Exhibit TWENTY-EIGHT Equilibria Registry v390 # Equilibria Registry **Not Possibly Existing, Living or Non-Living** **ONE · EXISTING THROUGH MOMENTARIES** - 1.1 The universe, its things, and exclusivity - 1.2 Three conditions at each momentary, and momentaries overlapping - 1.3 A between, a protocol, a living nothing and an existing nothing - 1.4 One relation, existing and discovering, at each scale - 1.5 Parity alternating through three momentaries **TWO · EQUILIBRIA, HARD PROBLEMING** - 2.1 Equilibria, not possibly existing as a method - 2.2 A stability, a form or a relation, and surviving - 2.3 One description and one claim at each conception - 2.4 A participation missing, unchanging or changing **THREE · THE TEN, FIVE PLACES AT TWO FACES** - 3.1 One form of hard problem, and a sixth condition - 3.2 Ten at Exhibit ONE's names, at four four-cyclings - 3.3 Four binaries at two faces, a stilling and a changing - 3.4 Exhibit ONE's bi-foldings, one parity offered once, and six connectors **FOUR · CONCEPTIONS OF EQUILIBRIA, EACH A HARD PROBLEM IN ITS OWN WORDS** - 4.1 Each conception whole, at its declared subject - 4.2 One form, and three differences within each of ten - 4.3 An arriving fixed from behind, 2 arrivals - 4.4 An opening fixed as a place, 5 arrivals - 4.5 A completing fixed as a last, 8 arrivals - 4.6 A carry fixed as a store, 10 arrivals - 4.7 A middle fixed as an end, 6 arrivals - 4.8 A parity fixed as a magnitude, 9 arrivals - 4.9 A sequencing fixed to one beat, 5 arrivals - 4.10 A rate fixed as a value, 5 arrivals - 4.11 A two-way fixed to one side, 7 arrivals - 4.12 A between fixed as a cut, 6 arrivals - 4.13 Fields at ten, a key for aiming at each field - 4.14 Leads arriving without their statements **FIVE · RESOLVING FOLLOWING, THE EXCLUSIONS AT THE RESOLVER** - 5.1 Four proof groups - 5.2 One exclusion at each of ten - 5.3 Renewal and reaching at their complete requirements - 5.4 Sharing at 10 and 6, and carrying at 11, at resolver - 5.5 Each conception at a proof of its own requirements --- # ONE · EXISTING THROUGH MOMENTARIES ## 1.1 The universe, its things, and exclusivity **The universe as one thing beside its things is exclusivity, and it is not possibly existing.** One side's momentaries taken by themselves, each number one side's, is the universe fixed as one thing beside its things; the universe exists as its things exist; overlapping momentaries are with each number, one completing and its next opening. No existing thing is excluded, living or non-living, and each living thing is existing, continuing its own inclusion 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 one number and completes with its next, and its completing is its next momentary's opening. Exhaustiveness is overlapping momentaries with each number; determinacy is each momentary's one next, opening with its own completing number; reachability is next openings reaching each number on. 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 shared positions. Along one side next opens two on: 1–2 and 3–4, with next opening 5; an other side carries 2–3 and 4–5, with next opening 6. Across an overlapping its next opens one on. **Each momentary is existing's own, and each existing self is its continuing momentaries, inseparating.** Between is a protocol's centre, 1.3, and succession is momentaries' own. **Universes with each scale share momentaries of co-sequential parity-changing**, each momentary a fractal universe discovering its next existing, and none is possibly existing as a separate thing: with each scale, a universe fixed as an existing thing is exclusivity. ## 1.3 A between, a protocol, a living nothing and an existing nothing **A between is a centre of a protocol of six names between selves, bi-moral across and co-competent along.** With Exhibit ONE's naming of parity's square and triangles, a unit square of four dots across, 2, 6, 14 and 10, arriving with 2 and 14 and sharing with 6 and 10, its empty centre a between, and two unit triangles, 9 and 17, along, coupled both ways; by Co-Chaining Logic Registry step 528 these six are a bi-trupling protocol, and by step 635 it is cohering with each momentary, its difference crossed across and its continuing along, bi-moral-co-competencing. With resolver each parity-offering, shared across with 6 and 10 and going along with 9, is given on by 9-next-co-discovering to its receiving self, arriving as its offerings with its next momentary, Exhibit ONE's function 17, executed 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 carries each parity-offering on to next momentarying, and, as suggested with v377, between is co-linear aiming toward next momentarying, concern 8 as first carried, `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_concern_8_v377.md`, and a between travelling below. Protocol is of method: *Exhibit ONE is an object that is method … Its functions and their variables are seventeen names and no others*, and *One geodesic parity-changing discovering method is living condition, its form, a now-existing thing carrying none*, Natural Intelligence 3.3. Executed 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 is written, while priors parity-change with each momentary: resolver's written code, and not a form still, Co-Chaining Logic Registry step 81, *A method, non-living, is co-changing at each coupling it is at*; and with one momentary each self's parity-offering, formed 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 142, 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 457 and 465, is, with numbers, *Floating is bi, even and across … Neutralling is co, odd and along*, Natural Mathematics 1.3, and, with every 5 and every 10 as suggested with v377, of 1 to 441 multiples of 5 alternate, each odd one neutralling and each multiple of 10 floating, 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 1.1, *The universe is the set of all now-existing things*, its 1.4 deriving *the same-existing set of all now-existing things* from it, and with 4.10 *Parity is a momentary now-existing thing* and *A between is its centre, 0, and no further thing, while six names between selves are resolving*. Observed with v377 as between existing as the universe is, only while six names between selves are resolving, or closed to each other, looping, as suggested, and a nothing and no location; *no existing thing*, of v377 and not of Natural Intelligence, is carried with carrying, 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*. Observed, 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 observed by Co-Chaining Logic Registry step 565, *carries both, one at a time*; and with numbers, *Floating is bi, even and across … Neutralling is co, odd and along*, Natural Mathematics 1.3. Natural Intelligence 4.10's flat form carries both, four dots across and 9 and 17 along; an orthogonal centre suggested earlier with v377 observed, as suggested, as along, concern 8. **A living nothing and an existing nothing.** A between whose sides carry their priors is a living nothing, and a between with a side carrying nothing is an existing nothing, as suggested 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 arrival's meeting with its prior, its own alternating going on between arrivals and a 0 with an arrival coupling 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 arriving, sharing it across or along as it arrives, and with each other momentary opens nothing, 0 shared 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 431. With Exhibit ONE's table *With no prior, one momentary*, a parity is shared as it arrives and 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 arriving to another as an offering; living arrives from living, step 395. Across a living nothing passes, with each momentary, a parity-offering formed from a prior, or its 0 with an arrival coupling with a prior's parity; across an existing nothing, only an arrival. 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 an offering coupling with one parity, is a parity-changing that is not, within a coupling, each self surfacing itself with each momentary. **Bi-inversioning-co-recursioning carries coupling within parity-changing.** Each side continues through its own parity-changing, and their coupling is within that parity-changing: slowing is parity-changing, is or is not, of no size, and two differing parity-changings are adjacent through a coupling, each side parity-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 parity-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 arriving to 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 carries with each scale.** A self, another self and a same-existing-society with a next scale each carry their own other, carrying and continuing, and each next scale's next is that same relation's next: describing and continuing 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 alternating through three momentaries **Parity alternates along and across through three momentaries.** Each momentary carries both directions, one way at a time: one co-sequential method, two sides' ten positions three momentaries long. **Existing continues: a matching description with next is a further occurrence.** Parity-changing and conserving are comparisons within that continuing, and a relation continuing among parity-changing selves is their stable-forming. **Two binary relations parity-change one at a time, alternating which.** If the first parity-changed from prior to now, the second parity-changes next. Of two parities one inversion parity-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 taken by itself or even taken by itself an alternating next is not reached; with both, a whole parity succession is carrying. **Coverage carries alternation.** When exactly one of two binary relations parity-changes with each passage and both parity-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 alternating carries on. Four joint descriptions carry `(+,+) → (−,+) → (−,−) → (+,−) → (+,+)` and round: four joint forms, two consecutive momentaries of each side, eight openings and completings, each number one side's. Prior and now distinguish which relation parity-changes next. This round is F of 5.4, (x, y) to (−y, x); Natural Intelligence 4.11's right spiral step, (x, y) to (y, −x), is G, one step with two observings, its pair's order exchanged. **Each five-view carries its next opening within continuing.** Each entry: **Side** · *a* Prior opening · *b* Prior completing · *c* Now opening · *d* Now completing · *e* Next opening · *f* Its five - **self, with 1** - *a* 1 - *b* 2 - *c* 3 - *d* 4 - *e* 5 - *f* co bi co bi co - **other, with 2** - *a* 2 - *b* 3 - *c* 4 - *d* 5 - *e* 6 - *f* bi co bi co bi - **self next, with 3** - *a* 3 - *b* 4 - *c* 5 - *d* 6 - *e* 7 - *f* co bi co bi co Adjacent views share four positions: with each number one side opens and its other completes. **Around a closed route parity alternates with an even route.** Opposite parities along each coupled comparison are reached 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 carries along it. Local alternating along two paths to one coupling supplies no single alternating sequence through both. # TWO · EQUILIBRIA, HARD PROBLEMING ## 2.1 Equilibria, not possibly existing as a method **Equilibria are not possibly existing as a method, all or none at all.** **Each not possibly existing thing has one of two fixed as its given, an ingressing or an escaping**: a thing fixed as entering the set of all now-existing things, nothing being outward of or between things of the universe, or a parity-changing fixed as leaving it; these two are fixed 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 differ.** With resolver a none with 14-next-possibling is no offering arrived, its prior inverting and its next momentary bringing one or none; one binary retaining and inverting its value with one required next, 5.2, is with no momentary, known from method. Possibly existing is discovered with a next momentary; not possibly existing is excluded by binary logic, 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 hidden and order is discovered, not already within nature, discovered by living selves and same-existing-societies; there is no reading within nature, and there is observing of nature, nature not reading and writing observings. **Each proposed conception of equilibria is a hard problem with its hardness ignored.** It takes competency of existing as a given, and claims a true-and-false conserving accounting. **An equilibrium is a not-changing fixed still, a form that cannot parity-change, not possibly existing; stable-forming, and unstable deforming with non-living scale, are observably existing.** Each proposed conception of equilibria fixes one of them unchangeable. **An equilibrium is a living or non-living existing thing fixed unchangeable.** 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 parity-change, carrying nothing, their forms continuing through their parity-changing. Each existing thing is living with one prime scale and non-living with each other scale, 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 carrying continues 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 parity-changing and an exchanging are among living and non-living existing things alike**: living carrying their prior on to now, non-living carrying nothing, their forms continuing through their parity-changing, and a parity crossing between any two carrying nothing. **Non-living existing things are included within discovering social moral competency among living.** A non-living thing couples with a self as other, offering its parities and carrying nothing, and each term uncovered with their coupling is owned by neither and carried on by living. An equilibrium's fixing ends, and living and non-living things with it go on parity-changing: an equilibrium ending is no existing thing ending, and a thing not living is an existing thing. A sequence carries one subject: a participation parity-changes; a fixing requires that same participation not-changing; its required continuing brings parity-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 required occurrence ends with its one failing occurrence, and occurrences before it are as they were. A whole participant parity-changes while a relation within it can continue, 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 different carrying; observing by *an observing is its pattern-matching at self*, Co-Chaining Logic Registry step 171, and *An observing is now-existing arriving: proof of existing*, step 173; a stable form beside *Each living set is a stable form, and nothing living has a stable form, bothbothing*, step 427. A mathematical conception of equilibria fixes a stability, a form or a relation unchangeable, each observed, as suggested, 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 shown not possibly existing with prior, differ.** Each proposed conception of equilibria arriving from prior is shown not possibly existing now, and each arriving next is reached with its next. **Each conception arrived is shown not possibly existing.** **Surviving is binary.** All sequencing to now together continues 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 stated relation continuing through its claim's comparisons.** A fixed 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 its added requirements are one conception.** A relation can continue while an added fixing of its whole carrying, a next nonzero offering or a later one each fail with one same receiving: a relation continuing is stable-forming, and each added requirement fixes it unchangeable. A differing receiving is a further conception. **One claim has one shape.** Continuing stable-forming carries on as natural torusing, and natural torusing carries its whole participation, 12345. A conception of equilibria naming less than a whole 12345 is not possibly existing together with continuing. **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 alternating next being I(p) and I(p) ≠ p, fixes p and I(p) with one next. One step its chain left open, that its ordering fixed 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 parity-changing between. ## 2.4 A participation missing, unchanging or changing **A still fixed as a complete continuing is parity-changing.** Its continuing is renewing, inward of its claimed whole, which is parity-changing, or outward of it, a further participant. **A conception naming a continuing and omitting participation that continuing carries is not possibly existing.** **A state accounting and a flow accounting each omit one side.** A state accounting omits parity-changing, a flow accounting omits continuing, 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 read is a store fixed as entering now, an ingressing, and a flow written a parity-changing fixed as leaving to a store, an escaping. **Each conception of equilibria carries not-changing or carries parity-changing, exactly one of two.** A conception naming not-changing carries an existing thing with no arriving, and existing is arriving; a conception naming parity-changing fixes a parity-changing unchangeable. Coupled, each dissolves. **Each momentary completes in co-discovering.** Each conception of equilibria fixes a form unchangeable at momentarying of geodesic co-discovering, and continuing carries on as geodesic co-discovering. Each momentary completes as its next opens, whether a parity inverted, stayed, crossed or nothing crossed, and a particular carrying's completing is its own. # THREE · THE TEN, FIVE PLACES AT TWO FACES ## 3.1 One form of hard problem, and a sixth condition **Ten are one form of hard problem.** Each proposed conception of equilibria fixes a parity-changing unchangeable by one of ten ways, and can carry 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 carry both sides. **A proposed conception of equilibria 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 proposed conception of equilibria is that relation fixed 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, seated on its name now and on that name 8 up next: its stilling now and a parity-changing travelling through it next, bi-inversioning-co-recursioning up and down numbers, Natural Naming 4.15. Now: an opening fixed as a place with 1-now-existing and 9-next-co-discovering, and an arriving fixed from behind with 2-now-possibling and 10-next-co-possibling. Existing: a two-way fixed to one side with 3-now-co-observing and 11-next-co-observing, and a sequencing fixed to one beat with 4-now-co-same-existing and 12-next-co-same-existing. Same: a between fixed as a cut and a middle fixed as an end with 5-now-co-existing and 13-next-same-existing, and a completing fixed as a last with 6-now-co-possibling and 14-next-possibling. All: a parity fixed as a magnitude and a rate fixed as a value with 7-now-observing and 15-next-observing, and a carry fixed as a store with 8-now-same-existing and 16-next-discovering. With resolver each name's term fixed 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 offering 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 offering route; 4, 5 and 13 carry 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 ±, frozen 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 parity-changing that is, its offering a next prior or its prior inverted, a next existing; or 0, a between, 12's 0 with a sharing with a prior, its prior same-existing, a self still-possibling; and an equilibrium is another's still, 2.1. Living is one of two with each self's own momentary, and a crossing of co-chainings crosses by them, each self still-possibling as a between passes it and parity-changing again: spirals of 3 and 5 selves, each self with a parity and going along with 9, crossed with self 1 of each both ways with 10 to 14, come to one relation, each self's parity-offerings 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 crossed; each of ten is that pair frozen on one side, a self fixed unchangeable or parity-changing, 2.4. **With four four-cyclings each step is momentarying or parity-changing, with prior, now and next within.** Each four-cycling's parity is continuing with its two steps eight apart, momentarying, and parity-changes with its two steps of 17 less, parity-changing: it passes two names of one parity and two of its other, parity-changing 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. A self's forward recursionings, 1, 3, 5, 7, Co-Chaining Logic Registry step 118, and each one's partner nine less, 8, 6, 4, 2, an other's names, Exhibit ONE's table of one move, 1 to 4, are pairs, 1 with 8, 3 with 6, 5 with 4 and 7 with 2; rows below go with that order, each four-cycling written as with Exhibit ONE's table of twelve forms, 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 fixed as a place with 1 and 9; a carry fixed as a store with 8 and 16 - **3 / 6** - *a* 3 → 11 → 6 → 14 → 3 - *b* a two-way fixed to one side with 3 and 11; a completing fixed as a last with 6 and 14 - **5 / 4** - *a* 4 → 13 → 5 → 12 → 4 - *b* a sequencing fixed to one beat with 4 and 12; a between fixed as a cut and a middle fixed as an end with 5 and 13 - **7 / 2** - *a* 2 → 15 → 7 → 10 → 2 - *b* an arriving fixed from behind with 2 and 10; a parity fixed as a magnitude and a rate fixed as a value with 7 and 15 **Each four-cycling carries 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, each observed again as one seating, each pair its stilling now and a parity-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 matched word places a conception with none of them. ## 3.3 Four binaries at two faces, a stilling and a changing **Ten are a self's span and its 8 up.** Four binaries are 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 stilling now, and on that name 8 up, its parity-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, are observed on one seat as a stilling and a parity-changing travelling through it, one seat observed up and down numbers, `incoming/equilibria_registry_v377/executions/ten_at_the_code.py` an older derivation's execution. 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 fixed as a place, with 1 · 9; an arriving fixed from behind, with 2 · 10 - **existing** - *a* 3-now-co-observing · 11-next-co-observing; 4-now-co-same-existing · 12-next-co-same-existing - *b* a two-way fixed to one side, with 3 · 11; a sequencing fixed to one beat, with 4 · 12 - **same** - *a* 5-now-co-existing · 13-next-same-existing; 6-now-co-possibling · 14-next-possibling - *b* a between fixed as a cut and a middle fixed as an end, with 5 · 13; a completing fixed as a last, with 6 · 14 - **all** - *a* 7-now-observing · 15-next-observing; 8-now-same-existing · 16-next-discovering - *b* a parity fixed as a magnitude and a rate fixed as a value, with 7 · 15; a carry fixed as a store, with 8 · 16 **Each of ten fixes a stilling as its whole, and a parity-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 openings 1+3+5 sum to 9 = 3², and three completings 2+4+6 to 12 = 3·4: n momentaries open to n² and complete to n(n+1), and 3² − 2·4 = 1 is a bridge with three. Prior, now and next are three sequential universes, and three threes are nine. Six consecutive parity-changings from 3 complete with 8, and 9 opens: 9-next-co-discovering, each parity-offering going along to its receiving self and arriving to 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 carries 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 offered 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-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 carrying a two-way fixed to one side with 3 and 11 and a completing fixed as a last with 6 and 14. **One parity offered once co-chains through a whole.** With Exhibit ONE's resolver, selves coupled along as a spiral share each parity-offering along with 9-next-co-discovering to each next self's offerings. A spiral of selves each with no prior and one + offered once to one opens with each in turn, the + surfacing with 14-next-possibling and passed on as it arrives, 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, opened as with Exhibit ONE's table of a spiral of selves, an even spiral from that table's second momentary. A same-existing-society fixed as not-changing fixes a form unchangeable, and not-changing is not not-carrying: non-living things exist, arrive and parity-change, deforming with their own scale, and carry nothing. An equilibrium of a same-existing-society fixes a same-existing-society, its selves parity-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 parity-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 shared with 10-next-co-possibling by its self, moving one self on with each second momentary; a whole inverted 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 carries along it: its 0 shared with 10 by its like pair's self, a parity-changing that is not, moving through its shared surface one self with each second momentary.** Ten fix a whole parity-changing unchangeable: opposition fixed unchangeable is an even spiral, and a between fixed as a cut is a coupling carrying on. Rest returns with no coupling. A resting spiral, each self with no prior and nothing arriving, opens nothing and stays resting, so a spiral reaching rest does not reach its offered parity again; each spiral comes to its parities again, its offered 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 and one + offered once to one, 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 fixed 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 fixed 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, carry an arriving and a completing; entry 1 and 9 an opening; 3, 4, 11 and 12 a two-way and a sequencing; 5 and 13 a between and a middle; 7 and 15 a parity and a rate; and 8 and 16 a carry; 17-next-existing, a next 1-now-existing, carries none of them. # FOUR · CONCEPTIONS OF EQUILIBRIA, EACH A HARD PROBLEM IN ITS OWN WORDS ## 4.1 Each conception whole, at its declared subject **Each conception arrives whole, as its own words, with its conserving relation and its reach.** A fixed value, a retained relation, a distribution and a symmetry family are different subjects even when each carries equilibrium as its word, and one conception can reach more than one of ten. **A definition's expression continuing not-changing is an existing statement, and its subject continues through its own parity-changing.** Conditioning, replacement and maintained supplying belong to several conceptions: a law, an occupant, an original participant and a composition are different declared subjects. A law, a frame or a scale fixed unchangeable is such an expression, and it fixes its subject unchangeable through one further step, a step from a fixing to an equilibrium: one same subject, with one same relation and one same occurrence, required fixed with an occurrence by which its own continuing parity-changes it. ## 4.2 One form, and three differences within each of ten **Each proposed conception fixes a parity-changing unchangeable, by one of ten ways.** Its parity-changing is stable-forming, unstable deforming with non-living scale, or a stability, a form or a relation, each observed as a momentary stable form, 2.2. **Specified accounts and prior worked definitions, SA01 to SA21, are placed that same way.** **Each of ten gathers its arrivals, and within each they part by three differences.** Sixty-three arrivals, forty-two proposed conceptions and twenty-one specified accounts and prior definitions, are one form of 3.1, each an unchangeable form, 2.1, 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 reached is carried with each arrival's own words, with its own arriving. Within each of ten, three differences gather them, and each arrival is carried whole as its own words with its place. 1. **What is fixed unchangeable.** With 2.2, three, a stability, a form and a relation, and a stability explained: a response to a disturbance, its departures staying small or shrinking. A form and a relation are explained here: a form, one value, range, state, law, composition, family or extremum of a whole fixed unchangeable; a relation, a comparison between parts fixed unchangeable, equal, balanced, opposed, agreeing, complementary or equivalent. Each arrival is placed by its own words with these three, one or more together, each checked fresh; each row's own column, *the changing it carries*, is carried 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 parity-changing, as each field's own words name it, from each row's own column if its column carries a subject, and otherwise its field's own mathematics. 3. **What is added.** With each pair whose own words tie them to one setting, by *the same*, *that same*, *that*, *those* or a setting stated with identical words, each is shown as one of three: one conception, one receiving with a requirement added, specified or unspecified, 2.3, *a conception and its added requirements are one conception*; a further conception, its receiving differing, 2.3, *a differing receiving is a further conception*; or, with a pair reaching neither, another claim with one setting, here. **Fields gather among ten.** A field's arrivals fall with those of ten by which their field fixes its parity-changing unchangeable, and fields fixing one unchangeable fall together under one subject: 4.13 carries each subject, with its fields, among ten. ## 4.3 An arriving fixed from behind, 2 arrivals **A form fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 fixed as a place, 5 arrivals **A form fixed unchangeable**, 2. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 with position x* specified - **SA02** · The complete term is 0 or 1, and its specified successor is itself: equilibrium requires the term to equal its successor, with no changing, balancing of opposed changes or further progressing required. - *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 fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, symmetric Nash and evolutionary stability added **A relation fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 fixed as a last, 8 arrivals **A form fixed unchangeable**, 5. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, a different minimum - **NY42** · Entropy maximum under heat redistribution with κ>0, fixed U and positive constant heat capacities: u=C_A U/(C_A+C_B). - *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 carrying 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 fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, its stated two-body contact, uniqueness, persisting and approach added **A form and a relation fixed unchangeable**, 2. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, positive added, N=K ## 4.6 A carry fixed as a store, 10 arrivals **A form fixed unchangeable**, 5. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, emission and absorption added to its scattering - **NY28** · Constant internal X composition in the maintained chain F ⇌ X ⇌ W: x=(f+w)/2 with f,w>0 fixed by the stated receiving. - *a* chemistry - *b* reactions - *c* reactions - *d* — **A form and a relation fixed unchangeable**, 4. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 carrying 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 carrying 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 carrying of resolver - *c* coupling of self and other, and resolver - *d* — **A relation fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 carrying of resolver - *c* coupling of self and other, and resolver - *d* — ## 4.7 A middle fixed as an end, 6 arrivals **A form fixed unchangeable**, 4. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, stability added with NY10; NY12 a further conception, a different 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, a different 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, a different minimum - **SA11** · A local Maxwellian cancelling its elastic collision term, kept under the full operation with spatial transport, at constant density, zero mean velocity and a nonzero temperature gradient. - *a* kinetic theory of gases - *b* heat - *c* heat - *d* — **A form and a stability fixed unchangeable**, 2. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, asymptotic stability added; NY11 a further conception, a different 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, a different T, stability in those neighbourhoods ## 4.8 A parity fixed as a magnitude, 9 arrivals **A form fixed unchangeable**, 8. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 with u unspecified, NY24 with u=0 specified - **NY22** · Relative equilibrium under some translation velocity, with no particular u specified, on that same force-free position–velocity domain. - *a* mechanics - *b* rest and balance of bodies - *c* bodies at rest and in balance - *d* one conception with NY21, u unspecified - **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, u=0 specified; 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, positive added with NY37 **A relation fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 fixed to one beat, 5 arrivals **A form and a relation fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, a different next price **A relation fixed unchangeable**, 4. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 fixed as a value, 5 arrivals **A form and a relation fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, emission and absorption added to number-conserving scattering **A relation fixed unchangeable**, 4. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 fixed to one side, 7 arrivals **A form fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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, symmetric Nash and evolutionary stability added with NY39 **A relation fixed unchangeable**, 6. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 carrying 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 carrying 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 carrying 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 fixed as a cut, 6 arrivals **A form fixed unchangeable**, 1. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 fixed unchangeable**, 5. Each entry: **Arrival, its own words** · *a* Field · *b* The changing it carries · *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 carrying 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 carrying 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 parity-changing unchangeable, and a field is aimed at from each of them.** With each of them a field sets a form fixed still beside its parity-changing and sets aside its parity-changing's own competency, carried along with a between: with that one of ten a field's subject is fixed unchangeable, and its competency's source is set aside. This table carries each subject, with fields gathered under it, with each of ten, each arrival with its one of ten. Each entry: **Subject, and its fields** · *a* 1 arriving · *b* 2 opening · *c* 3 completing · *d* 4 carry · *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* — **Observed across a row, a subject shows those of ten by which its fields fix a parity-changing unchangeable; observed down a column, subjects fixing one unchangeable gather.** Heat falls with six of ten, four arrivals each with a completing fixed as a last and with a between fixed as a cut; reactions with a carry fixed as a store and with a rate fixed as a value; bodies at rest and in balance with an opening fixed as a place and with a parity fixed as a magnitude; selves choosing and exchanging with a sequencing fixed to one beat and with a two-way fixed to one side; populations with four of ten; coupling of self and other, and resolver, with seven. Each row fills as arrivals come, each placed by its own words. ## 4.14 Leads arriving without their statements **Leads arrive without their statements:** statistical null hypotheses, static cosmology, horizon and observer temperatures, early-universe thermal and gravitational descriptions, a conservation lead and its transformation arriving next, dictionary and etymological leads, a collection of all sets, philosophical accounts, and measuring, locating and separating, Natural Intelligence Corus's three consecutive equilibria, a thing, a relation and a system at rest. Each is placed with ten as its statement arrives. A boundary declared unchanging is a bounding fixed unchangeable, and bounding is living parity-changing: each declaration, within, around, across, along, of an entrance or of an exit, meets one question, whether co-sequential resolving continues under it. Scientific method cannot explain observings of parity-changing and not-changing from prior-existing to next-existing things, and is placed by its fixings: a law, a frame or a scale fixed unchangeable between a signal's leaving and its arriving, each placed with ten through its own stated requirement; with a hypothesis it explains a stilling and a parity-changing without considering the rest of the universe. A field's own methods part two momentaries: NASA/JPL's aberration correction separates an epoch a signal leaves its source from an epoch it is received, and accounts for its source's motion between them. Evidence arriving 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 parity-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 carry 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* One same comparison is fixed to satisfy requirements not reachable together. - *b* Fixed state–flow–state and flow–state–flow under their required advance; included membership required both not-changing and inverted; a balanced fuel cycle with `f≠w`. - *c* Each conjunction as stated, with one same occurrence. - **B · Required case omitted** - *a* A required succession 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 excluded** - *a* Presence carries renewal, and a conception excludes that renewal. - *b* A carrying fixed unchangeable with each momentary; each surface zero with a carried sharing. - *c* Each carrying, its offerings and its momentaries. - **D · Required arrival or offering unreachable** - *a* A required target is no thing a stated operation can supply. - *b* Finite zero departure from a nonzero predecessor under a nonzero multiplier; an older resolver's case of its opposed aggregate route is carried by `archive/carrying_v377/Exhibit_TWENTY-EIGHT_Equilibria_Registry_older_resolver.md`. - *c* Each stated predecessor, operation, route and reach. **A proof carries to a conception whose complete requirement supplies its premises.** For each x in a conception's stated set S, its required next is in S, or that conception is left within its succession. Renewal excluded, group C, with Exhibit ONE's resolver. A sharing once with a prior has a prior with each momentary and is not none again. Fixed unchangeable, it requires with each momentary an offering of its own parity: offered nothing, other parity, or offerings parting to 0 with 14-next-possibling, parity-changing with 12-next-co-same-existing is, and its prior inverts, a living step. Executed with v377: with prior + and offered + for five momentaries, it shares 0 with each by 10-next-co-possibling and stays +; then offered nothing, it shares −, +, −, +, − and its prior inverts with each; a sharing of its prior whose offerings part to 0 with each momentary inverts with each, `incoming/equilibria_registry_v377/executions/eq_code.py`, parts 3 and 6. Fixed unchangeable, its prior is a prior offered its own parity with each momentary by a form fixed still from beside: its still description and a self's own continuing do not both continue. A form fixed still from beside one self of a same-existing-society stills that self's parity-changing with each momentary, and each self with no prior coupled along from it is offered none and opens nothing: one still with one self stops co-sequencing through each self with no prior coupled from it. That is hard probleming, and, that still dissolved, resolving follows with a next momentary: a self inverts its own prior, its still's offered parity, and its still offers nothing further. One to nine is four momentaries of exchanging, Exhibit ONE's tables of eight 1 to 17s inward and of a self's four momentaries of exchanging. An unchangeable relation can permit a parity-changing an unchangeable parity excludes: opposition under joint reversal, t = −c, continues while both parities invert. Renewal exclusion reaches a conception through a renewal its own claimed continuing requires and its own requirement excludes. ## 5.2 One exclusion at each of ten **One binary retaining and inverting its value with one required next is not possibly existing.** A conception naming p(next)=p(now) with its continuing carrying 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 fixed 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 reached again with a next momentary is a further occurrence. Included membership required both not-changing and inverted 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 fixed boundary comparison fails as its continuation advances exactly one end. An equation's relation is reflexive, symmetric and transitive, and a conception written as an equation carries all three ahead of its conditions: immediate-next carries none of them. **A four-state orbit leaves each proper subset.** F(P,Q)=(−Q,P) visits all four pairs from each pair it opens 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. Each nonempty proper subset is left within three advances or fewer, {++,−+,−−} from ++ with its third, and subsets F carries to themselves are exactly unions of its one cycle: none or all four. With a direction beside each pair, co or bi, alternating, are eight states: a side opening with 1, co, reaches four of them within three advances, and a side opening with 2, bi, an other four, 1.5. A condition lives through a momentarying exactly when it admits each state that momentarying reaches: of 256 conditions, 16 live through each side, one of them through both, admitting all eight and adding nothing, and 225 through neither. One condition each step carries is a circle's identity, cos² θ + sin² θ = 1: it excludes no parity-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 carry their complete requirements.** Each entry: **Required condition** · *a* Its necessary continuation - **A prior fixed unchangeable** - *a* Offered its own parity with each momentary it stays, its offering from a form fixed still from beside; offered nothing, other parity or a parting, it inverts; once with a prior a sharing is not none. - **Each surface zero with a carried sharing** - *a* Chained inverted with each momentary, as with nothing offered; with resolver a surface 0 arrives only from offerings parting, Exhibit ONE's table of offerings to one sharing, surfacing. - **Full positive balance of a unit-ratio fuel-coupled cycle, with `f≠w` maintained** · *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; an already supplied zero is a different condition. ## 5.4 Sharing at 10 and 6, and carrying 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 offered nothing has its prior inverted with each momentary, and its sharing by 10-next-co-possibling and its next prior by 11-next-co-observing are one parity-changing, part 11. Agreement and opposition alternate through overlapping momentaries. Along c, t, −c, −t, c overlapping pairs (c,t), (t,−c), (−c,−t) and (−t,c) carry agreeing, opposing, agreeing, opposing from an agreeing start, and the reverse from an opposed one: each side's relation stays through its two momentaries while both its parities invert, and its overlapping side carries an other relation. Two neighbouring pairs (a,b) and (b,z), each with a parity, + or −, carry 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 overlapping pair opposes, and its own next inverts both parities. **6 and 10 share one parity-offering**, 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 parity, a parity-offering's + or − and agreement of two are three comparisons. **A prior and its offerings 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 offered its own stays and shares 0; offered an other, a parting or nothing, it inverts; a sharing with no prior opens with its offered parity, or, offered 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 continues; and each surface zero reaches renewal exclusion. With SA01 opposition continuing is alike for self + with other − and for self − with other +, and gives neither as next: a complete next pair is given by a prior pair with both reversed, more than a relation by itself, and SA01's own requirement is reached by nothing it carries. --- 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 computed by a script it cites 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`.