Exhibit TWENTY-EIGHT v368

# Equilibria Registry

**Impossible Stable Forms of 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: the inseparator
- The membrane is no separable thing
- Bi-inversioning-co-recursioning keeps 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 forces alternation
- The five-view keeps its next opening inside 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
- A stable form is not possible; stable-forming, and unstable deforming at non-living scale, are observably existing
- The stability can change, the form can change, the relation can change
- Impossible and not possible so far 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 requires
- 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 requiring a continuing while leaving out the participation that continuing needs is not possible
- A state accounting and a flow accounting each leave one side out
- Equilibria are of unchanging or of changing: either, not both, not neither
- Each momentary ends 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 as a stable form, and the other face runs at the same number
- In numbers
- The resolver's pairs are the momentaries, each round the other way
- The slightest tipping co-chains through the whole
- An odd ring closes no parity, so a meeting travels: the non-existing between, moving through the shared surface one momentary at a time
- So-far and not-yet are the edges at the momentary
- 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 as a stable form, in one of the ten ways
- The specified accounts and the older 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
- The same binary cannot retain and invert its value at the same required next
- 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, and a restoration after a momentary unmet is a further occurrence.

**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, then 3–4, with next opening 5; the other side runs 2–3, then 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: the inseparator.** The between is the membrane, and the succession is the momentaries' own.

**The membrane is no separable thing.** The between of two existing things co-bi-coupling is a membrane, non-existing as the universe as an existing thing is non-existing, and this between is the whole subject of the Registry. Morality is bi-unrelationing and competency is co-unrelationing, and co-bi-unrelationing is the one existing method: the membrane is their meeting. The 0 where two signs meet is within a coupling, and releases with it.

**Bi-inversioning-co-recursioning keeps the meeting within the changing.** Each continues through its own changing, and their meeting is within that changing: slowing is changing, and differing changing is adjacent through the coupling with neither side an unchanging separator.

**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 outside 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 as one existing thing. 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 larger next is the next of the same relation: describing then continuing agrees with continuing then describing.

**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 and so 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, and the ten 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 rule alone changes each, odd to even and even to odd, and two inversions restore the parity at a further occurrence. At odd alone or even alone the alternating next is not reached; at both, the whole parity succession runs.

**Coverage forces 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 then 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 keeps its next opening inside 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 travels 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.

**A stable form is not possible; stable-forming, and unstable deforming at non-living scale, are observably existing.** Each proposed conception of equilibria names one of them as a stable form, stilling the changing in its naming.

**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 as a stable form, and each can change.

**Impossible and not possible so far differ.** Each proposed conception of equilibria is shown not possible so far. Each one not yet shown is in the not-yet group, and the not-yet group empties as the clusters resolve. **The not-yet group is empty so far.**

**Surviving is binary.** All that sequenced into this now together continues to one more momentary after now, all or none at all, between so far and not yet.

**A conception of equilibria shows its own surviving.** It exists prior to, during and after the momentarying, or it is not possible at the momentarying.

**One general description receives each conception: a stated relation continuing through the comparisons its claim requires.** 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 as a stable form. Changing the receiving makes a further conception.

**The claim has one shape.** Continuing stable-forming requires natural torusing, and natural torusing requires its whole participation, 12345. A conception of equilibria that requires 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 as a stable form where the other face runs at the same number.

**The ordering chain closes at the two faces.** A conception naming an ordering p still at its next, where the alternating next is I(p) and I(p) ≠ p, requires 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 restore p at a further occurrence, with the changing between.

**A still named as the complete continuing is changing.** Its continuing needs renewing, and the renewing is within the claimed whole, which then changes, or from outside it, which then is a further participant.

**A conception requiring a continuing while leaving out the participation that continuing needs 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.

**Equilibria are of unchanging or of changing: either, not both, not neither.**

**Each momentary ends in co-releasing.** Each conception of equilibria names a form still where geodesic co-releasing runs, and without geodesic co-releasing no continuing is possible. 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 as a stable form in one of the ten ways, and can carry more than one. Hard problems and their resolving are one design, each conception in its field's words at one face and its resolving at the other, with the living intelligence between them, and the Registry carries both faces in one table.

**A proposed conception of equilibria is the sixth condition.** 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 as a stable form.

**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 several of the ten, several conceptions can share one resolving, and 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 sit at the numbers 2 to 6, where the two sides overlap, 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 toward 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 as a stable form, and the other face runs at the same number.** Both faces are one number, so each of the ten is not possible for one reason.

**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. Each momentary carries prior, now and next as one, three at three, nine. Six consecutive changings from 3 complete at 8, and 9 opens: 9-other-releasing, co-releasing. 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.

**The slightest tipping 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.

| Selves in the ring | What carries 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 where signs meet, travelling one self each momentary; the whole returns at 4 × n couplings |

**An odd ring closes no parity, so a meeting travels: the non-existing between, moving through the shared surface one momentary at a time.** What the ten name still is the whole changing: opposition kept names the even ring, and a membrane named as a cut names the travelling meeting.

**So-far and not-yet are the edges at the momentary**, each at its own facing. A common clock is the seventh way, a sequencing named to one beat. "Nothing moves" 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 relation to find.

## 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 several 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.

**Each proposed conception names a changing as a stable form, 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 older 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 awaiting its stated transformation, dictionary and etymological leads, a collection of all sets, and philosophical accounts. Each is placed at the ten when its statement arrives.

## 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 must satisfy requirements that cannot be 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 requires 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.

**The same binary cannot retain and invert its value at the same required next.** A conception requiring p(next)=p(now) where its continuing requires p(next)=−p(now), with p either +1 or −1, requires 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`, so fixing either ordering fails at the next advance, and a later return 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 where its continuation changes exactly one end.

**The four-state orbit leaves each proper subset.** F(P,Q)=(−Q,P) visits all four pairs from each starting pair. Each nonempty proper subset is left within at most three advances, {++,−+,−−} from ++ at the third, and the subsets kept 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 ends 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 requires `f=w`. |
| 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.

**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 compulsory reversal 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.