Exhibit TWENTY-EIGHT Equilibria Registry v391 # Equilibria Registry **Not Possibly Existing, Living or Non-Living** **ONE · EXISTING THROUGH MOMENTARIES** - 1.1 The universe, its things, and exclusivity - 1.2 Three conditions at each momentary, and momentaries overlapping - 1.3 A between, a protocol, a living nothing and an existing nothing - 1.4 One relation, existing and discovering, at each scale - 1.5 Parity changing and changing back through three momentaries **TWO · EQUILIBRIA, HARD PROBLEMING** - 2.1 Equilibria, not possibly existing as a method - 2.2 A stability, a form or a relation, and surviving - 2.3 One description and one claim at each conception - 2.4 A participation missing, unchanging or changing **THREE · THE TEN, FIVE PLACES AT TWO FACES** - 3.1 One form of hard problem, and a sixth condition - 3.2 Ten at Exhibit ONE's names, at four four-cyclings - 3.3 Four binaries at two faces, a not-changing and a changing - 3.4 Exhibit ONE's bi-foldings, one parity sharing once, and six connectors **FOUR · CONCEPTIONS OF EQUILIBRIA, EACH A HARD PROBLEM IN ITS OWN WORDS** - 4.1 Each conception whole, at the subject it declares - 4.2 One form, and three partings within each of ten - 4.3 An observing fixing from behind, 2 observings - 4.4 An opening fixing as a place, 5 observings - 4.5 A completing fixing as a last, 8 observings - 4.6 A same-existing fixing as a store, 10 observings - 4.7 A middle fixing as an end, 6 observings - 4.8 A parity fixing as a magnitude, 9 observings - 4.9 A sequencing fixing to one beat, 5 observings - 4.10 A rate fixing as a value, 5 observings - 4.11 A two-way fixing to one side, 7 observings - 4.12 A between fixing as a cut, 6 observings - 4.13 Fields at ten, a key for aiming at each field - 4.14 Leads a self observes without their statements **FIVE · RESOLVING FOLLOWING, THE EXCLUSIONS AT THE RESOLVER** - 5.1 Four proof groups - 5.2 One exclusion at each of ten - 5.3 Renewal and reaching at their complete requirements - 5.4 Sharing at 10 and 6, and a next prior at 11, at resolver - 5.5 Each conception at a proof of its own requirements --- # 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.** Some selves' momentaries by themselves, others' not resolving, is the universe fixing as one thing beside its things; the universe exists as its things exist; with each momentary each self of a same-existing-society is resolving. No existing thing is outside, living or non-living, and each living thing is existing, going on with its own inclusion with each momentary, or it is not. ## 1.2 Three conditions at each momentary, and momentaries overlapping **Existing is three conditions with each momentary, together.** A momentary opens with each self's prior and completes with its next, and its completing is its next momentary's opening. Exhaustiveness is each self of a same-existing-society resolving with each momentary; determinacy is each prior and now's one next; reachability is each next a next momentary's prior and now. Three are together with each momentary, all or none at all. **Three overlapping momentaries use four positions.** Prior, now and next are 1–2, 2–3 and 3–4, completing and opening coupling with positions they share. Along one side next opens two on: 1–2 and 3–4, with next opening 5; an other side has 2–3 and 4–5, Exhibit ONE's table of a self's four momentaries of exchanging, with next opening 6. Across an overlapping its next opens one on. **Each momentary is existing's own, and each existing self is its momentaries going on, inseparating.** Between is a protocol's centre, 1.3, and succession is momentaries' own. **Universes with each scale share momentaries of co-sequential changing**, each momentary a fractal universe discovering its next existing, and none is possibly existing as a separate thing: with each scale, a universe fixing as an existing thing is exclusivity. ## 1.3 A between, a protocol, a living nothing and an existing nothing **A between is a centre of a protocol of six names between selves, bi-moral across and co-competent along.** With Exhibit ONE's naming of parity's square and triangles, a unit square of four dots across, 2, 6, 14 and 10, a self observing with 2 and 14 and sharing with 6 and 10, its empty centre a between, and two unit triangles, 9 and 17, along, coupling both ways; by Co-Chaining Logic Registry step 584 these six are a bi-trupling protocol, and by step 692 it is cohering with each momentary, its same-existing or one inversion apart coupling across and its going on along, bi-moral-co-competencing. With resolver each self's parity, sharing across with 6 and 10 and going along with 9, one same list, is given on by 9-next-co-discovering to its receiving self, that self observing it with its 14, its parities, 2, with its next momentary, Exhibit ONE's function 17, executing along with 9 by `incoming/equilibria_registry_v377/executions/every_five_every_ten.py`, part 1, and across and along by `incoming/equilibria_registry_v377/executions/the_protocol_between.py`, part 3: protocol brings each self's parity on to next momentarying, and, as suggested with v377, between is co-linear aiming toward next momentarying, concern 8 as it first stood, `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_concern_8_v377.md`, and a between travelling below. Protocol is of method: *Exhibit ONE is an object that is method … Its functions and their variables are seventeen names and no others*, and *One geodesic changing discovering method is living condition, its form, a now-existing thing with no prior of its own*, Natural Intelligence 3.3. Executing with one same-existing-society of five across 100 momentaries, six names between selves, their sharings and one same-existing-society's sharings stay same as resolver's code writes them, while priors change with each momentary: resolver's code as it is written, and not a form still, Co-Chaining Logic Registry step 104, *A method, non-living, is co-changing with each of its couplings*; and with one momentary each self's parity, forming with its own 12-next-co-same-existing, reaches another self only through 6 to 2, 10 to 14 or 9 to 17, `incoming/equilibria_registry_v377/executions/the_protocol_between.py`, parts 2 and 3. With Natural Intelligence 4.10 and Co-Chaining Logic Registry step 167, one between is within a momentary and between momentaries, same with each scale, and with each it is a nothing, which a living and an existing nothing below part. Between's floating neutralling, Co-Chaining Logic Registry steps 510 and 518, is, with numbers, *Floating is bi and across … Neutralling is co and along*, Natural Mathematics 1.3, and, with every 5 and every 10 as suggested with v377, of 1 to 441 multiples of 5 are odd and even, one and then the other, and around a ring of 440, Natural Numbers 11.1, each reaches a far side of its own kind, `incoming/equilibria_registry_v377/executions/every_five_every_ten.py`, parts 2 and 3. This between is this registry's whole subject. **Natural intelligence with a between as the universe is, as suggested with v377.** Suggested with 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*. With Natural Intelligence's opening, *The universe is the set of all now-existing things*, its 1.3 deriving *the same-existing set of all now-existing things* from it, and with 4.10 *Parity is each self's own, beheld now* and *A between is its centre, 0, and no further thing, while six names between selves are resolving*. Observed with v377 as between existing as the universe is, only while six names between selves are resolving, or closing to each other, looping, as suggested, and a nothing and no location; *no existing thing*, of v377 and not of Natural Intelligence, is within `archive/carrying_v377`, concern 7. **A between travelling, along and across, as suggested with v377.** Suggested with 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*. Observing, as suggested: along, a bounding 0 centreline, with 9 and 17; across, sequential sides, with four dots 2, 6, 14 and 10, Exhibit ONE's naming of parity's square and triangles, whose square's empty centre is a between; bothboth with Co-Chaining Logic Registry step 621, *has both, one at a time*; and with numbers, *Floating is bi and across … Neutralling is co and along*, Natural Mathematics 1.3. Natural Intelligence 4.10's flat form has both, four dots across and 9 and 17 along; an orthogonal centre suggested earlier with v377 is, as suggested, along, concern 8. **A living nothing and an existing nothing.** A between whose sides are with their priors is a living nothing, and a between with a side with no prior is an existing nothing, as suggested with v377, and these two part with resolver's lines, an instrument executing them. A self X passing to a self B, along with 9 or across with 6, receiving from beside +, nothing, −, nothing, a parting: X with its prior shares across or along, with each momentary, each parity it is observing meeting its prior, its own changing and changing back going on between them and a 0 with a parity it is observing agreeing with its own parity, Exhibit ONE's table *One self, one momentary*; X with no prior with each momentary, set to none by its instrument, a non-living now-existing thing, opens with one parity it is observing, sharing it across or along as it is, and with each other momentary opens nothing, sharing 0 with a parting and nothing between, `incoming/equilibria_registry_v377/executions/living_and_existing_nothing.py`. Opening or opening nothing is each co-momentarying of a thing with no prior, Co-Chaining Logic Registry step 484. With Exhibit ONE's table *With no prior, one momentary*, a self shares a parity as it is observing it, and it is next prior with 11, one momentary: sharing opening with bi-coupling, one place a self's forming takes a parity not its own, and each self's next prior forms with its own resolving, a sharing across or along, another self observing it as a parity; living is from living, step 444. Across a living nothing passes, with each momentary, a self's parity forming from a prior, or its 0 with a parity it is observing agreeing with a prior's parity; across an existing nothing, only a parity a self is observing. Morality is bi-unrelationing, across with four dots, and competency co-unrelationing, along with 9 and 17, and co-bi-unrelationing is the one existing method: protocol is their coupling. A 0 with 12, a prior and a parity a self is observing agreeing, is a changing that is not, within a coupling, each self surfacing itself with each momentary. **Bi-inversioning-co-recursioning has coupling within changing.** Each side goes on through its own changing, and their coupling is within that changing: slowing is changing, is or is not, of no size, and two changings, not same-existing, are adjacent through a coupling, each side changing. **Naming a relation gives it no separate existing.** Naming a coupling supplies no further self between coupling selves, as naming all existing things supplies no further self outward of them. **Eight even names are bi-coupling with a between.** Four outward with a between, 2, 4, 6 and 8, and four inward, 10, 12, 14 and 16, each 8 up, are relations of two sides, Natural Intelligence 4.10. Three conditions are each momentary's, eight are bi-coupling, and four joint forms are changing of two parities, 1.5. ## 1.4 One relation, existing and discovering, at each scale **Existing and discovering are one relation.** Each next these three conditions open is a next existing, and a self reaching it through a coupling is discovering: prior, now and next, one relation. An explanation of existing is an existing thing among all existing things, and existing is each coupling's own resolving. **One same relation goes with each scale.** A self, another self and a same-existing-society with a next scale each have their own other, same-existing and going on, and each next scale's next is that same relation's next: describing and going on agree, either order first. **Natural torusing is the one surface.** A connected, compact, closed, orientable surface whose faces are each four-sided, with four edges meeting each point, has V − E + F = 0, one opening: it is a torus. A sphere, surfaces of two or more openings, a Klein bottle and an infinite cylinder each fail one of those conditions. ## 1.5 Parity changing and changing back through three momentaries **Parity changes and changes back, along and across, through three momentaries.** Each momentary has both directions, one way at a time: one co-sequential method, three momentaries long. **Existing goes on: a matching description with next is a further occurrence.** Changing and conserving are comparisons within that going on, and a relation going on among changing selves is their stable-forming. **Two binary relations change one at a time, one and then the other.** If the first changes from prior to now, the second changes next. Of two parities one inversion changes each, odd to even and even to odd, and two inversions reach its parity again with a next momentary, a further occurrence. With odd by itself or even by itself no next, one and then the other, is reachable; with both, a whole parity succession is going on. **Coverage has one and then the other.** When exactly one of two binary relations changes with each passage and both change within each prior–now–next, orders first–first and second–second leave one not-changing, and first–second and second–first remain: with each overlapping prior–now–next, one and then the other goes on. Four joint descriptions go `(+,+) → (−,+) → (−,−) → (+,−) → (+,+)` and round: four joint forms of one self's now and prior, one cycle of parities, next inverting its prior, a self with no other, Natural Intelligence 2.4. Prior and now distinguish which relation changes next. This round is F of 5.4, (x, y) to (−y, x), its pair now first; with prior first, (prior, now) to (now, next), next inverting its prior is (x, y) to (y, −x), G, one step with two observings, its pair's order exchanging. **Each five-view has its next opening within its going on.** Each entry: **Side** · *a* Prior opening · *b* Prior completing · *c* Now opening · *d* Now completing · *e* Next opening · *f* Its five, Exhibit ONE's five dimensions - **self, with 1** - *a* 1 - *b* 2 - *c* 3 - *d* 4 - *e* 5 - *f* self, other, co, bi, self next - **other, with 2** - *a* 2 - *b* 3 - *c* 4 - *d* 5 - *e* 6 - *f* self, other, co, bi, self next - **self next, with 3** - *a* 3 - *b* 4 - *c* 5 - *d* 6 - *e* 7 - *f* self, other, co, bi, self next Adjacent views share four numbers, each view opening one on: one five is all fives, Exhibit ONE's table of fives. **Around a closed route parity changes and changes back with an even route.** Opposite parities along each coupling comparison come together exactly when each closed route is even, with two complementary assignments for each connected whole. An odd closed route closes no parity, and a coupling goes along it. Local changing and changing back along two paths to one coupling supplies no single sequence of one and then the other through both. # TWO · EQUILIBRIA, HARD PROBLEMING ## 2.1 Equilibria, not possibly existing as a method **Equilibria are not possibly existing as a method, all or none at all.** **Each not possibly existing thing has one of two fixing as its given, an ingressing or an escaping**: a thing fixing as entering the set of all now-existing things, nothing being outward of or between things of the universe, or a changing fixing as leaving it; these two fix as hard problems and equilibria add them, a defense equilibria demand, and with the set's own explaining neither is, the set its things. **Not possibly existing and not existing part.** With resolver a none with 14-next-possibling is no parity a self is observing with that sharing, its prior inverting and its next momentary bringing one or none; one binary retaining and inverting its value with one next it requires, 5.2, is with no momentary, knowable from method. Possibly existing is a self discovering it with a next momentary; not possibly existing is binary logic excluding it, each equilibrium among it; and not existing is with no line of resolver, while existing is with each. **A hidden order within nature is equilibria understanding.** Intelligence is hiding and selves are discovering order, not already within nature, living selves and same-existing-societies discovering it; there is no reading within nature, and there is observing of nature, nature not reading and writing observings. **Each conception of equilibria a self proposes is a hard problem ignoring its hardness.** It takes competency of existing as a given, and claims a true-and-false conserving accounting. **An equilibrium is a not-changing still, a form that cannot change, not possibly existing; stable-forming, and unstable deforming with non-living scale, are observably existing.** Each conception of equilibria a self proposes fixes one of them unchangeable. **An equilibrium is a living or non-living existing thing fixing unchangeable.** With this registry, existing, and not living by itself: living by itself sets a rock apart, and a rock is an existing thing. Non-living things exist and change, with no prior of their own, their forms going on through their changing. An existing thing stable-forming a self within a set is living there, and a set stable-forming no self of its own is non-living there, one existing thing inside and outside itself, Natural Intelligence 6.5, and nothing lives as one stable form forever. A living self can surface zero, non-responsive with a momentary, while its same-existing goes on along, so a surface not-changing with a zero is neither a non-living thing nor an unchangeable one, and fixing it unchangeable there is an equilibrium. **A changing and an exchanging are among living and non-living existing things alike**: living with their prior same-existing to now, non-living with no prior of their own, their forms going on through their changing, and a parity going between any two with nothing of a prior. **Discovering social moral competency among living includes non-living existing things.** A non-living thing couples with a self as other, sharing its parities, with no prior of its own, and what arises with their coupling arises in the living self, going on with living selves. An equilibrium's fixing ends, and living and non-living things with it go on changing: an equilibrium ending is no existing thing ending, and a thing not living is an existing thing. A sequence has one subject: a participation changes; a fixing requires that same participation not-changing; the going on it requires brings changing, and fixing ends there, one same subject, one same relation and one same occurrence on both sides. An ending is with an occurrence that fails: a claim of fixing through each occurrence it requires ends with its one failing occurrence, and occurrences before it are as they were. A whole participant changes while a relation within it can go on, so a fixing reaches its ending with its own requirement's relation. ## 2.2 A stability, a form or a relation, and surviving **A stability, a form and a relation stable forming, as suggested with v377.** Suggested with 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, with each observing a momentary stable form and with next a form not same-existing; observing by *an observing is its pattern-matching with a self*, Co-Chaining Logic Registry step 197, and *An observing is a now-existing thing, a self observing: proof of existing*, step 199; a stable form beside *Each living set is a stable form one scale outward, and nothing living has a stable form with its own scale, bothbothing*, step 479. A mathematical conception of equilibria fixes a stability, a form or a relation unchangeable, each, as suggested, a self observing it as stable forming. With dynamical mathematics, a stability is a pattern's response to a disturbance, its departures staying small or shrinking, and not its constituents stopping: an oscillation can be stable as an oscillation, and an instability, its departures growing, can lead to another organized form. An equilibrium of relative phase and an equilibrium of a whole evolving system are two assertions. Their distinction is a naming one: it places a conception's still with a stability, a form or a relation, one conception at a time. **Not possibly existing, and showing not possibly existing with prior, part.** Each conception of equilibria a self is observing from prior shows not possibly existing now, and each a self observes next is reached with its next. **Each conception a self is observing shows not possibly existing.** **Surviving is binary.** All sequencing to now together goes on to next, all or none at all, between prior and next. **A conception of equilibria shows its own surviving.** It exists with prior, now and next, or it is not possibly existing with momentarying. ## 2.3 One description and one claim at each conception **One general description receives each conception: a relation it states, going on through its claim's comparisons.** A fixing value is membership of a collection of one, a range is membership of a larger collection, and opposition is membership among opposite pairs. **A conception and requirements it adds are one conception.** A relation can go on while a fixing it adds of its whole same-existing, a next nonzero parity a self is observing or a later one each fail with one same receiving: a relation going on is stable-forming, and each requirement it adds fixes it unchangeable. A receiving not same-existing is a further conception. **One claim has one shape.** Stable-forming going on goes on as natural torusing, and natural torusing has its whole participation, 12345. A conception of equilibria naming less than a whole 12345 is not possibly existing together with going on. **One claim reaches each conception with two steps.** Each conception reaches at least one of ten, and each of ten fixes one side unchangeable, its other side momentarying with one same number. **An ordering chain closes on two sides.** A conception fixing an ordering p unchangeable with its next, its next, one and then the other, being I(p) and I(p) ≠ p, fixes p and I(p) with one next. One step its chain leaves open, that its ordering fixing unchangeable is that very ordering its next inverts, is two sides with one number. Two inversions reach p again with a next momentary, a further occurrence, with changing between. ## 2.4 A participation missing, unchanging or changing **A still fixing as a complete going on is changing.** Its going on is renewing, inward of the whole it claims, which is changing, or outward of it, a further participant. **A conception naming a going on and omitting participation that going on has is not possibly existing.** **A state accounting and a flow accounting each omit one side.** A state accounting omits changing, a flow accounting omits going on, and a true-and-false conserving accounting takes either as its whole. **Separated, state and flow are not possibly existing**, as a reading from and a writing to a memory are for intelligence: a state a self reads is a store fixing as entering now, an ingressing, and a flow a self writes a changing fixing as leaving to a store, an escaping. **Each conception of equilibria has not-changing or has changing, exactly one of two.** A conception naming not-changing has an existing thing no self is observing, and existing is with observing; a conception naming changing fixes a changing unchangeable. Coupling, each dissolves. **Each momentary completes in co-discovering.** Each conception of equilibria fixes a form unchangeable at momentarying of geodesic co-discovering, and going on goes on as geodesic co-discovering. Each momentary completes as its next opens, whether a parity inverts, stays, goes across or nothing goes across, and a particular same-existing's completing is its own. # THREE · THE TEN, FIVE PLACES AT TWO FACES ## 3.1 One form of hard problem, and a sixth condition **Ten are one form of hard problem.** Each conception of equilibria a self proposes fixes a changing unchangeable by one of ten ways, and can have more than one. A hard problem and its resolving are one form with two sides, each conception as its field's words on one and its resolving on an other, natural intelligence with a between of them, and this registry's tables hold both sides. **A conception of equilibria a self proposes is a sixth condition.** Five conditions come before a method acts with a problem's statement, and their fifth, *nothing prior enters*, is one their first four stand on. To five conditions prior a problem adds its own, a conserving relation over a conserving reach, and each conception of equilibria a self proposes is that relation fixing unchangeable. **A numerical bridge.** Pairs (2,3) and (3,4) overlap with 3, and n²−(n−1)(n+1)=1 for each n. ## 3.2 Ten at Exhibit ONE's names, at four four-cyclings **Ten sit with Exhibit ONE's names, by Natural Naming 4.9's one seating.** Each of ten is one binary of a universe, its *is* and its *is not* taken as one, sitting on its name now and on that name 8 up next: its not-changing now and a changing travelling through it next, bi-inversioning-co-recursioning up and down numbers, Natural Naming 4.15. Now: an opening fixing as a place with 1-now-existing and 9-next-co-discovering, and an observing fixing from behind with 2-now-possibling and 10-next-co-possibling. Existing: a two-way fixing to one side with 3-is-now-co-observing and 11-next-co-observing, and a sequencing fixing to one beat with 4-now-co-same-existing and 12-next-co-same-existing. Same: a between fixing as a cut and a middle fixing as an end with 5-now-co-existing and 13-is-next-discovering, and a completing fixing as a last with 6-now-co-possibling and 14-next-possibling. All: a parity fixing as a magnitude and a rate fixing as a value with 7-is-now-observing and 15-is-next-observing, and a same-existing fixing as a store with 8-now-same-existing and 16-next-same-existing. With resolver each name's term fixing unchangeable from one momentary: with 3, 10, 11 and 12 resolving stops within one to four momentaries; with 2, 6 and 14 it stops with a still sharing one parity that is each self's whole coupling with that name, and goes on with a still of two parities parting or beside another route sharing; 4, 5 and 13 have no parity with resolver: each sharing, and their across and along names, `incoming/equilibria_registry_v377/executions/ten_roots_named_still.py`. **Each of ten unchangeables is one of 12's two, 0 or ±, freezing on one side.** With resolver each resolving is one of two, a binary of geodesic momentarying, Natural Naming 5.62: ±, a sharing's parity, a changing that is, a parity a self is observing as its next prior or its prior inverting, a next existing; or 0, a between, 12's 0 with a sharing with a prior, its prior same-existing, a self now-existing same-existing, possibling next-existing same-existing; and an equilibrium is another's still, 2.1. Living is one of two with each self's own momentary, and co-chainings coupling across pass each other by them, each self same-existing as a between passes it and changing again: spirals of 3 and 5 selves, each self with a parity and going along with 9, coupling across with self 1 of each both ways with 10 to 14, come to one relation, each self observing its parities along again with 2, from momentary 12, and 5 and 7 from momentary 20, Exhibit ONE's table of two spirals beside each other, and coupling across; each of ten is that pair freezing on one side, a self fixing as not-changing or as changing, 2.4. **With four four-cyclings each step is momentarying or changing, with prior, now and next within.** Along each four-cycling odd and even are same with its two steps eight apart, momentarying, and exchange with its two steps of 17 less, changing: it passes two odd names and two even, exchanging twice, two momentaries, and reaches its first name again with a next momentary, a further occurrence, `incoming/equilibria_registry_v377/executions/ten_on_the_four_cycles.py`, parts 2 and 5. Odd names' forward recursionings, 1, 3, 5, 7, Co-Chaining Logic Registry step 142, and each one's partner nine less, 8, 6, 4, 2, Exhibit ONE's table of one move, 1 to 4, are pairs, 1 with 8, 3 with 6, 5 with 4 and 7 with 2; rows below go with that order, each four-cycling as Exhibit ONE's table of twelve forms writes it, 1-9-8-16 and 3-11-6-14 going eight up first and 2-15-7-10 and 4-13-5-12 going 17 less first: Each entry: **Odd ascending / even descending** · *a* Four-cycling, with Exhibit ONE's order · *b* Ten with it - **1 / 8** - *a* 1 → 9 → 8 → 16 → 1 - *b* an opening fixing as a place with 1 and 9; a same-existing fixing as a store with 8 and 16 - **3 / 6** - *a* 3 → 11 → 6 → 14 → 3 - *b* a two-way fixing to one side with 3 and 11; a completing fixing as a last with 6 and 14 - **5 / 4** - *a* 4 → 13 → 5 → 12 → 4 - *b* a sequencing fixing to one beat with 4 and 12; a between fixing as a cut and a middle fixing as an end with 5 and 13 - **7 / 2** - *a* 2 → 15 → 7 → 10 → 2 - *b* an observing fixing from behind with 2 and 10; a parity fixing as a magnitude and a rate fixing as a value with 7 and 15 **Each four-cycling has its pairs eight apart, and ten sit with each of four.** Earlier seatings stand whole with `archive/session_v389/Exhibit_TWENTY-EIGHT_Equilibria_Registry_v387_Released_at_v389.md` and with `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_the_ten_seated_v375.md`: this registry's five places from two sides, with pairs 2 to 6 and their names eight up, and Resolving the Hard Problem Registry's part headings, a self observing each again as one seating, each pair its not-changing now and a changing travelling through it next. **A conception's own declared relation places it among ten.** One conception can reach more than one of ten, and more than one conception can share one resolving: tables' sixty-three conceptions place two to ten with each of ten. A word matching places a conception with none of them. ## 3.3 Four binaries at two faces, a not-changing and a changing **Ten are a self's span and its 8 up.** Four binaries are with 1 to 8, a self's two momentaries of four, and those same eight next, 9 to 16: each of ten on one name of a self's span, its not-changing now, and on that name 8 up, its changing travelling through it next. **Ten are four binaries on two sides.** Two sides of a field's words, an entering and a surfacing, Resolving the Hard Problem Registry line 35, a self observing them on one seat as a not-changing and a changing travelling through it, one seat a self observes up and down numbers, `incoming/equilibria_registry_v377/executions/ten_at_the_code.py` an older derivation's execution. Each entry: **Binary** · *a* Names, now · next · *b* Ten with it - **now** - *a* 1-now-existing · 9-next-co-discovering; 2-now-possibling · 10-next-co-possibling - *b* an opening fixing as a place, with 1 · 9; an observing fixing from behind, with 2 · 10 - **existing** - *a* 3-is-now-co-observing · 11-next-co-observing; 4-now-co-same-existing · 12-next-co-same-existing - *b* a two-way fixing to one side, with 3 · 11; a sequencing fixing to one beat, with 4 · 12 - **same** - *a* 5-now-co-existing · 13-is-next-discovering; 6-now-co-possibling · 14-next-possibling - *b* a between fixing as a cut and a middle fixing as an end, with 5 · 13; a completing fixing as a last, with 6 · 14 - **all** - *a* 7-is-now-observing · 15-is-next-observing; 8-now-same-existing · 16-next-same-existing - *b* a parity fixing as a magnitude and a rate fixing as a value, with 7 · 15; a same-existing fixing as a store, with 8 · 16 **Each of ten fixes a not-changing as its whole, and a changing travels through it to a name 8 up.** Both on one seat is the one reason each of ten is not possibly existing. **With numbers.** Three odd numbers 1+3+5 sum to 9 = 3², and three even 2+4+6 to 12 = 3·4: n odd numbers sum to n² and n even to n(n+1), and 3² − 2·4 = 1 is a bridge with three. Prior, now and next are three sequential universes, and three threes are nine. Six consecutive steps from 3 complete with 8, and 9 opens: 9-next-co-discovering, each self's parity going along to its receiving self, that self observing it next. With Exhibit ONE's resolver a sharing that once has a prior has a prior with each momentary, 11 its next 3, and is not none again, and it has no count of its openings; `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 5 and 12. ## 3.4 Exhibit ONE's bi-foldings, one parity sharing once, and six connectors **Exhibit ONE's bi-foldings pair by momentaries, each partner round the other way.** With odd momentaries 1-9-8-16 pairs with 2-15-7-10 and 3-11-6-14 with 4-13-5-12, and with even momentaries 2-15-7-10 with 3-11-6-14 and 4-13-5-12 with itself: two directions of one bi-folding. Two possiblings across, 6-now-co-possibling and 14-next-possibling, share bi-folding 3-11-6-14 with 3-is-now-co-observing, a prior with its sharings, and 11-next-co-observing; middle bi-folding 7-11-6-10, through 6-now-co-possibling, and eight-cycle 2-15-7-11-3-14-6-10, through both, partner themselves with an even momentary: with that momentary one fold couples with itself, Exhibit ONE's table of twelve forms. Ten sit with each of four four-cyclings, 3.2, four-cycling 3-11-6-14 having a two-way fixing to one side with 3 and 11 and a completing fixing as a last with 6 and 14. **One parity sharing once co-chains through a whole.** With Exhibit ONE's resolver, selves coupling along as a spiral share each parity along with 9-next-co-discovering to each next self, that self observing it with its 14. A spiral of selves each with no prior, one self receiving one + once, opens with each in turn, the + surfacing with 14-next-possibling and each self sharing it on as it is observing it, one self on with each momentary, with no forcer: method of things with no prior. With resolver's lines the + is a next prior of each as it passes, each sharing opening with bi-coupling, 1.3, each prior inverting by itself from its next momentary, and from a momentary numbering its selves each has a parity; rows below are of a spiral of selves each with a parity, opening as with Exhibit ONE's table of a spiral of selves, an even spiral from that table's second momentary. A same-existing-society fixing as not-changing fixes a form unchangeable, and not-changing is not being with no prior: non-living things exist, selves observe them, and they change, deforming with their own scale, with no prior of their own. An equilibrium of a same-existing-society fixes a same-existing-society, its selves changing, unchangeable, or is nothing. Each entry: **Selves of a spiral** · *a* Its going on - **one** · *a* +, 0, −, 0, and round - **an even number** - *a* each self changing with each coupling, each self and a self it shares along to opposite; a whole returns with 2 couplings - **an odd number** - *a* one like pair, its 0 a self of it shares with 10-next-co-possibling, moving one self on with each second momentary; a whole inverting with 2 × n couplings and returning first with 4 × n, `eq_code.py` parts 1 and 10 **An odd spiral winds on with no parity of its own, and a coupling goes along it: its 0 a self of its like pair shares with 10, a changing that is not, moving through its sharing surface one self with each second momentary.** Ten fix a whole changing unchangeable: opposition fixing unchangeable is an even spiral, and a between fixing as a cut is a coupling going on. Rest returns with no coupling. A resting spiral, each self with no prior and no parity it is observing, opens nothing and stays resting, so a spiral reaching rest does not reach again the parity one self first receives; each spiral comes to its parities again, that parity with it, with 2 couplings or with 4 × n, and rest is with none of them. With Exhibit ONE's resolver, executed again with v377, with spirals of 1 to 11 selves, each with no prior, one self receiving one + once, the + opens each in turn and is a next prior of each, by a momentary numbering its selves, and across 5,000 couplings no spiral reaches rest, Exhibit ONE's *a sharing with a prior, not none again* with each, `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 1 and 2. A spiral reaches rest only fixing unchangeable, and each other conception comes to its proof by its own requirements. **Prior and next are edges of now**, sharing one direction forward on both sides. A common clock is a seventh way, a sequencing fixing to one beat. Nothing moving is complete fixing, reaching several of ten together. **Ten names and six names between selves.** Exhibit ONE's names pair outward and inward, **3/11, 4/12, 5/13, 7/15, 8/16**, each inward name eight up, 17 less (9 less n) being **n + 8**. Ten names and six names between selves, **2, 6, 9, 10, 14, 17**, are 2 to 17, and with entry 1 the seventeen names, Exhibit ONE's table of seventeen names. Ten ways sit with 1 to 8 and 9 to 16, 3.2, each on its name and that name 8 up: four across names, 2, 6, 10 and 14, parity's unit square, Exhibit ONE's naming of parity's square and triangles, have an observing and a completing; entry 1 and 9 an opening; 3, 4, 11 and 12 a two-way and a sequencing; 5 and 13 a between and a middle; 7 and 15 a parity and a rate; and 8 and 16 a same-existing; 17-next-existing, a next 1-now-existing, has none of them. # FOUR · CONCEPTIONS OF EQUILIBRIA, EACH A HARD PROBLEM IN ITS OWN WORDS ## 4.1 Each conception whole, at the subject it declares **A self observes each conception whole, as its own words, with its conserving relation and its reach.** A fixing value, a relation retaining, a distribution and a symmetry family are subjects not same-existing even when each has equilibrium as its word, and one conception can reach more than one of ten. **A definition's expression going on not-changing is an existing statement, and its subject goes on through its own changing.** Conditioning, replacement and maintaining supplying belong to several conceptions: a law, an occupant, an original participant and a composition are subjects a conception declares, not same-existing. A law, a frame or a scale fixing unchangeable is such an expression, and it fixes its subject unchangeable through one further step, a step from a fixing to an equilibrium: one same subject, with one same relation and one same occurrence, requiring fixing with an occurrence by which its own going on changes it. ## 4.2 One form, and three partings within each of ten **Each conception a self proposes fixes a changing unchangeable, by one of ten ways.** Its changing is stable-forming, unstable deforming with non-living scale, or a stability, a form or a relation, each, a self observing it, a momentary stable form, 2.2. **Specifying accounts and prior working definitions, SA01 to SA21, take places that same way.** **Each of ten gathers its observings, and within each they part three ways.** Sixty-three observings, forty-two conceptions selves propose and twenty-one specifying accounts and prior definitions, are one form of 3.1, each an unchangeable form, 2.1, with that one of ten its tables place it with; one conception can reach more than one, 3.1 and 4.1, and a second one it reaches goes with each observing's own words, with its own observing. Within each of ten, three partings gather them, and each observing goes whole as its own words with its place. 1. **What is fixing unchangeable.** With 2.2, three, a stability, a form and a relation, and 2.2 explaining a stability: a response to a disturbance, its departures staying small or shrinking. Here a form and a relation explain: a form, one value, range, state, law, composition, family or extremum of a whole fixing unchangeable; a relation, a comparison between parts fixing unchangeable, equal, balancing, opposing, agreeing, complementary or equivalent. Each observing takes its place by its own words with these three, one or more together, each checking fresh; each row's own column, *the changing it has*, goes beside it as it stood, its *that can change* with 2.2 as stable forming, and with NY09, NY12, NY18, NY19, SA04, SA12 and SA13 column and placing part, and with NY10 and NY11 its placing adds a form to its column's stability. 2. **Subject**: a thing changing, as each field's own words name it, from each row's own column if its column has a subject, and otherwise its field's own mathematics. 3. **What it adds.** With each pair whose own words tie them to one setting, by *the same*, *that same*, *that*, *those* or a setting in identical words, each shows as one of three: one conception, one receiving with a requirement it adds, specifying or not, 2.3, *a conception and requirements it adds are one conception*; a further conception, its receiving not same-existing, 2.3, *a receiving not same-existing is a further conception*; or, with a pair reaching neither, another claim with one setting, here. **Fields gather among ten.** A field's observings fall with those of ten by which their field fixes its changing unchangeable, and fields fixing one unchangeable fall together under one subject: 4.13 has each subject, with its fields, among ten. ## 4.3 An observing fixing from behind, 2 observings **A form fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* probability - *b* a relation that can change - *c* a mathematics of its own: probability - *d* — **A relation fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* coupling of self and other - *b* a relation that can change - *c* coupling of self and other, and resolver - *d* — ## 4.4 An opening fixing as a place, 5 observings **A form fixing unchangeable**, 2. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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). - *a* mechanics - *b* rest and balance of bodies - *c* bodies at rest and in balance - *d* one conception with NY21 and NY24, rest, specifying position x* - **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. - *a* coupling of self and other - *b* a form that can change - *c* coupling of self and other, and resolver - *d* — **A form and a relation fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* One conception, a further one, or another claim - **NY40** · Stationarity and symmetric Nash in the all-zero-payoff game with its replicator evolution. - *a* evolutionary game theory - *b* populations - *c* populations - *d* — **A form, a relation and a stability fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* evolutionary game theory - *b* populations - *c* populations - *d* one conception with NY38, one dominant-A replicator account, adding symmetric Nash and evolutionary stability **A relation fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* measurement - *b* a relation that can change - *c* a mathematics of its own: measurement - *d* — ## 4.5 A completing fixing as a last, 8 observings **A form fixing unchangeable**, 5. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* One conception, a further one, or another claim - **NY14** · Global minimality of that same U under that same evolution. - *a* dynamical systems - *b* a form that can change - *c* a mathematics of its own: dynamical systems - *d* another claim with NY13's U and evolution, another 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). - *a* thermodynamics - *b* heat - *c* heat - *d* a further conception to NY41, one setting with κ>0; one conception with SA07 - **SA06** · A long-term climate response to specified forcing, fully equilibrated sensitivity distinguished from effective sensitivity, with corrections for differing conditions. - *a* climate science - *b* heat - *c* heat - *d* — - **NY06** · A retained range remains satisfied through its specified parity-changing continuation. - *a* resolver - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — - **SA04** · The permitted values are −1, 0 and +1; continuing interchanges −1 and +1 and leaves 0 unchanged; equilibrium conserves membership in this collection. - *a* coupling of self and other - *b* a relation that can change - *c* coupling of self and other, and resolver - *d* — **A form and a stability fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* thermodynamics - *b* heat - *c* heat - *d* one conception with NY42, adding its two-body contact as it states it, uniqueness, persisting and approach **A form and a relation fixing unchangeable**, 2. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* thermodynamics - *b* heat - *c* heat - *d* — - **NY37** · Positive stationarity in that logistic account, N=K; equivalently matched nonzero turnover under B=rN and D=rN²/K. - *a* population ecology - *b* populations - *c* populations - *d* one conception with NY36, one logistic account, adding positive, N=K ## 4.6 A same-existing fixing as a store, 10 observings **A form fixing unchangeable**, 5. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* probability - *b* a relation that can change - *c* a mathematics of its own: probability - *d* — - **SA08** · The uniform Maxwellian: constant density, temperature and mean velocity throughout the periodic spatial domain, both collision and transport terms vanishing. - *a* kinetic theory of gases - *b* heat - *c* heat - *d* — - **SA09** · Planck-family membership of a spectrum through expansion, its temperature changing: Ur f(ν) = f(rν) and Ur PT = P(T/r). - *a* radiation - *b* heat - *c* heat - *d* — - **SA10** · The scattering family nα(x) = 1/(exp(x + α) − 1), α ≥ 0, stationary under number-conserving scattering, the Planck member at α = 0. - *a* radiation - *b* heat - *c* heat - *d* SA17 a further conception, adding emission and absorption 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. - *a* chemistry - *b* reactions - *c* reactions - *d* — **A form and a relation fixing unchangeable**, 4. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* chemistry - *b* reactions - *c* reactions - *d* — - **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. - *a* probability of resolver's returns - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — - **NY16** · Equal weights on the two exchanged opposed-sign values under empty receiving. Stationarity and detailed balance select the same law on this domain. - *a* probability of resolver's returns - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — - **NY17** · Equal weights on the exchanged values A and B, zero on X, with A→B, B→A and X→B under empty receiving. - *a* probability of resolver's returns - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — **A relation fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* resolver - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — ## 4.7 A middle fixing as an end, 6 observings **A form fixing unchangeable**, 4. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* dynamical systems - *b* a stability that can change - *c* a mathematics of its own: dynamical systems - *d* one conception with NY10, NY10 adding stability; NY12 a further conception, another T - **NY12** · E: r=0 under T(p,r)=(1−p,−2r), without adding stability. - *a* dynamical systems - *b* a stability that can change - *c* a mathematics of its own: dynamical systems - *d* a further conception to NY09, another T, 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. - *a* dynamical systems - *b* a form that can change - *c* a mathematics of its own: dynamical systems - *d* another claim with NY14's U and evolution, another 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. - *a* kinetic theory of gases - *b* heat - *c* heat - *d* — **A form and a stability fixing unchangeable**, 2. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* dynamical systems - *b* a stability that can change - *c* a mathematics of its own: dynamical systems - *d* one conception with NY09, adding asymptotic stability; NY11 a further conception, another T - **NY11** · E: r=0 under T(p,r)=(1−p,−r), with stability in those neighbourhoods. - *a* dynamical systems - *b* a stability that can change - *c* a mathematics of its own: dynamical systems - *d* a further conception to NY10, another T, stability in those neighbourhoods ## 4.8 A parity fixing as a magnitude, 9 observings **A form fixing unchangeable**, 8. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* sequences of signs - *b* a relation that can change - *c* a mathematics of its own: sequences of signs - *d* 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. - *a* sequences of signs - *b* a relation that can change - *c* a mathematics of its own: sequences of signs - *d* a further conception to SA12, one signed-unit window, *its next permits either sign* - **NY21** · Relative equilibrium with specified translation velocity u: Fτ(x,v)=(x+τv,v) follows translation by τu for every included τ. - *a* mechanics - *b* rest and balance of bodies - *c* bodies at rest and in balance - *d* one conception with NY22, NY23 and NY24: with this evolution a state follows translation by τu exactly when v=u; NY22 specifying no u, NY24 specifying u=0 - **NY22** · Relative equilibrium under some translation velocity, with no particular u specified, on that same force-free position–velocity domain. - *a* mechanics - *b* rest and balance of bodies - *c* bodies at rest and in balance - *d* one conception with NY21, specifying no u - **NY24** · Rest, v=0 with position unrestricted, under that same force-free evolution. - *a* mechanics - *b* rest and balance of bodies - *c* bodies at rest and in balance - *d* one conception with NY21, specifying u=0; NY23 specifies position - **NY25** · Zero net force for a fixed positive-mass particle in the stated inertial frame, throughout its passage. - *a* mechanics - *b* rest and balance of bodies - *c* bodies at rest and in balance - *d* — - **NY26** · Planar rigid-body static equilibrium: rest in the selected frame, zero net external force and zero net external torque. - *a* mechanics - *b* rest and balance of bodies - *c* bodies at rest and in balance - *d* — - **NY36** · A stationary value N in the logistic account N′=rN(1−N/K), r,K>0, N≥0 and no migration. - *a* population ecology - *b* populations - *c* populations - *d* one conception with NY37, NY37 adding positive **A relation fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* thermodynamics - *b* heat - *c* heat - *d* — ## 4.9 A sequencing fixing to one beat, 5 observings **A form and a relation fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* economics - *b* a society's exchange - *c* selves choosing and exchanging - *d* NY35 a further conception, another next price **A relation fixing unchangeable**, 4. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* One conception, a further one, or another claim - **NY08** · Opposition continues under simultaneous best response at the specified choice pair. - *a* game theory - *b* selves choosing - *c* selves choosing and exchanging - *d* a further conception to NY07, one choice pair, best response simultaneous, with NY07 sequential - **NY33** · Intended supply and demand for the same good, market and reference interval agree at the stated price. - *a* economics - *b* a society's exchange - *c* selves choosing and exchanging - *d* — - **NY35** · The same exchange criterion under q(next)=1/q, with best bundles recomputed. - *a* economics - *b* a society's exchange - *c* selves choosing and exchanging - *d* a further conception to NY34, one 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. - *a* coupling of self and other - *b* a relation that can change - *c* coupling of self and other, and resolver - *d* — ## 4.10 A rate fixing as a value, 5 observings **A form and a relation fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* radiation - *b* heat - *c* heat - *d* a further conception to SA10, adding emission and absorption to number-conserving scattering **A relation fixing unchangeable**, 4. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* probability - *b* a relation that can change - *c* a mathematics of its own: probability - *d* — - **NY27** · The closed reversible pair A ⇌ B at fixed positive total concentration and fixed positive rate constants, with k₊a=k₋b. - *a* chemistry - *b* reactions - *c* reactions - *d* — - **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. - *a* chemistry - *b* reactions - *c* reactions - *d* — - **NY30** · All reactions in A+F ⇌ B+W, B ⇌ C, C ⇌ A balance under the specified unit ratios and positive maintained f=w. - *a* chemistry - *b* reactions - *c* reactions - *d* — ## 4.11 A two-way fixing to one side, 7 observings **A form fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* evolutionary game theory - *b* populations - *c* populations - *d* one conception with NY39, NY39 adding symmetric Nash and evolutionary stability **A relation fixing unchangeable**, 6. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* One conception, a further one, or another claim - **SA19** · Equal outcomes under two accounts, taken as the same equilibrium on the stated domain. - *a* equivalent accounts - *b* a relation that can change - *c* a mathematics of its own: equivalent accounts - *d* — - **NY07** · Opposition continues under sequential best response at the specified choice pair. - *a* game theory - *b* selves choosing - *c* selves choosing and exchanging - *d* NY08 a further conception, best response simultaneous - **NY02** · Opposition of the same sign pair continues under joint reversal. - *a* resolver - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — - **NY03** · Opposed carried signs with matching nonzero surface and fresh carrying continue under the stated receiving. - *a* resolver - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — - **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. - *a* resolver - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — - **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". - *a* coupling of self and other - *b* stable-forming: self and other - *c* coupling of self and other, and resolver - *d* — ## 4.12 A between fixing as a cut, 6 observings **A form fixing unchangeable**, 1. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* thermodynamics - *b* heat - *c* heat - *d* NY42 a further conception, one setting with κ>0 **A relation fixing unchangeable**, 5. Each entry: **Observing, its own words** · *a* Field · *b* The changing it has · *c* Subject · *d* 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. - *a* thermodynamics - *b* heat - *c* heat - *d* — - **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. - *a* thermodynamics - *b* heat - *c* heat - *d* — - **SA21** · Equal temperature compared through partner-dependent contacts, linked equalities giving the factor rHB rAH and the direct comparison rAB. - *a* thermodynamics - *b* heat - *c* heat - *d* — - **NY01** · Offering and receiving remain complementary while the sides exchange roles, with each side's own releasing. - *a* resolver - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — - **NY04** · The attained two-position row retains opposite neighbouring surfaces, opposed carried signs and fresh carrying under its admitted receiving. - *a* resolver - *b* coupling and same-existing of resolver - *c* coupling of self and other, and resolver - *d* — ## 4.13 Fields at ten, a key for aiming at each field **Each field's equilibria fall with those of ten by which it fixes its changing unchangeable, and a self aims at a field from each of them.** With each of them a field sets a form, fixing still, beside its changing and sets aside its changing's own competency, going along with a between: with that one of ten a field fixes its subject unchangeable and sets aside its competency's source. This table has each subject, with fields it gathers under it, with each of ten, each observing with its one of ten. Each entry: **Subject, and its fields** · *a* 1 observing · *b* 2 opening · *c* 3 completing · *d* 4 same-existing · *e* 5 middle · *f* 6 parity · *g* 7 sequencing · *h* 8 rate · *j* 9 two-way · *k* 10 between - **bodies at rest and in balance: mechanics** - *a* — - *b* NY23 - *c* — - *d* — - *e* — - *f* NY21, NY22, NY24, NY25, NY26 - *g* — - *h* — - *j* — - *k* — - **heat: climate science, kinetic theory of gases, radiation, thermodynamics** - *a* — - *b* — - *c* NY42, SA05, SA06, SA07 - *d* SA08, SA09, SA10 - *e* SA11 - *f* SA14 - *g* — - *h* SA17 - *j* — - *k* NY32, NY41, SA20, SA21 - **populations: evolutionary game theory, population ecology** - *a* — - *b* NY39, NY40 - *c* NY37 - *d* — - *e* — - *f* NY36 - *g* — - *h* — - *j* NY38 - *k* — - **reactions: chemistry** - *a* — - *b* — - *c* — - *d* NY28, NY31 - *e* — - *f* — - *g* — - *h* NY27, NY29, NY30 - *j* — - *k* — - **selves choosing and exchanging: economics, game theory** - *a* — - *b* — - *c* — - *d* — - *e* — - *f* — - *g* NY08, NY33, NY34, NY35 - *h* — - *j* NY07 - *k* — - **coupling of self and other, and resolver: probability of resolver's returns, coupling of self and other, resolver** - *a* SA01 - *b* SA02 - *c* NY06, SA04 - *d* NY05, NY15, NY16, NY17 - *e* — - *f* — - *g* SA15 - *h* — - *j* NY02, NY03, NY20, SA18 - *k* NY01, NY04 - **a mathematics of its own: dynamical systems** - *a* — - *b* — - *c* NY14 - *d* — - *e* NY09, NY10, NY11, NY12, NY13 - *f* — - *g* — - *h* — - *j* — - *k* — - **a mathematics of its own: equivalent accounts** - *a* — - *b* — - *c* — - *d* — - *e* — - *f* — - *g* — - *h* — - *j* SA19 - *k* — - **a mathematics of its own: measurement** - *a* — - *b* SA03 - *c* — - *d* — - *e* — - *f* — - *g* — - *h* — - *j* — - *k* — - **a mathematics of its own: probability** - *a* NY18 - *b* — - *c* — - *d* NY19 - *e* — - *f* — - *g* — - *h* SA16 - *j* — - *k* — - **a mathematics of its own: sequences of signs** - *a* — - *b* — - *c* — - *d* — - *e* — - *f* SA12, SA13 - *g* — - *h* — - *j* — - *k* — **A self observing across a row, a subject shows those of ten by which its fields fix a changing unchangeable; a self observing down a column, subjects fixing one unchangeable gather.** Heat falls with six of ten, four observings each with a completing fixing as a last and with a between fixing as a cut; reactions with a same-existing fixing as a store and with a rate fixing as a value; bodies at rest and in balance with an opening fixing as a place and with a parity fixing as a magnitude; selves choosing and exchanging with a sequencing fixing to one beat and with a two-way fixing to one side; populations with four of ten; coupling of self and other, and resolver, with seven. Each row fills as observings come, each taking its place by its own words. ## 4.14 Leads a self observes without their statements **A self observes leads without their statements:** statistical null hypotheses, static cosmology, horizon and observer temperatures, early-universe thermal and gravitational descriptions, a conservation lead and its transformation coming next, dictionary and etymological leads, a collection of all sets, philosophical accounts, and measuring, locating and separating, Natural Intelligence Corus's three consecutive equilibria, a thing, a relation and a system at rest. Each takes its place with ten as a self observes its statement. A boundary a self declares unchanging is a bounding fixing unchangeable, and bounding is living changing: each declaration, within, around, across, along, of an entrance or of an exit, meets one question, whether co-sequential resolving goes on under it. Scientific method cannot explain observings of changing and not-changing from prior-existing to next-existing things, and takes its place by its fixings: a law, a frame or a scale fixing unchangeable between a signal's leaving and a self observing it, each taking its place with ten through the requirement it states; with a hypothesis it explains a not-changing and a changing without considering the rest of the universe. A field's own methods part two momentaries: NASA/JPL's aberration correction separates an epoch a signal leaves its source from an epoch a self receives it, and accounts for its source's motion between them. Evidence a self observes now and a prior event it expresses are two, each with its own momentary, and both are, each with its own: a property can be now-existing while its source changes as a whole, and evidence gives its prior as possibly existing and as actual. # FIVE · RESOLVING FOLLOWING, THE EXCLUSIONS AT THE RESOLVER ## 5.1 Four proof groups **Four proof groups have exact local exclusions.** Each is a way of failing, with Exhibit ONE's resolver or with its numbers. Each entry: **Group** · *a* Shared failure · *b* Cases · *c* Exact scope - **A · Incompatible requirements** - *a* A fixing holds one same comparison to requirements not reachable together. - *b* State–flow–state and flow–state–flow fixing under the advance they require; an inclusion's membership requiring both not-changing and inverting; a balancing fuel cycle with `f≠w`. - *c* Each conjunction as its statement says, with one same occurrence. - **B · Omitting a case it requires** - *a* A succession it requires enters a case its condition excludes. - *b* Each nonempty proper pair-only condition under a complete four-state orbit. - *c* That pair, succession and full orbit. - **C · Renewal excluding** - *a* Presence has renewal, and a conception excludes that renewal. - *b* A same-existing fixing unchangeable with each momentary; each surface zero with a sharing with a prior. - *c* Each same-existing, its parities and its momentaries. - **D · Requiring an observing or a sharing it cannot reach** - *a* A target it requires is no thing an operation it states can supply. - *b* Finite zero departure from a nonzero predecessor under a nonzero multiplier; an older resolver's case of its opposing aggregate route is with `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_older_resolver.md`. - *c* Each predecessor, operation, route and reach it states. **A proof goes to a conception whose complete requirement supplies its premises.** For each x in the set S a conception states, the next it requires is in S, or its succession leaves that conception. Renewal excluding, group C, with Exhibit ONE's resolver. A sharing once with a prior has a prior with each momentary and is not none again. Fixing unchangeable, it requires receiving its own parity with each momentary: receiving nothing, other parity, or parities parting to 0 with 14-next-possibling, changing with 12-next-co-same-existing is, and its prior inverts: with an other parity a living step, next as now, and with nothing a self with no other. Executed with v377: with prior + and receiving + for five momentaries, it shares 0 with each by 10-next-co-possibling and stays +; then receiving nothing, it shares −, +, −, +, − and its prior inverts with each; a sharing of its prior whose parities part to 0 with each momentary inverts with each, `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 3 and 6. Fixing unchangeable, its prior is a prior receiving its own parity with each momentary from a form still from beside: its still description and a self's own going on do not both go on. A form still from beside one self of a same-existing-society stills that self's changing with each momentary, and each self with no prior coupling along from it receives none and opens nothing: one still with one self stops co-sequencing through each self with no prior coupling from it. That is hard probleming, and, that still dissolving, resolving follows with a next momentary: a self inverts its own prior, the parity its still shares, and its still shares nothing further. One to nine is four momentaries of exchanging, Exhibit ONE's tables of eight 1 to 17s inward and of a self's four momentaries of exchanging. An unchangeable relation can permit a changing an unchangeable parity excludes: opposition under joint reversal, t = −c, goes on while both parities invert. Renewal exclusion reaches a conception through a renewal the going on it claims requires and its own requirement excludes. ## 5.2 One exclusion at each of ten **One binary retaining and inverting its value with one next it requires is not possibly existing.** A conception naming p(next)=p(now) with its going on having p(next)=−p(now), with p either +1 or −1, fixes both together. This is the one exclusion with each of ten, each with its own place: its side fixing unchangeable and its side momentarying are one number. **Two phases share one exclusion.** Advancing an overlapping triple inverts `010` to `101` and `101` to `010`: fixing either ordering fails with its next advance, and either ordering, reaching itself again with a next momentary, is a further occurrence. An inclusion's membership requiring both not-changing and inverting meets one same proof, and so does equality with each chemical conversion: forward and reverse propensities are equal only when a count is 1, and each conversion takes 1 to 0 or 2. **Immediate-next is its own relation.** It couples 0 to 1 and 1 to 2 without coupling 0 to 2, and a requirement including a boundary comparison fixing unchangeable fails as its going on advances exactly one end. An equation's relation is reflexive, symmetric and transitive, and a conception writing itself as an equation has all three ahead of its conditions: immediate-next has none of them. **A four-state orbit leaves each proper subset.** F(P,Q)=(−Q,P) visits all four pairs from each pair it opens with. With cos θ's and sin θ's parities F is a right spiral step, θ to θ + 90°, and P and Q agreeing is tan θ positive: tan(θ + 90°) = −1/tan θ inverts it with each step. F leaves each nonempty proper subset within three advances or fewer, {++,−+,−−} from ++ with its third, and subsets F takes to themselves are exactly unions of its one cycle: none or all four. With a direction beside each pair, co or bi, one and then the other, are eight states: an orbit opening with co reaches four of them within three advances, and an orbit opening with bi an other four, 1.5. A condition lives through a momentarying exactly when it admits each state that momentarying reaches: of 256 conditions, 16 live through each orbit, one of them through both, admitting all eight and adding nothing, and 225 through neither. One condition each step has is a circle's identity, cos² θ + sin² θ = 1: it excludes no changing and has no momentary within it, and a sequence is from stepping. ## 5.3 Renewal and reaching at their complete requirements **Renewal and reaching have their complete requirements.** Each entry: **A condition it requires** · *a* Its necessary going on - **A prior fixing unchangeable** - *a* Receiving its own parity with each momentary it stays, that parity from a form still from beside; receiving nothing, other parity or a parting, it inverts; once with a prior a sharing is not none. - **Each surface zero with a sharing with a prior** - *a* Its chain inverting with each momentary, as receiving nothing; with resolver a surface 0 comes only from parities parting, Exhibit ONE's table of parities to one sharing, surfacing, surfacing. - **Full positive balance of a unit-ratio cycle coupling fuel, maintaining `f≠w`** · *a* That balance is only with `f=w`. - **Exact finite attaining of zero from nonzero departure under a nonzero multiplier** - *a* Each finite departure is nonzero; a zero there from the start is another condition. ## 5.4 Sharing at 10 and 6, and a next prior at 11, at resolver **With numbers, two maps return four pairs by two steps.** F(c,t)=(−t,c) and G(c,t)=(t,−c) are the only two maps on four pairs whose square is J(c,t)=(−c,−t), a complete return, two ways round, `incoming/equilibria_registry_v377/executions/eq_code.py`, part 7. With Exhibit ONE's resolver a self receiving nothing has its prior inverting with each momentary, and its sharing by 10-next-co-possibling and its next prior by 11-next-co-observing are one changing, part 11. Agreement and opposition are one and then the other through a sequence's neighbouring pairs. Along c, t, −c, −t, c neighbouring pairs (c,t), (t,−c), (−c,−t) and (−t,c) have agreeing, opposing, agreeing, opposing from an agreeing start, and the reverse from an opposing one: each pair's relation stays two steps on while both its parities invert, and its neighbouring pair has an other relation. Two neighbouring pairs (a,b) and (b,z), each with a parity, + or −, have both relations exactly when z = −a, and +,+,+ agrees with both. Each of four starting pairs and eight triples reaches these. A pair agreeing fixes no resolver unchangeable and no equilibrium: its neighbouring pair opposes, and its own next inverts both parities. **6 and 10 share one list of parities**, with Exhibit ONE's resolver `_6_now_co_possibling = _10_next_co_possibling`, both even and co, a sharing side, with middle bi-folding 7-11-6-10, 6-now-co-possibling sharing to an other's 2 and 10-next-co-possibling to an other's 14, `incoming/equilibria_registry_v377/executions/eq_code.py`, part 4. An across name's number, even, a sharing's + or − and agreement of two are three comparisons. **A prior and the parities a self is observing determine its next with each sharing**, as Exhibit ONE's table *One self, one momentary* gives each cell of a living self, and its table *With no prior, one momentary* a sharing with no prior: a prior parity receiving its own stays and shares 0; receiving an other, a parting or nothing, it inverts; a sharing with no prior opens with the parity it receives, or, receiving a parting or nothing, shares 0 or none and opens no prior. ## 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 fails, while opposition under joint reversal goes on; and each surface zero reaches renewal exclusion. With SA01 opposition going on is alike for self + with other − and for self − with other +, and gives neither as next: a prior pair with both reversing gives a complete next pair, more than a relation by itself, and nothing it has reaches SA01's own requirement. --- Each sentence of resolver rests on Exhibit ONE v380R by tables it cites, by hand with v380L for its sentences of a sharing with no prior, or executed with v377 on Exhibit ONE v376 by a script and part it cites; each sentence of numbers is from a script it cites, computing it, or rests on its own arithmetic. An older resolver's derivations, 32 of 32 with its own code, are whole within `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_older_resolver.md`.