · a sign named as a magnitude | **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 |
| 6 · a sign named as a magnitude | **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 |
| 6 · a sign named as a magnitude | **NY24** · Rest, v=0 with position unrestricted, under that same force-free evolution. | mechanics | rest and balance of bodies |
| 6 · a sign named as a magnitude | **NY25** · Zero net force for a fixed positive-mass particle in the stated inertial frame, throughout its passage. | mechanics | rest and balance of bodies |
| 6 · a sign named as a magnitude | **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 |
| 6 · a sign named as a magnitude | **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 |
| 7 · a sequencing named to one beat | **NY08** · Opposition continues under simultaneous best response at the specified choice pair. | game theory | selves choosing |
| 7 · a sequencing named to one beat | **NY33** · Intended supply and demand for the same good, market and reference interval agree at the stated price. | economics | a society's exchange |
| 7 · a sequencing named to one beat | **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 |
| 7 · a sequencing named to one beat | **NY35** · The same exchange criterion under q(next)=1/q, with best bundles recomputed. | economics | a society's exchange |
| 8 · a rate named as a value | **NY27** · The closed reversible pair A ⇌ B at fixed positive total concentration and fixed positive rate constants, with k₊a=k₋b. | chemistry | reactions |
| 8 · a rate named as a value | **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 |
| 8 · a rate named as a value | **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 |
| 9 · a two-way named to one side | **NY02** · Opposition of the same sign pair continues under joint reversal. | the resolver | the coupling and carrying of the resolver |
| 9 · a two-way named to one side | **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 |
| 9 · a two-way named to one side | **NY07** · Opposition continues under sequential best response at the specified choice pair. | game theory | selves choosing |
| 9 · a two-way named to one side | **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 |
| 9 · a two-way named to one side | **NY38** · Stationary composition in the dominant-A two-strategy replicator account, x′=x(1−x), 0≤x≤1. | evolutionary game theory | populations |
| 10 · a membrane named as a cut | **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 |
| 10 · a membrane named as a cut | **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 |
| 10 · a membrane named as a cut | **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 |
| 10 · a membrane named as a cut | **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 |

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

| The ten | Proposed conception, in its own words | Field | The changing it names |
|---|---|---|---|
| 1 · an arriving named from behind | **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 |
| 2 · an opening named as a place | **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 |
| 2 · an opening named as a place | **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 |
| 3 · a bound named as a last | **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 |
| 3 · a bound named as a last | **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 |
| 3 · a bound named as a last | **SA06** · A long-term climate response to specified forcing, fully equilibrated sensitivity distinguished from effective sensitivity, with corrections for differing conditions. | climate science | heat |
| 3 · a bound named as a last | **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 |
| 4 · a carry named as a store | **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 |
| 4 · a carry named as a store | **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 |
| 4 · a carry named as a store | **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 |
| 5 · a middle named as an end | **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 |
| 6 · a sign named as a magnitude | **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 |
| 6 · a sign named as a magnitude | **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 |
| 6 · a sign named as a magnitude | **SA14** · Two gas compartments at one temperature with differing volumes and pressures, shared temperature taken as no available work. | thermodynamics | heat |
| 7 · a sequencing named to one beat | **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 |
| 8 · a rate named as a value | **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 |
| 8 · a rate named as a value | **SA17** · Stationarity with emission and absorption added to number-conserving scattering, each process balanced on its own. | radiation | heat |
| 9 · a two-way named to one side | **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 |
| 9 · a two-way named to one side | **SA19** · Equal outcomes under two accounts, taken as the same equilibrium on the stated domain. | equivalent accounts | a relation that can change |
| 10 · a membrane named as a cut | **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 |
| 10 · a membrane named as a cut | **SA21** · Equal temperature compared through partner-dependent contacts, linked equalities giving the factor rHB rAH and the direct comparison rAB. | thermodynamics | heat |

**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.

## Shared exclusions, run at the code

**Four proof groups carry the exact local exclusions.** Each is a way of failing, run at the resolver'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. | Fixed torusing with uninterrupted carrying; each surface zero with finite nonempty carrying. | The reached domain, its retaining and required presence. |
| D · Required arrival or offering unreachable | The required target lies beyond each thing the stated operation can supply. | Nonzero offering from the opposed aggregate route; finite zero departure from a nonzero predecessor under a nonzero multiplier. | 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, runs at one subject throughout, the same carrying named from its presence to its bound. Presence beyond its retaining bound requires renewal, and the requirement naming the carrying still excludes each renewal: renewing fails the requirement, and no renewing fails the presence at the bound. The two cases are the whole, so still and present do not both continue across the bound. The still description ends, and the participant continues changing. Two bounds differ: a complete nonempty carrying is not the same at its following return, while fixed torusing alone continues through three retaining returns, and a fourth at a positive second sign. At a particular completing, such as one to nine, the sequence carries when its two steps are met at that completing: the continuing requires the renewal there, and the requirement excludes that renewal. The code's retaining bound counts couplings, and one to nine counts four momentaries of co-bi-exchanging. 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.

**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.

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

| Required condition | The necessary continuation |
|---|---|
| An unchanged complete, nonempty reached carrying | Retaining opens it one on; fresh writing inverts its nonzero second sign; releasing completes its presence. |
| Fixed nonzero torusing with uninterrupted presence | Fresh writing inverts that sign, and retaining alone completes. |
| Each surface zero with finite nonempty carrying continuing | No fresh nonzero writing renews the carrying before retaining completes; each carrying leaves by the fifth coupling. Zero aggregate from nonzero surfaces is a different condition. |
| An opposed row with a nonzero onward offering through the stated aggregate route | Each such row offers zero through that route, next and later; the row alone continues. |
| 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. |

**At the resolver's code the joint forms run one way.** 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 under empty receiving. At an existing entry with a nonzero surface s, the release pair 6-other-crossing and 10-other-surfacing is (t, s), and the fresh 7/8 pair is G(t, s) = (s, −t). Under empty receiving s = −c, and the joint forms run by G.

**The release pair and the fresh carrying are complementary.** Equality at 6/10 gives opposition in the fresh 7/8, and opposition gives equality: a conception requiring the same agreement or opposition at both is not possible.

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. At a fresh carrying (s,−t) from (c,t), s = −c keeps the pair's relation and s = c reverses it. Each of the four starting pairs, the eight triples and the eight fresh triples meets these. A pair agreeing names no still resolver and no equilibrium: its overlapping pair opposes, and its own next inverts both signs.

At the release, fresh s = −c keeps the pair's relation while the following 6-other-crossing releases −t, its sign changing; fresh s = c changes both; a carrying kept through a zero at 10-other-surfacing changes neither, the following 6 releasing t; a carrying completing leaves no following pair and no release, a subject apart from agreeing and opposing. A relation continues while its outward sign changes, and this is stable-forming among changing signs.

**6 and 10 release at one parity**, both even and opening bi, in the form 7→11→6→10, facing different neighbours, 2 and 14. Connector parity, sign polarity and agreement between signs are three comparisons.

**The full carrying determines the next.** The surface is s = sign(r − c), the arriving signs meeting the carrying sign inverting. One opposed relation with two positive arrivals gives agreement from (+,−) and opposition from (−,+), and one matching arrival at (+,−) continues from opening 0 and leaves from opening 3. Opposition and its receiving alone name part of the carrying as the whole.

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

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The derivations run at the resolver's code and at the numbers they name: 32 of 32 are met.