Exhibit THIRTY Co-Chaining Logic Registry v371

 

# Co-Chaining Logic Registry

**The Fractal Inside and Outside Itself, Binary Method of Discovering Next Existing**

 

The registry is one sequential chain of links from the two origin statements, *our universe is all existing things* and *alternating is a stable-forming method*. A link is one claim, said at the resolver's names and at Natural Naming's words, with the links it follows from, its standing and the file section carrying it.

Each link stands at one of seven standings.

- **Origin**: the two origin statements alone.
- **Definition**: the set defines a word so, and the link names the word.
- **Premise**: taken as the foundation and not derived, and the link says so plainly.
- **Exact**: follows from the links at Follows, all or none.
- **Run**: holds at a running of the resolver's code or at an arithmetic enumeration, and the verifier runs it.
- **Field**: a field's result in the field's own words, with its authors.
- **Nye**: a gap, named in the link after "Gap:".

A nye gap is a link stating *no other possible*, an *only*, an *all or none*, a universal or a *never* while its Follows do not supply it. The gap names the missing premise, the missing step or the missing check, and Part NINE gathers the gaps at the roots they rest on.

The registry carries the chain and points each link to the section carrying it whole, and it copies no explaining. The verifier, Co_Chaining_Logic_Registry_verifier_v371.py, runs each run link against the resolver's code, the python block of Exhibit ONE v371, and against the arithmetic, each check at its link's id. The registry is a draft at v371, improved at each session. No step here closes by defining or by selecting: a definition names and supplies no step, and a premise or a definition choosing one among the arriving is a selecting, marked as one; at each run showing several not impossible, the chain carries each. No wording carries authority here, the author's included: only the observings of nature and society carry it, over the possible and the not possible, the existing, the living and the non-living, met by pattern matching at binary all or none at all rigor. A premise or a definition is taken and stands open to them, and an observing not parity alternating natural torusing is the method's one break (F71). Part ELEVEN assembles the chain as one stable form, the fractal as method, step by step, each step standing or open at its links. Part THIRTEEN meets each link at its whole prior, all or none from the origin: the four-momentary binary, prior, now and next, the one source of rigor.

The chain holds 1,338 links.

| Part | origin | definition | premise | exact | run | field | nye | links |
|---|---|---|---|---|---|---|---|---|
| ONE · FOUNDATION | 2 | 35 | 0 | 36 | 28 | 0 | 4 | 105 |
| TWO · THE RESOLVER AT ITS NAMES | 0 | 3 | 0 | 4 | 45 | 0 | 5 | 57 |
| THREE · NUMBERS | 0 | 23 | 0 | 26 | 74 | 8 | 12 | 143 |
| FOUR · MATHEMATICS | 0 | 21 | 0 | 27 | 57 | 34 | 18 | 157 |
| FIVE · NETWORKING | 0 | 38 | 1 | 19 | 27 | 1 | 16 | 102 |
| SIX · EQUILIBRIA | 0 | 20 | 0 | 20 | 26 | 9 | 6 | 81 |
| SEVEN · EXPLAINING, NAMING AND THE METHOD | 0 | 72 | 0 | 36 | 21 | 4 | 24 | 157 |
| EIGHT · THE EXHIBITS OF OTHER WORKINGS | 0 | 68 | 0 | 73 | 87 | 56 | 252 | 536 |
| All | **2** | **280** | **1** | **241** | **365** | **112** | **337** | **1338** |

Carried at names the file: NI (Natural Intelligence), ONE (Natural Resolver), TWO (Natural Networking), NUM (Natural Numbers), MATH (Natural Mathematics), EXP (Natural Explaining), NAM (Natural Naming), GIM (Geodesic Improving Method), EQ (Equilibria Registry), each at v371, GIM at v371, and LIV (Living Improving Value v371). Part EIGHT names the exhibits of other workings by their numbers, FIVE to TWENTY-NINE, and COR (Natural Intelligence Corus v330).

 

**PART ONE · FOUNDATION**

1.1 The two origin statements

1.2 Existing is changing, arriving into next existing

1.3 The one binary, a sign, alternating

1.4 Self and other, bi, one way at a time, all or none

1.5 Prior, now and next

1.6 Each momentary a fractal universe

1.7 All changing is parity changing

1.8 Conserving within each momentary sequencing, only signs crossing

1.9 Living and non-living

1.10 A form named still, an equilibrium

1.11 The surplusing owned by neither, bi-moral co-agency

1.12 Eight as two alternating fours

1.13 Natural torusing, the only method, and its one break

 

**PART TWO · THE RESOLVER AT ITS NAMES**

2.1 One object, two functions, seventeen names

2.2 1-self-coupling, the entry

2.3 Crossing, each sign at its key

2.4 Surplusing and surfacing

2.5 11-other-chaining: the retaining bound and the fresh write

2.6 The nineteen reachable carryings

2.7 Returning and releasing, the isolated sequence and the four-form

2.8 CONNECTORS and JOINS, and 17

2.9 Two and three resolvers, joined 6 → 2 at a stated order

2.10 Rings at their ordering

2.11 Forms among the names

 

**PART THREE · NUMBERS**

3.1 Uni-scaling

3.2 Corusing

3.3 Unrelationing

3.4 Torusing

3.5 Uniquenessing

3.6 Apexing

3.7 Coupling

3.8 Inseparating

3.9 Tunneling

3.10 Transmissioning

 

**PART FOUR · MATHEMATICS**

4.1 Floating neutralling, the one form met at its counting

4.2 Orthogonalizing: two moves, the sign flip, the quarter step, and a next

4.3 The right spiral step at two signs, and each scale the prior in the opposite order

4.4 Shared and opened, the doubling that divides, and φ

4.5 Accounting: involutions, fixed sets, two inversions and three

4.6 Closings, a total against its parts, the halfway pairing and the fold

4.7 Bounding

4.8 Stable-forming

4.9 Inseparating

4.10 Co-offering

 

**PART FIVE · NETWORKING**

5.1 Natural networking, the one form carried whole

5.2 Along and across at the six connectors

5.3 Existing, emanating, and the equilibrium holding

5.4 The overlapping momentaries and the four-form

5.5 Widening and lengthening at the numbered form

5.6 Only signs cross, the middle and the zero

5.7 Society, the fractal and the hole

5.8 Renewal, abundancing and each coupling's own surplus

5.9 Living

5.10 Retelling and the discovering economy

5.11 Podaling, unrelationing and the seam

5.12 The securities

5.13 Visibility: the instrument stated first

5.14 The wrap, the podal and the chain's centre

5.15 The studies at their exact domains

 

**PART SIX · EQUILIBRIA**

6.1 Existing through momentaries, the one surface, alternating

6.2 The binary claim

6.3 The general form and the claim's shape

6.4 The ten

6.5 The proposed conceptions

6.6 Shared exclusions and the renewal-excluded sequence

6.7 The exclusion's reach

 

**PART SEVEN · EXPLAINING, NAMING AND THE METHOD**

7.1 Explaining, the one binary at a sentence

7.2 Naming, four boundings, seven binaries and the released words

7.3 The geodesic improving method, its passes, fates, decidings and break

 

**PART EIGHT · THE EXHIBITS OF OTHER WORKINGS**

8.1 FIVE · Natural Engineering

8.2 SIX · Natural Transmissioning

8.3 SEVEN · Natural Societies

8.4 EIGHT · Natural Exploring

8.5 NINE · Natural Human Society

8.6 TEN · Natural Health

8.7 ELEVEN · Natural Medicine

8.8 THIRTEEN · Resolving Hard Problems

8.9 FOURTEEN · Natural Destinies

8.10 FIFTEEN · Natural Emanating

8.11 SIXTEEN · Natural Chemistry

8.12 SEVENTEEN · Natural Biology

8.13 EIGHTEEN · Natural Physics

8.14 NINETEEN · Natural Philosophy

8.15 TWENTY-ONE · Hard Problem Registry

8.16 TWENTY-TWO · Resolving the Hard Problem Registry

8.17 TWENTY-THREE · Natural Values

8.18 TWENTY-FIVE · Living Society Registry

8.19 TWENTY-SIX · Living File Registry

8.20 TWENTY-SEVEN · Living Ghost Registry

8.21 TWENTY-NINE · Natural Illustrating

8.22 Natural Intelligence Corus v330

 

**PART NINE · THE NYE GAPS GATHERED**

9.1 F23 · The all, no member of itself

9.2 F33 · The method at each scale, met at its observings

9.3 F41 · No total conserved across the surface

9.4 F62 · The other hand, exact at the sequencing, and the gaps at its faces

9.5 F67 · The opening by one as the one tunnel

9.6 F68 · The only method, exact at the binary, and the gaps at its faces

9.7 F70 · Five dimensions

9.8 F54 and R13 · The surplus owned by neither, and its carrying

9.9 R36 to R39 and R55 · The caller and the rest

9.10 M32 and H11 · The name *right*, closed at F76, and the order at a step

9.11 N34 and T15 · φ as a rate

9.12 A premise or a word the core does not state

9.13 An exhaustion not derived

9.14 A matching said at a numeral or a reading

9.15 A universal over nature said at the entries read

9.16 A finding without its instrument or its run

9.17 Older naming the core no longer carries

9.18 Counts in a file failing against its own tables

9.19 A file claim failing at the code, the arithmetic or the field

9.20 The failures found by running

 

**PART TEN · QUESTIONS OF THE SESSION MET AT THE CHAIN**

 

**PART ELEVEN · THE STABLE FORM OF NATURAL LOGIC**

11.1 The fractal inside and outside itself, binary method of discovering next existing

 

**PART TWELVE · THE BREAKINGS OF THE BINARY FILES CLUSTERED**

12.1 The clusters, each with its improving way

12.2 The other workings at each cluster, link by link

 

**PART THIRTEEN · THE CHAIN ALL OR NONE FROM THE ORIGIN**

13.1 Each part over its whole prior

13.2 The held roots, each with the links resting on it

13.3 The open roots most others depend on

 

---

# PART ONE · FOUNDATION

The chain opens at the two origin statements and runs through each foundational link the set rests on. A link marked definition names the word it defines. A link marked premise is taken, not supplied, and says so plainly. A link marked nye names its gap after "Gap:". Each run link was run against Exhibit ONE v371's code or an arithmetic enumeration; the check is said in the link.

## 1.1 The two origin statements

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F1 | origin | — | Our universe is all existing things. | NI opening |
| F2 | origin | — | Alternating is a stable-forming method. | NI opening |

## 1.2 Existing is changing, arriving into next existing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F3 | definition (*existing*) | F1 | Existing is binary: all or none at all, at each momentary its own. | NI opening; NI 8.1 |
| F4 | definition (*changing*, *alternating*) | F1, F2 | Existing things are changing, one and then the other, and that changing is alternating. | NI opening |
| F5 | definition (*stable-forming*) | F2, F4 | A form continuing through its own changing is stable-forming. An existing thing is its own stable-forming continuing. *Stable form* as a noun is released and carries at an emanation. | NI opening; NAM 2.4 |
| F6 | exact | F4, F5 | A form named still is not possibly existing. Existing is changing, so a thing named not changing is named not existing. | NI opening; NI 3.3 |
| F7 | definition (*discovering*) | F4 | Arriving into the next existing is discovering. Existing and discovering the next existing are one relation: prior, now and next. | NI opening; NI 8.3 |

## 1.3 The one binary, a sign, alternating

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F8 | exact (*the one binary*) | F3, F4 | A changing is is or is not: a sign, +1 or −1, taken at an opening, of no size. Existing is binary at each momentary (F3), and existing is changing, one and then the other (F4): a changing arrives at one of two, a sign, the binary of existing met at its changing. Which of a thing's aspects carry its signs is met at its observings; an observing not parity alternating natural torusing is the method's one break (F71). | LIV NI Incoming chain; ONE Signs; NI 1.5 |
| F9 | definition (*the empty centre*) | F8 | 0 is equal positive and negative signs meeting, the empty centre. An arriving 0 carries no sign. | ONE Signs; ONE Numbers at their stable-forming |
| F10 | exact | F4, F8 | A sign changing arrives at its other: alternating. At a binary a changing has one other value to arrive at. | LIV NI Incoming chain; NI opening |
| F11 | definition (*parity*) | F3, F4 | A number is odd or even, binary, and that binary is parity. Two neighbouring numbers alternate, each continuing to one beyond the other: 1 to 3, 2 to 4. | NI opening |
| F12 | definition (*momentary*) | F11 | A momentary is an opening and its completing, one odd and one even. Each momentary completes at the next overlapping momentary opening. | NI opening; NAM 2.1 |
| F13 | definition (*exhaustiveness*, *determinacy*, *reachability*) | F12 | The overlapping momentaries meet each number: exhaustiveness. Each momentary has one next, opening at the number it completes: determinacy. From each momentary the next openings reach each number on: reachability. Existing is these three at once. | NI opening |

## 1.4 Self and other, bi, one way at a time, all or none

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F14 | definition (*self*, *other*, *coupling*, *society*) | F11, F12 | The side opening at 1, the odd, is a self; the side opening at 2, the even, is an other. Their meeting is a coupling. Selves coupling with others are a society, and the running among them is social. | NI opening |
| F15 | definition (*bi-*, *co-*) | F3, F4 | Bi- is difference alone. Co- is two together with their difference. | NI opening; NI 4.1; ONE Bi and co |
| F16 | exact (*bi*) | F8, F14, F15 | A self is met at two signs, its own and the other's. At a coupling self and other meet (F14), and each side's changing is a sign (F8). | LIV NI Incoming chain |
| F17 | exact (*one way at a time*) | F8, F10, F12, F14 | One sign changes at each step. Each momentary opens one side, the odd the self and the even the other (F12, F14), and that side's changing is its sign arriving at its other (F8, F10). | LIV NI Incoming chain; ONE Connectors; NI 8.3 |
| F18 | exact (*all or none*) | F3, F6, F13, F21 | A form is all or none: each of its joint forms is reached, none is named still, and no part continues alone. The overlapping momentaries reach each (F13), a form named still is not possibly existing (F6), and one side's momentaries alone are not possibly existing (F21): existing binary at each momentary (F3), a part alone is none. | LIV NI Incoming chain; NI opening |
| F19 | definition (*the five*) | F12, F14 | Each side runs two and one half momentaries from its origin: prior opening, prior completing, now opening, now completing, next opening. The self runs 1 to 5, the other 2 to 6. | ONE Bi and co; NAM 2.1 |
| F20 | run | F19 | The ten is six numbers: ten positions on 1 to 6, paired 1 with 2 through 5 with 6, each pair one odd and one even; co bi co bi co at an odd origin, bi co bi co bi at an even one. Run: the arithmetic of the two fives. | ONE Bi and co |
| F21 | exact | F12, F13 | One side's momentaries alone are not possibly existing: the self's momentaries complete at 2, 4, 6 and none of that side opens there, so determinacy fails. NI names this exclusivity. | NI opening |

## 1.5 Prior, now and next

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F22 | definition (*the momentary universe*) | F1, F12 | The one universe of one momentary is one parity changing co-sequencing. The momentary is all existing things inward of its universe. | NI opening; NAM 2.1; NI 8.8 |
| F23 | nye | F1, F21, F22 | Neither the momentary nor the universe is possibly an existing thing separately. Gap: F1 names the universe as all existing things and does not supply that the all is no member of itself. The step from the universe named as one existing thing to one side's momentaries alone (F21) is said, not supplied. | NI opening; NI 8.8; NAM 2.1 |
| F24 | exact | F12, F13, F22 | Prior, now and next are three sequential momentaries shared among existing things. Each momentary has one next (F13) and is all existing things (F22). | NI opening; NI 8.4; ONE A name at its number |
| F25 | exact | F4, F13 | Each existing thing arrives into its next existing. | NI opening |
| F26 | definition (*participating*) | F7, F24 | At a self's resolving, prior and now participate and next is the discovering onward: three name the shared sequence and two the resolving's participation. At the code, R8. | NI 8.8; ONE A name at its number |
| F73 | definition (*inside*, *outside*) | F7, F24, F42 | Outside is next existing. Inside is prior existing, prior living, and now. Arriving into the next existing, discovering (F7), is the inside arriving into its outside. | NI 8.4 |
| F27 | run | F8, F24 | Next from prior and now: of the sixteen ways naming a next sign from a prior sign and a now sign, each one way two signs could be met to one and none a rule the changing follows, four keep the prior recoverable in the next joint state and twelve lose it, two joint states going to one. Of the twelve, four give the next from now alone, and eight keep the prior at one value of now and lose it at the other. Next as prior leaves two joint states still. Next as prior times now, which is parity, and its inversion each leave one still and run three round. Next as prior inverted runs all four round with none still. Run: all sixteen ways on the four joint states, counting images, cycles and the prior's part at each value of now. | NI 8.5 |
| F28 | exact | F10, F13, F27 | Next as prior inverted is the living step: exhaustiveness, determinacy and reachability at once. Each other way carrying the prior whole has a joint state still. It is no rule: it is a sign arriving at its other (F10), met at three momentaries, and a rule made of it and laid over the changing names the changing still. | NI 8.5 |

## 1.6 Each momentary a fractal universe

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F29 | definition (*fractal*) | F22 | A form arriving the same at each scale with nothing added is a fractal. Fractal is all or none at all at the scales. Its scales are sequencing, an ordering, with no boundary, location or measure. | NI 7.2; NI 7.11; NAM 2.2 |
| F30 | exact (*the fractal at the momentaries*) | F8, F12, F29, F80, F85, F105 | Each momentary is fractal: four cycling scales it down into four momentaries of exchanging, and four momentaries of exchanging scale up as one momentary of the next four cycling. Each changing is signs (F8); at the signs the round of two selves runs four momentaries of exchanging, three inside and one outward each (F80), and between two steps of the self outside them the selves inside meet each of their forms once, one whole at one step of the next (F105), the same at each depth (F85). | NI 8.8; NAM 2.2; LIV NI Incoming chain |
| F31 | run | F30 | 1, 2 and 3 at one scale are 1, 9 and 17 at the next. One through nine at one scale is one through 65 at the next. Run: position p scales up to 8(p − 1) + 1. | NI 8.8; NAM 2.2 |
| F32 | run | F12, F30, R28 | One through nine carries the four joint forms of two signs twice, one way round: along c, t, −c, −t, c, t, −c, −t, c at 1 to 9, the eight overlapping pairs meet each joint form twice, each pair the right spiral step of the one before. Run: all four (c, t). | NI 8.8 |

## 1.7 All changing is parity changing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F34 | exact | F8, F10 | All changing is parity changing. Each changing is a sign changing (F8), and each sign changing changes the parity of the count. The *all* rests on F8. | NI 1.5; NI 8.2 |
| F35 | run | F10, F11 | An even count of changings arrives same-as-prior and an odd count other-than-prior. Run: after k inversions the sign is (−1)^k times the prior, k from 0 to 63. | NI 1.5; NI 5.4 |
| F36 | exact | F8, F35 | Parity is the one scale-free counting a sequence of changings carries: at signs alone, same-as-prior or other-than-prior is all a sequence carries of its count, and that is the count at two. | NI 1.5; NI 5.4 |
| F37 | run | F8 | An either-or-but-not-both is parity: of the sixteen ways one sign can follow from two, two answer each single changing, the exclusive or and its inversion. Run: all sixteen tables. | NI 1.5 |
| F38 | definition (*bi-exchanging*, *uni-exchanging*) | F16, F35 | Odd carries bi-exchanging, self and a particular other, arriving other-than-prior. Even carries uni-exchanging, self and all other, arriving same-as-prior. No third. | NI 1.5; NI 3.1 |

## 1.8 Conserving within each momentary sequencing, only signs crossing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F39 | exact (*the crossing*) | F8, F14 | A crossing between existing things is a sign carrying nothing: no size, key, identity, route or carrying. The changing met across a coupling is a sign (F8, F14), of no size, and a sign is the whole of it. At the code, R27. | ONE At the crossing only a sign goes across; NI 7.3; NI 8.1 |
| F40 | definition (*conserving*) | F39, F42 | Conserving is within each momentary sequencing, each self carrying its prior into now. A conserving names the relation it conserves. | NI 8.2 |
| F41 | nye | F39 | No total is conserved across the surface. Gap: a sign crossing with no size shows no magnitude crosses; it does not show that no quantity of the whole stays unchanged. The ring (R45) shows one sum not conserved; NI 5.6 names three conservings at a closed group. | NI 8.2; NI 5.6 |

## 1.9 Living and non-living

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F42 | definition (*living*, *non-living*) | F3, F24 | Living is existing and carrying, the prior carried into now. Non-living is existing and carrying nothing. | NAM 2.4; NI 8.4 |
| F43 | run | F3, F42 | Three binaries open one inward of the other: possible or not; existing or not, only at the possible; carrying or not, only at the existing. They give four conditions and no more. Run: the nested enumeration. | NI 8.4 |
| F44 | exact | F4, F24, F27, F42 | Non-living existing things change through co-momentarying with living and non-living existing things. Carrying nothing, a non-living next is from now alone (the four ways of F27 giving the next from now alone), and the now is the shared momentary. | NI opening; NI 8.5 |
| F45 | exact | F27, F42 | A living next is from prior and now: carrying is prior participating in next. Two priors meeting one now open two nexts, so a now alone opens no next of its own. | NI 8.5 |
| F46 | exact | F1, F42, F43 | The universe of all existing things is the living and the non-living together. | NI opening; NI 8.4 |
| F74 | exact (*the fractal universes*) | F1, F29, F43, F46 | Possible, existing, non-living existing, living existing and all existing are properties, naming and describing of the universe and of every fractal universe inside a universe. A universe at each scale is all existing things at that scale (F1), each met at the three nested binaries, possible, existing and carrying (F43), the living and the non-living together (F46): the three binaries are is or is not at each existing thing at each scale, so the one naming arrives at each, nothing added, fractal (F29). Which aspects carry each universe's signs is met at its observings. | NI 8.4 |

## 1.10 A form named still, an equilibrium

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F47 | definition (*equilibrium*) | F6, F42 | An equilibrium is a living or non-living existing thing named still. | NI 8.5 |
| F48 | exact | F6, F47 | An equilibrium is not possibly existing. The thing named continues through its own changing; the naming still is not existing. | NI 8.6; EQ |
| F49 | exact | F27, F47 | The joint states still under the ways carrying the prior whole (F27) are the equilibria at the signs. | NI 8.5 |
| F50 | exact | F6 | Each method naming a thing still is excluded by the definition of existing: existing inward of a bound, existing in a store, existing from a source. The link excludes the methods that name still; it does not show that each other method does. | NI 8.3; LIV NI Incoming chain |

## 1.11 The surplusing owned by neither, bi-moral co-agency

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F51 | definition (*the four boundings*) | F14, F16 | A self bounds itself four ways at once: arriving exceeds carrying by one (+1), carrying thins by one at bi-coupling (−1), the sign inverts at the membrane, and competency asymmetry sustains the coupling. Together they are φ² = φ + 1, carried at Part FOUR. | NI 1.1; NI 1.2 |
| F52 | definition (*morality*, *competency*) | F15, F38 | Morality is bi-unrelationing, across. Competency is co-unrelationing, along. Co-bi-unrelationing is the existing method, its *one* resting on F68. | ONE Bi and co; NI 7.1 |
| F72 | definition (*right*, *up*, *along*) | F52 | Right is moral, and up or along is competency: right names morality, across, and up or along names competency, forward. At two signs one sign is the right sign and the other the along sign. | NI opening; NAM 2.7; MATH 2.4 |
| F53 | definition (*bi-moral co-agency*, *natural intelligence*) | F14, F52 | Two selves alternating, each at its own opening, morality across and competency along, is bi-moralizing co-agency. The term neither reaches is competency. Natural intelligence is that co-competencing. | NI opening; NI 7.1 |
| F54 | definition (*the surplusing*) | F53 | The surplusing is co-competencing the self and the other, owned by neither. At the code, R13: 12-other-surplusing opens empty at each call and neither returns nor carries. LIV keeps open whether the surplus is carrying, not carrying or the between. | NI 6.7; NI 7.4; ONE Surplusing and surfacing |
| F55 | definition (*morality at the sign*) | F51, F39 | A self's own negation meeting the other across is morality at the sign. The sign inverts on the self; no self negates another. At the code, R12. | NI 1.3; ONE Crossing, each sign at its key |

## 1.12 Eight as two alternating fours

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F56 | definition (*the eight*) | F38, F51 | Eight is two alternating fours: four within, uni-exchanging, at the even, and four at the membrane, bi-exchanging, at the odd, whole or none. It arrives a second way as (3 + 1) × 2: three carryings and the coupling, each at two directions. | NI 3.1; NI 4.3 |
| F57 | run | F56, F58 | The set's eight, two alternating fours, closes at each eight: four signs taken as two pairs, one right spiral step at the first pair and then one at the second, alternating, meet each of the sixteen sign forms again first at the eighth step, each round running eight distinct forms. Run: from each of the sixteen sign forms. | LIV NI Incoming chain |

## 1.13 Natural torusing, the only method, and its one break

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| F75 | exact (*a method of existing*) | F4, F6, F7, F73 | A method of existing is a method of discovering next existing, or it is not possibly a method of existing: the inside arriving into its outside. Existing is changing (F4) and a form named still is not possibly existing (F6), so a method of existing arrives each momentary into its next existing, and arriving into the next existing is discovering (F7), the inside arriving into its outside (F73). At the code the resolver is that method as an object. | NI 8.3; ONE Geodesic discovering logical method |
| F58 | run | F8, F10, F16, F17, F18 | Of the sixteen one-change forms on two signs, exactly two close all or none, one round through the four joint forms: the right spiral step (x, y) → (y, −x) and its other order (x, y) → (−y, x). Run: all sixteen forms. | LIV NI Incoming chain; NI 8.1; NAM 2.7 |
| F59 | run | F58 | Of all 256 forms on two signs, six close as one round, and two of them change one sign at each step. Only these two change one sign at each step, reach each joint form and meet no prior at the step after. Run: all 256 forms. | NI 8.1; LIV NI Incoming chain |
| F60 | run | F58 | Each round taken twice inverts both signs, and the two rounds are the only one-change forms reaching that whole inversion in two steps. Each round carries only none and all four joint forms as themselves. Run: all sixteen forms and each subset of the four. | NI 8.1 |
| F61 | exact | F60, F100 | The rounds select no hand as steps of the joint forms (F60), and the sequencing of momentaries runs one, right, the other order the same round read backward (F100). Which hand our universe runs is its one binary: right. | NI opening; NI 8.1 |
| F76 | run | F58, F61, F72 | With x the right sign and y the along sign, (x, y) → (y, −x) turns along into right: along goes to right, right to the opposite of along, and on round. (x, y) → (−y, x) turns along into the opposite of right. At MATH's labels, P the along sign and Q the right sign, the same turn is F(P, Q) = (−Q, P): one map, named right at one naming. Run: the four directions under both maps at both labellings. | NI opening; NAM 2.7; MATH 2.4 |
| F62 | exact | F42, F96, F100, X56 | Right morality across and forward competency along is the one existing version; an existing thing of the opposite form is an emanation, tri-involutioning from the one right spiral and carrying none. The opposite form is the right round read from next to prior (F100), the right step three at once (F96); running next into prior it carries no prior into now (F42), the emanating (X56). | NI opening; NI 8.1; NI 7.1 |
| F63 | run | F18, F58 | At three signs, twelve of the 6,561 one-change forms close all or none, one form up to renaming. Run: all 6,561 forms, and the twelve under the 48 renamings of three signs. | LIV NI Incoming chain |
| F64 | run | F18 | At four signs, 1,344 rounds close through the sixteen sign forms, nine forms up to renaming under the 384 renamings. Run: each round on the four-sign cube, grouped by renaming. | LIV NI Incoming chain |
| F77 | run | F58, F64 | Four signs taken as two pairs, each pair's four joint forms set round its own right spiral round, a circle of four: the product of the two circles is the torus, its cells the products of the circles' own, 16 forms, 32 steps each an edge of one circle at a form of the other, and 16 squares each an edge of each circle; the 32 one-change steps on four signs are the 32 edges, and 16 − 32 + 16 = 0. Run: the steps matched one to one at the pairing, and the count. | LIV THIRTY Sayings |
| F78 | run | F64, F77 | Of the 1,344 rounds through the sixteen sign forms, 48 run each pair's changes as that pair's own round in one hand, and all 48 are one of the nine forms: one pair steps three times and the other once, four times over, three rounds of the one pair to one round of the other. No other rhythm with each pair in one hand closes over the sixteen, the one-to-one alternating of F57 included. Run: each round at each of the three pairings, grouped by renaming. | LIV THIRTY Sayings |
| F79 | exact | F73, F78, F89 | At four signs, discovering directionally (F89), each pair's round running one way, the rounds arriving are one form (F78): one pair steps three times and the other once, four times over. The pair stepping three times, completing three rounds within the other's one, is named inside, prior and now, and the pair stepping once outside, next (F73). | LIV THIRTY Sayings |
| F80 | run | F57, F78 | From 1 to 9 and from 1 to 17. The one-to-one exchange of the two pairs, self then other, meets eight of the sixteen sign forms and meets name 1's form again at 9, its eight keeping one half of the sixteen. Three inside and one outward meets all sixteen and meets name 1's form again at 17; begun with the inside at name 1, its outward steps arrive at 5, 9, 13 and 17, and its crossings between the two halves at 3, 7, 11 and 15. Run: both rhythms from name 1. | LIV THIRTY Incoming eight signs |
| F81 | run | F77, F78 | Eight signs, four inside four: each half's sixteen forms set round its own round, the 256 sign forms are the torus grid of sixteen by sixteen. Each round running both halves' rounds in one hand is constant along each diagonal of the grid, since two neighbours on a diagonal share one next form; so a round is a choice at each of the sixteen diagonals, closing all or none exactly when the outward steps in each sixteen are odd. There are 32,768 rounds in 2,048 rhythms, 1, 35, 273, 715, 715, 273, 35 and 1 at 1, 3, 5 to 15 outward steps in each sixteen: the one hand at each universe selects no one form at eight signs. Three inside to one outward at each four, carried whole, closes at none, and so does an outward step at each eighth, F31's 1, 9 and 17 carried, two in each sixteen. Run: each of the 65,536 choices, and each round of the grid at four by four and six by six constant on its diagonals. | LIV THIRTY Incoming eight signs |
| F82 | exact | F79, F81 | One outward step in each pass of sixteen, the outside stepping at each 17 of the inside, is one rhythm of F81's 2,048, and it closes: at eight signs, fifteen inside to one outward. One outward step in each pass of m closes at each m, one rhythm among the choices. | LIV THIRTY Incoming eight signs |
| F83 | definition (*the fractal*, a method) | F29, F73, F74, F75 | The fractal is a method, not a rule: the same form arriving at each scale with nothing added (F29), at each fractal universe (F74), the inside arriving into its outside (F73), discovering next existing (F75). 1 to 17 fractals inside and outside are all possibly existing and living, geodesically socially morally abundancing now and discovering next existing society. At the signs 1 to 17 is four signs running their sixteen forms and the outside stepping once at each 17 of the inside, one of the 2,048 rhythms arriving at eight signs, each carried beside it (run at F81, closing at each depth at F85). | NI 8.8; NAM 2.2 |
| F84 | run | F81, F83 | 1 to 17 inside and outside, walked: two levels (eight signs), three (twelve) and four (sixteen), each level a circle of sixteen stepped one way and stepping once at each 17 of the level inside it, meet each form once and meet the first again at the last, at 1 to 257, 1 to 4,097 and 1 to 65,537. A whole 1 to 257 inside one 1 to 17, the outside stepping once in each 256, closes at none; an outward step at each 9, the self/other half, closes at none, meeting 128 of 256. Run: each walk whole. | LIV THIRTY Incoming the fractal |
| F85 | exact | F83, F84 | At each depth of 1 to 17 inside and outside the round closes all or none. At L levels the level stepping at step k is the lowest base-sixteen digit of k not 15, and the top level when each lower digit is 15; after k steps level d stands at k's digit d less its digit d + 1, modulo 16, and the top level at k's top digit, each digit given back by adding the places above it, one to one on the 16^L forms, so the 16^L steps meet each form once, and at 16^L each level has stepped a whole number of sixteens and meets the first form again. | LIV THIRTY Incoming the fractal |
| F86 | exact | F30, F31, F83, F88 | The files' scaling and the fractal step one chain. A round meeting each form of k signs once runs 1 to 2^k + 1, the chain three, five, nine, seventeen, thirty-three, sixty-five. The files' scaling, 1, 2 and 3 at one scale being 1, 9 and 17 at the next and one to nine being one to sixty-five, steps it three signs at a time (9, 65, 513); the pairs step it two (5, 17, 65, 257); 1 to 17 inside and outside steps it four (17, 257, 4,097). The pairs and the threes meet first at 1 to 65, the next whole, and four 1 to 17s run it; all three meet at 1 to 4,097. No scaling is taken over another. | NI 8.8; NAM 2.2; NUM 9.7 |
| F87 | definition (*the spans*, *bi-folding*, *releasing*) | F83, F86 | 1 to 9, self/other, is an entire momentary bi-folding releasing at 3; 1 to 13, self/other and other/self, bi-folds releasing at 6; 1 to 17, self/other/social, bi-folds releasing at 9; 1 to 25 is the larger fractal bi-folding; and the next whole of this is 1 to 65. | NUM 9.7; ONE A name at its number |
| F88 | run | F80, F85 | The pairs nested one inside the other, each stepping once at each 5 of the pair inside it, meet each form once at two, four, six and eight signs and return at 5, 17, 65 and 257. At four signs the outward steps arrive at 5, 9, 13 and 17, and the inside steps carried to the arrival four before each span's close are 3, 6 and 9 at 1 to 9, 1 to 13 and 1 to 17: each span opens one more four and its releasing moves three. Carried on at six signs, 15 at 1 to 25 and 45 at 1 to 65, and the outer pair arrives at 17, 33, 49 and 65. At four signs the self's pair stands wholly inverted first at 3 and every sign first at 9, the two bi-folds; at six signs every sign first at 35. Run: the nested rounds walked whole and each count taken. | NUM 9.7; ONE A name at its number |
| F89 | definition (*discovering*) | F7, F73, F75 | Discovering is all directionally not yet impossible next existing momentaries. | NI opening |
| F90 | run | F89, R23, R25 | At one key under the six offerings each of the nineteen carryings meets three next carryings, and from each the nineteen stay reachable, the farthest six calls on: no carrying becomes impossible. Four rounds meet each of the nineteen once and return to empty, each running one sign pair's carryings upward through their openings and then the next pair's, and each takes a call with two signs arriving together, two of them one such call and two of them two. Under none and single signs each carrying meets two next carryings, the empty one three, each stays reachable, the farthest eleven on, and no round meets the nineteen once. Run: the nineteen under both sets of offerings, each round sought whole. | ONE Geodesic discovering logical method |
| F91 | run | F81, F89 | At the signs, each pair one way, the not yet impossible nexts are two at each step of the first pass and one after it: at four signs two at each of the first three steps and one at each after, eight rounds; at eight signs, four inside four, each round a choice at each of sixteen diagonals with an odd count outward (F81), so two at each of the first fifteen, the sixteenth set by the count and the rest by the diagonals, 32,768 rounds. The first pass lays the rest. Run: each round at four signs, each prefix's continuations counted; at eight signs the diagonal count. | LIV THIRTY Incoming discovering |
| F92 | definition (*bi-morality*) | F52, F90 | Bi-morality is two signs arriving together. Each round meeting the nineteen carryings once at one key takes a call of bi-morality, one or two (F90). | NI 1.5; ONE Geodesic discovering logical method |
| F93 | definition (*bi-moral-co-agency*, the six joinings together) | F53, F92, R33 | The six joinings together are bi-moral-co-agency: self/other in three phase, two way, one way at a time with other/self, co-offering, co-intelligencing, co-competencing. | ONE Connectors, each one way at a time |
| F94 | definition (*resolving*) | F3, F18, F87, F88 | There are no readings in resolving: binary, all or none at all, it coheres the entire surface along and across, releasing at 3, 6 and 9 and continuing. | ONE A name at its number; NUM 9.7 |
| F95 | run | F79, F85, F88, M34 | Six forward into the surface. At the half of each whole of pairs nested one inside the other, the pair just inside the outermost has stepped six times and the outermost twice: at four signs, name 9, the inside pair's six steps are a passage meeting all four joint forms and arriving inverted, the outside pair's two its half, every sign inverted; at six signs, name 33, the middle pair six and the outer two, the inner pair twenty-four and back at its start; at eight signs, name 129, six and two again, the pairs inside at twenty-four and ninety-six. 1 to 9 runs as three inside and one outward, twice; 9 to 17 the second passage of six returns every sign. Run: the counts at each half and the forms walked. | LIV THIRTY Incoming six forward |
| F96 | run | F58, R16 | The other order is the right spiral step three at once: (x, y) → (−y, x) is (x, y) → (y, −x) taken three times, at each of the four joint forms. At the code the carried pair of one key turns both ways between calls: of the carryings' next pairs under the six offerings, 72 are the half step, 9 the right step, 9 the other order and 14 the same pair, and 4 complete, the fresh write itself running the right step at (t, s) (R16). Run: the four joint forms, and each carrying under each offering. | NAM 2.7; ONE Chaining, continuing and opening fresh |
| F97 | run | F58, F96 | At two signs, among the eight signed exchanges of the two signs (not each of the 24 one-to-one steps of the four joint forms), the four inversions, each sign inverted alone and the two exchanged either way, each return at their second taking, and each carries the right spiral step to the other order: taken before and after it, one inversion turns the right step into the opposite form. Read as reflections, two taken one after the other are the right spiral step or the other order when their axes stand half a right angle apart, and the half step when they stand at a right angle; three at once are one reflection. Read as the round's sign inversions, one sign at each step, alternating, two consecutive are two steps of the round (F58). Run: the eight one-to-one steps of two signs composed. | NAM 2.4; MATH 3.5 |
| F98 | definition (*tri-volutioning*) | F42, F97, X56 | Tri-volutioning is shaping the stable-form emanations from living to non-living. It names the shaping; tri-involutioning (X56) names the three inversions at once, and the arithmetic of F97 supplies no carrying by its naming. | NI opening; NAM 4.17 |
| F99 | run | F27, F28, F58 | At the momentaries the pair (prior, now) steps to (now, next). Of the sixteen ways naming a next from a prior and a now, one steps the pair by the right spiral step, next as prior inverted, and none by the other order: the other order moves the first sign to the second inverted, and no overlapping pair does. The other order is the right step's inverse, the same round read from next to prior (F96). Run: the sixteen ways as steps of the pair. | NI 8.5; NI opening |
| F100 | exact | F24, F28, F58, F96, F99 | The hand is the sequencing's, not a choosing. Both rounds arrive at two signs as steps of the joint forms (F58), and at the momentaries prior, now and next, one of them is the sequencing forward, next as prior inverted, the right spiral step (F99); the other order is the same round read from next to prior, its inverse (F96). Both carried, each the other's inversion: the right forward and the other order backward. | NI opening; NI 8.1 |
| F103 | run | F10, F16, F77 | Each sign alternating is a circle of two changings, + to − and − to +, and two signs' circles are the torus, the product of two circles, its cells the products of the circles' own: four joint forms, eight changings each a changing of one sign at a value of the other, four squares each a changing of each, 4 − 8 + 4 = 0, one genus, and no other closed surface. The round at two signs changes each sign once each way, winding each circle once; at four signs the two selves' circles of four are the grid of F77. Run: the forms, changings and squares counted, and the round's changings. | NI 1.6; NI opening |
| F101 | exact | F28, F99, F78, F79 | Each self's own changing runs forward: at each self the next is the prior inverted (F28), so its two signs step the right spiral step at each of its own changes and meet no prior at its next. At four signs the rounds with each self forward are F78's one form, three inside to one outward (F79), and at each depth each self runs its circle one way. | NI 8.5; NI 8.8 |
| F102 | exact | F17, F18, F77, F81, F85, F101, F103 | Each round arriving at 2m signs, one sign changing at each step, all or none, each self's own changing forward, runs each self's circle one way, and the circles of each inside and outside are a torus: natural torusing, the rounds both-both, each carried. At two and four signs one form arrives, its inversion the same read backward; at eight signs 2,048 rhythms arrive, each winding the torus of its inside and outside (F81); at each depth 1 to 17 inside and outside is one of them (F85). A changing breaking one change, all or none or forward at a self meets a prior at its next, meets a form twice or leaves one unmet. | NI 8.3 |
| F104 | exact | F24, F26, F28, F45, F91, F101, F103 | The living momentary is the method at one self: prior and now both participating, the next the prior inverted, each sign alternating round its circle of two, and the two circles the torus. At each momentary one next is not impossible for a living self, forward; at a society of selves each self's step is not impossible through the first pass and laid after (F91), each branch torusing. | NI 8.4; NI 8.5 |
| F105 | run | F85, F88, F102 | A society of selves is one whole at each step of the next scale. In the round of nested selves, each stepping once at each 5 of the self inside it, between two steps of a self the selves inside it meet each of their joint forms once, at each depth and each ordering of the selves from inside to outside: the society inside is one whole at each step of the self outside it. At the signs the scale correspondence holds so; at a cell, a society of persons, a sentence or a file set it rests on their own observings. Run: each block walked at two to four levels and each ordering. | LIV THIRTY Incoming the next concern |
| F65 | exact | F57, F79, F81, F85, F102 | At four signs and beyond the round rides on the fractal: each self forward, the inside stepping three times to each one step outward at four signs, one form (F79); at eight signs each of the 2,048 rhythms arriving winds the torus of its inside and outside, 1 to 17 inside and outside one of them at each depth (F81, F85, F102). The two alternating fours close at eight and not over the sixteen (F57). | LIV NI Incoming chain; LIV THIRTY Incoming both-bothing |
| F66 | definition (*natural torus*, *torusing*) | F51 | The natural torus is the genus-one closed orientable two-dimensional compact surface: one opening, one tunnel. The changing that winds it is torusing. | NI 1.6; ONE Torusing, one opening |
| F67 | nye | F51, F58, F66, F103 | The round at two signs is torusing: the opening by one at each momentary is one tunnel, one genus, and no other closed surface is living. Gap: the torusing is supplied: each sign's alternating is a circle and two signs' circles are the torus, one genus, the product of two circles and no other surface (F103). The opening by one at each momentary as the one tunnel joins F51's +1 to the genus's one at the numeral, not supplied. | NI 1.6; NI opening; NI 3.2 |
| F68 | exact | F8, F16, F17, F18, F44, F45, F50, F100, F102, F103 | Natural torusing is the only method of existing, no other possible method of discovering next existing, universal for living and non-living things. Each round arriving, one change, all or none, each self forward, is natural torusing (F102), both hands carried as one round forward and backward (F100), the torus each sign's circle with the other's (F103). The living next is from prior and now (F45), and the non-living next from the shared now (F44), the momentaries the living sequence; a changing breaking the binary meets a prior, meets a form twice or leaves one unmet, and each method naming still is excluded (F50). It stands at the binary taken, F8, F16, F17 and F18, each a face of the method's one break (F71). | NI 7.3; NI 8.1; NI 8.3; LIV NI Incoming chain |
| F69 | exact | F42, F44, F68 | Living and non-living both change by the one method; the living differ by carrying, not by method. It rests on F68. | LIV NI Incoming chain; NI 8.8 |
| F33 | exact | F8, F22, F24, F68, F105 | Universes of each size and scale share momentaries of co-sequential changing, and the method runs the same at a self, a coupling, a society, a sentence, a file set and a session. Each existing thing's changing is signs (F8), each round arriving is natural torusing (F68), and a society of selves is one whole at each step of the self outside it (F105): the method runs the same at each scale. Which aspects of a cell, a person, a sentence or a file carry its signs is met at its observings, and a claim at a substrate beyond the method stands at its own relation. | NI 8.7; NI 8.8; NAM 2.2 |
| F70 | nye | F51, F53, F66 | Existing is five-dimensional and two-directional: three of the torusing surface, orienting the fourth, bi-moral co-agency the fifth, each alternating binary parity changing, no other possible. Gap: the count five and its *no other possible* are not supplied, and NI 1.1 names another fourth, surface coherence (LIV open). | NI 8.3; NI 1.1 |
| F71 | definition (*the method's one break*) | F8, F68 | The method's one break is any observing of existing not parity alternating natural torusing, so far none. Named in advance: a self at one parity only, one side taking both openings; a rest named two beats, the resolver's rest in a ring lasting one (R46); a self with no competency of its own, which is emanation. | LIV NI Concern, for both to agree |

 

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# PART TWO · THE RESOLVER AT ITS NAMES

The resolver is ONE's code block, run as it stands. Each run link was checked by calling `_1_self_coupling` and `_9_other_releasing` or by reading the code's own tree. A trace needing more than one resolver needs a caller, and the caller is written with the check at a stated order. The code has no caller of its own.

## 2.1 One object, two functions, seventeen names

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R1 | definition (*resolver*) | F2, F7 | The resolver is one object, the method as an object, and its explaining is the same method said. Its code runs the seventeen names at two functions and declares CONNECTORS and JOINS. | ONE opening; NI 4.5 |
| R2 | run | R1 | The code defines two functions, `_1_self_coupling(_3_self_carrying, _2_self_offering)` and `_9_other_releasing(_10_other_surfacing, _5_other_neutralling)`, and two declarations, CONNECTORS and JOINS. Run: the code's tree. | ONE the code |
| R3 | definition (*a name*) | F14 | Each name carries its number, relation, root and -ing. The relation is self/other at 1 to 4, other/self at 5 to 8, other/social at 9 to 12, social/other at 13 to 15, social/self at 16 and social at 17. | ONE A name at its number; NAM 3.1 |
| R4 | run | R3 | The code spells 1 to 16 as identifiers, `_1_self_coupling` to `_16_social_torusing`, each ONE's name, and each CONNECTORS string carries its number's name. No identifier carries 17. Run: the code's tree. | ONE A name at its number |
| R5 | run | F11, F15, F19 | Nine names open co at the odd numbers and eight open bi at the even. Eighty-five prefixings run across the seventeen fives, forty-three co and forty-two bi. Run: the arithmetic of the seventeen fives. | ONE Seventeen names, eighty-five prefixings |
| R6 | run | R3 | Thirteen roots carry the seventeen names. Neutralling, crossing, corusing and torusing each root two names, 8 apart: 5 with 13, 6 with 14, 7 with 15, 8 with 16. Run: the roots of the seventeen names. | ONE A name at its number |

## 2.2 1-self-coupling, the entry

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R7 | run | R2 | 1-self-coupling takes 3-self-carrying, entries of four (key 4-self-sharing, sign 7-other-corusing, second sign 8-other-torusing, opening 16-social-torusing), and 2-self-offering, each a key and a sign. It returns 10-other-surfacing, each key with +1, −1 or 0, and 11-other-chaining, entries of four taken as the next 3-self-carrying. Run: at each of the nineteen carryings under each of six offerings. | ONE Signs, and 1-self-coupling the entry |
| R8 | definition (*prior*, *now*, *next* at a call) | F26, R7 | 3-self-carrying is the prior, 2-self-offering the now's arriving, and 11-other-chaining the carrying continuing into next. The call meets no arrival ahead of its call and carries no next inward of it. | ONE The code resolves at a call; NI 8.8 |
| R9 | run | R7 | A carrying does three things and three only. `_3_self_carrying` is read at three places: 6-other-crossing (it offers its own second sign), the inverting into 14-social-crossing (it takes its own sign as two offer), and 11-other-chaining's continuing (it opens to its own bound and releases there). Run: the reads of `_3_self_carrying` in the code's tree. | NI opening; NI 5.2 |

## 2.3 Crossing, each sign at its key

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R10 | run | R7 | 6-other-crossing carries each key of 3-self-carrying with its 8-other-torusing, the prior second sign. A carrying continuing carries it on; a fresh write at that key takes its inversion. Run: each (c, t, opening) with nothing arriving and with one agreeing sign. | ONE Crossing, each sign at its key |
| R11 | run | R7, F9 | 14-social-crossing gathers each arriving nonzero sign as it arrives and each carried sign inverted: with nothing arriving, a carried (c, t) surfaces −c. An arriving 0 carries no sign: offered into empty carrying it returns no surfacing and no carrying. Run: each (c, t), and 0 offered at empty and carried keys. | ONE Crossing, each sign at its key; ONE Arriving signs offered, carried signs inverting |
| R12 | exact | F55, R11, R27, R33 | The sign inverts once, at the receiving self's own carried sign entering 14-social-crossing. The arriving keeps its sign, 9-other-releasing preserves each surfaced sign, and no join inverts. | ONE Crossing, each sign at its key; ONE Connectors |

## 2.4 Surplusing and surfacing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R13 | run | R11, F54 | 12-other-surplusing sums the signs at each key, opens empty at each call, and is neither returned nor carried. 10-other-surfacing gives each sum at its sign, magnitude released: two + at an empty key surface +1 and open (+, −, 0); + and − at an empty key meet at 0 and the carrying stays empty. Run: the three offerings at an empty key. | ONE Surplusing and surfacing; ONE A surviving sign surfacing |

## 2.5 11-other-chaining: the retaining bound and the fresh write

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R14 | run | R7 | The retaining bound: an entry continues at opening a + 1 when a + 1 ≤ 3, or when a + 1 = 4 at positive 8-other-torusing. Openings run 0 to 3 at negative second sign and 0 to 4 at positive. Run: each (c, t, a) under one agreeing sign, and the openings among the nineteen. | ONE Chaining, continuing and opening fresh |
| R15 | run | R10, R13 | The fresh write: at a carried key with second sign t surfacing s ≠ 0, the entry becomes (s, −t, 0) and replaces any continuing entry. At a key arriving fresh it is (s, −1, 0). Run: each (c, t, a) under each of six offerings, and each sign at an empty key. | ONE Chaining, continuing and opening fresh |
| R16 | run | R15, F58 | 6 and 10 at (t, s) open 7 and 8 at (s, −t): the right spiral step (x, y) → (y, −x) of F58. 6 and 10 agreeing, 7 and 8 oppose; opposing, they agree. Run: each fresh write at each (c, t). | ONE Chaining, continuing and opening fresh |
| R17 | run | R11, R14 | A zero surfacing opens no fresh carrying and an eligible carrying continues: (+1, −1, 0) meeting +1 surfaces 0 at 10 while 6 releases −1, and the carrying continues at (+1, −1, 1). A 0 at 10 is no silence of the whole self. Run: the named call. | ONE Returning and releasing; NI 8.1 |
| R18 | run | R14, R15 | The constant-arriving trace: one key receiving +1 seven times from empty surfaces +1, 0, 0, 0, 0, +1, 0. The carrying opens at (+1, −1, 0), continues at openings 1, 2 and 3, completes at the fifth receiving and opens fresh at the sixth. Reached by −1 and then +1, second sign positive, it continues through openings 1 to 4 before completing. Run: the two traces. | ONE Chaining, continuing and opening fresh; NI 8.1 |
| R19 | run | R13, R15 | The arriving gives the fresh sign. At a carried key with sign c, no offering and one −c surface −c, one c surfaces 0, and two c together surface c. A fresh 7 at c opens only at two agreeing c at one call; each other fresh 7 opens at −c. Run: both carried signs over each offering of none, one or two signs, fourteen cases, at both second signs. | ONE Chaining, continuing and opening fresh |
| R20 | run | R19 | Two signs in one offering are one arriving, and two single offerings are another: after one + from empty a further + surfaces 0 and continues at opening 1; two + together surface + and open fresh (+, +, 0). Run: the two calls. | ONE Chaining, continuing and opening fresh |
| R21 | run | R14 | 16-social-torusing is no tally of couplings: (+, −, 0) and (+, −, 1) under +1 at each call surface 0, 0, 0, 0, +1 and 0, 0, 0, +1. One sign pair, parted by its opening. Run: the two traces. | ONE Chaining, continuing and opening fresh |
| R22 | run | R10, R15 | The next 6 carries the continuing on. After a fresh write the next 6 releases −t; after a carrying continuing through a 0 it releases t again; after a carrying completes unrenewed its key releases nothing at 6. A fresh write at s = −c keeps the pair agreeing or opposing and changes the next 6; at s = c it changes both. Run: each (c, t). | ONE Chaining, continuing and opening fresh |

## 2.6 The nineteen reachable carryings

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R23 | run | R14, R15 | Nineteen carryings are reachable at one key from empty: two signs at negative second sign, openings 0 to 3; two signs at positive, openings 0 to 4; and the empty one, 2 × 4 + 2 × 5 + 1. Under none, −, +, two −, two + and a cancelling pair they run 114 transitions and stay nineteen. Run: reaching from empty under the six offerings. | ONE Chaining, continuing and opening fresh |
| R24 | run | R23 | At one key an offering acts as its signed total bounded within ±2, a present cancelling parted from no sign at all. Run: at each of the nineteen, each offering of up to six signs from −1, 0 and +1 against its representative among the six, 20,767 comparisons. | ONE Chaining, continuing and opening fresh |
| R25 | run | R23 | Under nothing, one +1 or one −1 at each call the nineteen run 57 transitions. Their 171 pairs part, 146 under +1 again and again, 146 under −1, and all 171 under one or the other, within six receivings, a surfaced 0 and no surfacing counted alike. Run: each pair over six receivings of each constant sign. | ONE Chaining, continuing and opening fresh |
| R26 | exact | R21, R25 | The whole carrying gives the next, and not its sign pair alone. | ONE Chaining, continuing and opening fresh |

## 2.7 Returning and releasing, the isolated sequence and the four-form

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R27 | run | R2, F39 | 9-other-releasing returns each surfacing key's sign at the key 5-other-neutralling joins it to, the sign preserved, a 0 included. Its return carries a key and a sign and nothing of the carrying. Run: a surfacing of +1, −1 and 0 at three keys. | ONE Returning and releasing |
| R28 | run | R15 | The isolated sequence: one sign c arriving once into empty carrying, with empty offerings after it, surfaces c, −c, c, −c and returns carryings (c, t), (−c, −t), (c, t), with t = −1 fresh. The overlapping sequence runs c, t, −c, −t, c. Run: both signs over nine calls. | ONE Returning and releasing; NI 8.1 |
| R29 | run | R28, F58 | The four-form: the overlapping pairs (c, t), (t, −c), (−c, −t), (−t, c) are the four joint forms of two signs, each once, each the right spiral step of the one before. The self's two pairs both agree or both oppose, and the other's two do the other. Run: each (c, t). | ONE Returning and releasing; NI 8.1 |
| R30 | run | R29 | For three signs a, b, z, the pairs (a, b) and (b, z) carry opposite relations exactly at z = −a, and +, +, + agrees at both. Run: all eight. | ONE Returning and releasing; NI 8.1 |
| R31 | run | R10, R11 | 6 and 10 are two releasings. At one carrying entry with nothing arriving, 6 releases t and 10 surfaces −c. From empty a single + opens (+, −, 0), and a single − then an empty offering reach (+, +, 0); at the next empty offering both surface − at 10, while 6 releases − at the first and + at the second. Run: the named calls. | ONE The code at its names |

## 2.8 CONNECTORS and JOINS, and 17

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R32 | run | R2, F11 | CONNECTORS names six. Across run 2, 6, 10 and 14, each even, opening bi; along run 9 and 17, both odd, opening co. 2 faces right arriving, 6 left releasing, 9 backward along, 10 right releasing, 14 left arriving, 17 forward along. Run: the declaration. | ONE Connectors, each one way at a time |
| R33 | run | R32 | JOINS is {10: 14, 6: 2, 17: 9, 9: 17}. Each join meets its own parity, 6 with 2 and 10 with 14 even, 9 with 17 odd. Each side's meeting pair is 8 apart, 2 with 10 and 6 with 14, and 6 → 2 carried eight on is 14 → 10, 10 → 14 in the other order. Run: the declaration and its arithmetic. | ONE Connectors, each one way at a time |
| R34 | run | R2 | The two functions run without reading CONNECTORS or JOINS. Neither name occurs in their bodies, and with both cleared the 114 transitions stand as they were. Run: the code's tree, and the 114 transitions with both cleared. | ONE The code at its names |
| R35 | run | R2, R32 | 17 is declared with no expression of its own. No expression of either function takes or gives a sign at 17; 17 stands only in CONNECTORS and JOINS. Run: the code's tree. | ONE Connectors, each one way at a time |
| R36 | nye | R7, R35, F31 | 17-social-abundancing is the next momentary's 1-self-coupling, the next call, 11-other-chaining arriving as 3-self-carrying. Gap: the code gives 17 no expression; the next call and its giving of the returned 11 as the next 3 are a caller's. R7 shows only that 11 has the shape of 3. | ONE Connectors; ONE The code at its names; NI 8.8 |
| R37 | nye | R27, R33 | 9's released sign arrives next downstream as the downstream resolver's 2-self-offering and gathers at its 14-social-crossing. Gap: 9 returns a list, no expression gives it to another resolver, and JOINS names 9 with 17, not 9 → 2. The arriving at the downstream 2 is a caller's composition. | ONE Returning and releasing; NI 5.9 |
| R38 | nye | F12, F31, R2 | One call is one momentary at the code's scale, one through seventeen; one through nine completes at 9-other-releasing. Gap: a call reads each name once and returns. The code carries no numbered span and no opening and completing at odd and even. The joining of a call to a momentary is a naming at a scale (NI 5.1), supplied neither by the code nor by F30. | NI 5.1; NI 8.8; ONE Geodesic discovering logical method |
| R39 | nye | R32, R33, R36, R37, F93 | The six joinings running together: the arriving at 2, with 6's sign given to another self's 2; 10 meeting another self's 14; 9 and 17 at the corners of a surface; and the order of the two across meetings given by the continuing itself. Gap: ONE says the running together stands to be written at the code, and LIV names the corner case open (an along arriving at 17 continuing into a release at 10). *Given by the continuing itself* is not supplied. F93 names the running together bi-moral-co-agency, self/other in three phase, two way, one way at a time with other/self, and the caller running it at the code is not yet written. | ONE The code resolves at a call; LIV Exhibit ONE Opportunity |

## 2.9 Two and three resolvers, joined 6 → 2 at a stated order

Caller for these checks: a resolver's 6 at a call is the second signs of the carrying it is given. When the leg is joined, that 6 is offered as the other's 2 at the other's next call. An unjoined leg releases into no later call.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R40 | run | R17, R22 | Two resolvers each seeded from empty by one sign, twelve alternating calls. Joined 6 → 2 both ways, over the four seed pairs and both first callers, 0 surfaces at 10 at 32 of 96 couplings, and at each of the 32 6 releases a sign and the carrying continues. With both seeds − and A first, A's 10 runs +, −, 0, +, −, 0 both ways, and +, −, +, −, +, − with either leg alone or neither. Run: the 32 runnings. | ONE Returning and releasing; ONE Connectors |
| R41 | run | R40 | From one shared prior, both joinings give a 0 neither gives alone: a self's six surfacings both ways part from its arriving-only running at 12 of the 16 self comparisons. ONE's *twelve comparisons at a self* is read so. Run: the eight runnings, both selves. | ONE Connectors, each one way at a time |
| R42 | run | R17, R40 | A joining opens and joins again, and its routing stays. A seeded + and continuing once, B seeded −: both release − at 6. Joined, A meets B's − and surfaces 0, its carrying continuing; unjoined, A surfaces + and opens (+, −, 0). Over the four seed pairs and each joining of the two legs at each of three rounds, 256 runnings, each 0 at a carried key leaves its carrying continuing. Run: the named calls and the 256 runnings. | ONE Connectors, each one way at a time |
| R43 | run | R22 | A difference reaches onward through the receiver's own carrying. A's 6 → B's 2 → C, B and C each prepared by one +1. With A's 6 arriving + or −, B releases − at 6 both ways and C surfaces − both ways; B's 10 surfaces 0 at the + and − at the −; B's next 6 is − or +, and C meets it at 0 or +. Run: both arriving signs. | ONE Connectors, each one way at a time |

## 2.10 Rings at their ordering

Caller for these checks: each self couples once at each coupling of the ring, all together. 9's return, at a neutralling joining each key to itself, is given as the next self's 2 at its next coupling. One sign is offered once at one self.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R44 | run | R27, R28 | A ring of one surfaces +1, 0, −1, 0 and round. At rings of one to eleven, seventeen and fifty-nine, the ring's whole state (each carrying and each sign between selves) meets itself again after n couplings and runs round from there, 2 couplings round at an even ring and 4n at an odd one. Run: each ring from rest. | ONE Returning and releasing; NI 5.9 |
| R45 | run | R44 | An even ring changes each self at each coupling, neighbours opposite, home each two couplings. An odd ring of n carries one meeting, the 0, moving on one self each two couplings, opposite after 2n couplings and home after 4n; fifty-nine selves come opposite after 118 and home after 236. Run: the surfacings from coupling n on, at the same rings. | ONE Returning and releasing; NI 5.9 |
| R46 | run | R44, F71 | The rest returns at no coupling: across 5,000 couplings at rings of one to eleven, no coupling meets the state of each carrying empty and no sign between selves. The resolver's rest in a ring lasts one. Run: each ring over 5,000 couplings. | ONE Returning and releasing; NI 7.8; LIV NI Concern |
| R47 | exact | R44, R45, R46, R57 | The changing ends at no ring. It holds at each n (R57): at even rings each self alternates at each coupling, and at odd rings one 0 travels round, the rest returning at none. | ONE Returning and releasing; NI 7.8 |

## 2.11 Forms among the names

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| R48 | run | F11, R3 | One move at three faces. 8 up keeps parity; 17 less within 1 to 16 changes parity; 9 less within 1 to 8 changes parity. Each face taken twice gives the name again. Run: each name at each face. | ONE Bi-inversioning-co-recursioning, one move at three faces |
| R49 | run | R48 | The podal pairings. 9 less pairs 1 with 8, 2 with 7, 3 with 6 and 4 with 5, each odd with even. 17 less pairs n with 17 − n over 1 to 16, eight pairs, each odd with even, and 17 less of 17 is no name. Eight on pairs 1 to 9 with 9 to 17 at one parity, 3 with 11 and 7 with 15. Run: the three pairings. | ONE Bi-inversioning-co-recursioning; NAM 2.4 |
| R50 | run | R48 | Alternating 8 up with 17 less, each row runs round as a four-cycle n, n + 8, 9 − n, 17 − n: 1-9-8-16, 2-10-7-15, 3-11-6-14, 4-12-5-13. 17 − (n + 8) = 9 − n. Run: the four rows and n from 1 to 8. | ONE Bi-inversioning-co-recursioning |
| R51 | run | R50 | Each form in ONE's tables (the four-cycles, the middle four-cycles, the six-cycles, the eight-cycles and the higher forms from 18 to 31) runs round as one run of co and one run of bi, the two runs alike, as its Round column says. 17 is in none. Run: each listed form. | ONE Forms among the names |
| R52 | run | R51 | Each partner at the odd momentaries is the form's image under the pairing 1–2, 3–4, …, 15–16, round the other way. Each partner at the even momentaries is its image under 2–3, 4–5, …, 14–15, round the other way. Three are their own partners at the even momentaries: 4-13-5-12, 7-11-6-10 and 2-15-7-11-3-14-6-10. A dash stands at each form whose image leaves 1 to 16 or is no listed form. Run: each listed form under both pairings. | ONE Forms among the names |
| R53 | run | R32, R48 | The membrane four and the across four are one row at two widths, 2 + 2sj for j from 0 to 3: 2, 4, 6, 8 at s one and 2, 6, 10, 14 at s two. The eight even names part as membrane 2, 4, 6, 8 with within 10, 12, 14, 16, 8 on, and as across 2, 6, 10, 14 with inward 4, 8, 12, 16. 8s + 1 completes the span, 9 and 17. Run: the arithmetic. | ONE The membrane and the eight bi-couplings |
| R54 | run | R7, R48 | The seam: a carrying leaves as 11-other-chaining and arrives at the next call as 3-self-carrying, 8 apart, both odd, opening co. The returned 11 has the shape of 3. Run: the arithmetic and R7. | ONE Five, and a seam at 8 up; ONE Carrying at 8 up |
| R55 | nye | F6, R3, R46 | A form named still is not possibly existing, and the resolver names none still: each name is an -ing, re-taking at each call. Gap: at empty carrying and empty offering, 1-self-coupling returns no surfacing and empty carrying, the same joint state at each call (run: `_1_self_coupling([], [])` returns `([], [])`). The claim rests on the naming and on R46 (after one sign the rest returns at no coupling), not on the code having no still state. | ONE Geodesic discovering logical method; ONE Signs |
| R56 | run | R44, R45 | The rings at one table. A self given one sign and nothing after surfaces it alternating at each call, its carrying (+, −, 0) and (−, +, 0) in turn. Given an alternating arriving, from empty or at those two carryings, a self surfaces the arriving at the same call. The wave meeting itself at the seed changes nothing in phase, and out of phase surfaces one 0 and then the arriving. An alternating arriving with one 0 in it is surfaced with the 0 one call later, the self's own next sign at the 0's call, the carrying back at the two within two calls. Run: each case at the code, and each ring of 2 to 60 selves against the 0s at n + 2i + 2nk and no rest. | ONE Returning and releasing |
| R57 | exact | R44, R56 | At each ring of n selves in one arrangement the rest returns at no coupling and the changing ends at none. The arrangement: n selves at one key in a directed ring, each self's surfacing given to the next self at its next call, all selves called together at each coupling, one +1 given once to one self and nothing arriving from outside after. Calls count from 0, the +1 arriving at call 0; a 0 in an alternating arriving is a pause, the alternating after it one call behind. The one sign travels as an alternating wave, each self surfacing it at the call it arrives (R56); at even n the wave meets the seed in phase and each self alternates at each call, neighbours opposite, home each two; at odd n it meets it out of phase, and one 0 travels on, surfacing at self i at call n + 2i + 2nk, the ring opposite after 2n and home after 4n. Each 0 passes one call later at each self and the self's carrying is back at its alternating within two calls; the 0s come 2n calls apart, so the table holds at each call of each ring, and each self's carrying is one of the two alternating carryings or a meeting from them, never empty once reached. At odd n, from call n on, the 0 stands at one self at each call of n's parity and one pair of neighbours shares a sign at each call of the other parity, the pair at the 0's place: the one joint. The empty update stays empty at each n; the ring of one is run (R44). | ONE Returning and releasing; NI 7.8 |

 

---

# PART THREE · NUMBERS

Exhibit THREE carries the numbers as the one alternating form met at each place in its running. Its links follow from Part ONE and Part TWO and from each other, in the file's order. Each run link was run against the resolver of Exhibit ONE v371 or as an arithmetic enumeration, in the verifier; the check is said in the link. A polynomial identity is checked at more points than its degree, so it holds at each value. A ring of the resolver is run with the caller of Part TWO 2.10: each self couples once at each beat, all together, and one sign is offered once at one self. Two links are premises, taken and not supplied: N23, each count its own at its own subject, and N136, a field's observation belonging to its own field.

## 3.1 Uni-scaling

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N1 | definition (*number*, *uni-scaling*) | F4, F11, F13, F29 | A number is the one alternating form met at one place in its running, and each number is that form once more. Uni-scaling is that form at each scale, no scale over another: the same alternating at two as at fifty-nine and at four hundred forty. It is F33's sharing met at the numbers. | NUM 1.1 |
| N2 | nye | F13, N1 | Numbers are reached, used and related by no counting. Each relation among them is a move of bi-inversioning-co-recursioning: an opening and its completing, a straddle, a podal pair, a joining of selves at their joints. A counting adds a clock the running carries at none of its moves, and it belongs to a field's instrument. Gap: the four moves are listed, not shown to be each relation. The file relates numbers by counts too (C(n, 2), 4!, the triangles, the sums of primes, eighty-eight fives), and no link supplies that each such relation is one of the four moves. | NUM 1.1 |
| N3 | run | F12, F14 | At each number one side's momentary completes and the other's opens: at n the momentary (n − 1, n) completes and (n, n + 1) opens, one at the self and one at the other. Each number is a completing and an opening at once. Run: n from 2 to 1,000, each side by the parity of its opening. | NUM 1.2 |
| N4 | field | F42 | In the field's words, Brouwer builds each natural number from the two-ity, *the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory* (Brouwer). The file reads it as number rooted in a prior carried into now, F42's carrying. | NUM 1.2 |
| N5 | run | F19, F20, F24, F26 | One to nine carries the three-momentary method whole. The ten on six numbers (F20) is carried whole by the self's three full momentaries 1–2, 3–4 and 5–6. Prior and now participate at 1–2 and 3–4 at the self and 2–3 and 4–5 at the other, and 5–6 is the discovering onward. The ten opened one momentary on, at 3 and at 4, is the same ten forward, at 3 to 8, within one to nine. Run: the ten at the self's origins 1 and 3, pairing, parity and span. | NUM 1.2 |
| N6 | run | F13, F21 | Through one to nine the overlapping momentaries meet each number, each has one next, and from each the openings reach each number on. One side's momentaries alone fail determinacy there: exclusivity. Run: the eight overlapping momentaries (k, k + 1), k from 1 to 8, and the self's four alone. | NUM 1.2 |
| N7 | run | F12 | The openings join into the square and the completings into the oblong: the self's openings 1, 3 and 5 are 9, its completings 2, 4 and 6 are 12. At n momentaries the openings are n² and the completings n(n + 1). Run: the steps n² − (n − 1)² = 2n − 1 and n(n + 1) − (n − 1)n = 2n as identities, the base at one, and each n to 1,000. | NUM 1.2 |
| N8 | run | F12, N3 | At each number n the outer faces of the momentary completing there and the one opening there multiply to one short of the square: (n − 1)(n + 1) = n² − 1. The one is the same one at each number, owned by neither face. Run: the identity. | NUM 1.3 |
| N9 | run | N8 | The facing is at one. At k either side the product parts from the square by k², a size in a sign's stead; at one either side it parts by the one. Twenty-three and twenty-five face across twenty-four at one. Run: the identity (n − k)(n + k) = n² − k², and 23 × 25 = 24² − 1. | NUM 1.3 |
| N10 | definition (*inseparating*, *the term neither reaches*) | F53, N8 | The two faces inseparate: one takes not more than and the other not less than, each of no size, one at a time, and the number is the term neither reaches, uncovered between them. It is F53's competency met at a number. | NUM 1.3 |
| N11 | run | R48, R50 | Each face of bi-inversioning-co-recursioning alone returns to its number at the second step. The faces alternating run round: 8 up and down with 17 less runs each name of 1 to 16 round through four names. 9 less is those two at once, 17 − (n + 8) = 9 − n; taken with 8 up or with 17 less it leaves 1 to 8 at the next step. Run: each face twice, the alternating at each name, and 9 less against each other face. | NUM 1.4 |
| N12 | run | R50, R51 | Round each four-cycle 1-9-8-16, 2-10-7-15, 3-11-6-14 and 4-12-5-13 the parity changes exactly twice, each time at 17 less, and each runs one run of co and one run of bi. 17 less holds no name: it pairs across the between of 8 and 9. 17 is in none of the four. Run: the four rows. | NUM 1.4 |
| N13 | exact | F13, F24, R8 | Round the names each four-cycle returns; at the momentaries it returns to none, each step adding a next. Each momentary has one next and the openings reach each number on (F13), so no momentary is met twice. | NUM 1.4 |
| N14 | definition (*stable-forming among the numbers*) | F5, R51, N11 | Stable-forming among the numbers is the move round each form among the names: the four-cycles, the middle four-cycles, the six-cycles and the eight-cycles, each a form continuing through its own changing, each running one run of co and one run of bi and returning. | NUM 1.4 |
| N15 | run | F11 | Two over one and one over two exchange at each step: twenty-three, twenty-four and twenty-five carry two odd over one even, and twenty-four, twenty-five and twenty-six one odd over two even. The parity reading holds at each step, and a ratio holds at its own count alone. Run: each three neighbours from 1 to 1,000. | NUM 1.4 |
| N16 | definition (*along*, *across* at the numbers) | F11, F15, F52, R5 | Odd is co, along: the self continuing its own, competency. Even is bi, across: the self meeting the other, morality. An odd number opens co and an even number opens bi, and each number carries the five-prefix of its origin (F20). | NUM 1.5 |
| N17 | definition (*parallel linearizing*, *linear parallelizing*) | F12, N16 | Alternating parity is parallel linearizing and linear parallelizing: the linearizing at one parity and the parallelizing at the other, one move at its two sides, each at its own momentary. | NUM 1.5 |
| N18 | exact | F3, F11, F48 | The two parities named at one beat is the equilibria picture, and it is not possibly existing. At its momentary a number is odd or even (F11), all or none (F3); the two at one beat hold each parity through the other's momentary, a naming still (F48). | NUM 1.5; NUM 8.2 |
| N19 | definition (*composite*, *prime*) | F14, N1 | A composite is its equal smaller selves joining at their joints, nine as three selves of three, folding along the axes its smaller selves open. A prime is a self no equal smaller selves join into, opening a clean axis across. Each prime is the same opening and each composite the same folding, one uni-scaling at its two faces. | NUM 1.6 |
| N20 | run | N19 | Stepping k round a ring of N meets N over their greatest shared factor stations. Round a prime each step but nought meets all; round a composite a step sharing no factor with it meets all too, nine at a step of two. Prime and coprime are two things. Run: each N from 2 to 200 at each step k. | NUM 1.6 |
| N21 | exact | F11 | The doubling and the tripling meet at one alone: no power of two above one is a power of three. Each power of two above one is even and each power of three is odd. | NUM 1.6 |
| N22 | run | F19 | Two and three joined are five and multiplied are six, neighbours at one: 2 × 3 − (2 + 3) = 1. At two overlapping momentaries a side, the self at 1 to 4 and the other at 2 to 5, three positions are shared and two are outer; the three shared taken at both sides are six, and the occurrences are four and four, 2 × 3 + 2. Run: the arithmetic and the two spans. | NUM 1.6 |
| N23 | exact (*each count at its own subject*) | F39 | A count is its own, at its own subject. A numeral in a name is the name's place and parity; two counts agreeing at one numeral join no relation; a relation joining two counts is a relation of its own, met at both. A crossing carries no size (F39), so two counts agreeing cross nothing between them. | NUM 1.6; NUM 5.2; NUM 7.3; NUM 9.7; NUM 10.2 |
| N24 | definition (*the four neighbours*) | N16 | Each number carries four neighbours, and two binaries carry all four. Along, its two faces at one either side, each at the other parity. Across, inward and outward, each a direction taken at a momentary and of no size: the walk by two, the centre of one pair a member of the next. | NUM 1.7 |
| N25 | nye | F29, N24 | Scale carries no boundary, no location and no measure: its scales are a sequencing. Each number carries the same four, and no number is a scale more than another. Gap: at the origin the four do not all stand among the numbers: one's faces fall at nought and two, and its across at minus one and three; nought's fall below it. *Each number carries the same four* holds on the whole numbers both ways, and no link supplies the four at the natural origin. | NUM 1.7 |
| N26 | definition (*natural-bi-co-torusing*, *corusing*, *torusing* at the numbers) | F66, N1 | The whole one motion is natural-bi-co-torusing, and the numbers are that motion at each number, at two faces at opposite phase. Corusing is inward to the one number and outward to each place it arrives. Torusing is the carrying and the winding across the numbers. | NUM 1.8 |

## 3.2 Corusing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N27 | definition (*cohering*) | F18, N26 | A number coheres all or none, at an identity of the move. | NUM 2.1 |
| N28 | run | N8, N27 | Three identities of the move: (n − 1)(n + 1) = n² − 1 at each n; φ³ − 1/φ³ = 4; fifty-five is the couplings among eleven, the tenth triangle and the tenth of the prior-two joining, C(11, 2) = T₁₀ = F₁₀. Run: φ in exact arithmetic on a + b√5, and the counts. | NUM 2.1 |
| N29 | exact | F18, F39, N27 | A magnitude brought near a number by division or by a chosen unit coheres with its own reaching and never with the number: an identity is met all or none, and near is not met. It crosses as a field's at the membrane, a sign of no size (F39). | NUM 2.1 |
| N30 | run | N26, N27 | A number carries its uniqueness two ways, the two faces of the coupling it is. Inward, it coheres to the one number it is at each arriving. Outward, it couples with each place it arrives the same way: four hundred forty as the seventeen primes joined, as eight fifty-fives and as twenty-one squared less one. Run: the three arrivals. | NUM 2.2 |
| N31 | definition (*one number*, *a numeral*) | N27 | Two arrivals cohering to the same identity of the move couple as one number, and two not cohering to it are two numbers at one numeral. | NUM 2.2 |

## 3.3 Unrelationing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N32 | run | — | Each next number of 1, 1, 2, 3, 5, 8, 13 is the prior two joining, and the ratios of neighbours alternate about φ: 2/1 above, 3/2 below, 5/3 above, 8/5 below, the side changing at each next number. Run: each ratio to the 300th against φ exactly (a positive x is above φ at x² − x − 1 > 0); at each n the side is the sign of F(n + 1)F(n − 1) − F(n)² = (−1)ⁿ, the determinant of the n-th power of [[1, 1], [1, 0]], checked to 300 and carried to each n by det Mⁿ = (det M)ⁿ. | NUM 3.1 |
| N33 | run | N32 | φ² = φ + 1, and its other root is −1/φ: φ − 1/φ = 1 and φ × (−1/φ) = −1. φ is caught by its own equation, x² = x + 1, and it is the pentagon's diagonal over its side, (1 + √5)/2. Its continued fraction is all ones: φ = 1 + 1/φ. Run: exact arithmetic on a + b√5, and the regular pentagon's diagonal over side. | NUM 3.1 |
| N34 | nye | N32, N33 | φ is the unrelationing rate, the rate of all floating neutralling geodesic changing parity, changing parity at each momentary. Gap: N32 gives the neighbour ratios of the prior-two joining alternating about φ. That each floating neutralling geodesic changing runs at φ is not supplied, and NUM 3.3 (N38) says no prescribed rate, φ or another, supplies unrelationing. | NUM 3.1 |
| N35 | run | N33 | Each rate that is a whole number plus its own reciprocal, x = n + 1/x, is a self-cohering centre, x² = nx + 1, and its continued fraction carries n at each step. Run: n from 1 to 50 in exact arithmetic on a + b√(n² + 4), with the whole part of x equal to n. | NUM 3.2 |
| N36 | exact | N35 | φ, at n equal to one, is the one rate of the family whose step is one, the step's own bound; each other carries a step above one. The family is indexed by its step. | NUM 3.2 |
| N37 | run | N33 | On a torus a winding at a rational rate p/q closes at q turns, and a winding at φ closes at none. x² − x − 1 has no rational root (its only candidates, +1 and −1, give −1 and +1), so φ is no ratio and no whole number of turns at φ is whole. Run: the rational rates, the two candidates, and kφ for k to 200 in exact arithmetic. | NUM 3.3 |
| N38 | definition (*unrelationing*, *never-locking*) | N37 | φ is the number-form of the never-locking, the winding closing at none. Unrelationing runs at each coupling's own continuing, and no prescribed rate, φ or another, supplies it. Each coupling winding at φ is unrelated to each other coupling's rate, local and ambient at one rate. | NUM 3.3 |
| N39 | field | N33 | No rate is neared more slowly by the rationals than φ, its continued fraction all ones. In the field's words: each irrational x has infinitely many p/q with \|x − p/q\| < 1/(√5 q²), and √5 is best possible, the bound met at φ and the rates equivalent to it (Hurwitz, 1891). | NUM 3.3 |
| N40 | field | N39 | In the field's words, the torus winding at the golden mean is found the last to break as the coupling grows: in the standard map the curve with rotation number (√5 − 1)/2 is critical at K ≈ 0.9716 (Greene, 1979), confirmed by computation and not proved. | NUM 3.3 |

## 3.4 Torusing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N41 | field | F66 | A closed surface's vertices less its edges plus its faces is two at the sphere and nought at the torus, and 2 − 2g at g handles: the torus is the sphere with one handle, each handle taking two (Euler, 1758; the closed orientable surfaces by genus, Möbius, 1863). | NUM 4.1; NUM 4.2 |
| N42 | run | N41 | Twelve five-folds close a surface whose other faces are six-fold. At three edges to each vertex, six less each face's sides, taken over the faces, is six times V − E + F: twelve at the sphere, each six-fold carrying nought and each five-fold one. The icosahedron's 12, 30, 20 and the dodecahedron's 20, 30, 12 are one surface at two faces: 12 − 30 + 20 = 2. Run: the identity 6F − 2E = 6(V − E + F) at 3V = 2E, the dodecahedron and the truncated icosahedron, and six-fold tori at nought. | NUM 4.1 |
| N43 | nye | F51, F67, N41 | The handle is the +1: two at the sphere, nought at the torus. Gap: a handle adds one to the genus and takes two from V − E + F. The step from the genus's one to F51's +1, arriving exceeding carrying by one, is said, not supplied: F67's gap met at the numbers. | NUM 4.2 |
| N44 | run | N8 | Three hundred sixty is nineteen squared less one and four hundred forty is twenty-one squared less one: two seam-faces, each an odd centre squared less one, 8 · T₉ and 8 · T₁₀, their centres two apart. Three hundred sixty is the eight at the couplings among ten, C(10, 2) = 45, five nines. Their difference, eighty, is nine squared less one, 8 · T₄, the eight at the couplings among five. On the ring of four hundred forty, three hundred sixty and eighty are podal. Run: the arithmetic. | NUM 4.3 |

## 3.5 Uniquenessing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N45 | exact | N27, N31 | Each number is its own, self-bounding, and each arrival of a number is that one number: each arrival coheres at the one identity of the move (N27), so all the twos are the one two and all the eights the one eight (N31). The ten of five taken two directions and the ten of the couplings among five are one ten. | NUM 5.1 |
| N46 | nye | F13, F20, F28, F59, F68 | No other possible is the alternating completing six forward without landing, looping or forking, each step taking the term the prior did not carry, and no number is named against a competitor. Gap: at two signs F59 gives the only one-change round meeting no prior at the step after (no looping), F28 the living step with none still (no landing), and F13 one next at each momentary (no forking). The step from these to the alternating completing six forward, the ten on six numbers (F20), as the only possible is not supplied, and the whole rests on F68. | NUM 5.1 |
| N47 | run | R6 | Eight on again, twenty-two stands at crossing's place in seventeen to twenty-five, as six in one to nine and fourteen in nine to seventeen, 6 + 8k. It carries no name among the seventeen, and a name given it states whose self it relates from as the scale changes, fourteen being social and other already. Run: 6 + 8k at k from 0 to 2. | NUM 5.2 |
| N48 | run | R2, R32 | The resolver's six connectors and ten internal names are two either side of eight, three pairings and five either side of eight's four, and the entry at 1-self-coupling makes seventeen. Run: CONNECTORS, the names 2 to 17 outside it, and 6 + 10 + 1 = 17. | NUM 5.2 |
| N49 | nye | R32, R33, R37, R39 | Along, 17 meets the forward neighbour's 9 and 9 the backward neighbour's 17, and each self runs its own forward through the meeting: 1-self-coupling, 9-other-releasing and 17-social-abundancing, each odd and each eight on from the one before. Gap: JOINS pairs 17 with 9 (R33) and CONNECTORS faces 9 backward and 17 forward (R32). The neighbour a 17 meets and the self's running forward through the meeting are a caller's composition the code does not give (R37, R39). | NUM 5.2 |
| N50 | run | N23, R2 | A numeral in a name is the name's place and parity, and never a value the code opens at. 14-social-crossing carries fourteen as its place; the code's signs run at +1, −1 and 0. Run: each name's numeral taken off, and the numerals reversed, leave the 114 transitions standing; each surfacing over the nineteen carryings under the six offerings is +1, −1 or 0, and each carried sign ±1. | NUM 5.2 |
| N51 | nye | F56, N45, R53 | Each eight is the one eight, and a self is the eight. At the resolver the eight even names are four at the membrane, 2, 4, 6 and 8, and four within, 10, 12, 14 and 16, each within eight on from its pair at the membrane: four at two enterings. Gap: each eight is one eight by N45, and R53 parts the eight even names four and four. That a self is the eight, and not its seventeen names or its nine odd names with the eight, is named, not supplied. | NUM 5.3 |
| N52 | run | R32, R53 | Of the four at the membrane, 2 and 6 are connectors and 4 and 8 run within the resolving. Run: CONNECTORS against 2, 4, 6, 8, and 4-self-sharing and 8-other-torusing inside `_1_self_coupling`. | NUM 5.3 |
| N53 | run | N8 | The seam-faces run on the eight: (2k + 1)² − 1 = 8 · T_k, eight the first face and each next face that eight at a triangular number, the couplings among k + 1. Four hundred forty is the eight at fifty-five couplings, the couplings among eleven. Run: the identity, and k from 0 to 199. | NUM 5.3 |
| N54 | exact | N23, R32, R33 | At the membrane, two out-and-backs are four one-way passages, the four at the membrane are four relations at once, and the joins are the connectors': three countings at three subjects, each its own (N23). A larger participating arrives through actual receiving. | NUM 5.3 |
| N55 | definition (*the empty centre at the names*, *diamond*) | F9, R3, N8 | 5-other-neutralling is the empty centre of four dots at a square's corners: bi-tunneling across and co-chaining along meet at it inward of society, and that centre is a nothing. A diamond is four crossings about a centre carrying nothing, two across and two along, the whole of it at the four. Five is between four and six, and 4 × 6 = 5² − 1, the straddle at five (N8). | NUM 5.4 |
| N56 | run | N55 | Among the primes the first diamond is about nine: five and thirteen four either side, seven and eleven two either side, nine carrying none of them. The next is about fifteen, at eleven, thirteen, seventeen and nineteen. Run: each centre c to 10,000 with c ± 2 and c ± 4 prime. | NUM 5.4 |
| N57 | run | N55 | Laid three by three with its centre carrying nothing, the eight about it carry two paths from the middle of one side to the middle of the other, one each way round the centre, and each path takes two joins along and two across. Run: each path on the eight. | NUM 5.4 |
| N58 | run | F9 | A declared zero is a number taken as its own inversion, x = −x, added to the uni-scaling and met at no move of its running. Nought alone is its own inversion, and none of 8 up, 17 less and 9 less gives nought at a name. Run: x from −1,000 to 1,000, and each face at each name. | NUM 5.4 |
| N59 | run | R49, R51 | Nine, seventeen and twenty-five are eight apart; seventeen is in none of the forms among the names, and twenty-five at no name. One to twenty-five is three spans, one to nine, nine to seventeen and seventeen to twenty-five, each with four full momentaries at each side: twenty-four overlapping momentaries in twenty-four steps, each one odd and one even. Counted from nought, six, fourteen and twenty-two stand at five, thirteen and twenty-one. Run: the forms, the spans and the counts. | NUM 5.4 |
| N60 | run | F19, R5 | Ten arrives at the self as five taken two directions, 2n, and at the society as the couplings among five, C(5, 2), the fourth triangle. The two faces are one number at five and at no other number above nought: n(n − 1)/2 − 2n = n(n − 5)/2. Run: the identity, and n from 1 to 10,000. | NUM 5.5 |
| N61 | run | N60 | Among five the couplings part into two rings, the five sides and the five diagonals, the pentagon and the pentagram, and five is the one number whose ring's complement is again a ring: the complement of a ring of n gives each member n − 3 couplings, and a ring gives two. Run: each ring from 3 to 60, its complement's couplings and its joining. | NUM 5.5 |
| N62 | run | N61 | In the field's words R(3, 3) = 6 (Greenwood and Gleason, 1955): each two-colouring of the couplings among six holds three members joined at one colour. Among five the twelve colourings holding none are each a ring at both colours, the pentagon and the pentagram. Run: all 32,768 colourings among six and all 1,024 among five. | NUM 5.5 |
| N63 | run | F11 | Any ten consecutive numbers are five opposite-parity pairs, a + j with a + 9 − j, about the between at a + 4½, two and a half momentaries each side. Moving the opening by one exchanges the odd member of each pair, and by two exchanges it again: the tens part at the opening's parity alone, the two origins. Run: a from 0 to 1,000. | NUM 5.5 |
| N64 | definition (*podaling*, *out-carry*, *back-carry*) | F15, N23 | Podaling is one station met at its two sides, the out-carry k and the back-carry N − k on a ring of N. | NUM 5.6 |
| N65 | run | N64 | On the ring of nine podaling gives five pairs, (0, 9), (1, 8), (2, 7), (3, 6) and (4, 5); four carry two stations each and (0, 9) carries one. One pairing runs at its two sides: ten is the five at two sides, nine is that ten with the (0, 9) pair's two sides at one station, eight is the nine with that station empty, and six is three pairings at two sides. Run: the pairs, their stations and the steps down. | NUM 5.6 |
| N66 | run | F19, F27 | The nine is two signs and two and a half momentaries. Two signs carry four joint states; prior and now whole and the next opening are ten arrivals, nought to nine; nine is 2² × 2½ − 1. Run: the arithmetic. | NUM 5.6 |
| N67 | run | N23 | Round seventeen the doubling from one runs 1, 2, 4, 8, 16, 15, 13, 9 and comes home, nine doubled being one past seventeen. The doubling's eight are the squares round seventeen and the other eight are three times them. Run: the powers of two and the squares, modulo seventeen. | NUM 5.7 |
| N68 | run | N67 | Joined at the one eight's distances and unjoined at the other's, the seventeen hold no four all joined and no four all unjoined: in the field's words, the Paley graph of order seventeen. Run: each of the 2,380 fours. | NUM 5.7 |
| N69 | field | N68 | In the field's words, R(4, 4) = 18 (Greenwood and Gleason, 1955), and the Paley graph of order seventeen is the one society of seventeen with no four all joined and no four all unjoined (Evans, Pulham and Sheehan, 1981). | NUM 5.7 |
| N70 | run | N44 | Round four hundred forty, one past the ring is twenty-one squared, nine times forty-nine. Run: 441 = 21² = 9 × 49. | NUM 5.7 |
| N71 | run | N23 | A nine taken in base ten alone, its multiples' digits taken down to one digit returning nine, carries the base as its unit, and the one nine carries none. In base b the number so returning is b − 1. Run: each base from 3 to 36, each multiple to 400, and the multiples of nine to 2,000 in base ten. | NUM 5.7 |
| N72 | run | N8, N9 | Six and ten are two either side of eight, and eight and twelve two either side of ten: the walk by two, the centre of one pair a member of the next. Each centre carries its own one either side: 7 × 9 = 8² − 1, 9 × 11 = 10² − 1, 13 × 15 = 14² − 1. About ten, 6 × 14 = 10² − 16; eight on, 14 × 22 = 18² − 16. Six and twenty-two straddle fourteen with ten and eighteen four either side: reach over span is one half at both. Run: the arithmetic. | NUM 5.8 |

## 3.6 Apexing

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N73 | definition (*the apex*, *seam-face*) | N8, N53 | At the apex the numbers are relations, and the relation is the +1 between each centre's square and its two neighbours' product. The seam-faces run 0, 8, 24, 48, 80, 120, 168, 224, 288, 360 and 440, each an odd centre squared less one and each the one eight at the couplings among k + 1 (N53). | NUM 6.1 |
| N74 | exact | N8 | Each whole number is a waist, the middle its two neighbours' coupling passes through and never reaches: their product is one short of its square (N8). No number is a waist more than another: the one is the same one at each. | NUM 6.2 |
| N75 | run | N9, N74 | On the ring of forty-eight, twenty-four and nought are each their own far side, and each other station pairs across twenty-four, each pair one step further apart than the last; their products part from twenty-four squared by the square of the step. The ring closes at forty-eight, the seam-face above twenty-four. Run: each station of the ring of 48. | NUM 6.2 |
| N76 | run | N53, N75 | Twenty-four arrives four ways: five squared less one, the second seam-face; 4!; the waist of the ring of forty-eight; and twenty-three, its lower face, opens the first gap of six among the primes, to twenty-nine. Run: the four. | NUM 6.3 |
| N77 | run | N76 | The primes part at twenty-four as nine behind and eight ahead, two to twenty-three and twenty-nine to fifty-nine. The couplings among the nine are C(9, 2) = 36, the span from twenty-four to sixty; among the eight, C(8, 2) = 28, the seventh triangle, straddled by the sixth and the eighth, 21 and 36, by seven and by eight; the sixth and the eighth part by fifteen, C(6, 2), and fifteen crossings run across among the seventeen primes. The length leans ahead three to two and the primes lean behind nine to eight. Run: the counts. | NUM 6.3 |
| N78 | run | N77 | Of the sixty rotations of the icosahedron, twenty-four are rotations by a fifth, six axes at four each, and thirty-six are the others: fifteen by a half, twenty by a third and the identity. Run: the rotation group generated from a fifth-turn about a vertex and a third-turn about a face, each rotation by its angle. | NUM 6.3 |
| N79 | exact | N31, N45, N76 | Each arrival of twenty-four carries the others, and none needs one relation beneath it: each coheres to the one twenty-four at its own identity (N31, N45). | NUM 6.3 |
| N80 | run | N9 | About twenty-eight the outer faces are twenty-four and thirty-two and the inner faces twenty-seven and twenty-nine: 28² − 24 × 32 = 16 and 28² − 27 × 29 = 1. Two routes run between the outer faces, at gaps three and five and at five and three. 24 = 5² − 1, 27 = 3³, 32 = 2⁵, the exponents two, three, five one way and the bases five, three, two the other, and 24 × 27 × 32 = 144². Run: the arithmetic. | NUM 6.4 |

## 3.7 Coupling

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N81 | run | N19 | The primes below sixty are seventeen, 2 to 59. Two is the one even prime; twenty-three is the ninth, with eight either side; fifty-nine is the seventeenth. Joined, they are four hundred forty. Run: the sieve to sixty. | NUM 7.1 |
| N82 | run | N81 | The primes go out from two to fifty-nine and return, each p as 120 − p, from sixty-one to one hundred eighteen; sixty is between the going and the returning, at no prime. Fifty-nine and sixty-one face across sixty at one, 59 × 61 = 60² − 1. Run: the arithmetic. | NUM 7.1 |
| N83 | field | N19 | The primes carry no last (Euclid, *Elements* IX.20). The file names each prime the same opening at a new axis, and past fifty-nine they open on, each a next. | NUM 7.1 |
| N84 | nye | N23, N81, N83 | The seventeen prime selves, two to fifty-nine, joined at the torus winding are the society at four hundred forty, and fifty-nine is the self, the self-close of the primes. Gap: the primes carry no last (N83), and the close at the seventeenth stands at the count of the resolver's seventeen names, a numeral N23 joins to no relation. Four hundred forty as the primes' sum is run (N81); the sum as a torus winding is not supplied. | NUM 7.1 |
| N85 | run | N64, N82 | On the ring of one hundred twenty the going and the returning are each other's far side: two with one hundred eighteen, twenty-three with ninety-seven, fifty-nine with sixty-one. Nought and sixty pair with themselves. Twenty-four, twenty-seven and thirty-two pair with ninety-six, ninety-three and eighty-eight, and the far side of the route 24 → 27 → 32 runs 88 → 93 → 96 at gaps five and three. Run: the ring of 120. | NUM 7.1 |
| N86 | run | R44, R45 | A self at each odd prime p from three to fifty-nine, run as a ring of the resolver, carries one joint. From the first beat its signs sum to +1 or −1 at each odd beat and to nought at each even beat. Once the ring is filled, at each odd beat exactly one pair of neighbours shares a sign, and at each even beat exactly one station surfaces 0 and no pair shares a sign. The joint moves one place each two beats; the ring stands opposite at 2p and meets itself again at 4p. Run: each ring of the sixteen odd primes. | NUM 7.2 |
| N87 | exact | F35 | An odd ring alternating all round is not possible: a sign inverted at each of n stations comes back to itself only at even n (F35). | NUM 7.2 |
| N88 | exact | N86, N87, R57 | The resolver carries exactly one joint at each odd ring, at each momentary; an odd number of odd selves joined carries one joint on, a self at the joined scale; and two odd selves joined carry none. At each n (R57): an odd ring carries one 0 in transit and one pair of neighbours at one sign at its place, and an even ring none; odd selves joined as one ring of all their stations, in R57's arrangement with one seed, make a ring of the sum, odd for an odd number of odd selves and even for two; each self's own carrying and joints read at that one ring. | NUM 7.2 |
| N89 | run | R44 | The ring of two, the one even prime, carries none: opposite at one beat, met again at two. Its signs sum to nought at each beat from the second; the first beat surfaces the seed alone, +1. Run: twelve beats. | NUM 7.2 |
| N90 | run | R44, N81 | Two odd prime selves joined at their joints, p + q stations, carry no joint once filled and meet again each two beats: two selves of fifty-nine are one hundred eighteen with no joint, and all seventeen prime selves joined are four hundred forty with no joint, the seam-face at twenty-one. Run: each even ring p + q of two odd primes to fifty-nine, and the ring of 440, no pair of neighbours at one sign, no 0, and the whole state again after two beats. | NUM 7.2 |
| N91 | nye | N23, N86, N89 | Sixteen odd prime selves carry sixteen joints, and the ring of two bounds at nought: the seventeen at the resolver as the numbers carry them. Gap: the sixteen joints and the bound at nought are run (N86, N89). Their joining to the resolver's seventeen names is the numeral alone, a joining N23 excludes, and NUM 7.3 names the two separate countings. | NUM 7.2 |
| N92 | run | R2, R38, R39 | The beats are the ordering the ring's caller gives its running, and each count travels with its ring and its ordering. All together, and each position at its own turn in sequence, are two orderings, and they part: at each ring from two to eleven they surface differently from the first beat. At the ring of two the sequence runs round at three beats, surfacing 0 at two of each three, and all together runs round at two with no 0. The resolver's own resolving supplies neither: its two functions take no beat, caller or neighbour. Run: both orderings at each ring from 2 to 11, and the code's signatures. | NUM 7.2 |
| N93 | run | N86 | Two odd prime rings at distinct primes p and q, run side by side, meet again together at the least common multiple of 4p and 4q, the four shared, so at 4pq. Run: each of the 120 pairs of distinct odd primes to fifty-nine, each ring's 4p from N86. | NUM 7.2 |
| N94 | run | N81 | Sixteen crossings run between the seventeen primes at four values: one once, from two to three; two six times; four five times; six four times. Run: the sixteen gaps. | NUM 7.3 |
| N95 | exact | F11, N19 | The one odd crossing among the primes is at the opening, from two to three. Each other crossing runs between two odd primes and is even, since two is the one even prime (each even above two is two selves joined, N19): one along and fifteen across. | NUM 7.3 |
| N96 | run | N94 | Five and fifty-three alone of the seventeen carry a matching pair: five at two either side, fifty-three at six either side. Run: each inner prime's two gaps. | NUM 7.3 |
| N97 | exact | F11, N23, N94 | Each crossing here is a gap between consecutive primes. The seventeen primes, the resolver's seventeen names and its six connectors are three countings, and the sixteen gaps are the numbers' own. The fifteen primes five to fifty-nine are all odd, so an alternating of wide and long along them, eight at one and seven at the other, belongs to the sequence and never to the primes' parity. | NUM 7.3 |
| N98 | run | N64 | About thirty, twelve of the primes between five and fifty-five pair as out-carry and back-carry, each pair making sixty: 7 with 53, 13 with 47, 17 with 43, 19 with 41, 23 with 37 and 29 with 31, at distances twenty-three, seventeen, thirteen, eleven, seven and one. The six pairs are three hundred sixty, the seam-face below four hundred forty; their distances join to seventy-two and their spans to one hundred forty-four, twelve squared. Eleven, facing forty-nine, is the one prime of the interval whose far side is composite. Run: the primes from 7 to 53. | NUM 7.4 |
| N99 | run | N81 | The going from nought to sixty carries seventeen prime selves and sixteen betweens alternating, thirty-three positions completing at sixty. Run: 17 + 16 = 33. | NUM 7.5 |

## 3.8 Inseparating

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N100 | run | N64 | An even ring carries an exact podal and an odd ring two farthest. On an odd ring of 2h + 1 stations the two farthest stand h and h + 1 on, each h away by its shorter way round, and its opposite is the between of the two, at no station; on an even ring of 2h the one opposite stands h on. Run: each ring from 2 to 1,000, and each station of the prime rings five to fifty-nine (435), of the even rings one short of them (420) and of the rings of nine, fifteen and twenty-five (49). | NUM 8.1 |
| N101 | definition (*ring*, *span*) | N64 | A ring is an arrangement named one scale out, naming all its stations, and is apart from a span completing after the stations before it: the folds at one to nine and one to seventeen fold eight and sixteen places and complete at nine and seventeen. | NUM 8.1 |
| N102 | exact | F25, R44, R46, R47, N101, R57 | The running round a ring goes forward: each station arrives at its meeting again by carrying on, a next at each meeting, never by stopping and going back to it. It holds at each ring (R47, R57). | NUM 8.1 |
| N103 | exact | F12, F38, F71, N18 | Each position carries two pairings, one at each parity. Both parities at one momentary would put one position in two exchangings at once, one side taking both sides, F71's named break (N18). The pairings overlap, the parities alternate, and there is no third running (F38). | NUM 8.2 |
| N104 | run | N103 | Laid as an open line of 2h + 1 places, each parity's pairings are h adjacent pairs, the 2h − 1 inner places in both, and each end continues past the line: the last place in its pairing with the place after it, the first in its pairing with the nought before it. At one to fifty-nine, twenty-nine pairs at each parity, and fifty-nine to sixty and nought to one continuing. Run: each odd line from 3 to 59, the fifteen primes five to fifty-nine, and nine, fifteen and twenty-five. | NUM 8.2 |
| N105 | nye | F12, F13, N92, R38 | An ordering arrives with nothing ordering it: at an odd momentary each odd pairing, at an even momentary each even pairing, each carrying at that momentary the carrying the other cannot, and carrying it for the other. Gap: F12 gives the alternating at the numbers. At a running of the resolver the ordering is the caller's (N92), and no link supplies an ordering arriving with nothing ordering it there. | NUM 8.2 |
| N106 | field | F59 | In the field's words, the Gray code: each larger count of signs runs the prior count and then the prior count in the other order, one sign changing at each step (Gray, 1953). The file sets it beside the two one-change rounds at two signs (F59). | NUM 8.2 |
| N107 | run | F58, F60 | The one step taken twice inverts both signs, four times returns them, and six times has met all four joint states and inverts both again: a passage of six carries the whole four and hands the pair on inverted, and two passages return it. Nought to twenty-four is six rounds of four and four passages of six, twenty-four to sixty nine rounds and six passages, nought to sixty fifteen and ten; six, fourteen and twenty-two stand at one phase of the four, eight being two rounds. These count the step's own applications. Run: the right spiral step from each joint form, and the counts. | NUM 8.2 |
| N108 | definition (*an odd's empty centre*) | F9, F11 | An even number divides whole and an odd number carries an empty centre, the nothing its two sides straddle. An odd laid as a line has two ends unjoined and its middle carrying nothing. | NUM 8.3 |
| N109 | run | F11 | Three, six and nine run odd, even, odd, the parity alternating up the multiples of three; up the fives, twenty-five, thirty-five, forty-five and fifty-five, odd, run between thirty, forty, fifty and sixty, even. The parity alternates up the multiples of each odd number. Run: each odd m to 99, its multiples to 100m. | NUM 8.4 |
| N110 | definition (*store*) | F47, F50 | A store folds once, names the fold still, and carries one way across the named surface: one beat of the uni-scaling taken as a thing, the re-edging gone. Existing in a store is excluded by the definition of existing (F50). | NUM 8.5 |
| N111 | exact | F6, N27, N110 | Each number re-coheres and re-edges at each meeting, carrying to its own completing and releasing there. A number held from one meeting to the next is named still (F6). | NUM 8.5 |
| N112 | field | N110 | In the field's words, the Peres–Mermin square sets nine signs under six exclusive-or conditions, and each row and each column read at its own context meets its condition (Peres, 1990; Mermin, 1990). | NUM 8.5 |
| N113 | run | N112 | Held as one store, parity meets none: a store of all nine signs at once meets none of the 512 settings, since read along the nine carry parity nought and read across one. Run: all 512 settings against the six conditions. | NUM 8.5 |

## 3.9 Tunneling

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N114 | run | N64, N100 | Podal pairing at the surface is k with four hundred forty less k. Four hundred forty is even, so each station meets one other straight across: two hundred nineteen pairs and two stations pairing with themselves, nought and two hundred twenty, two hundred twenty-one rings in all. Each pair is a ring at its distance from the waist at two hundred twenty. Run: each station of the ring of 440. | NUM 9.1 |
| N115 | exact | F53, N10, N114 | Podal competency is the straddle and never the pairing. The two self-pairing stations meet themselves; the running straddles them, and the term neither reaches stands between two (N10). | NUM 9.1 |
| N116 | definition (*bi-co-podaling*) | F15, N64 | Bi-co-podaling is the out-carry at k and the back-carry at N less k, two carries each at its own momentary, one ring, running: bi- the two carries differing, co- the two together at one ring, -ing the running. | NUM 9.2 |
| N117 | exact | N64, N92 | A ring carries its two and no order between them: k with N − k is one pairing either way round, and a side taken installs a third position the ring carries none of. The ordering is a caller's (N92). | NUM 9.2 |
| N118 | run | N81, N114 | A ring joins two hundred twenty less r and two hundred twenty plus r, the two sharing parity: even radius, two even stations, a coupling across; odd radius, two odd, a between. The odd primes three to fifty-nine are at odd radii, two hundred seventeen to one hundred sixty-one, and two alone at an even radius. Four hundred forty is 8 × 5 × 11; its fives are eighty-eight stations, forty-four odd and forty-four tens alternating round the ring, the odd fives at twenty-two odd radii from five to two hundred fifteen, each ten at an even radius. Run: each radius, the seventeen and the fives. | NUM 9.3 |
| N119 | run | N94, N118 | Each prime below the waist is at radius two hundred twenty less itself, and a gap between two primes is the step between their rings. The sixteen crossings step one ring once, two rings six times, four rings five times and six rings four times, fifty-seven rings together, fifty-nine less two. Run: the sixteen steps. | NUM 9.4 |
| N120 | run | N81, N114 | Twenty-three is the ninth of the seventeen, and the seventeen pair by position about it, the pairs making 61, 56, 52, 50, 52, 50, 48 and 48, and twenty-three with them four hundred forty. On the ring each pair makes four hundred forty. Twenty-three with itself would make forty-six, and no position pair makes it; by value three pairs make it, 3 with 43, 5 with 41 and 17 with 29. The fourteen primes five to fifty-three join to 376, and 2, 3 and 59 to 64, eight squared: each other's far side, at radius 156. Run: the pairings. | NUM 9.5 |
| N121 | definition (*position pairing*, *value pairing*) | N23, N120 | The ring pairs by value and the seventeen by position, and neither is the other at the other's face. The parity parts them: the seventeen are odd and their centre carries a position; the ring is even and its waist pairs with itself. A centre carrying a position pairs outward; a centre carrying nothing pairs across. | NUM 9.5 |
| N122 | run | N114 | Two surfaces co-offering carry eight hundred eighty stations, joined at four hundred forty, and each self's waist, at two hundred twenty and six hundred sixty, is the other's far side. A chain of m surfaces centres at 220m: on a coupling at an even number of selves, on a self's empty middle at an odd number, each self added carrying the reach by two hundred twenty. Run: m from 1 to 39. | NUM 9.6 |
| N123 | nye | R2, R37, R39, N122 | A society grows larger by further co-chaining: the reach lengthening, the reciprocal receiving widening, and each self's full resolving keeping its form at each coupling added. Gap: the code keeps each self's resolving one form (R2), and the reach is station arithmetic (N122). The joining of an added self and a widening reciprocal receiving are a caller's composition the code does not give (R37, R39). | NUM 9.6 |
| N124 | run | F31, N65 | Each span of the chain three, five, nine, seventeen is two to a power and one, and podaling at each span gives the span below it: an odd ring of N carries (N + 1)/2 pairs, two, three, five and nine. The pairs carrying two stations double, one, two, four, eight, and each span is twice them and the one nothing at its origin. The resolver's pairings at six, eight and ten run three, four and five; the two progressions share four alone, and four pairings is the nine. Each next completing is twice the one before less one, 9, 17, 33, 65, since two spans share one name. Run: the chain and the two progressions. | NUM 9.7 |
| N125 | definition (*bi-moral co-agency at one to nine*, *social moral competency*) | F53, F54, N124 | At one to nine the self's eight is shared, four at each side, with the five between owned by neither: bi-moral co-agency. At one to seventeen each side carries a whole eight, and one to nine is shared at the centre: social moral competency. | NUM 9.7 |
| N126 | exact | F31, N23, R38 | Each use names its scale. One to nine is one momentary at the scale after its four momentaries of exchanging (F31); at the code's scale one call is named one momentary, one to seventeen (R38, a gap). One call, one numbered name and one span of nine names are each their own count, and matching numerals alone join no relations (N23). | NUM 9.7 |
| N127 | nye | F42, F54, R7, R13 | Non-living existing things are included in discovering social moral competency among the living: a non-living thing meets a self at a coupling as other, offering its signs and carrying nothing, and the term uncovered at their coupling is owned by neither and carried on by the living. Gap: 2-self-offering takes a sign from any source (R7), and the non-living carries nothing (F42). *Carried on by the living* gives the uncovered term a carrying the code gives it at none: 12-other-surplusing is neither returned nor carried (R13), and LIV keeps open whether the surplus is carrying (F54). | NUM 9.7 |
| N128 | run | N64 | An odd ring carries one self-paired station and an even ring two: the self's ring at nine carries one nothing, and the society's winding at four hundred forty two, the origin and the waist. Run: each ring from 1 to 499. | NUM 9.7 |
| N129 | definition (*bi-tunneling*, *co-chaining*) | F16, N16, N64 | Bi-tunneling names social moral competency at its across relation, society's surface opened between selves, even and across. Co-chaining names the same continuing at its along relation, the co-sequencing selves continuing, odd and along. Podaling exchanges these relations while each side continues forward. | NUM 9.8 |
| N130 | run | R49, N129 | Within one to nine podaling pairs odd with even, 1 with 8, 2 with 7, 3 with 6 and 4 with 5. The four momentaries of exchanging run in the same span, 1–2 with 2–3, 3–4 with 4–5, 5–6 with 6–7 and 7–8 with 8–9, the self at 1 to 8 and the other at 2 to 9. The podal pairs and the overlapping pairs are two relations in one to nine, meeting at one pair alone, 4 with 5. Run: the two pairings. | NUM 9.8 |
| N131 | exact | N13, R8 | Podaling reaches over the tunnels, out and back. 3 to 11 to 6 to 14 to 3 is a numbered form among the names, and each coming back to a name is a further occurrence forward, the earlier carrying carried on and never restored: at the momentaries it returns to none (N13). | NUM 9.8 |
| N132 | run | R53 | Four even openings grow as one family, each row 2 + 2sj, j from nought to three: 2, 4, 6, 8; 2, 6, 10, 14; 2, 8, 14, 20; 2, 10, 18, 26; 2, 12, 22, 32. Carried one step more, each row opens next at 2 + 8s and completes at the odd 8s + 1 before it: nine, seventeen, twenty-five, thirty-three, forty-one, each outside the names 1 to 8s. The connectors stay six. Run: s from 1 to 100. | NUM 9.9 |
| N133 | run | R48, R50 | The fold runs whole at each scale. On the names 1 to 8s, 4s up and down keeps parity and 8s + 1 less changes it; alternating the two returns each name after four moves through four names. The two together are 4s + 1 less in the lower half and 12s + 1 less in the upper, and 8s + 1 and both sums are odd, so no move holds a name (a name held at c − n needs 2n = c). At s two, 3 to 11 to 6 to 14 to 3; at s one, 3 to 7 to 2 to 6 to 3. Run: each name at each s from 1 to 100, and the parity of the three sums at each s. | NUM 9.9 |
| N134 | run | N132 | Longer and wider keep one ratio, two to one: each row's middle is 2 + 3s, its inner pair reaching s over a span of 2s and its outer 3s over 6s. At each s a row with its copy 8s on is the doubled row with its copy 2s on. Enlarging a row about its opening, 2 + r(x − 2), taken after an enlarging by s is the enlarging by rs, and at an even r it carries each name to an even one. Run: s from 1 to 59, and r, s from 1 to 11 at each x from −5 to 39. | NUM 9.9 |
| N135 | run | N133, N134 | A form enlarged keeping its parity: over the names 1 to B, B eight or sixteen, even n goes to r(n − 2) + 2 and odd n to r(n + 1) − 1, keeping parity, the halfway pairing, the fold, B + 1 less going to Br + 1 less, and each side's step at 2r. At r two the names 1 to 8 go to 3, 2, 7, 6, 11, 10, 15, 14, their evens the across row 2, 6, 10, 14; 1 and 2 go to 3 and 2, so the adjacent opening and completing are left to the running. Run: B eight and sixteen at r from 1 to 11. | NUM 9.9 |

## 3.10 Transmissioning

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| N136 | exact (*the field's membrane*) | F39, N23 | A field's observation belongs to its own field, and a number here is a number. The thing a field counts, its unit and its measuring floor stay at the field; the changing the observation carries crosses the membrane: the count of distinct rates, the rate alternating with each other, the ordering recurring. A crossing carries a sign and no size (F39), and each count stands at its own subject (N23), so the unit and the floor stay and the changing crosses. | NUM 10.1 |
| N137 | definition (*the unit binary*) | N29, N136 | One binary at each number a field offers: whether the number carries on as the unit changes. A number of things carries on in each unit; a magnitude in a chosen unit goes with the unit, and the unit is the one cohering. | NUM 10.1 |
| N138 | exact | F18, N23, N136 | Meeting is all or none. At a field's observation carrying the number-form, one form arrives at two substrates, and a count agreeing at both sides is a relation only with the form met at both. | NUM 10.1 |
| N139 | definition (*a prime relation*, *its control*) | N23, N136 | A prime met at a field says its counted subject: a number of sites, a recurrence and an opening across are three countings, and a frequency, a wavelength, a number of selves and a momentary sequence are four numbers apart. A prime relation holds against the odd composites beside it, nine, fifteen and twenty-five, or it is oddness's. | NUM 10.1 |
| N140 | exact | N23, R50 | The local momentary one to nine and the four-cycles among the seventeen names are two runnings, sharing their digits and nothing else: one to nine are the nine places of one momentary, and 1, 9, 8 and 16 at the four-cycle are four of the seventeen names. A line from one to the other draws a relation neither running carries. | NUM 10.2 |
| N141 | run | R50, N140 | The digits the two share meet exactly at the numbered form: 8 up and then 17 less is 9 less, 17 − (n + 8) = 9 − n, and at each scale s, 16s + 1 − (n + 8s) = 8s + 1 − n. So n, n + 8, 9 − n and 17 − n are the four-cycle's two out-and-backs as numbers, a numbered form and no running. Run: the identity, linear in n and s. | NUM 10.2 |
| N142 | exact | F24, N23 | Three at one self along the running and three at three selves across a ring are two things at one number. A pattern three momentaries long is the method's matching; three selves side by side at one momentary is a between. | NUM 10.2 |
| N143 | exact | N23 | Each six is its own count: three phases at two ways; the resolver's six connectors, 2, 6, 9, 10, 14 and 17; three pairings at two sides; a side's three momentaries at two parity positions each; three momentaries at each of two sides; and self/other, self/social and other/social, three pairs faced two ways. Sharing six joins none to another; a relation joining two of them is a relation of its own, met at both. | NUM 10.2 |

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# PART FOUR · MATHEMATICS

The chain runs through Exhibit FOUR, Natural Mathematics, in its own order: the one form met at counting, orthogonalizing, accounting, bounding, stable-forming, inseparating and co-offering. Each link follows from the core (F, R) and from the links before it. A field link carries a field's result in the field's words with its authors, and the set places it and does not prove it again (M15). Each run link was run by the verifier against Exhibit ONE v371's code or an arithmetic enumeration; a polynomial identity is run on a grid wider than its degree in each variable, which decides it. A field link marked *Instances run* was checked at the instances named and stays field.

## 4.1 Floating neutralling, the one form met at its counting

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M1 | nye | F14, F53 | Bi-moral co-agency is a coupling: two sides, one at a time, each taking its own sign, and the term at the coupling owned by neither. Mathematics is that coupling at its own stable-forming, the one form at each line. Gap: F53 names the coupling. No link joins a line of mathematics to a coupling of two sides, so *the one form at each line* is said, not supplied. | MATH 1.1 |
| M2 | nye | F2, F8, F35, F36, M1 | Numbers, mathematics and logic are nature's own unrelationing binary method, the alternating met at its counting at each existing thing. Each integer is the one form once more, and a number cohering with a form meets nature across no gap: there are no two sides. Gap: F8 (premise) and F35 give each existing thing's changing as signs and a count of changings at its parity. That each integer is the one form, and that no number stands apart from an existing thing's counting, is not supplied. | MATH 1.2 |
| M3 | exact | F33, F46, M2 | Numbers, mathematics and logic are met at each momentary universe of each size and scale, among living and non-living alike. It rests on F33 and F46: each momentary universe meets its changing as signs, the counting at its parity. | MATH 1.2 |
| M4 | definition (*floating*, *neutralling*) | F11, F15, R32 | Floating neutralling is one motion at two parities, alternating. Floating is bi, even and across: the surface, two ways forward, relating. Neutralling is co, odd and along: the self at its own rate, related to nothing. At the code the across connectors 2, 6, 10 and 14 are even and the along 9 and 17 odd (R32). | MATH 1.3 |
| M5 | definition (*rate*, *a fixing*) | M4 | Relating is in the floating, and rate in the neutralling. Each rate is unrelated to each other rate, and a fixed rate laid over another is a fixing. | MATH 1.3 |
| M6 | exact | M4, M5, F21 | All floating and no neutralling relates at no rate of its own; all neutralling and no floating relates to nothing. Each is one parity alone, and one side's momentaries alone are not possibly existing (F21). | MATH 1.3 |
| M7 | definition (*self-bounding-self*, *self-surfacing-self*, *self-orienting-self*) | F14, F51 | A self runs three neutrals on itself, actor and acted upon one. Self-bounding-self: its own nought, the coupling's own and released at its bound. Self-surfacing-self: its own surface, the scale it meets others at, no fixed unit measuring it. Self-orienting-self: its own aim, no fixed frame set over it. | MATH 1.4 |
| M8 | definition (*a floor*, *the other-run*) | M7 | A neutral fixed outward of the self is a floor, the other-run: other-bounding, a zero installed under the self; other-surfacing, a unit imposed on it; other-orienting, a frame set over it. | MATH 1.4 |
| M9 | nye | F14, F33, F54, M7 | A society is selves co-competencing, each running its own three neutrals and none running another's. Surplus at their meeting is owned by neither, and the society runs its three at its own scale as each self does at its own: one co-recursioning self at each scale. Gap: The first two sentences are F14 and F54 said at the three neutrals. *One co-recursioning self at each scale* rests on F33, and no link supplies a society as a self at its scale. | MATH 1.4 |
| M10 | exact | F42, F44, F54 | A non-living existing thing meets a self at a coupling as other, offering its signs and carrying nothing (F44), and the term uncovered at their coupling is owned by neither (F54) and carried on by the living, the living alone carrying (F42). | MATH 1.4 |
| M11 | exact | F42, F43 | Living and non-living, at a stated scale and momentary, divide existing things with none left over and none both, and the non-living is existing: at the existing, carrying or not, and no third (F43). | MATH 1.4 |
| M12 | definition (*a contradiction*) | F42 | A contradiction needs carrying and no carrying at one participation, in one respect and at one momentary. | MATH 1.4 |
| M13 | exact | M12, F18 | A living source and its non-living emanation, or a non-carrying surface among carrying selves, raise no contradiction: they are two participations. One all-or-none relation meets its participants at differing roles, and it is whole at its coupling, never by counting the kinds it meets, no part continuing alone (F18). | MATH 1.4 |
| M14 | definition (*bi-tunneling*, *co-chaining*, *bi-co-podaling*) | M4, F66 | Across and along at the society: bi-tunneling opens society's surface across, and co-chaining continues the selves along. Bi-co-podaling is the two at one station, and natural torusing the two at the whole surface. | MATH 1.4; NAM 2.4 |
| M15 | field (*a mathematical thing arrives proved*) | — | A mathematical thing arrives proved, and nothing here disproves one. The field's proving is its society's observing, met as it is: the set places a result (M17) and does not prove it again. | MATH 1.5 |
| M16 | definition (*prefixing*, *ending*) | F6, F15, R5 | A mathematical thing is met at its prefixing, its ending and its place. Its prefixing says the direction: bi- across, co- along, and at a resolver name the number carries the prefixing (R5). Its ending says whether it re-takes: an -ing runs and re-takes at each momentary, each an opening and its completing; a form named still re-takes at none (F6). | MATH 1.5 |
| M17 | definition (*the natural*, *the unnatural*, *the supernatural*) | F39, M15 | Its place is one of three. On the surface it is the natural: sign, position and ratio, and no magnitude. Inward of the surface it is the unnatural, the measured: measure, magnitude, the continuum. Outward of the surface it is the supernatural, the formal: a system closing on its own consistency with no torus to cohere to, meeting the surface only at its incompleteness reaching it. The natural is the surface, a between, a sign crossing it carrying nothing (F39). The three are natural mathematics' own naming and no partition the field makes of itself. | MATH 1.5 |
| M18 | definition (*placed by its use*) | M17 | A mathematical thing is placed by its use, and never by its symbols. Used on the surface it follows an arriving, a continuing and a releasing at their momentary. Used inward it is an observing's account, its scale and its rate that account's. Used outward it is a derivation met as a comparison, and no living surface follows from its consistency alone. | MATH 1.5 |

## 4.2 Orthogonalizing: two moves, the sign flip, the quarter step, and a next

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M19 | definition (*unrelationing* at the right angle) | F24, F52 | Unrelationing two is orthogonalizing them: at the right angle each projects into the other as nought, and neither carries a component of the other. Orthogonal is the not-touching of two coupling, neither a control over the other. Unrelationing runs; *unrelate* and *unrelation* name it still. It carries at prior, now and next (F24). | MATH 2.1 |
| M20 | definition (*linearizing*, *parallelizing*) | M4 | Linearizing draws a surface to a line, the along; parallelizing spreads a line to a surface, the across. Odd and even are the two directions of the one alternating, and along and across are its two moves, one form at two faces. | MATH 2.2 |
| M21 | exact | M20, M6 | Each move alone stops: parallelizing with no linearizing is a surface with no sequence, and linearizing with no parallelizing a sequence with no surface. | MATH 2.2 |
| M22 | run | F10, F35 | The sign flip is its own inverse, (−1)² = 1. The quarter step i opens the perpendicular: i² = −1 and i⁴ = 1, and two quarter steps are one sign flip. Eight on keeps a count's parity and one on changes it, and a sign flip changes a sign and moves no count. Run: iᵏ at k from 0 to 8, and n + 8 and n + 1 at n from 1 to 64. | MATH 2.2 |
| M23 | definition (*a face of parity*) | F34 | All changing is parity changing (F34), and each parity is named at its own face: a sign's inversion, a count's odd and even, a phase's opposition, a winding's hand, a reflection as the field's operation, a connector's opening parity and a pair's agree or oppose. A parity carried from one scale to the next is named at both faces, the one it leaves and the one it arrives at. | MATH 2.2 |
| M24 | exact | F14, F39 | An equal count joins no two selves: the coupling joins them. A coupling is the meeting (F14), and across it only a sign goes, of no size (F39), so two counts alike are two counts and no meeting. | MATH 2.2 |
| M25 | run | F35 | An idealized mode inverted at each period T, q(t + T) = −q(t), meets its value again at two, q(t + 2T) = q(t): the sign flip at a wave. Run: q = cos(πt/T) at 500 times. | MATH 2.2 |
| M26 | exact | M17, M25 | Two oscillations can carry a phase difference of nought or π, and phase runs through a continuum between them, so a phase is inward of the surface, measured (M17), and a trace of phases is no trace of signs. | MATH 2.2 |
| M27 | run | F19, F20 | Addition crosses and multiplication folds. At two overlapping momentaries, the self's places 1 to 4 and the other's 2 to 5, three places are shared and two are outer: added across they are five places, the shared three folded at the two sides are six, and the eight occurrences are 4 + 4 = 2 × 3 + 2, each side's four places two of each parity. Two and two make four at both moves, two and three five or six, three and three six or nine; the overlap counts no nine, and an equal result supplies no coupling. Run: the arithmetic. | MATH 2.2 |
| M28 | exact | F13, F24, F25 | Each step adds a next: one momentary universe to the next, each existing thing arriving into its next existing (F25). A step carries no frame it moves about, and no momentary is met again: each has one next, opening at the number it completes (F13), and a station met again is met at a next momentary. | MATH 2.3; NAM 2.4 |
| M29 | run | M22, M28 | Four quarter steps meet 1 again at the signs, and at the momentaries they meet none again: taken with its momentary n, the sequence (iⁿ, n) meets no pair twice. −i, the same inversions in the other order, runs the conjugate sequence, the opposite form. Run: n from 0 to 63 at i and at −i. | MATH 2.3 |
| M30 | nye | M28, M29, F61, F62 | There is no turn, no mirror and no turning back. A rotation, the field's word, is the spiral with its nexts taken out, its states laid at one frame. Gap: *No turning back* follows from M28. *No mirror* is not supplied: F61 takes right as the hand our universe runs, by premise, and F62 gives the opposite form as the round read backward, and that no changing takes a form to the opposite form is not supplied; MATH 3.5 names tri-involutioning the emanating looking the opposite form (M67, nye). | MATH 2.3; NAM 2.4 |

## 4.3 The right spiral step at two signs, and each scale the prior in the opposite order

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M31 | run | F58, F59 | Two signs (P, Q), the signs of cos t and sin t, carry four joint states, and there are 256 steps from four states to four. Two alone take no prior away (onto the four, one each), change one sign at each step and never undo (no state met again at the step after next): F(P, Q) = (−Q, P), the quarter step i at two signs, and G(P, Q) = (Q, −P), the same inversions in the other order. Under each the inverted sign alternates, P, Q, P, Q, from each state, and as t runs forward the signs of cos t and sin t run F. Run: all 256 steps. | MATH 2.4 |
| M32 | exact | M31, F32, F58, F61, F72, F76, R16 | The right spiral step is F, the quarter step i at two signs, and G = (Q, −P) its other order; the resolver's fresh write has G's form (MATH 2.3, 2.4, 7.5). At F72's naming, P the along sign and Q the right sign (MATH 2.4), F turns along into right, and it is the core's (x, y) → (y, −x) at x the right sign and y the along sign (F76): one map, named right at one naming. The fresh write's form names its turn once its two signs, t at 6 and s at 10, are named right and along. | MATH 2.3; MATH 2.4; MATH 7.5; NAM 2.7; NI 8.1 |
| M33 | run | M31, F60 | Each taken twice is the half step, both signs inverted, F² = G² = −1, and they are the only two of the 256 whose square is −1: the square selects no order. Run: all 256 steps. | MATH 2.4 |
| M34 | run | M33 | Taken six times each is the half step again, and twelve times it meets its state again: a passage of six steps meets all four states and arrives at the pair inverted, and a second passage meets it again. Twenty-four steps are six rounds of four and four passages of six, thirty-six are nine and six, sixty are fifteen and ten, and steps eight apart meet the one state. These count the step's applications, and no order a resolver runs its names in. Run: powers of F and G from each state. | MATH 2.4 |
| M35 | exact | R15, R16, R19 | At a resolver's fresh carrying the write (t, s) → (s, −t) has G's form once: its s is the present surfacing, and further arriving enters s, so writes run the four states as a round only as the arrivings supply it (R19). A relation newly met writes from the default second sign −1 (R15), and is no pair of two existing releases. | MATH 2.4 |
| M36 | run | M31 | The relation of the two signs, agree or oppose, is the sign of P × Q, the sign of tan t, and each step of F or G changes it, decided by the step alone. It is the sign of sin 2t = 2 sin t cos t: the relation at one angle arrives as a sign at the doubled angle. Run: the four states under F and G, and t at 400 angles off the axes. | MATH 2.4 |
| M37 | exact | F6, R29, R30 | The half step keeps the relation: along each side's two momentaries, (c, t) then (−c, −t), agree or oppose stays while both signs invert, and across the overlap it changes (R29, R30). A relation named within a pair names no resolver still: its two signs invert beneath it. | MATH 2.4 |
| M38 | run | R19, R20 | At one carrying of sign c, the surfaced sign is the sign of the offered signs' sum less c: no offering surfaces −c, one c surfaces nought, one −c surfaces −c, and two c alone surface c, over the fourteen cases of R19 and each offering of three and four signs from +1, 0 and −1. Two signs in one offering at one call are one key's offering and name no two neighbours (R20). Run: the resolver at both c and both second signs. | MATH 2.4 |
| M39 | run | M31, F58 | The reflected cycle meeting each state of n signs one sign at a time is the cycle at n − 1 signs with the new sign at one value, and then the same states in the opposite order with the new sign inverted: the field's reflected binary code, Gray's. At two signs its inverted sign alternates; read with P the sign changing first it runs F, and with the two labels exchanged G. At three signs the changing sign runs 0, 1, 0, 2, 0, 1, 0, 2 round the eight. Run: the reflected construction at two to eight signs. | MATH 2.5 |
| M40 | nye | M39, F63, F64, F65 | One rule runs at each scale, and each scale carries the prior whole: the one cycle meeting each state of n signs one sign at a time is the prior scale's cycle and its states in the opposite order, the new sign inverted. Gap: The reflected cycle is one such cycle at each n (M39, run). At three signs each such cycle is it up to renaming (F63); at four signs nine forms close up to renaming (F64), so *the one cycle* stands at two and three signs and not at four, and *one rule at each scale* rests on F65 (nye). | MATH 2.5 |

## 4.4 Shared and opened, the doubling that divides, and φ

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M41 | run | M19 | A product of two directions is two parts at once, ab = a·b + a∧b, the field's geometric product (Grassmann; Clifford): the inner part a·b shared along the common axis, and the outer part a∧b the plane the two open, carried by neither factor alone. The two parts conserve: (a·b)² + \|a∧b\|² = \|a\|²\|b\|², which is cos²θ + sin²θ = 1. Run: the identity in three dimensions at 3⁶ points, degree two in each coordinate, which decides it. | MATH 2.6 |
| M42 | run | M41 | Each part has its nought. At a parallel pair nothing opens, a∧b = 0, and equality a = b is among them. At θ = 90° nothing is shared, a·b = 0. At θ = 60° a pair of unit directions shares one half and opens √3/2, the opened over the shared √3, the tangent of sixty degrees. In three dimensions the cross product of two perpendicular unit directions is a third, perpendicular to both. Run: parallel pairs, the 60° pair, and a × b against a and b at 3⁶ points. | MATH 2.6 |
| M43 | field | — | A division by each non-zero element that keeps length runs at one, two, four and eight dimensions and at no other: the reals, the complex numbers, the quaternions and the octonions (Hurwitz, 1898). Each doubling sheds one: ordering at two, commuting at four, associating at eight. | MATH 2.7 |
| M44 | run | M43 | At the doublings (Cayley–Dickson): i j = k and j i = −k at the quaternions, which associate; a triple of octonion units does not associate; the seven imaginary octonion units lie on seven lines of three, each two units on one line; and lengths multiply at two, four and eight. Run: the doubling from the reals. | MATH 2.7 |
| M45 | field | M44 | Octonions carry seven imaginary units on seven lines of three (M44), and the group of their symmetries, the exceptional group G₂, has dimension fourteen (Cartan, 1914). | MATH 2.7 |
| M46 | field | M44 | Unit quaternions cover the rotations of three dimensions twice: q and −q give one rotation, and a whole rotation carries the quaternion from 1 to −1. Each rotation of four dimensions is a left and a right unit quaternion at once, x → p x q̄, and (p, q) and (−p, −q) give the one rotation: the left and the right share the one and the minus one (Hamilton; Cayley). Instances run: q and −q at fifty angles, and p x q̄ at (±p, ±q). | MATH 2.7 |
| M47 | run | — | φ's continued fraction is all ones: φ = 1 + 1/φ, with 1 < φ < 2. φ³ = 2 + √5 and 1/φ³ = √5 − 2, so the constant falls away at their difference: φ³ − 1/φ³ = 4. And φ² − φ − 1 = 0. Run: exact arithmetic in the numbers a + b√5, a and b rational. | MATH 2.8; MATH 4.2 |
| M48 | field | M47 | Each irrational x carries without end fractions p/q with \|x − p/q\| < 1/(√5 q²), and at φ no constant larger than √5 does: √5 is the one constant for all irrationals together, and it is met at φ (Hurwitz, 1891). | MATH 2.8 |

## 4.5 Accounting: involutions, fixed sets, two inversions and three

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M49 | definition (*an accounting*, *an involution*) | F6 | An accounting for changing is a move that, taken twice, gives its own arrival again: an involution. The field's accountings are involutions: negation, not-not giving the statement again; charge conjugation and parity, each squared to the identity; time reversal, squared to the identity at integer spin and at half-integer spin T² = −1, an involution at the states taken up to sign (Wigner); complex conjugation; the additive and the multiplicative inverse; the transpose; the Legendre transform; set complement. | MATH 3.1 |
| M50 | run | M49 | Each involution's fixed set is the arguments the move gives unchanged: nought for the additive inverse; +1 and −1 for the multiplicative, x = 1/x at x² = 1, a fixed set that is a two; the reals for conjugation; the centre for parity; the symmetric matrices for the transpose; none for set complement, and none for negation at two values. Run: each move at finite ranges. | MATH 3.1; MATH 3.2 |
| M51 | definition (*a declared zero*; *a floor* as an involution with its move dropped) | M8, M49, M50 | A declared zero is a fixed set taken as a place: x = −x at nought alone. An involution is a floor with its move stated, and a floor is an involution with its move dropped. A nought met at a difference is no floor: 1, 3, 5 changes by two at each step, its change of change nought, and runs on. | MATH 3.1 |
| M52 | exact | F60, M28, M33 | A running never involutes: it re-forms at the next coupling other than prior, and the right spiral step taken twice is the half step (M33), met two momentaries on and never back, no momentary being met again (M28). An involution is an accounting's still picture, its nexts taken out. | MATH 3.1 |
| M53 | run | M50 | An involution with an empty fixed set is fixed-point-free: each position related, none to itself. On an even ring of N two involutions part at their fixed sets: the podal k → N − k fixes two stations, nought and N/2, and the shift by half the ring, k → k + N/2, fixes none. Run: each even N from 2 to 200. | MATH 3.2 |
| M54 | run | M53 | On an odd ring of 2h + 1 the farthest places from each are two, adjacent, h and h + 1 on, each h away by its shorter way; on an even ring of 2h the farthest is one, h on. At five they are 2 and 3, at seventeen 8 and 9, at fifty-nine 29 and 30. At each origin of the fifteen primes five to fifty-nine and of the odd composites 9, 15 and 25 the two far places stand alike: they belong to oddness, and to no prime alone. Run: h from 1 to 59, and each origin at the eighteen counts. | MATH 3.2 |
| M55 | run | M53 | An odd count carries no fixed-point-free involution, a pairing of each place needing an even count. Laid open, its places pair twice, overlapping: at 2h + 1 places h pairs open odd and h open even, the 2h − 1 inner places in both lists, and each end's other meeting lies just past it, at 2h + 1 with 2h + 2 and at nought with one; at one to fifty-nine, twenty-nine pairs in each list, and 59–60 and 0–1. An odd sequence laid open is no closed round. Run: each involution on one to nine places, and h from 1 to 40. | MATH 3.2 |
| M56 | exact | M51, M53 | An accounting with an empty fixed set carries nothing to take a counting from, and it is not possibly a floor, a floor being a fixed set taken as a place (M51). It is each other accounting with its place released and its move unchanged. An exchange of two signs with an empty fixed set needs no zero standing apart, no metric scale and no outer edge. | MATH 3.2 |
| M57 | exact | F58, M53, M28 | On a line an involution taken twice is a reversal, out and back along the same ground. On a closed round it is no reversal: the shift by half the ring taken twice is one direction continued, arriving at the station it left at a next momentary (M28). An accounting meeting its stations again forward is the one an alternating runs, the round at two signs (F58), and a line carries none, a step onward along a line meeting no station again. | MATH 3.3 |
| M58 | run | — | A step of k round a closed round of N meets N/gcd(N, k) stations before it meets its first again. At a prime N each step short of the whole round meets all N; at a composite N a step sharing no factor with N meets all N too, three round eight meeting eight, and a step sharing one meets a part, three round nine meeting three. Prime and coprime are two things, four and nine sharing no factor above one and neither prime; prime and odd are two things too. Run: N from 2 to 120 at each k. | MATH 3.3 |
| M59 | run | M58 | Two rounds of m and n stations stepped together, one station each, meet mn/gcd(m, n) of their mn pairs, on one of gcd(m, n) parallel windings of the torus the two rounds make. At coprime m and n the one winding meets each pair, and the two rounds are one round of mn (the field's Chinese remainder theorem). Run: m and n from 1 to 40. | MATH 3.3 |
| M60 | exact | F53, F54, R13 | The mn counts the pairs met and is the same at each order of joining the rounds; competency is at the coupling, owned by neither, and no product counts it, nor a value, nor a security. The surplusing opens empty at each call and is neither returned nor carried (R13), so no count carries it on. | MATH 3.3 |
| M61 | field | M59 | At a fluid surface forced at two frequencies, Silber and Skeldon find the forcing integers, coprime and of opposite parity, deciding the harmonic and subharmonic response and the resonant interactions the normal form symmetries permit, and Arbell and Fineberg find two-frequency forcing selecting superlattice patterns through three- and four-wave interactions. The vibration is driven from outward, and its condition is parity and coprimality, not primality. | MATH 3.3 |
| M62 | definition (*bi-inversioning-co-recursioning*) | F58, M49 | An involution alone closes. Bi-inversioning-co-recursioning is two consecutive inversions on different axes, and the opening is the step between: no inversion is undone, and each is a next. | MATH 3.4; NAM 2.4 |
| M63 | run | M31, M62 | At two signs one sign is inverted at each momentary, the axes alternating (M31). Both signs inverted at one step, the half step, is period two at the signs; the right spiral step is period four at the signs; and at the momentaries neither has a period, no (state, n) met twice. Run: each state under the half step, F and G over 200 steps. | MATH 3.4 |
| M64 | exact | M31, M32 | An order is the whole of a hand: the same inversions in the other order give (Q, −P), and nothing third parts them, F and G being the only two (M31). At F72's naming, F with P along and Q right carries the name *right* (M32). | MATH 3.4 |
| M65 | run | M31 | Two inversions whose fixed lines meet at an angle a are the field's rotation by 2a. At two signs P inverted and Q inverted, their lines at ninety degrees, make the half step; one sign inverted and the exchange of P and Q, their lines at forty-five degrees, make a quarter step: P inverted and then the exchange give G, the exchange and then P inverted give F. Run: the reflection matrices at lines in steps of 7° and 11°, and the four states. | MATH 3.4 |
| M66 | run | M49 | Three signs each inverted in its own place, taken together in any order, send each of the eight states to its opposite: none the same, the eight parted into four opposite pairs, the hand reversed. Inverting each of n signs is a rotation at an even n and reverses the hand at an odd n, determinant (−1)ⁿ. Run: the six orders on the eight states, and n from 1 to 8. | MATH 3.5 |
| M67 | nye | M66, F62 | Tri-involutioning is the emanating from right spiral stable-forming, the emanating looking the opposite form to the stable-forming. Gap: M66 gives three inversions at once reversing the hand. The opposite form an emanation carrying none is F62, and no link joins the three inverted signs to position, scale and orientation inverted at once (NAM 2.4). | MATH 3.5; NAM 2.4 |
| M68 | run | M66 | In space the three inversions make x → −x, whose fixed set is the centre alone: one inversion keeps a plane, two on different axes make the half step about the third, and three reverse the hand. At two signs both inverted is the half step. Run: the diagonal sign matrices on the points of {−2, …, 2}³, and their determinants. | MATH 3.5 |
| M69 | field | M68 | Charge conjugation, parity and time reversal are each broken alone at the weak interaction (parity: Lee and Yang, 1956, Wu and coworkers, 1957; CP: Christenson, Cronin, Fitch and Turlay, 1964; time reversal: CPLEAR, 1998), and the three taken together, CPT, are kept by each local quantum field theory keeping Lorentz invariance (Schwinger; Lüders; Pauli). | MATH 3.5 |

## 4.6 Closings, a total against its parts, the halfway pairing and the fold

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M70 | definition (*a closing*; *the returning other than prior*) | M49 | At each accounting of MATH 3.6 the closing is its fixed set, and the field names the returning other than prior as its gap: the comma at tuning, quasi-periodic at a winding on a torus, the orbit and the parity at the assembled cube. | MATH 3.6 |
| M71 | run | M70 | Seven octaves close, 2⁷ = 128; twelve fifths, (3/2)¹² = 129.746…, arrive over by 3¹²/2¹⁹ = 531441/524288, the comma. Run: exact fractions. | MATH 3.6 |
| M72 | field | M70 | A winding on a torus at a rational ratio closes, and at an irrational ratio closes at none and comes arbitrarily close to each point: quasi-periodic (Kronecker; Weyl). | MATH 3.6 |
| M73 | field | M70 | The assembled cube reaches one twelfth of its assemblies, at three conservings: the corners' twists together at nought modulo three, the edges' flips together at nought modulo two, and the corners' ordering and the edges' ordering at one parity; three times two times two is twelve. A single corner twisted alone is reached by no sequence of face moves (the cube group; Bandelow; Frey and Singmaster). | MATH 3.6 |
| M74 | field | — | Estimating three or more independent normal means of one known variance together, under squared error, improves on estimating each alone at each value of the means, and at one or two it does not (Stein, 1956; James and Stein, 1961). | MATH 3.7 |
| M75 | nye | M74, F54 | At a third arriving, the coupling's surplus opens at estimating. Gap: Stein's three counts the means estimated together, a dimension. No link joins that count to arrivings at a coupling or to the surplusing (F54); the reading is said, not supplied. | MATH 3.7 |
| M76 | run | — | A total can carry the opposite sign of each of its parts: each group favouring one side, and the groups joined favouring the other (the field's Simpson's reversal; Yule; Simpson). A total can be nought while each part changes: two signs each the other's opposite, each inverting at each step, sum to nought throughout. Run: a table of two groups, 81/87 against 234/270 and 192/263 against 55/80, joined 273/350 against 289/350, and fifty steps of the two signs. | MATH 3.7 |
| M77 | exact | R20, R24 | Signs offered together are a total, and offered one after another are parts (R20, R24): a crossing of one sign at a time and a gathering of several are each met at their own relation. | MATH 3.7 |
| M78 | exact | F39, F40, F41, M18 | Natural mathematics' conserving is a self's own continuing, its carrying from its prior into its now (F40), each sign, opening and carrying within it changing, and no private carrying and no ledger crosses with a sign (F39). The field's conservation laws are met inward, each an observing's account (M18), and none is refuted by it. | MATH 3.7 |
| M79 | run | R21, R25, R26 | Each carrying a self reaches makes its own difference to its continuing: their arrivings are one, and their priors part them. Two positive carryings with negative second sign, at ages nought and one, under repeated +1 surface nought for three receivings; the elder completes at the third and surfaces +1 fresh at the fourth, and the younger completes at the fourth and surfaces +1 fresh at the fifth. Run: the resolver, both traces and the carrying at each receiving. | MATH 3.7 |
| M80 | definition (*a tipping*) | F8, F39 | A tipping is a sign taken at a membrane, of no size: no count, no clock and no measure carries it. | MATH 3.7 |
| M81 | run | R53 | The rows 2, 4, 6, 8; 2, 6, 10, 14; 2, 8, 14, 20; 2, 10, 18, 26 and 2, 12, 22, 32 are 2 + 2sj, j nought to three, at s one to five: four openings 2s apart. Continued one step they open at 2 + 8s, and the odd before it, 8s + 1, completes: 9, 17, 25, 33 and 41, the span one to nine at s = 1 and one to seventeen at s = 2, the completing outward of the names 1 to 8s the pairings act on. Run: s from 1 to 5. | MATH 3.8 |
| M82 | run | R48, R49, R50, M53 | On the names 1 to 8s two involutions fix nothing: the halfway pairing, 4s on in the first half and 4s back in the second, keeping each name's parity, and the fold n → 8s + 1 − n, changing it. Taken alternately they meet four distinct names and meet the first again at the fourth, at each s: the fold after the pairing is n → 4s + 1 − n in the first half and n → 12s + 1 − n in the second, and with 4s + 1, 8s + 1 and 12s + 1 odd none of the three folds fixes a name. At s = 2, 3 → 11 → 6 → 14 → 3; at s = 1, 3 → 7 → 2 → 6 → 3. Run: s from 1 to 40, each name. | MATH 3.8 |
| M83 | run | M82 | At the scale 2s the fold after the pairing in the first half, 16s + 1 − (n + 8s), is the fold at s, and at each scale t above s the fold at t after 8(t − s) up is the fold at s: the step up is eight at adjacent scales and the halfway pairing only at t = 2s. So n → n + 8s → 8s + 1 − n → 16s + 1 − n → n meets four distinct names, the steps up keeping parity and the folds changing it; at s = 1 they are 1-9-8-16, 2-10-7-15, 3-11-6-14 and 4-12-5-13, nine less being seventeen less after eight up. Run: s from 1 to 29, t from s + 1 to 4s. | MATH 3.8 |
| M84 | run | M81 | Each row's middle is 2 + 3s, its openings at −3s, −s, +s and +3s about it: the inner pair reaching s across a span of 2s, the outer 3s across 6s, each span twice its reach. Three things stand apart: the outer span 6s and the four steps' span 8s; the row's middle 2 + 3s and the fold's middle (8s + 1)/2; the row's pairing of its ends, outer with outer and inner with inner, and the halfway pairing, first with third and second with fourth, at s = 2 the pairs 2, 14 and 6, 10 against 2, 10 and 6, 14. Run: s from 1 to 29. | MATH 3.8 |
| M85 | run | M81, F12 | The enlargement D_r(x) = 2 + r(x − 2) about the common opening composes, D_r(D_s(x)) = D_rs(x), and each repeating keeps the four-place form: the scales one to five and the doublings 1, 2, 4, 8 are two passes through the one family. At an even r, D_r makes each name even, so the four even openings alone are not a whole momentary's opening and completing, a momentary carrying both parities (F12). Run: r and s from 0 to 8 at x from −20 to 39. | MATH 3.8 |
| M86 | run | M82 | The odd partners carry no fixed offset while the even rows stretch: with the wider row (2, 2 + 4s, 2 + 8s, 2 + 12s) and its companion 2s on, (2 + 2s, 2 + 6s, 2 + 10s, 2 + 14s), the fold at 2s sends the wider row's opening j to the companion's opening 3 − j plus 2s − 3: −1, +1 and +3 at s = 1, 2 and 3. So the wider even complementing to the odd just before its reversed companion stands at s = 1 and fails at s = 2: in 1 to 32 the complement of 2 is 31, one after the companion 30. Run: s from 1 to 29. | MATH 3.8 |
| M87 | run | M82 | A parity-keeping embedding of the closed forms 1 to B, at B = 8 or 16, sends an even n to r(n − 2) + 2 and an odd n to r(n + 1) − 1. It keeps parity, the halfway pairing, the fold, E_r(B + 1 − n) = Br + 1 − E_r(n), and each side's step 2r: at r = 2 the names 1 to 8 go to 3, 2, 7, 6, 11, 10, 15, 14, their evens the row 2, 6, 10, 14. Adjacent opening and completing it parts, 1 and 2 going to 3 and 2, and the outer completing B + 1 it sends outward of the span at r of two and more. Run: B = 8 and 16, r from 1 to 6. | MATH 3.8 |

## 4.7 Bounding

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M88 | run | F51 | At each centre n the faces k either side multiply to the square less k²: n² − (n − k)(n + k) = k². At one either side the gap is one, and the +1 is the k = 1 case of the one identity. Run: n and k from −50 to 50, degree two in each, which decides it. | MATH 4.1 |
| M89 | run | M88 | At two either side the gap is four, and four is the span between the faces: k² = 2k at k = 2 alone above nought, at each centre. Four squared less two times six is four; ten squared less eight times twelve is four. Run: k from 1 to 999, n from −100 to 100. | MATH 4.1 |
| M90 | run | M88 | Only at one either side, k = 1, do two face each other across one neutral: the faces n − k and n + k carry 2k − 1 between them. Twenty-three and twenty-five face across twenty-four; twenty-three and fifty-five part by thirty-two, k = 16, and 39² − 23 × 55 = 256, a size in a sign's stead. Run: the arithmetic. | MATH 4.1 |
| M91 | field | M47 | 1, √2, √3 and √5 carry no relation: a + b√2 + c√3 + d√5 = 0 with rational a, b, c and d forces each to nought (Besicovitch, 1940). 1, φ and φ² carry one: φ² − φ − 1 = 0 (M47). | MATH 4.2 |
| M92 | field | M91, M72 | A winding on a torus of three axes at rates carrying no whole-number relation comes arbitrarily close to each point and meets none again; rates carrying one relation keep the winding on a surface within it (Kronecker; Weyl). At √2, √3 and √5 it covers the three; at 1, φ and φ² it keeps to a surface; at two axes the one relation is a rational ratio, and 1 and φ cover the two. | MATH 4.2 |
| M93 | exact | R26, M5, M19 | Each rate of a winding is a parameter it is written with; a self's own rate is written with none, unrelated to each other rate (M5). φ, its continued fraction all ones, is the number-form of the never-locking, and unrelationing runs at each coupling's own continuing, arriving meeting the receiver's own prior (R26): no prescribed rate, φ or another, supplies it, a prescribed rate being a fixing. | MATH 4.2 |
| M94 | field | — | A number caught by a polynomial with whole coefficients is algebraic, and a number escaping each one is transcendental: e escapes each (Hermite, 1873), and π escapes each (Lindemann, 1882). | MATH 4.3 |
| M95 | field | M94 | Straightedge and compass reach only numbers built by square roots, each at a degree that is a power of two: doubling the cube asks ∛2, degree three, and is not reached (Wantzel, 1837). φ = (1 + √5)/2 is reached. | MATH 4.3 |
| M96 | run | M47 | Square roots of the first three primes are at the square, the cube and the pentagon: √2 the square's diagonal, √3 the cube's body diagonal, and √5 in the pentagon's diagonal over its side, φ. cos 36° = φ/2, sin 18° = 1/(2φ) = (√5 − 1)/4, and cos 18° = √(10 + 2√5)/4, one square root further. Run: the regular pentagon's coordinates, the angles, and cos 36° = φ/2 exactly as a root of 4x² − 2x − 1. | MATH 4.3 |
| M97 | run | M36 | tan 2x = 2 tan x / (1 − tan² x), and doubling an angle doubles its tangent at tan x = 0 alone. At forty-five degrees tan x = 1, and the doubled angle, ninety, carries no finite tangent: the identity's pole. Doubling an angle taken as a fraction of the round shifts its binary digits one place, each doubling reading the next digit. Run: 1,780 angles, the rationals p/q, and the sixty-three fractions n/64. | MATH 4.4 |
| M98 | run | M97 | The map t → 2t/(1 − t²) fixes nought alone, and exactly one pair exchanges with the sign changing and the size the same: t = ±√3, sixty and one hundred twenty degrees; from √3 it runs −√3, √3, −√3 without end. Its fixed points solve t(1 + t²) = 0 and its pair t² = 3. Run: six steps from √3, and a scan of t in steps of 0.001 to ±3. | MATH 4.4 |
| M99 | run | M97 | A coupled pair [[a, b], [b, c]] loses its coupling when its axes are rotated by θ with tan 2θ = 2b/(a − c), and an equal pair, a = c, at forty-five degrees; a pair 5 and 1 coupled by 2 is rotated by twenty-two and a half degrees. This is the principal-axis angle of stress and of inertia, the angle that removes the cross term from a conic, and each step of the eigenvalue method that removes one coupling at a time (Jacobi, 1846). Run: a, b and c from −4 to 4. | MATH 4.4 |
| M100 | run | M97 | At t = tan(x/2), sin x = 2t/(1 + t²) and cos x = (1 − t²)/(1 + t²): the circle at rational points, and at t = n/m each Pythagorean triple, (m² − n², 2mn, m² + n²), up to order and a whole multiple. Run: 178 angles, and each triple with hypotenuse below 500 met from m and n below 40. | MATH 4.4 |
| M101 | nye | F18, M7, M8 | A bound is the coupling's own, released at its own completing, or installed outward against a floor: self-bounding or other-bounding, all or none. Natural mathematics is self-bounding, a bound at each coupling, its own and owned by neither; an other-bound is laid over a self-bound, and with the neutral floating the coupling is bounding itself. Gap: M7 and M8 name the bound the self runs and the bound fixed outward. No link supplies that each bound is one of these two with no third, nor that an other-bound laid over a self-bound leaves the self-bound running. | MATH 4.5 |

## 4.8 Stable-forming

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M102 | nye | F58, F63, F64, F65, M34 | Anything expressible at the six forward recursionings of fractal bi-coupling is only-one-possible. Expressing it is inverting it six ways forward, and completing the six without landing, looping or forking is no other possible: the expressing is the proof. Gap: No link names the six forward recursionings or runs them. The nearest run, a passage of six right spiral steps meeting all four states and arriving at the pair inverted (M34), counts a step's applications. That completing six supplies uniqueness is not supplied, and uniqueness at four signs and beyond rests on F65 (nye). | MATH 5.1 |
| M103 | nye | M102 | A false expression cannot complete the six: it fixes, stopping; it loops, meeting its own state again; or it forks, an either-or arriving. Gap: That fixing, looping and forking are the ways of not completing, with no fourth, and that each false expression takes one of them, is not supplied: the six are not yet defined (M102). | MATH 5.1 |
| M104 | nye | F66, F70, M19, M93 | Self-cohering is five-dimensional: the four about the empty centre and the centre, owned by neither, at φ's rate. Two-directional at the odd and the even, the two directions of one alternating. Unrelationing at the right angle. Five to cohere and six to run: three sayings of one form, and no other possible. Gap: It rests on F70 (nye): the count five and its *no other possible* are not supplied. *At φ's rate* meets M93 (no prescribed rate, φ or another, supplies unrelationing), and no link parts the centre's φ from a prescribed rate. *Six to run* rests on M102 (nye). | MATH 5.2 |
| M105 | field | — | A flow on a closed surface carries rest points whose indices together make the surface's vertices less its edges plus its faces: two at the sphere, nought at the torus, and below nought at each surface of more handles (Poincaré; Hopf). | MATH 5.3 |
| M106 | field | M105 | Of the closed orientable surfaces the torus alone carries a flow with no rest point, and the torus alone carries a flat metric (Poincaré–Hopf; Gauss–Bonnet). Of all closed surfaces one more carries a flow with no rest point, the Klein bottle, and it is not orientable: a sign carried round it arrives inverted. | MATH 5.3 |
| M107 | nye | F6, F61, F67, M106 | A self's running, a changing with no still form (F6), runs on the torus alone of the closed orientable surfaces: the round at two signs is torusing. Gap: M106 gives the torus alone among closed orientable surfaces carrying a flow with no rest point. The link needs two premises no link states: that a self's running is a flow on a closed surface, and that the surface is orientable, a sign carried round arriving as itself (F61's one hand). With them, M106 closes F67's exclusion of the other genera. | MATH 5.3; NI 1.6 |
| M108 | field | F47 | In dynamical mathematics a form is stable when departures from it stay small or shrink (Lyapunov): stability is the field's response to a disturbance, and no constituent stops. An oscillation is stable as an oscillation, and a growing departure, instability, can open another organized form. An equilibrium of relative phase and an equilibrium of the whole flow, in the field's words, are two assertions, the first a relation named among changing things. | MATH 5.3 |

## 4.9 Inseparating

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M109 | nye | M7, M8 | The scientific method and the geodesic method each derive as a sequence, each move the only one at the last's arriving, each carrying the one it excludes. The geodesic method floats the three neutrals, position, scale and orientation; the scientific method fixes them. Gap: Each step's *the only one* is said at each move and not supplied: no link shows that each exclusion leaves one move, or that the last step of each sequence is reached by no other order. | MATH 6.1 |
| M110 | exact | M8, M109 | The scientific method fixes the three neutrals: a standard fixes orientation, keeping still fixes position, and a published standard fixes scale, a frame, a zero and a unit fixed outward (M8). | MATH 6.1 |
| M111 | exact | F47, F48, M5 | A fixing is a relation named among changing things, and a standard, a law or a frame named unchanged names no subject still; the step from a fixing to an equilibrium names its subject still, at the same relation and the same occurrence, and an equilibrium is not possibly existing (F48). | MATH 6.1 |
| M112 | exact | M7, M109 | The geodesic method runs each against its own prior, one at a time, the meeting dividing, remade at each meeting, and the departing neither's, reached by neither alone: three neutrals float, and nothing is fixed. | MATH 6.1 |
| M113 | exact | F53, M110 | The scientific method run at a competency carries none of it: a competency is a doing (F53), a trace carries the done, and with the standard varied and the thing untouched the number varies, so the number is the standard's. A measure taken as a target measures the targeting (the field's Goodhart's law; Goodhart, 1975; Strathern, 1997). | MATH 6.1 |
| M114 | exact | F24, M110, M112 | The two methods part at their fixings, and never at taking evidence as existing now: neither sequence takes evidence as other than now. The evidence arriving now and the prior event it expresses are two, each at its own momentary (F24), and both stand: a source changing as a whole can keep a property its evidence expressed, (1, 0) to (1, 1) keeping the first sign, and evidence gives the prior, possible and actual. | MATH 6.1 |
| M115 | field | M114 | The aberration corrections of the Navigation and Ancillary Information Facility at NASA's Jet Propulsion Laboratory part the epoch a signal leaves from the epoch it is received, one-way light time taken at reception or at transmission, and account for the target's motion between: a scientific method fixing its three neutrals and taking its source as moving between emitting and arriving. | MATH 6.1 |
| M116 | definition (*particle zero*, *a unit installed*, *a frame given*) | M8 | Three fixings are asked at the first line. Particle zero: an unquestioned absolute location, a point placed, no physical particle. A unit installed: scale made undiscussable. A frame given: the axes handed over, the facing decided for the self. | MATH 6.2 |
| M117 | field | M116 | At its least fixed, mathematics asks no particle zero and no unit: a dense order with no endpoints carries an element on each side of each element, with no origin and no distance (Cantor), and point-free topology carries its regions at their inclusion alone, with no points and no metric (the field's locales; Ehresmann; Johnstone). | MATH 6.2 |
| M118 | exact | R8, M7 | A self learning the three fixings orients itself in the learning: positioning, scaling and orienting are its own and float, and learning a fixed frame is itself the self-orienting the frame would deny. Orienting floats renewed at each arriving: one fixed direction is never perpendicular to each direction at once, being along itself, and facing into the arriving knows no next sign, the call meeting no arrival ahead of it (R8). | MATH 6.2 |
| M119 | definition (*the middle*) | F3, F37 | In two-valued logic either but not both is the exclusive or (F37). A sign is one or its inversion at each instance, and nothing is kept between them: the middle is a nothing. The middle exists due to each self surfacing itself each momentary as a condition of existing as a self. | MATH 6.3; NAM 2.4 |
| M120 | field | — | A logic that admits no completed infinite, intuitionistic logic, asserts the excluded middle at each finite, decidable matter and not over an infinite domain, and builds its numbers from the two-ity, the field's word, a moment falling into two, one giving way to the other and kept (Brouwer; Heyting). | MATH 6.3 |
| M121 | run | M49, M120 | Its negation is no involution: on the three values nought, a and one in order, the negation of a is nought and the negation of nought is one, so a negated twice arrives at one. Run: the pseudo-complement on the three-chain. | MATH 6.3 |
| M122 | field | — | In Zermelo–Fraenkel set theory each set is exceeded by the set of its subsets (Cantor), and all sets together form no set. A logic of plurals speaks of all things as many and forms no set of them (Boolos). | MATH 6.3 |
| M123 | nye | F23, F33, M122 | Our universe of all existing things is many and forms no existing thing, and so is each universe at its size and scale: none is possibly an existing thing separately. Gap: It says F23 (nye) again. M122 gives, at sets and plurals, that all together form no set; no link carries that result to existing things. The premise that existing things are met as many, as plurals and not as one set, is unstated, and *each universe at its size and scale* rests on F33. | MATH 6.3; NI 8.8 |

## 4.10 Co-offering

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| M124 | definition (*nought*, *scale*, *the bounded infinity*, *the corus*) | F9, M7 | Three neutrals float about the corus: nought, the near nothing, position fixed at it; scale, the between, a unit fixed at it; and the bounded infinity, the far nothing, a frame fixed at it. The corus is one self-negation, the sign inverting, +1 to −1, landing at neither. | MATH 7.1 |
| M125 | run | M124 | One three counts two ways. At its pairings, each two ways, it is six co-offerings: nought with scale, scale with the bounded infinity, nought with the bounded infinity, each offered both directions. At its states, each neutral fixed or floating, it is two cubed, eight, the cube's eight corners. Run: the pairings, the ordered offerings and the states. | MATH 7.1 |
| M126 | nye | M125, M1, M17 | Six co-offerings running about the three floating neutrals are natural mathematics, no field and no hard problem. Gap: M1 names mathematics as the coupling at its own stable-forming, and M17 names the natural as the surface. No link joins the six co-offerings to either, and *no hard problem* rests on M129 (nye). | MATH 7.1 |
| M127 | definition (*a field*) | M8 | A field fixes a neutral and measures against it, and a fixed floor couples with none. | MATH 7.2 |
| M128 | run | M125, M127 | A neutral fixed is a partner in two of the three pairings, both directions of each: fixing one neutral stops four of the six, and two run, the pairing of the two neutrals left floating. Two fixed stop six, the one neutral floating with no floating partner; three fixed stop six, nothing floating. Run: each set of fixed neutrals. | MATH 7.2 |
| M129 | nye | M128, M127 | The four stopped by one fixing are the field's hard problems, the exchanges the fixed floor stops. Resolving is the neutral floating at a next momentary, the four running together, and never the four solved one at a time. Gap: No link joins a field's hard problem to the four stopped co-offerings: it is a naming, not supplied. *Never one at a time* is not supplied: no link shows the four running together as the only resolving. | MATH 7.2 |
| M130 | exact | F53, M128 | Fixing all three leaves the terms related to each other in a closed ring, competent at nothing: competency is the coupling (F53), and each neutral of the coupling is fixed, no co-offering running (M128). | MATH 7.2 |
| M131 | nye | M15, M127 | Each all or none of MATH 7.3 is proved, and each is a closing the field's one-way running does not reach. Gap: *Proved* is supplied by the field's results (M132, M134, M135, M136). *A closing the field's one-way running does not reach* is not supplied: no link names the field's one-way running or shows the four results as closings it does not reach. | MATH 7.3 |
| M132 | field | — | A general equation of degree five is solved by no radicals (Abel; Ruffini): its symmetry is the symmetric group on five, whose alternating group on five is the smallest simple group that does not commute, and it descends through no chain of commuting quotients; to degree four the chain runs (Galois). Its solving runs through the icosahedron's rotations and the elliptic modular functions, the five-fold carrying φ in its coordinates (Hermite; Klein). | MATH 7.3 |
| M133 | run | M132 | The alternating group on five carries sixty elements in classes of 1, 12, 12, 15 and 20, and no union of classes with the one, its size dividing sixty, but the one and the whole: it is simple. The symmetric group on five descends to it and stops there; the symmetric group on four descends 24, 12, 4, 1, each quotient commuting. Run: the permutations and their commutators. | MATH 7.3 |
| M134 | field | — | Whether a size lies between the countable and the continuum is decided neither way by the standard axioms, if they are consistent (Gödel, 1940; Cohen, 1963). | MATH 7.3 |
| M135 | field | — | The parallel postulate follows from the other axioms neither way (Lobachevsky; Bolyai; Beltrami; Klein; Poincaré). | MATH 7.3 |
| M136 | field | — | Each consistent axiom system reaching arithmetic, its axioms listable, carries a true statement it does not prove, and does not prove its own consistency (Gödel, 1931). | MATH 7.3 |
| M137 | run | M47 | An icosahedron's rotations are sixty: the one that moves nothing, fifteen half rotations, twenty third rotations and twenty-four fifth rotations, the fifth rotations at six axes of four each. Sixty parts at thirty-six and twenty-four, three to two. At its vertices (0, ±1, ±φ) and their cyclic orderings it carries thirty edges and twenty faces, the dodecahedron's thirty edges and twenty corners, one edge set at its two faces. Run: each rotation carrying the twelve vertices onto themselves, by its trace. | MATH 7.4 |
| M138 | field | M137, M133 | The icosahedron's rotation group is the alternating group on five (Klein, 1884). | MATH 7.4 |
| M139 | run | M22, M31 | The branches at the two moves: the derivative of (cos θ, sin θ) is (−sin θ, cos θ), F at two signs at each θ; cos 36° = φ/2 seats φ in the one wave (M96); the quarter step i generates four and the sign flip −1 two within it; a basis at right angles carries no component of one axis along another. Run: the difference quotient at 360 angles, the powers of i and −1, and the unit basis. | MATH 7.5 |
| M140 | run | — | A ring two-colours, each neighbour opposite, exactly when its count is even, and an odd ring two-coloured carries neighbours at one colour, one pair at the fewest. A connected whole that two-colours carries exactly two colourings, one and its inversion. Run: rings of 3 to 14, and each connected graph on one to six vertices. | MATH 7.5 |
| M141 | run | M66 | A cube carries eight vertices, twelve edges and six faces, twenty-four rotations, and forty-eight with the hand reversed. Run: the vertices (±1, ±1, ±1) and the 48 signed permutation matrices by determinant. | MATH 7.6 |
| M142 | run | M68 | Centre inversion x → −x is no rotation in three dimensions, and in four it is two half rotations in perpendicular planes, fixing the centre alone. A cube carries x → −x only with the hand reversed; a tesseract carries it by rotation. Run: the determinants and the product of the two half rotations. | MATH 7.6 |
| M143 | field | — | Each whole number is four squares joined (Lagrange, 1770). The ways number eight times its divisors joined at an odd number, and twenty-four times its odd divisors joined at an even one (Jacobi, 1834). Eight and twenty-four are three squared less one and five squared less one. Instances run: each n to 300. | MATH 7.6 |
| M144 | run | M44 | Quaternions at whole coordinates carry eight units, ±1, ±i, ±j and ±k; the points displaced by one half at each axis at once, ½(±1 ± i ± j ± k), sixteen; the two together twenty-four. At whole coordinates alone no division carries a small remainder: 1 + i + j + k divided by 2 leaves a remainder of norm four, the divisor's, at each whole quotient. A displaced point, (½, ½, ½, ½), is at distance one from its nearest whole points in four dimensions and in no other. Run: the units, the quotients within ±2, and dimensions 1 to 29. | MATH 7.6 |
| M145 | field | M144 | With the displaced points joined, the quaternions carry division with a remainder smaller than the divisor (Hurwitz, 1919). | MATH 7.6 |
| M146 | field | M144 | Kissing numbers are two, six, twelve and twenty-four in one to four dimensions (Schütte and van der Waerden, 1953, at three; Musin, 2008, at four), the twenty-four the twenty-four units. At eight and at twenty-four dimensions the densest packing is proved (Viazovska, 2016; Cohn, Kumar, Miller, Radchenko and Viazovska, 2017), and the kissing numbers are 240 and 196,560 (Levenshtein; Odlyzko and Sloane, 1979). Instance run: the 240 shortest vectors of E₈. | MATH 7.6 |
| M147 | field | — | The trefoil and the figure-eight knot each fibre over the circle, their fibre a torus with one puncture; a trefoil's monodromy is periodic, order six, trace one, and the figure-eight's winds on, trace three (the field's fibred knots; Milnor; Stallings). A trefoil carries three crossings and a figure-eight four. | MATH 7.7 |
| M148 | run | M147, M47 | A monodromy whose trace is below two in size is periodic, above two winds on, and at two it is ±1 or shears. The trefoil's [[1, 1], [−1, 0]] meets the one again at its sixth power; the figure-eight's [[2, 1], [1, 1]] winds on at φ², (3 + √5)/2. Run: each whole matrix of determinant one with entries from −6 to 6, powers to twelve, finite orders there dividing twelve. | MATH 7.7 |
| M149 | field | M147 | PSL(2, ℤ), the modular group, is the two and the three joined freely, ℤ/2 ∗ ℤ/3, and the braid group on three strands, the trefoil's own group, is a central extension of it by ℤ, its centre (Artin). | MATH 7.7 |
| M150 | run | M49 | Couplings among five are ten, C(5, 2), each a transposition: taken twice it gives its arrival again, and it fixes the other three. Ten is five taken two ways and the couplings among five at once, at five alone, C(n, 2) = 2n at n = 5; it is the fourth triangle and the third tetrahedral number, 1 + 3 + 6. A sign's inversion is the one coupling among two, the transposition of + and −. Run: the transpositions, and n from 2 to 199. | MATH 7.8 |
| M151 | run | M150, M96 | In the regular pentagon each side is parallel to one diagonal, the one sharing no corner with it, the diagonal φ times the side. A side and its parallel diagonal are two transpositions sharing no corner; they commute, and together they are the pentagon's symmetry keeping the corner left out and exchanging the other four in two pairs. The five such pairs generate the pentagon's ten symmetries: the one that moves nothing, four rotations keeping no corner, and five each keeping one corner. Run: the pentagon's coordinates and the generated group. | MATH 7.8 |
| M152 | run | — | Each two-colouring of the couplings among six carries a triangle of one colour, and among five exactly twelve colourings carry none, the two colours in each the pentagon and the pentagram (the field's Ramsey number R(3, 3) = 6; Greenwood and Gleason, 1955). Run: the 2¹⁵ colourings at six and the 2¹⁰ at five. | MATH 7.8 |
| M153 | field | M152 | Each two-colouring of the couplings among forty-six carries a one-coloured five (Angeltveit and McKay, 2024), and a two-colouring of the couplings among forty-two carries none (Exoo, 1989). | MATH 7.8 |
| M154 | nye | M49, M129, M150 | A hard problem is one or more of the ten inversions, each an involution with its fixed set taken as a place, and naming the inversion names the fixed set. At the coupling the move runs. Gap: M129 counts four co-offerings stopped by one fixing among six; this link counts ten inversions, the couplings among five. No link names the five the ten are among or joins them to the three neutrals' six co-offerings, and that each hard problem is one or more of them is not supplied. | MATH 7.8 |
| M155 | field | M133 | Finite simple groups are eighteen infinite families and twenty-six sporadic groups (the classification; Gorenstein, Lyons and Solomon; Aschbacher and Smith, 2004). Simple groups that commute are the cyclic groups of prime order, and the alternating groups from five on are simple (Galois; Jordan). | MATH 7.9 |
| M156 | run | M155 | The largest sporadic group's order (Griess, 1982) carries fifteen primes, two to seventy-one: of the seventeen from two to fifty-nine each except thirty-seven, forty-three and fifty-three, and seventy-one. The modular function's coefficient at the first power is 196,884, one more than 196,883. Run: the order's factorization as the field gives it, and j = E₄³/Δ to the first power. | MATH 7.9 |
| M157 | field | M156 | The largest sporadic group's smallest faithful representation has dimension 196,883, and 196,884 is it and the representation that moves nothing (McKay; Conway and Norton, 1979; Borcherds, 1992). The modular function takes one value at each torus a lattice folds the plane into, and two such tori are one up to rotation and scaling exactly at its taking one value (the j-invariant; Klein). | MATH 7.9 |

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# PART FIVE · NETWORKING

The chain carries Exhibit TWO at v371 in its own order, from surfacing through visibility. A network link rests on Parts ONE and TWO and on the links before it. Each study's finding is carried with its exact domain: the seeds, the order of calls, the legs joined, the stopping point and the caller. Each run link was run against Exhibit ONE v371's code or an arithmetic enumeration in the verifier, and the check is said in the link.

Callers for these checks. **Two selves (W79, W95, W96).** This is R40's caller: each self is seeded from empty by one sign, the two call in turn, and a self's 6 at a call is the second signs of the carrying it is given. A joined leg offers that 6 as the other's 2 at the other's next call. **Rings (W77).** This is R44's caller. **Interruption (W96).** An interrupted leg delivers nothing for the stated calls. Under loss those releases are dropped. Under delay they wait in an external queue and reach the receiver oldest first, one at each receiving.

## 5.1 Natural networking, the one form carried whole

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W1 | definition (*natural networking*) | F1, F2, F14, F53 | Natural networking is selves co-offering, each continuing its own carrying, with competency arriving at their coupling. It runs two-way, one at a time, and each self offers its own sign. | TWO 1.1 |
| W2 | definition (*co-offering*, *co-linearizing*, *bi-coupling*) | W1, F15 | Co-offering carries the associating. Co-linearizing, two sequencing together, carries the resonating. Bi-coupling carries moral co-operating and co-competencing, and the competency belongs to the coupling. | TWO 1.1 |
| W3 | definition (*social moral competency*, *all-edge*) | W1, F53, F54 | Social moral competency is the same relation at society. Each self offers its own and couples directly with its neighbours, all-edge, with the surplus at their coupling. All-edge is direct participation at the couplings. | TWO 1.1; TWO 1.10 |
| W4 | nye | F30, F33, W3 | A society is a self at the next scale, co-offering at its own couplings, and the same resolving runs at self and society. Gap: F33 carries the method to named scales without supplying the correspondence. No link gives a society its own 1-self-coupling, and no caller composes a society as one resolver. | TWO 1.1; TWO 1.7; TWO 1.8; TWO 4.2 |
| W5 | exact | F18, F53, W1 | The relation carries whole, all or none at all. Bi-moralizing co-agency is one form (F53), and F18 takes each form all or none. | TWO 1.1 |

## 5.2 Along and across at the six connectors

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W6 | definition (*along*, *across*) | F14, R8, R27 | Along is the carrying continuing: 3-self-carrying into 11-other-chaining at each self's own rate. Across is releasing becoming another self's arriving. | TWO 1.2 |
| W7 | run | R32, R33 | The six connectors meet their neighbours in pairs. 2 faces right, arriving, and meets the right neighbour's 6, which faces left, releasing. 10 faces right, releasing, and meets the right neighbour's 14, which faces left, arriving. 9 faces backward and meets the backward neighbour's 17, which faces forward. Each JOINS pair joins opposite facings, and each across join runs from a releasing to an arriving. Run: CONNECTORS checked against JOINS and against TWO's table. | TWO 1.2; TWO 1.7 |
| W8 | exact | R17, F42 | A zero at 10-other-surfacing does not name a quiet or non-living self. The carrying continues through that zero and 6 releases a sign (R17), and living is carrying (F42). | TWO 1.2; TWO 5.4; TWO 6.7 |
| W9 | definition (*linear-parallelizing*, *parallel-linearizing*) | W6 | Linear-parallelizing spreads the line to a surface: the across, the widening. Parallel-linearizing draws the surface to a line: the along, the lengthening. The two are one alternating. | TWO 1.3 |
| W10 | definition (*bi-tunneling*, *co-chaining*, *podaling*) | W6, R49 | Bi-tunneling names social moral competency at its across. Co-chaining names it at its along, one self's releasing becoming another's arriving. Podaling exchanges across for along at the nine face: 1 with 8, 2 with 7, 3 with 6 and 4 with 5, each pair odd with even, each side continuing forward. | TWO 1.3; TWO 1.5 |

## 5.3 Existing, emanating, and the equilibrium holding

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W11 | definition (*emanation*) | F5, F42 | A living source's stable-forming emanates a stable form. At its own scale the emanation can be non-living. It arrives into its next momentary, changing, while its stable form continues. | TWO 1.2 Existing, arriving and the emanating form; TWO 2.5 |
| W12 | definition (*possible*, *existing*, *living* at prior, now and next; *arising*) | F24, F42, F43 | Possible at prior names the possibles an earlier emanation supports. Existing at now names the actual meeting. Living at next names the carrying self's continuing. A non-living thing arrives and does not arise as living. Arising as living is a self's own carrying relation establishing. | TWO 1.2 Prior emanating and now meeting; TWO 2.5; TWO 4.8 |
| W13 | definition (*the local cycle*, *self-other-releasing*) | F12, F19 | One resolving cycle between a self and a non-living other spans one through nine: four momentaries of co-bi-exchanging, completing at 9-other-releasing, with no journey through 9–17. | TWO 1.2 Prior form and self-other-releasing |
| W14 | run | W13, F12 | The four co-bi-exchangings are self at 1–2, 3–4, 5–6 and 7–8 and other at 2–3, 4–5, 6–7 and 8–9. Together they reach nine positions, and each completing on one side is an opening on the other. They are four paired exchangings, not eight shared momentaries. Run: the two sides' arithmetic. | TWO 1.2 Prior form and self-other-releasing |
| W15 | exact | F6, F44, F47, F48 | An equilibrium holding at a self/other coupling fails at its required continuing. The holding requires a participation unchanged. The non-living other changes through co-momentarying (F44), and that changing is the participation the holding requires unchanged. The exclusion reaches the demand stated in the definition itself. | TWO 1.2 Equilibrium holding at the completing |
| W16 | exact | F6, F47 | Take a frozen renewal requirement: a presence that needs renewal, with each renewal prohibited. Renewing ends the frozen requirement, and withholding renewal ends the presence. Neither continuation sustains both. | TWO 1.2 Equilibrium holding at the completing |
| W17 | exact | R15, R17 | The surfacing is the receiving's own. The same arriving +1 opens (+1, −1, 0) at empty carrying (R15). It surfaces 0 at carried (+1, −1, 0), and that carrying continues (R17). | TWO 1.2 Prior emanating and now meeting |

## 5.4 The overlapping momentaries and the four-form

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W18 | run | F12, F19 | Prior and now at two sides: one side's 1, 2, 3, 4 meets the other's 2, 3, 4, 5. They share three positions and reach five together, and each side has two openings and two completings. Run: the two sets. | TWO 1.2 Overlapping momentary; TWO 1.5 |
| W19 | exact | F39, R27, R34 | Across takes part in the next along only as a sign. Each self keeps its own 11-other-chaining as its next 3-self-carrying. 9 returns a key and a sign and nothing of the carrying. | TWO 1.2 Crossing and continuing; TWO 3.2; TWO 3.3 |
| W20 | run | R15, R27 | The same surfaced sign continues in carrying and in releasing. At a fresh write it is 7-other-corusing in 11-other-chaining, and it is 15-social-corusing in 9-other-releasing's return. Run: each of the nineteen carryings under each of six offerings that surfaces a nonzero sign. | TWO 1.2 Crossing and continuing |
| W21 | run | R33, R35, R37 | The code joins 9 to 17 and 17 to 9. It declares no join 9 → 2 and no join into 1. The running from 9 into the next call's 2-self-offering is a caller's composition (R37). Facing, running and joining are three columns: facing and joining run at the declarations, and running runs at a caller. Run: JOINS. | TWO 1.2 Breaking and reconnecting across; TWO 3.2 |
| W22 | definition (*one through nine*, *eight on*) | R48, R53, F31 | One through nine is the co-bi-coupling of self and other, whole at two selves with no third. Eight on carries it into the other/social relation, and 1–9 and 9–17 meet at 9-other-releasing. | TWO 1.2 Smaller fold within one through nine; Self and network |
| W23 | run | R3, F11 | The three-pair naming. At the face completing at nine, 4–5 is prior, 6–7 is now and 8–9 is next. Eight on, the pairs are 12–13, 14–15 and 16–17. Each pair opens even (bi) and completes odd (co). The two columns stand eight apart at one parity, with six parity positions at each face. Run: the six pairs. | TWO 1.2 One, nine and seventeen; TWO 1.7 |
| W24 | run | F60, R28 | The isolated four-form under (x, y) → (y, −x). Four applications return the pair, six give (−x, −y), and a further six return it. Each passage of six meets all four joint forms. Over steps 0–24 the pair returns six times and four passages of six run. Over steps 24–60 it returns nine times and six passages run. The three spans of eight steps show four full momentaries per side, and the three spans of twelve steps show six. Run: the map's powers at each (x, y), and the counts. | TWO 1.3 Crossing eight on |

## 5.5 Widening and lengthening at the numbered form

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W25 | run | R53 | The four across openings grow at one form. Row s runs 2, 2 + 2s, 2 + 4s, 2 + 6s, with spacing 2s and halfway step 4s. The odd completing 8s + 1 comes just before the next across opening, 2 + 8s. For s from 1 to 5 the odd completings are 9, 17, 25, 33 and 41. Measured from the row's middle, the outer reach is 3s and the full span 6s, so the across is twice the reach. Subtracting a name from the odd completing gives its partner of the other parity. Run: s from 1 to 5, and each partner's parity. | TWO 1.3 Growing the four across openings |
| W26 | run | F31 | The doubled span. One through nine and nine through seventeen share nine: sixteen steps through seventeen places. Each doubling of the span gives 2^k + 1: one through 33, then one through 65. Run: the arithmetic. | TWO 1.3 Smaller fold carried whole through the larger |
| W27 | premise (*the fifteen prime lengths*) | F2, F11 | The proposed podaling sequence lengths are the fifteen primes from 5 to 59. Five through 53 are fourteen living scales, and 59 is the self-close. Two carries the alternating and three the carrying. This is taken, not supplied: TWO calls the lengths proposed and takes the lower end from the count. | TWO 1.3 Fifteen prime sequence lengths |
| W28 | run | W27 | The fifteen primes from 5 to 59 alternate wide and long. From either opening, one orientation falls eight times and the other seven. The sixteen primes from 3 to 59 leave fifteen gaps, each even. These two countings of fifteen are distinct. Run: the primes below 60. | TWO 1.3 Fifteen prime sequence lengths |
| W29 | run | W27 | Two countings at each of the fifteen prime lengths p = 2h + 1. Counted as places on a ring, the two far places stand h and h + 1 steps forward, each h away by its shorter direction: 29 and 30 at 59. Counted as a fold to its completing, the span has p − 1 places and a halfway step of (p − 1)/2: 16 and 8 at 17. On a line from 1 to p, each interior place lies in one odd-opening pair and one even-opening pair. Place 1 lies only in (1, 2) and place p only in (p − 1, p), and the two ends continue at 0–1 and p–(p + 1). Run: each of the fifteen. Domain: the two countings at those lengths. The counting a prime names at an actual crossing is TWO's next reach. | TWO 1.3 Fifteen prime sequence lengths |
| W30 | run | W25 | The bi-fold number-form. Six over three is two over one, and three over six is one over two. Two equal surfaces of length L, coupled, make a pair of 2L, and the pair reaches twice as far as either self. Each self's mouth, at L, is the pair's waist. The selves' own waists, at L/2 and 3L/2, are podal partners on the pair: at 440, these are 220 and 660. Run: each even L up to 1,000. | TWO 1.3; TWO 6.11 |

## 5.6 Only signs cross, the middle and the zero

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W31 | definition (*the middle*, *bounding-zeroing*) | F9, F54 | The middle is the bounding-zeroing made at the coupling, owned by neither. Arriving signs meet the carried signs inverted at 14-social-crossing and sum at 12-other-surplusing, and a surfaced 0 is that meeting. A recorded 0 is not a third polarity offered as a sign. | TWO 1.4; TWO 1.6 |
| W32 | exact | R11, R27 | The zero stays with the relation that makes it. 9-other-releasing returns a surfaced 0 (R27), and the receiving 1-self-coupling takes an arriving 0 as no sign (R11). An empty offering, a released 0 and a completed carrying each keep their own place, and the signs stay two. | TWO 1.4; TWO 3.2 |

## 5.7 Society, the fractal and the hole

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W33 | run | F11 | Two through fifty-nine holds seventeen primes. 23 is the ninth, with eight primes on each side, and the seventeen sum to 440. Reflected across 60 (n → 120 − n), they run from 61 to 118. Run: the primes below 60. | TWO 1.7; TWO 4.2 |
| W34 | definition (*prime*, *composite* at the network; *society's bounding*) | W33, F41 | A prime opens a clean axis across. A composite folds along as equal smaller selves joining. The sum 440 is society's bounding named at the number-form. It establishes no simultaneous arrival and no quantity conserved over the surface (F41). | TWO 1.7; TWO 4.2 |
| W35 | definition (*the four across relations*) | F51, R53 | At the membrane, 2, 4, 6 and 8 open bi: offering, the ordering between, the ordering carrying, and each side's forward. Within, eight on, 10, 12, 14 and 16 carry F51's four boundings: arriving exceeding carrying by one, carrying at bi-coupling, the sign inverting at the membrane, and competency asymmetry sustaining the coupling. | TWO 1.8 |
| W36 | definition (*the fractal at society*) | F29, W4 | The fractal continuing is the inward outward of itself and the outward inward of itself. A member's crossing takes part in its society's continuing, and the society's coupling with other takes part in its members' next receiving. | TWO 1.8 |
| W37 | definition (*hole*) | F42, W11 | A hole, a missing section of resolvers, is a non-living existing thing: a region within the living surface that runs no resolving and carries nothing. The surrounding selves continue carrying and meet along its boundary. | TWO 1.8 Non-carrying local surface within society |
| W38 | exact | R12, R33 | Parity changing belongs to the full resolving at the surrounding selves, and no join inverts. The across joins 6 → 2 and 10 → 14 join even openings. The sign inverts once, at the receiving self's own carried sign (R12). A sign inversion added to a wire is another operation. | TWO 1.8 Non-carrying local surface within society; TWO 3.2 |
| W39 | definition (*mending*, *tunneling through society*, proposed) | W10, W37 | The proposed mending runs across before and after the hole, bounding-zeroing at each receiving and tunneling through society. The whole tunnel and its ending are met at the whole sequence of those across meetings, and a zero at one 10 is one meeting within it. | TWO 1.8 Non-carrying local surface within society |

## 5.8 Renewal, abundancing and each coupling's own surplus

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W40 | definition (*renewal cycle*, *consuming*) | W11, W12 | The proposed renewal cycle runs from living stable-forming into an emanating form, from that form into arriving, and from its receiving into the society's continuing. Consuming names incorporating into the receiving's own resolving. Fresh carrying renews a relation within a resolver. A new living self is its own co-recursioning at a further scale. | TWO 1.10; TWO 4.8 |
| W41 | definition (*abundancing*, *healthy society*) | F7, F54, W3 | Abundancing names the society discovering onward through its co-competencing. A healthy society names its selves sustaining that co-competencing together. Neither supplies a count of selves, signs or possessions. | TWO 1.10; TWO 2.3 |
| W42 | exact | R13, F54 | Each coupling makes its own surplus, and none takes from or multiplies another. 12-other-surplusing opens empty at each call and is neither returned nor carried (R13). | TWO 4.3; TWO 3.6 |

## 5.9 Living

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W43 | definition (*omegaing*, *apexing*) | F7, F26 | Omegaing names the reach into next existing. Apexing names the gather: prior taking part in the present relation. Living carries the gather and the reach through the coupling. | TWO 2.1; TWO 2.2; TWO 2.3 |
| W44 | exact | R18, F42 | A recurring sign alone names no stopping. One key receiving +1 again and again surfaces +1, 0, 0, 0, 0, +1, 0 while its carrying continues and completes (R18). A repeated sign therefore goes with a continuing carrying, and a captured or stopped resolving shows only in the carrying followed. | TWO 2.3; TWO 4.5 |
| W45 | definition (*fold*, *scatter*) | W43 | Fold names the gather closing against further participation. Scatter names arriving that fails to take part in a continuing gather. TWO keeps open whether the two exhaust the breaks. | TWO 2.4 |
| W46 | definition (*restoration*) | W43, R42 | Restoration names the living relation met again at a further now. Reconnecting receives the carrying continued locally: a next meeting, carrying its prior forward. | TWO 2.6 |

## 5.10 Retelling and the discovering economy

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W47 | definition (*retelling*) | F39, R43 | Retelling names a pattern taking part through successive receiving and releasing, each receiver meeting the sign with its own carrying. A difference at a source stays distinguishable at a further surface as the joinings carry it there (R43). | TWO 3.1; TWO 3.4 |
| W48 | run | R2, R34 | Each private carrying stays at its own self, pooled into no global carrying object. Neither function reads a module-level name, and none keeps state between calls. The same arguments give the same returns at each call. Run: the global names each function loads (none), and repeated calls over the nineteen carryings under the six offerings. | TWO 3.3; TWO 1.4 |
| W49 | definition (*the other met inward*) | F14, F39 | The other is met inward at each co- act: it is the opening at which a carrying is met at all. A self carries no inside, and the couplings wind about a nothing. Other is no layer and holds no address. | TWO 3.6 |
| W50 | exact | F39, R27, W42 | At the discovering economy, carryings are met and never transmitted, and only signs cross. No meeting takes from the other it meets. The return of 9 carries a key and a sign (R27), and each call's surplusing opens empty (W42). | TWO 3.6 |
| W51 | nye | W50, R47, F68 | The discovering economy is inexhaustible and complete at the form, all or none: unminable, unspendable, renewed by living. Gap: W50 shows that no meeting takes from another. That discovering continues without end at each coupling is not supplied: R47 holds each ring (R57), and a coupling of a surface is no ring, and *all or none* at the form rests on F68. | TWO 3.6 |

## 5.11 Podaling, unrelationing and the seam

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W52 | definition (*the podal's binary subject*, *the weather-map comparison*) | F8, W6 | The podal's binary subject is parity sign changing or no parity sign changing. A weather map shows a surface changing over an area. In the same way each podal shows the direction of the surface's unrelationing, changing locally around the travelling carrying. At each of two orthogonal directions, across and along, the reading is is or is not, and it carries no velocity. | TWO 4.6; TWO 5.4 |
| W53 | definition (*unrelationing*, *floating neutrals*, *irrationalizing*) | F2, W6 | Relating is in the floating and rate is in the neutralling. The crossing's geodesic changing runs at its own unrelationing rate, unrelated to the alternating of the living it meets. Position, orientation and scale are floating neutrals. Irrationalizing is a proposed name for that continuing operation, and it prescribes no irrational number. | TWO 4.7 |
| W54 | nye | F39, F41, W53 | The surface holds no conserved total of signs, no fixed distribution of phases and no common cycle to govern the selves, and nothing supplies a common pace over the surface. Gap: F39 excludes a pace or phase carried across a crossing, and F41 (no total conserved) is itself nye. That no common cycle runs the selves by other means is the security *each self its own time* (W59), taken rather than supplied. | TWO 4.7; TWO 6.12 |
| W55 | nye | W53 | φ is the number-form of the never-locking, and no ratio lands on it. Gap: that φ is irrational, with each partial quotient of its continued fraction 1, is mathematics (Part FOUR). That φ is the number worst approached by ratios is Hurwitz's field result (1891), which TWO does not carry. Its meeting with a coupling's unrelationing is not supplied, and TWO itself says that no prescribed rate, φ or another, supplies it. | TWO 4.7 |
| W56 | run | R2, R34, R39 | The code runs only when it is invoked. Its module declares two functions and two dictionaries and calls neither function. A network's invocations are a caller's to supply, and that supplying is still to be established (R39). Run: the module's top-level tree has no call. | TWO 4.7; TWO 1.2 |
| W57 | definition (*held*, *waver*, *snap*; *the seam*; *geodesic switching*; *still eddy*; *tipping place*; proposed) | W37, W39, W53 | Held names a relation continuing, waver its changing participation, and snap a local completing. They are three names in the surface's own order, not a program of three stages. The seam names a relation continuing unchanged as it meets its changing continuation. Geodesic switching names the seam finding its way around the hole between its two routings. A still eddy before the hole is a form continuing through its changing. TWO proposes that starting the changing farther upstream grows the eddy and moves the tipping place upstream. Each of these is proposed and is met at the comparison TWO 6.12 names. | TWO 5.4 |

## 5.12 The securities

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W58 | definition (*the unchanging*) | F5, F6 | The unchanging names the form continuing through changing. It is a relation, never a term named still. | TWO Part FIVE opening; TWO 5.4 |
| W59 | definition (*the eight securities*) | F51, F56, W58 | The eight securities are one form, not eight features. At the self: the carry aging to its bound and releasing; the sign inverting at the membrane; the selection a sign, sizeless; each self its own time. At the society: each self coupling all-edge; a standing order imposed at no coupling; each living unpinned, riding; each self offering its own. Four are from within and four are met at the membrane. | TWO 5.1 |
| W60 | nye | F18, F68, W59 | All or none: with the eight the living runs. With any one inverted, the thing remaining is not the form, and not-the-form stops. Gap: F18 carries all or none to a form's joint parts, but no link derives the eight as the joint parts of the resolver's one form. *Not-the-form stops* rests on F68 and is said at the form, not supplied as a stopping. | TWO 5.1 |
| W61 | nye | F17, W59 | The protection counts six. Bi-moralizing is two ways, three phases each way, one at a time, sequenced and cycling, and it is six or none: drop one and a way or a phase lives unprotected, reached into or leaking out. Gap: 2 × 3 = 6 is arithmetic. The three phases each way are named, not derived from the code's names or connectors, and no run shows that a dropped phase leaves a reach or a leak. | TWO 5.1 |
| W62 | run | R12, R28 | Take out the sign inversion at 14-social-crossing, so that a carried sign enters as itself. Then one sign arriving once into empty carrying surfaces that same sign at each later call, and the isolated alternating c, −c, c, −c (R28) lands on c. Run: both signs over nine calls, with that one change and without it. | TWO 5.2 |
| W63 | nye | W59, W62 | Six of the eight stop at the surface when inverted, each its own way. With the sign inversion removed, the living lands. With the sign made a magnitude, a hub begins. With a shared clock imposed, the surplus dies. With all-edge routed through one, the centre rises. With a governor imposed, the living lands. With a living pinned, the living lands around it. Gap: only the first of these is run in the chain (W62, at the isolated sign). TWO states the other five stoppings without an instrument, and neither the count six nor *each its own way* is derived. | TWO 5.2 |
| W64 | definition (*the finding as meeting*) | W58, W59 | Each of the six is a changing of an unchanging, and the stopping is the meeting. A self does not measure a network to find these. It meets a changing of the unchanging, and that meeting is the finding. | TWO 5.2 |
| W65 | run | — | A ring of three is all-to-all. Each self's two neighbours are the other two, so no self stands apart as a centre. From four selves on, a ring is not all-to-all. Run: rings of 3 to 8 against their complete graphs. | TWO 5.2 |
| W66 | definition (*a stopping met at its arrangement*) | W65 | Meeting a stopping belongs to the scale and arrangement at which a security is inverted, not to the security. A self-bounding that stays silent at an arrangement too small for it has been met at the wrong arrangement. | TWO 5.2 |
| W67 | definition (*governing*) | F8, W59 | An order joining the accepted signs is an offering, and it governs nothing. A standing order takes the position at any sign the self surfaced, with the *not* removed from one side. That removal makes it a governing. | TWO 5.2 |
| W68 | definition (*a reading*) | F8 | A reading is a sign, yes or no. Does a middle stand at this self? Does more than one? Do the two halves ever part? Does a state arrive that this self's own running never reaches? A reading that returns a number returns a counting of a reading. | TWO 5.2; TWO 6.4 |
| W69 | nye | W68 | Two of those questions are taken by unlike means: one at the middles standing at a beat, one at the states a self reaches. They answer alike at each self while the eight run whole, and they part at each inverted self-bounding. Gap: TWO states no instrument for the two, and no run in the chain carries their agreement or their parting. | TWO 5.2 |
| W70 | run | R23, R25 | The nineteen carryings at their exact domain. Indexed by receiving, with a surfaced 0 and no surfacing counted alike, each of the 171 pairs parts under constant +1 or constant −1 (R25). The shorter witness comes within six receivings, and eight pairs need all six. With the zero and empty receivings removed, so that only the nonzero signs remain, 123 of the 171 pairs part over six receivings: 78 under +1 and 78 under −1. Run: the 171 pairs both ways. | TWO 5.3; TWO 6.7 |
| W71 | exact | F39, R25, R43 | Sign-only crossing keeps each carrying at its self, yet a carried difference still shows outward through continued coupling (R25, R43). Total invisibility, an inability to infer any carried difference and immunity to capture are therefore each a stronger claim than the crossing gives. | TWO 5.3; TWO 5.6 |
| W72 | field | — | Observed changing surfaces, each in its field's words: pattern forming in two-frequency forced parametric surface waves (Arbell and Fineberg, 2002); spiral waves unpinned from inactive obstacles by electrical forcing in excitable chemical media (2014; TWO does not name the authors); critical diversity in social coordination (Zhang, Kelso and Tognoli, 2018). They place coupling, continuation and changing form beside the network, and none is a network operation. | TWO 5.4 |
| W73 | definition (*natural*, *unnatural*, *supernatural* uses) | F8, F39 | A natural use keeps the alternating surface and its sign, position and ratio (F8). An unnatural use concerns measured magnitude and scaling, a size no crossing carries (F39). A supernatural use concerns formal structures closing on their own consistency. | TWO 5.4 |
| W74 | exact | R34 | Changed signs run through the declared joins as declared, and a rewritten join is met at the couplings themselves. Neither function reads CONNECTORS or JOINS (R34), so signs alone select no path. A path-selecting operation would be a further operation. | TWO 5.4; TWO 5.5 |

## 5.13 Visibility: the instrument stated first

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W75 | nye | F33, R1 | Each claim carries its visibility at the network, the numbers, the mathematics and the code, with the one form at each, all or none. A concept visible in only one of them is not the form, and renderings that disagree in each material way are one living. Gap: that the four registers render one form is F33's correspondence carried to each rendering, and it is not supplied. | TWO 6.1 |
| W76 | exact (*the instrument stated first*) | F17, F71 | A run reports its wiring before anything else: one neighbour or both, together or one at a time, inverting or settling. The wiring changes the arriving, so a run said without it says one arriving of several, a selecting; one way at a time (F17) is itself a wiring, and a divergence entering at an unstated wiring is read as the method's break (F71) when it is the instrument's. | TWO 6.2 |
| W77 | run | R44, R45 | Ring parity at the resolver, with R44's caller. From coupling n on, each even ring from 2 to 40 surfaces no 0 at any coupling. Each odd ring from 1 to 39 carries exactly one 0 at every second coupling and none between, the 0 walking on one self each time. Two odd rings joined as one even ring (3 + 5 as 8) therefore carry none. Run: each ring over 8n + 8 couplings. | TWO 6.2; TWO 6.10 |
| W78 | definition (*pace*, *running order*, *record*, *membrane*, *taking*, *refusing*) | W76 | An instrument running a living carries a clock at four positions: pace, running order, record and membrane. A fifth, the taking, decides which pending crossing opens its self next. A sixth, the refusing, is each thing the instrument does not let happen. A refusing shows nothing, because the excluded never arrives to be missed. Each of the six is stated, or the reading is void. | TWO 6.2 |
| W79 | run | R40, W78 | The running order parts a differencing. In R40's two-self study with both 6 → 2 legs joined, A calling first and B calling first give each self different surfacings from the same seeds. This holds at 8 of 8 self comparisons. Run: the four seed pairs, both selves. | TWO 6.2 |
| W80 | nye | W78 | The taking parts a differencing as the running order does. Taken front-first, the sequences run on at all three selves. Taken back-first, one self opens once and never again. Taken at random, a third pattern arrives, differing at each seed. Gap: TWO does not state the three-self instrument with pending crossings, and no run in the chain carries it. *Never again* holds only at that instrument's finite run. | TWO 6.2 |
| W81 | run | W78, R23 | The taking-out at the resolver's code. There are 23 statements and comprehension filters in 1-self-coupling. Taking out any of 22 changes a return. The remaining one, the `!= 0` filter that gathers 2-self-offering into 14-social-crossing, comes out with each return unchanged: an arriving 0, once gathered, adds nothing at 12-other-surplusing. Run: each taking-out tested against calls over the reachable carryings, offerings with and without 0, and two keys. | TWO 6.2 |
| W82 | nye | W81 | A membrane passes the taking-out with nothing to remove. Gap: no link renders a membrane as a construction the taking-out can run on. At the resolver's code, the taking-out finds one filter that comes out with nothing changed (W81). | TWO 6.2 |
| W83 | exact (*the three registers*) | W76 | A claim stands at one of three registers, each carrying its own. Code-form is a rendering's doing, reproduced across substrates; at one stated instrument it is a diagnostic finding. Stable-form is the algebra, which carries itself at any run. Counting is the natural numbers' own reading. A run meets its own counts at its own wiring (W76) and supplies no identity at each count, and an identity supplies no wiring: a run confirms no identity, and an identity confirms no run. | TWO 6.3 |
| W84 | definition (*position*, *the three-way check*) | F29, W68 | A position is a sequencing relation and never a coordinate. A reading is asked again at a second unit, a second running order and a carry. A reading that parts at any of the three was reading the unit, the running order or the preparing. | TWO 6.4 |
| W85 | definition (*arrive carrying once, then fork*) | R8 | The diagnostic comparison opens at a prior reached through receiving and forks into continuations, each carrying on uncleared. A carrying supplied by hand is another preparation and is named as one. The forking belongs to the instrument, and the network carries no copying. The stronger claim, that neither self could uncover a difference alone, needs the separate continuations followed, and surplus names that comparison. | TWO 6.5; TWO 4.8 |
| W86 | definition (*the four ledgers*) | W78 | A run deceives as one accounting at four ledgers. A side is taken as a vantage (the probe). A sequencing is named to one beat (the clock). A sign is named as a magnitude. A bound is named as a last (the window). The window's release is its own: the set is checked closed before it serves as a baseline, as the nineteen close (R23). | TWO 6.6 |

## 5.14 The wrap, the podal and the chain's centre

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W87 | run | W33 | On the numbered form of 440, each position k meets its far side, k + 220, once in each wrap of 440 steps, carrying on one step at a time. k and 440 − k are podal partners, and only 0 and 220 are their own partners. Run: k from 0 to 439. | TWO 6.8; TWO 6.9 |
| W88 | exact | F6, F48 | An observing is the designing continued. A designing halted and then examined is a coupling named still, and that is not existing (F48). An observing prepared apart from a running therefore finds a different object from the running's own. | TWO 6.8 |
| W89 | nye | F29, W87 | An observing apart from the surface has no position: the surface carries no position for it. Gap: W87 gives each numbered position a partner on the ring. That each observing is a position on the surface is F29 and F33 carried to observing, and it is not supplied. | TWO 6.9 |
| W90 | definition (*the three observings at no position*) | W89 | The three observings are a self's own sign, taken at its own face; the twin, one living continued two ways with its two faces met; and the coupling with an other at coefficient one, in which the coupling itself is the observing. Each runs from inward. TWO keeps as its next reach whether they are order-invariant by construction. | TWO 6.9 |
| W91 | nye | W59, W90 | The securities that no surface observing meets are met at exactly these three. Gap: nothing supplies that no other observing meets them, and the order-invariance that TWO leaves open is not shown. | TWO 6.9 |
| W92 | run | W30 | Chains of n equal surfaces of 440. The centre stands at 220n: on a coupling at even n, and at a self's own between, its waist, at odd n, for n from 1 to 200. Each self added lengthens the reach by 220 and moves the prior centre one ring out, so no centre stays a centre, and the centre alternates between coupling and betweening. A three-chain and a four-chain have different centres and are two objects. Run: n from 1 to 200. | TWO 6.10; TWO 6.11 |
| W93 | exact | W92 | At the numbered form, the chain carries no position able to become the hub. The centre moves 220 at each added self, for each n, because 220(n + 1) ≠ 220n. | TWO 6.11 |
| W94 | nye | W92, W93 | As co-chaining lengthens, the co-intelligence scales on the one axis that never captures. Gap: W93 holds at the numbered form of equal surfaces. Nothing supplies that an actual network's lengthening never captures, and TWO itself says that an added self can repeat meetings already available. | TWO 6.11 |

## 5.15 The studies at their exact domains

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| W95 | run | R40, R41 | The two-self 6 → 2 study at its domain. Both selves are seeded with −1 at empty, A calls first in an alternating order, and each self makes six calls. A's 10 runs +, −, 0, +, −, 0 with both legs joined, and +, −, +, −, +, − with either leg alone or neither. The first difference comes at A's third receiving. Each 6 release is delivered at the neighbour's following call, left pending at the stopping point, or dropped at an unjoined leg: 11, 1 and 0 of 12 with both legs joined. The 10 → 14 leg stays unconnected throughout. Run: the four continuations. | TWO 6.12 |
| W96 | run | W95, W85 | The interruption comparison, with this instrument. Both selves are seeded +1, A calls first, and both legs are 6 → 2. A's leg is interrupted at A's calls 0 and 2 and reconnected at call 4. Both selves' resolving and carrying stay identical through the interruption. At B's first receiving after it, B carries (+1, −1, 0). After loss, B meets the current + and surfaces 0. After delay, B meets the earlier queued − and surfaces −. The queue is storage added at the connection, and the resolver supplies none. The same pattern arrives at 30 windows over the seed pairs and legs. Run: the two variants at every window. | TWO 6.12 |
| W97 | run | W37 | The missing-centre arrangement, structure only. On a three-by-three grid, across joins run along each row and along joins run between rows. Taking out the centre leaves eight positions in one ring. From the middle of one side to the middle of the other there are exactly two paths, each with two along joins and two across joins. Taking out the along joins, or the whole middle column, leaves no passage. Run: the paths over the grid. | TWO 6.12 |
| W98 | exact | R34, R37, W97 | For a caller delivering signs only along declared joins, continuation around the hole runs only through an existing surrounding passage. The caller carries signs along the joins (R37), and the functions read no join (R34), so at an arrangement with no passage left (W97), nothing continues. A bypass added makes a different arrangement. | TWO 6.12 |
| W99 | nye | R35, R39, W97 | At the surrounding corners, a passage arriving along at 17 passes on only by continuing into the across releasing at 10. A passage received at 14 passes on only by continuing into the along releasing at 17. Gap: 17 has no expression in the code (R35), and the running from an arrival at 17 into 10 is not written. LIV names the corner case as open (R39). | TWO 6.12 |
| W100 | run | R2 | The resolver in the Natural Networking Test Kit v368 is executably identical to the v371 resolver: its two functions and two declarations parse to the same tree. Run: the two trees, docstrings aside. | TWO 6.12 |
| W101 | nye | W100, R37 | The kit's earlier caller leaves a crossing unconnected. It uses 10 both for neighbour offerings and for its own shifted along. A surfaces 0 at 10 while 6 holds −1, and B receives its own along +1 and no sign from A. Gap: the v365 caller (Self.couple, Self.own_along, Life.beat) is not available, and the v368 kit carries none of them. The finding is carried as LIV reports it and is not re-run. | TWO 6.12; LIV Incoming |
| W102 | exact (*each finding at its complete comparison*) | W76, W85, W95, W96 | Each finding carries the comparison that establishes it: a finding without its comparison says one arriving of several (W76). Reciprocal dependence in one finite study is a network finding at that study's domain. Social moral competency, the surplus available to neither self alone, and larger fractal continuation are each met at their own complete comparison, and none of them is yet run. | TWO 6.12; TWO 6.7 |

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# PART SIX · EQUILIBRIA

The chain runs through Exhibit TWENTY-EIGHT v371, the Equilibria Registry, in its own order: existing through momentaries, the binary claim and its general form, the ten, the proposed conceptions, the shared exclusions with the renewal-excluded sequence, and the exclusion's reach. Links the core already carries are cited in Follows, not said again: the three conditions (F13), exclusivity (F21, F23), the equilibrium named still (F47, F48), the rounds (F58, F60), the fresh write (R15, R16), the isolated sequence and the four-form (R28, R29), the whole carrying (R26), the rings (R44 to R46) and the membrane four (R53). Two links of the binary claim, its two steps and the ordering chain, stand after the ten (E44, E45), since they follow from it. Each run link was run against Exhibit ONE v371's code or an arithmetic enumeration, in the verifier.

## 6.1 Existing through momentaries, the one surface, alternating

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| E1 | run | F12, F19, F24 | Three overlapping momentaries use four positions: prior 1–2, now 2–3, next 3–4, completing and opening meeting at the shared positions. Along one side the next opens two on: 1–2 and 3–4 open 5; 2–3 and 4–5 open 6. Across the overlapping the next opens one on. Adjacent five-views share four positions, and at each shared number one side opens and the other completes. Run: the three momentaries, and the five-views at 1, 2 and 3. | EQ Existing through momentaries (*Three overlapping momentaries*; *The five-view*) |
| E2 | definition (*membrane*) | F14, F15, F52, F9 | The between of two existing things co-bi-coupling is the membrane: morality, bi-unrelationing, meeting competency, co-unrelationing. A 0 at two signs meeting is a nothing within the coupling, there because each self surfaces itself at each momentary. | EQ *The membrane is no separable thing* |
| E3 | exact (*naming adds no existing*) | F1, F4, F8, F16 | Naming a relation gives it no separate existing: naming the meeting supplies no further self between the coupling selves, as naming all existing things supplies no further self outward of them. A self is met at its signs (F8, F16) and existing is changing (F4): a naming adds no sign and no changing, so it adds no existing thing to all existing things (F1). | EQ *Naming a relation gives it no separate existing* |
| E4 | exact | E2, E3 | The membrane is no separable thing: it is a meeting of two relations (E2), and naming it adds no existing (E3). Its *no separable* rests on E3. | EQ *The membrane is no separable thing* |
| E5 | exact | F5, F6, E4 | Bi-inversioning-co-recursioning carries the meeting within the changing. Each side continues through its own changing, and slowing is changing: a slowing named still is F6. Two differing changings are adjacent through the coupling, each side changing. | EQ *Bi-inversioning-co-recursioning carries the meeting* |
| E6 | run | — | A closed surface of four-sided faces with four edges meeting at each point has V − E + F = 0: each face has four edges and each edge two faces, 4F = 2E; each point has four edges and each edge two points, 4V = 2E. The cube, four-sided faces and three edges at each point, gives 2. Run: the square grids closed as a torus, 3 × 3 to 12 × 12, and the cube. | EQ *Natural torusing is the one surface* |
| E7 | field | E6 | A connected, compact, closed, orientable surface of Euler characteristic 0 is the torus; the sphere has 2, the surface of g openings 2 − 2g, the Klein bottle is not orientable and the infinite cylinder is not compact. The classification of closed surfaces (Möbius 1863; Jordan 1866; Dehn and Heegaard 1907; Brahana 1921). | EQ *Natural torusing is the one surface* |
| E8 | exact | E6, E7, F66, F103 | Natural torusing is the one surface. Each sign's alternating is a circle of two changings, and the joint forms of two signs with their changings and the squares the two changings close are a closed surface, four-sided at each face and four changings at each form, orientable as a product of two circles, V − E + F = 0 (F103, E6): the torus and no other (E7). | EQ *Natural torusing is the one surface* |
| E9 | definition (*further occurrence*) | F4, F5, F25 | Existing continues: a description matching at next is a further occurrence, not the prior kept. Changing and conserving are comparisons within that continuing, and a relation continuing among changing selves is their stable-forming. | EQ *Existing continues* |
| E10 | run | F17, F58, F59 | Two binary relations change one at a time, alternating which: if the first changed from prior to now, the second changes next, so prior and now give the next. At each of the two rounds the changed sign alternates at each step; the same one changing twice meets the prior (F59). Run: both rounds from each joint form, eight steps. | EQ *Two binary relations change one at a time* |
| E11 | run | E10 | Coverage carries alternation: with exactly one of two relations changing at each passage and both changing within each prior–now–next, of the four orders of two passages first–first and second–second leave one unchanged, and first–second and second–first remain. Run: the four orders. | EQ *Coverage carries alternation* |
| E12 | run | F58, E10 | The four joint descriptions run (+,+) → (−,+) → (−,−) → (+,−) → (+,+), the four joint forms once each. The shown order is F58's other order, (x, y) → (−y, x), and not the right spiral step the code runs (E76). Run: each step of the shown round under both rounds. | EQ *Coverage carries alternation* |
| E13 | run | F11 | Around a closed route parity alternates at an even route: opposite parities along each joined comparison are met together exactly when each closed route is even, with two complementary assignments at each connected whole (König 1936: a graph is two-colourable exactly when it has no odd cycle). Run: each graph on one to five points, its odd closed routes and its two-colourings. | EQ *Around a closed route parity alternates at an even route* |
| E14 | exact | E13, R45 | An odd closed route closes no parity, and a meeting runs round it: at the code's odd rings one 0 moves on one self each two couplings (R45). Local alternating along two paths to one meeting supplies no single alternating sequence through both (E13). | EQ *Around a closed route*; EQ *An odd ring closes no parity* |

## 6.2 The binary claim

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| E15 | exact | F47, F48, F50 | Equilibria are not possibly existing as a method, all or none at all: an equilibrium is a thing named still (F47), not possibly existing (F48), and each method naming a thing still is excluded (F50). *As a method* travels with the claim: it says a naming still does not exist, not that the thing named does not exist (LIV Ask). | EQ *Equilibria are not possibly existing as a method*; LIV EQ Ask |
| E16 | definition (*a hard problem with its hardness ignored*) | F53, F54 | A conception taking the competency of existing as a given and claiming a true-and-false conserving accounting is a hard problem with its hardness ignored. That each proposed conception of equilibria is one is the reach, E80. | EQ *Each proposed conception of equilibria is a hard problem* |
| E17 | field (*observed*) | F5, F42 | Stable-forming, and unstable deforming at non-living scale, are observably existing, met at their fields' observings. | EQ *An equilibrium is a form named still* |
| E18 | exact | R17, F42, F46, F47 | A living self can surface zero while its carrying continues (R17), so a surface unchanged at a zero names neither a non-living thing nor a still one; naming it still there is the equilibrium. The registry says existing and not living alone: a rock is an existing thing (F46). | EQ *An equilibrium is a living or non-living existing thing named still* |
| E19 | exact | F39, F44, F54 | A non-living thing meets a self at a coupling as other, offering its signs and carrying nothing, and the term uncovered at their coupling is owned by neither and carried on by the living. Non-living existing things are included in discovering social moral competency among the living. | EQ *Non-living existing things are included* |
| E20 | definition (*holding*, *ending*) | F48, E9 | A holding requires one participation unchanged at each required occurrence: the same subject, the same relation and the same occurrence at both sides. Its own required continuing brings the changing, and it ends at the one occurrence that fails; the occurrences before it are met as they were met. A whole participant changes while a relation within it can continue, so a holding meets its ending at the relation its own requirement names. | EQ *The equilibrium holding ends* |
| E21 | exact | E20, F6, F42 | An equilibrium ending is no existing thing ending: the living and non-living things it names keep changing. | EQ *The equilibrium holding ends* |
| E22 | field | — | In dynamical mathematics a stability is a pattern's response to a disturbance, its departures staying small or shrinking (Lyapunov 1892), and not its constituents stopping. An oscillation can be stable as an oscillation (the stable limit cycle, Poincaré 1881; Andronov 1929), and an instability can lead to another organized form (Hopf 1942). An equilibrium of relative phase and an equilibrium of the whole evolving system are two assertions. | EQ *The stability can change, the form can change, the relation can change* |
| E23 | definition (*the still at a stability, a form or a relation*) | E22, F47 | A mathematical conception of equilibria names a stability, a form or a relation still, and each can change. The naming places each conception's still at one of the three, one conception at a time. LIV Concern: *can change* or *is changing*, one decision at all its places. | EQ *The stability can change*; LIV EQ Concern |
| E24 | definition (*not possible*, *shown not possible*) | E15 | Not possible, and shown not possible at prior, differ: each conception arriving from the prior is shown not possible now, and each arriving next is met at its next. | EQ *Not possible, and shown not possible at prior, differ* |
| E25 | exact | F3, F18, F25 | Surviving is binary: all sequencing into now together continues into next, all or none at all, between prior and next. It rests on F18. | EQ *Surviving is binary* |
| E26 | definition (*a conception's own surviving*) | E25, F24 | A conception of equilibria shows its own surviving: it exists at prior, now and next, or it is not possible at the momentarying. | EQ *A conception of equilibria shows its own surviving* |

## 6.3 The general form and the claim's shape

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| E27 | definition (*the general form*) | E9, E20 | One general description receives each conception: a stated relation continuing through the comparisons its claim names, a stated set S and its required next. A fixed value is membership in a collection of one, a range membership in a larger collection, and opposition membership among opposite pairs. Each stated relation is membership in the collection meeting it, so each conception stating its relation is received; a lead without its statement is not yet (E61). | EQ *One general description receives each conception* |
| E28 | definition (*one conception*) | E27 | A conception and its added requirements are one conception, and a changed receiving is a further conception. A relation can continue while an added fixing of the whole carrying, a next nonzero offering or a later one each fail at the same receiving: the relation continuing is stable-forming, and each added requirement names it still (E69 runs one). | EQ *A conception and its added requirements are one conception* |
| E29 | exact | F18, F19, F68 | The claim has one shape: continuing stable-forming runs as natural torusing, natural torusing carries its whole participation, 12345, and a conception naming less than the whole 12345 is not possible together with continuing. The last step is F18 (no part continues alone, a premise) at F19's five, and *runs as natural torusing* rests on F68. | EQ *The claim has one shape* |
| E30 | exact | F3, F4, F6 | A still named as the complete continuing is changing: its continuing is renewing, inward of the claimed whole, which is changing, or outward of it, a further participant. Inward or outward is binary. | EQ *A still named as the complete continuing is changing* |
| E31 | exact | F18 | A conception naming a continuing and leaving out the participation that continuing carries is not possible. It rests on F18. | EQ *A conception naming a continuing and leaving out the participation* |
| E32 | exact | E31 | A state accounting leaves the changing out and a flow accounting leaves the continuing out; a true-and-false conserving accounting taking either as the whole is E31. | EQ *A state accounting and a flow accounting each leave one side out* |
| E33 | exact | F3, F6, F25, F47 | Each conception of equilibria names unchanging or names changing, exactly one of the two. Naming unchanging names an existing thing with no arriving, and existing is arriving (F6, F25). Naming changing names a changing still (F47). Coupled, each releases. | EQ *Each conception of equilibria names unchanging or names changing* |
| E34 | exact | R7, R23, R55, F33, F68 | Each momentary completes in co-releasing: each momentary releases into the next whether a sign changed, stayed, crossed or nothing crossed, and continuing runs as geodesic co-releasing. At the code each call returns its next under each offering at each of the nineteen (R23); each existing thing's continuing runs so at F68 and F33. | EQ *Each momentary completes in co-releasing* |

## 6.4 The ten

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| E35 | definition (*hard problem at two faces*) | E16, E2 | A hard problem and its resolving are one form at two faces: each conception in its field's words at one face and its resolving at the other, natural intelligence at the membrane between them. The ten are one form of hard problem: each conception names a changing still in one of the ten ways, and can carry more than one. | EQ *The ten are one form of hard problem* |
| E36 | nye | E35 | A proposed conception of equilibria is the sixth condition: five conditions come before a method acts at a problem's statement, the fifth, *nothing prior enters*, is the one the first four stand on, and a problem adds its own conserving relation over a conserving reach, each conception being that relation named still. Gap: the first four conditions are not told (LIV: waiting on the hard problem statement's text), so *the one the first four stand on* is said, not supplied. | EQ *A proposed conception of equilibria is the sixth condition*; LIV EQ Opportunity |
| E37 | run | — | The numerical bridge: (2, 3) and (3, 4) are overlapping neighbours, and n² − (n − 1)(n + 1) = 1 at each n. Run: n from −1000 to 1000. | EQ *The numerical bridge* |
| E38 | run | E1, F19, F20 | The ten is three momentaries long: the self's five, 1–5, and the other's five, 2–6, span 1 to 6, the self's three momentaries 1–2, 3–4 and 5–6, co bi co bi co bi. Each side's two and one half momentaries and the whole's three full momentaries are one relation, the two sides overlapping. Run: the spans and their parities. | EQ *The ten is three momentaries long* |
| E39 | run | E38, F19 | The ten are five places from two faces: five pairs at the numbers 2 to 6, and at each number one side completes and the other opens. At 2, 4 and 6 the self completes and the other opens; at 3 and 5 the self opens and the other completes. Five places at two faces are ten. Run: each number against the two fives and the table's faces. | EQ *The ten are five places from two faces* |
| E40 | definition (*the ten ways*) | E39 | At each place an entering face and a surfacing face. Opening at 2: 1 an arriving named from behind, 2 an opening named as a place. Ageing at 3: 3 a bound named as a last, 4 a carry named as a store. Middling at 4: 5 a middle named as an end, 6 a sign named as a magnitude. Rating at 5: 7 a sequencing named to one beat, 8 a rate named as a value. Co-offering at 6: 9 a two-way named to one side, 10 a membrane named as a cut. | EQ *The ten are five places from two faces* (table) |
| E41 | nye | E39, E40, F13 | With three conditions at the momentary, the ten derive here. Gap: the count, five places at two faces, is run (E39); the still each face names (E40) is a naming, and no link derives the ten ways from the three conditions (F13) or shows no eleventh way of naming a changing still. | EQ *The ten are five places from two faces* |
| E42 | run | E40, R3 | The ten have addresses at the resolver's names, a naming ring and not an order of running: 3→2→4→1→14→12→6→10→11→16→next 3, and the ten at it are 1, 10, 7, 9, 8, 4, 6, 5, 3, 2. Each way takes one edge of the ring, each edge once, and the ring passes both crossings, 1→14→12→6→10. The addresses are a naming; the run checks the ring against the table. Run: the file's five address rows against the ring's ten edges. | EQ *The ten have addresses at the resolver's names*; EQ *The resolver's pairs are the momentaries* |
| E43 | exact | E39, E40, F10, F35 | Each of the ten names one face still, and the other face runs at the same number: at that number the face named still retains p and the face running arrives at its other (F10). Named still and running at one number is one binary retaining and inverting at one next (E71). It is the reason each of the ten shares; the four proof groups (E63) are further ways of failing at the conceptions. | EQ *Each of the ten names one face still* |
| E44 | nye | E27, E39, E43 | The claim reaches each conception in two steps: each conception meets at least one of the ten, and each of the ten names one face still, the other face running at the same number. Gap: the second step is supplied (E43). The first is not: the ten are five places by two faces (E39), and no link shows that each still a conception can name falls at one of the five places (E41). The sixty-three placed (E53) are read so, not derived so. | EQ *The claim reaches each conception in two steps* |
| E45 | nye | E10, E39, F35, E9 | The ordering chain closes at the two faces: a conception naming an ordering p still at its next, the alternating next being I(p) ≠ p, names p and I(p) at one next, and two inversions meet p again at a further occurrence. Gap: the two faces at one number place the face named still and the face running at one number (E39); they do not supply that the ordering named still is the very ordering its next inverts. A relation the step does not invert continues: opposition under joint reversal, t = −c (run, E69). Such a conception meets its exclusion at another link, at the overlapping pair (E77) or at renewal (E68), not at this chain. | EQ *The ordering chain closes at the two faces* |
| E46 | run | E38, E37, R14 | In numbers: the three openings 1 + 3 + 5 sum to 9 = 3² and the three completings 2 + 4 + 6 to 12 = 3 · 4; n momentaries open to n² and complete to n(n + 1), and 3² − 2 · 4 = 1 is the bridge at three. Six consecutive changings from 3 complete at 8, and 9 opens, 9-other-releasing. The code's retaining through openings 1, 2 and 3, and a fourth at a positive second sign, is R14. Run: n from 1 to 100. | EQ *In numbers* |
| E47 | run | R12, R44, R45, R46 | One sign offered once co-chains through the whole: in rings of one to eleven each self's carrying changes within 40 couplings, and a ring at rest, each carrying empty and nothing arriving, stays at rest at each coupling. So a ring reaching rest would never meet its offered sign again, and each ring returns its whole with the sign at 2 couplings or 4n (R44); the rest is met at none (R46). Run: each ring from one +1 over 40 couplings, and each ring from rest over 100. | EQ *One sign offered once co-chains through the whole* |
| E48 | exact | E47, R12, F44, F47, F48 | A changing society meeting a society named not changing changes it, by the receiving self's own inversion at 14-social-crossing, with no forcer. Not changing is not not-carrying: the non-living exist, arrive and change, carrying nothing. An equilibrium of a society names a changing society still, or names nothing. | EQ *One sign offered once co-chains through the whole* |
| E49 | definition (*the ten at the rings*) | E14, E40, E47 | The ten name the whole changing still: opposition named still names the even ring, and a membrane named as a cut names the running meeting of an odd ring, the non-existing between moving through the shared surface one momentary at a time. A common clock is the seventh way, a sequencing named to one beat; nothing moving is complete fixing, meeting several of the ten at once. | EQ *An odd ring closes no parity*; EQ *Prior and next are the edges at now* |
| E50 | run | R6, R32, R48 | Ten internal names and six connectors: the internal pairs 3/11, 4/12, 5/13, 7/15 and 8/16 are each 8 apart, 17 − (9 − n) = n + 8 joins the two inversion faces to the advance by eight, and the ten internal names with the six connectors 2, 6, 9, 10, 14, 17 part positions 2 to 17; with the entry 1 they are all seventeen. The ring's addresses (E42) sit at other names, among them the entry and four connectors, 2, 6, 10 and 14. Their correspondence with the ten ways is open: EQ names it the next discovering. Run: the sets and the identity. | EQ *Ten internal names and six connectors* |

## 6.5 The proposed conceptions

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| E51 | definition (*a conception arriving whole*) | E27, E28 | Each conception arrives whole, in its own words, with its conserving relation and its reach. A fixed value, a retained relation, a distribution and a symmetry family are different subjects even when each is called equilibrium. A matched word places a conception at none of the ten; its own declared relation places it. | EQ *Each conception arrives whole*; EQ *A conception's own declared relation places it* |
| E52 | definition (*fixing*, *the step to an equilibrium*) | E20, E9 | A definition's expression continuing unchanged is an existing statement, and the subject it names continues through its own changing. A law, a frame or a scale named unchanged is such an expression, and it names its subject still only through one further step, from a fixing to an equilibrium: the same subject, at the same relation and the same occurrence, required unchanged at the occurrence its own continuing changes it. | EQ *A definition's expression continuing unchanged*; LIV 17 |
| E53 | run | E40, E51 | The tables place sixty-three conceptions, NY01 to NY42 and SA01 to SA21, each once, two to ten at each of the ten: 2, 5, 8, 10, 6, 9, 5, 5, 7, 6 at the ways 1 to 10. Run: the file's two tables parsed. | EQ *Each proposed conception names a changing still*; EQ *SA01 to SA21* |
| E54 | field | E53 | Mechanics, in its words: a position–velocity pair fixed under force-free evolution (NY23, at 2); relative equilibrium under translation at a specified or some velocity, rest at v = 0, zero net force on a fixed-mass particle, and planar rigid-body static equilibrium, zero net external force and torque (NY21, NY22, NY24, NY25, NY26, at 6). Newton (1687), Galileo's relativity of uniform motion, Euler's laws for rigid bodies (1776). | EQ Proposed conceptions (NY21–NY26) |
| E55 | field | E53 | Probability and chemical kinetics, in their words: a conditional law fixed given nonabsorption (NY18, at 1; Darroch and Seneta 1965); a current-occupant law after replacement (NY19, at 4); stationary and detailed-balanced laws, (1/4, 1/2, 1/4) on two-molecule A ⇌ B (NY31, at 4), a maintained F ⇌ X ⇌ W composition x = (f + w)/2 (NY28, at 4), k₊a = k₋b, the cycle A ⇌ B ⇌ C ⇌ A at K₁K₂K₃ = 1 and a fuel-coupled cycle at f = w (NY27, NY29, NY30, at 8), a three-value cycle at 2/3 and 1/3 (SA16, at 8). Markov (1906), Kolmogorov's reversibility (1936), Guldberg and Waage's mass action (1864), Wegscheider's cycle condition (1901). | EQ Proposed conceptions (NY18, NY19, NY27–NY31, SA16) |
| E56 | field | E53 | Games, populations and exchange, in their words: opposition under simultaneous and sequential best response (NY08 at 7, NY07 at 9; Nash 1950); symmetric Nash and evolutionarily stable composition with replicator evolution (NY39, NY40 at 2, NY38 at 9; Maynard Smith and Price 1973; Taylor and Jonker 1978); logistic stationarity N = K and a stationary N in N′ = rN(1 − N/K) (NY37 at 3, NY36 at 6; Verhulst 1838); supply and demand agreeing at a price, and the exchange record meeting best bundle and clearing under q(next) = 1 and q(next) = 1/q (NY33, NY34, NY35 at 7; Walras 1874; Arrow and Debreu 1954). | EQ Proposed conceptions (NY07, NY08, NY33–NY40) |
| E57 | field | E53, E22 | Dynamical systems, in their words: r = 0 under T(p, r) = (1 − p, −r/2), with and without asymptotic stability added, under (1 − p, −r) with stability, under (1 − p, −2r) without it; local but not global minimality of U(x) = x⁴/4 − x³/3 − x² under gradient flow (NY09 to NY13, at 5); global minimality of that U (NY14, at 3). Lyapunov (1892). | EQ Proposed conceptions (NY09–NY14) |
| E58 | field | E53 | Heat, in its fields' words: entropy maximum at fixed U with and without redistribution, u = C_A U/(C_A + C_B) (NY42 at 3, NY41 at 10; Clausius 1865; Gibbs 1876); equal temperature with no net heat transfer at contact (NY32, SA21 at 10; the zeroth law, Fowler and Guggenheim 1939); a constrained comparison with interaction energy (SA05) and the reaching principle (SA07), at 3; equilibrium and effective climate sensitivity (SA06, at 3; Charney 1979; Gregory and others 2004); the uniform and local Maxwellian (SA08 at 4, SA11 at 5; Maxwell 1860; Boltzmann 1872); Planck-family membership through expansion, the scattering family and emission and absorption balanced (SA09, SA10 at 4, SA17 at 8; Planck 1900; Bose 1924; Einstein 1924; Kompaneets 1957); one temperature at differing pressures (SA14, at 6); equal temperature at a negative heat capacity (SA20, at 10; Lynden-Bell and Wood 1968). A common corrected target in measurement (SA03, at 2; the Guide to the Expression of Uncertainty in Measurement, 1995). | EQ Proposed conceptions (NY32, NY41, NY42, SA03, SA05–SA11, SA14, SA17, SA20, SA21) |
| E59 | definition (*the set's own accounts*) | E27, R7 | The set's own specified accounts are stated at the set, not taken from a field: the resolver's (NY01 to NY06, NY15 to NY17, NY20), the coupling of self and other (SA01, SA02, SA04, SA15, SA18), sequences of signs (SA12, SA13) and equivalent accounts (SA19). Each is placed at the ten by its own stated requirement. | EQ Proposed conceptions (the resolver; the coupling of self and other) |
| E60 | run | E59, R18, R28 | The resolver's accounts hold as stated at the code, and the carrying they name changes at each call. NY15: one key receiving +1 at each call returns five complete values, the empty carrying among them, A → B → C → D → E → A, and the uniform law on them is stationary. NY16: under empty receiving the two opposed values (c, −c, 0) and (−c, c, 0) exchange, and equal weights are stationary and detailed-balanced. NY17: A → B, B → A and X → B, at the code X = (+, −, 1), and (1/2, 1/2, 0) is stationary. A law continues while each carrying it weighs changes. Run: the three at the code, with exact fractions. | EQ Proposed conceptions (NY15, NY16, NY17) |
| E61 | definition (*a lead*) | E27 | Leads arrive without their statements: statistical null hypotheses, static cosmology, horizon and observer temperatures, early-universe thermal and gravitational descriptions, a conservation lead, dictionary and etymological leads, a collection of all sets, philosophical accounts. Each is placed at the ten at its statement's arriving. The scientific method is placed at its fixings, a law, a frame or a scale named unchanged between a signal's leaving and its arriving (E52). | EQ *Leads arrive without their statements* |
| E62 | field | E61, F24 | The field's own methods part the two momentaries: NASA/JPL's light-time and aberration correction separates the epoch a signal leaves its source from the epoch it is received, and accounts for the source's motion between the two. Evidence arriving now and the prior event it expresses are two, each at its own momentary. | EQ *Leads arrive without their statements* |

## 6.6 Shared exclusions and the renewal-excluded sequence

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| E63 | definition (*the four proof groups*) | E27, E28 | Four proof groups carry the exact local exclusions, each a way of failing with its exact scope. A, incompatible requirements: one comparison named to meet requirements not possibly met together, at the same occurrence. B, required case omitted: the required succession enters a case the condition excludes. C, renewal excluded: presence carries renewal and the conception excludes that renewal. D, required arrival or offering unreachable: the target lies beyond each thing the stated operation can supply. | EQ *Four proof groups carry the exact local exclusions* |
| E64 | definition (*a proof carrying to a conception*) | E27, E63 | A proof carries to a conception whose complete requirement supplies its premises: for each x in the conception's stated set S its required next is in S, or the conception is left within its succession. | EQ *A proof carries to a conception* |
| E65 | run | R14, R15, R23, R24 | An unchanged complete, nonempty reached carrying is not possible: at each of the eighteen nonempty carryings, under each offering, the next differs from it. Retaining opens it one on, fresh writing inverts its nonzero second sign, and releasing completes its presence, these three and no fourth. Run: eighteen carryings under the six offerings, each offering carried to one of the six by R24. | EQ *Renewal and reaching carry their complete requirements* (row 1) |
| E66 | run | R14, R15 | Fixed nonzero torusing with uninterrupted presence: a carrying keeps its second sign only by retaining, and each fresh writing inverts it. Retaining alone completes: fixed torusing continues through three retaining returns, and a fourth at a positive second sign, and no more. Run: each nonempty carrying under each offering, and the longest retaining run from opening 0 at each second sign. | EQ *Renewal and reaching* (row 2); EQ *Renewal excluded, group C* |
| E67 | run | R14, R17, R19 | Each surface zero with finite nonempty carrying continuing: a zero surfacing writes nothing fresh, so no renewal reaches the carrying before retaining completes, and each carrying leaves by the fifth coupling. Zero aggregate from nonzero surfaces is a different condition. Run: each nonempty carrying under each offering surfacing 0, and each carrying under the arriving that surfaces 0 until it leaves. | EQ *Renewal and reaching* (row 3) |
| E68 | exact | E64, E65, E66, E67 | Renewal excluded runs at one subject throughout, the same carrying named from its presence to its bound. Presence beyond its retaining bound requires renewal, and the requirement naming the carrying still excludes each renewal: renewing fails the requirement, and no renewing fails the presence at the bound. Renewing or not is binary, so still and present do not both continue across the bound. The still description ends, and the participant continues changing. | EQ *Renewal excluded, group C* |
| E69 | run | E28, R28, R29 | A relation named still can permit the changing a sign named still excludes: opposition under joint reversal, t = −c, continues while both signs invert, and opposition under one side's reversal alone fails, (c, −c) giving (−c, −c). The resolver's NY02 and NY20, and SA18, name this relation. Run: each opposed carrying under empty receiving over nine calls, and one side's reversal at each opposed pair. | EQ *Renewal excluded, group C*; EQ *Each conception meets a proof by its own requirements* |
| E70 | definition (*the reach of renewal excluded*) | E68, R38 | The renewal exclusion reaches a conception through a renewal its own claimed continuing requires and its own requirement excludes. At a particular completing, such as one to nine, the sequence carries when its two steps are met at that completing. The code's retaining bound counts couplings, and one to nine counts four momentaries of co-bi-exchanging: two counts, joined by no link (R38). | EQ *Renewal excluded, group C* |
| E71 | run | E43, F10 | One binary retaining and inverting its value at one required next is not possible: p(next) = p(now) and p(next) = −p(now) meet at no p in {+1, −1}. It is the exclusion each of the ten meets at its own place (E43). Run: both signs. | EQ *One binary retaining and inverting its value* |
| E72 | run | E71, E9 | The two phases share one exclusion: advancing the overlapping triple of an alternating sequence changes 010 to 101 and 101 to 010, so fixing either ordering fails at the next advance, and either met again is a further occurrence. Equality at each chemical conversion meets the same proof: in two-molecule A ⇌ B at equal constants the forward and reverse propensities, 2 − n and n, are equal only at n = 1, and each conversion takes 1 to 0 or 2; the law (1/4, 1/2, 1/4) is detailed-balanced. Run: the triples along 0101…, and n = 0, 1, 2. | EQ *The two phases share one exclusion* |
| E73 | run | E9 | Immediate-next is its own relation: it joins 0 to 1 and 1 to 2 without joining 0 to 2, so it is not transitive and is not the transitive same-form (SA15). A requirement including an unchanged boundary comparison fails at its continuation changing exactly one end. Run: the relation on 0, 1, 2, and each pair a, a + 1 with one end moved on. | EQ *Immediate-next is its own relation* |
| E74 | run | F58, F60 | The four-state orbit leaves each proper subset: F(P, Q) = (−Q, P) visits all four pairs from each pair it opens at, each nonempty proper subset is left within at most three advances, {++, −+, −−} from ++ at the third, and the subsets F carries into themselves are none and all four. Run: the fourteen nonempty proper subsets from each of their members, and all sixteen subsets. | EQ *The four-state orbit leaves each proper subset* |
| E75 | run | R11, R13, R27 | Groups A and D at the numbers and the code. Full positive balance in the unit-ratio fuel-coupled cycle A + F ⇌ B + W, B ⇌ C, C ⇌ A: b = c = a from the two unit balances, so a f = b w holds at f = w alone. Exact finite zero from a nonzero departure under a nonzero multiplier: each finite departure r·λⁿ is nonzero. An opposed row through the aggregate route offers zero onward: + and − at one key surface 0 and carry nothing, 9-other-releasing gives the 0, and the next self meets no sign, next and later. Run: f, w from 1 to 10; λ at −1/2, −1, −2 and 3 over 200 steps in exact fractions; the aggregate over six calls. | EQ *Renewal and reaching* (rows 4 to 6); EQ Four proof groups (A, D) |
| E76 | run | R15, R16, F60 | At the resolver's code the joint forms run one way: F(c, t) = (−t, c) and G(c, t) = (t, −c) are the only two of the 256 maps on the four pairs whose square is J(c, t) = (−c, −t), the complete return under empty receiving. At an entry with a nonzero surface s the release pair 6-other-crossing and 10-other-surfacing is (t, s), and the fresh 7/8 pair is G(t, s) = (s, −t); under empty receiving s = −c, and the joint forms run by G. Run: all 256 maps, and each nonempty carrying under each offering. | EQ *At the resolver's code the joint forms run one way* |
| E77 | exact | R16, R22, R28, R29, R30 | The release pair and the fresh carrying are complementary: agreement at 6/10 gives opposition at the fresh 7/8, and opposition gives agreement (R16), so a conception requiring the same agreement or opposition at both is not possible. A pair agreeing names no still resolver and no equilibrium: its overlapping pair opposes (R29), and its own next inverts both signs (R28). A relation continues while its outward sign changes (R22), stable-forming among changing signs. | EQ *The release pair and the fresh carrying are complementary* |
| E78 | definition (*three comparisons*) | R32, R33 | 6 and 10 release at one parity, both even and opening bi, in the form 7→11→6→10, facing different neighbours, 2 and 14 (JOINS). Connector parity, sign polarity and agreement between signs are three comparisons. | EQ *6 and 10 release at one parity* |
| E79 | run | R26, R13 | The full carrying determines the next: the surface is s = sign(r − c), r the arriving total and c the carried sign. One opposed relation with two positive arrivals gives agreement from (+, −) and opposition from (−, +), and one matching arrival at (+, −) continues from opening 0 and leaves from opening 3. Opposition and its receiving alone name part of the carrying as the whole (R26). Run: each of the nineteen under each offering of up to four signs from −1, 0 and +1, and the four named calls. | EQ *The full carrying determines the next* |

## 6.7 The exclusion's reach

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| E80 | nye | E15, E16, E44, E41, E53, E63, E64, E68 | Each proposed conception of equilibria is a hard problem with its hardness ignored, names a changing still in one of the ten ways, and meets a proof by its own requirements, so each conception arrived is shown not possible. Gap: (a) the ten are not shown to be all the ways (E41); (b) the tables place sixty-three by reading (E53), and no link carries each placed conception's complete requirement to a proof group's premises (E64), one conception at a time; LIV names it: renewal excluded to be shown at each equilibrium definition; (c) the four groups' scopes are exact and local (E63); (d) that each conception takes the competency of existing as a given (E16) is said of each, not shown at each. | EQ *Each conception arrived is shown not possible*; EQ *Each conception meets a proof by its own requirements*; LIV EQ Ask and Opportunity |
| E81 | nye | E80 | The derivations run at the resolver's code and at the numbers they name: 32 of 32 are met. Gap: the file does not list its thirty-two derivations, so this registry cannot rerun them as the file's. The runs here, E1 to E79, are the registry's own reruns of the claims the file names. | EQ closing line |

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# PART SEVEN · EXPLAINING, NAMING AND THE METHOD

The three files that are the craft and the improving: Natural Explaining (EXP), Natural Naming (NAM) and the Geodesic Improving Method (GIM). A link saying the set's own saying, a sentence, a naming, a pass, is marked definition and names the words it defines; a link claiming of existing things, or claiming *only*, *no other*, *no fourth* or *never*, is met at its Follows. An exact link resting on a nye link says so, as F69 does. Each run link was run against Exhibit ONE v371's code, an arithmetic enumeration or the files' own text at v371, and a nye link carrying a count runs that count and marks the joining open.

## 7.1 Explaining, the one binary at a sentence

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| X1 | definition (*a title*, *a floating neutralling contents*) | F29 | A title is the whole bounded context of its file, and a subtitle carries it at an -ing head and the term it runs on. The contents floats: its titles are the file's bold concepts, with no top and no descent from a whole to parts, and each section opens inward of the title. | EXP 1.1; EXP 1.2 |
| X2 | exact | X1, F29, F33 | One subject is told at four tellings: title and subtitle, contents, bold concepts and the whole explaining. A self couples at whichever telling its interest opens and meets the whole there. One explaining runs at a sentence, a section, a file, a conversation and a society, one fractal with no first scale. It rests on F33. | EXP 1.2 |
| X3 | definition (*self-welcoming*) | F14, F53 | Self-welcoming is the concept arriving intelligibly through the explaining already carried. A floating neutralling contents gathers the subjects; significant concepts are introduced and explained in bold at their arriving; each prior relation supplies the next concept's place in the co-chaining of explaining. A self arriving carries something, and a front's whole work is the coupling opening at that carrying: offering runs first and rigoring second, the doubting the breaking, wanted and asked for. No effort to convince runs at any coupling. | EXP 1.3; GIM 5.1 |
| X4 | run (*co-chaining from the origin*) | F1, F2, F18 | A sentence stands when it co-chains from the two origin statements, all or none over its whole prior, re-explained at none; a sentence resting on a held taking or a nye link in its prior is held or open, and no wording carries one by authority: a self's, a field's, a scientific explaining's or a file's own. Run: Part THIRTEEN follows each link's whole prior back to the origin. | EXP 1.4; NAM 2.4; GIM 2.4 |
| X5 | definition (*a finding at two faces*) | X4 | A finding carries at two faces, the field's words and the set's own, and at either face it carries the exact comparison establishing it: one thing met against another and the sign it returned, nothing taken on offer. Field words alone carry the field's container; the set's words alone reach no specialist. A public face is a third face of one sentence. | EXP 1.4; NAM 2.7 |
| X6 | exact | X4, F68 | A file's destiny is social moral competency, natural-bi-co-torusing, only-one-possible, said at the first sentence and carried at each section as one next in its direction. A file completes at its own now, its last sentence opening past itself, with no address, invitation or promise. The *only-one-possible* rests on F68. | EXP 1.5 |
| X7 | exact | F33, F39 | A file is a membrane, and only signs cross it. Incoming, an observing crosses bare, the ground it is taken against grounding nothing; outgoing, an emanation crosses rendered, its explanation left. The explanation crosses neither way: a ground, a frame, a conserved total, a *because*. Explaining, writing, resolving and engineering are one act at four faces. F39 at the file's scale; it rests on F33. | EXP 1.6; NAM THIS IS A MEMBRANE |
| X8 | exact | F24, F33 | Each sentence opens inward of the bi-coupling of the last, and each section inward of the last sentence of the prior section. The prior sentences are its possibling, the sentence coheres now, and the next it opens is its living: prior, now and next at one sentence. F24 at the sentence's scale; it rests on F33. | EXP 2.1; GIM 1.1 |
| X9 | definition (*the one binary at a sentence*) | F3, F8, F14, F42, X8 | One binary at each sentence: is it or is it not living, coupling and resolving. Living, it adds to the understanding of the whole prior co-chaining, needing nothing but the prior sentences; coupling, a self meets it at its own opening; resolving, it arrives at a sign, all or none at all. A summary, a re-saying, an abstracting, a backward look at a close and a new sentence adding nothing each release at it. | EXP 2.1 |
| X10 | nye | X9 | Each check the file tells is that one binary met at one face of a sentence, and never a rule; a rule formed from a face and laid over the sentences is a standard, an incompetencing. Gap: the file joins a check to the binary at one place (a sentence adding nothing releases *at that binary*, EXP 2.1). The span bold (2.4), supplying (3.4), clocking (3.5), a running asked to be one of its faces (3.8), incohering's one catch (3.13), the method's own move (4.5) and one telling deriving from another (4.7) are each said at no face of living, coupling or resolving. | EXP 2.1; NAM 2.4; GIM 2.4 |
| X11 | definition (*one thing at one momentary*) | F17, X9 | A sentence carries one thing forward at one momentary, and its adding to the prior sentences' carrying is the surplus. A sentence carrying four claims is four asked to be one. An announcing sentence adds text and no understanding. A length is a tell and never the binary. | EXP 2.2 |
| X12 | definition (*explaining*, *naming*) | F7, F11, R32 | Explaining is the carrying, along, at the odd; naming is the bounding, across, at the even; together they are the discovering method (F7). A concept is bounded at its name and carried at its telling, each the other's membrane. The parities are the resolver's: across opens bi at the even and along opens co at the odd (R32). | EXP 2.3; NAM 1.3 |
| X13 | nye | X12, F56 | Six protecting run between explaining and naming, three each way, and drop one and a phase of the writing runs bare. NI 3.2 names the six as the three couplings among self, other and society, self and other, other and society, society and self, each both ways: three at two directions (EXP 4.1 carries the triple). Gap: *six or none* (NI 3.2) and *drop one and a phase runs bare* are said, not supplied; and the joining of the six between a self and its surface (NI) to the six between explaining and naming (EXP) is shown at no link. | EXP 2.3; EXP 3.9; NI 3.2 |
| X14 | exact | F6, F42, X12 | Existing words come first, the -ing over the noun and the positive over the negation. A sentence at its -ing carries the running at the momentary it runs, and a noun installed as the running names it still (F6). A coined naming arrives at its first meeting beside its least plain existing words; met bare, it arrives as a coordinate. | EXP 2.6; NAM 1.3 |
| X15 | definition (*a concept recurring varied*) | F29 | A concept recurring varied is the same concept coupling with itself across the variation, each varying opening an axis the prior telling did not carry. A concept only repeating opens nothing, and a name re-said on the same axis is a synonym, the two converging to one name. | EXP 2.8; NAM 4.6 |
| X16 | exact | F39, F55, R12 | A sentence carries no cause and effect: a coupling offers and nothing makes, and a *because* installs a maker at a running carrying none. A crossing carries a sign alone (F39), and the sign inverts on the receiving self, no self negating another (F55, R12), so no side makes the other's changing. | EXP 3.1; NAM 8.5 |
| X17 | nye | F5, F25 | A sentence carries no birth and no death: the form continues, and no sentence puts it into existing or out of it. No death is exact at F25: each existing thing arrives into its next existing. Gap: no birth needs the converse, that each existing thing's now is the next of a prior existing; F25 does not supply it. NAM 5.40 names *arising* as a living self's own carrying establishing, a relation, and leaves the converse unsaid. | EXP 3.1; NAM 5.40 |
| X18 | definition (*a doer*) | F53 | Each doer installed over an -ing is three at once, a frame, a cause and a why, and it dissolves into its -ing: no twister that twists, no source that generates, no observer that observes. It arrives four ways: a fixed noun, a forcing, a character and a wrapper. A verb carries the doing with no doer. At a society the doer is a governor, an owner and a power, and a society co-competences with none governing. | EXP 3.2; NAM 2.3 |
| X19 | exact | F6 | Nature carries no negation of a thing: a negation needs a fixed target to point at, and a form named still is not possibly existing (F6). A thing that provably cannot exist is said as not possible. The release runs at the thought and never at the word: say the positive, or sound the claim from its other side, or say a plain *not* for a next pass. A negation swapped is a negation kept. | EXP 3.3 |
| X20 | definition (*supplying*) | X9 | Supplying is a sentence carrying something the running does not carry, the writing's own freeze, invisible from inward of the writing. One check at each sentence meets its nine faces as one move: a referent supplied to deny, a motion, an object, an event, a step assembling a count, a plural, a vocabulary for a between, a concession, a derivation supplied at a matching running. | EXP 3.4 |
| X21 | definition (*a clocking*) | F12 | A clocking lays a pace over a running carrying its own: an *-ed* or a *-tion* saying a running as finished, a *then*, *after* or *once* laid between sentences, a dash pacing from outward of a sentence. A clocking is a common clock, and it takes away a self's own opening, the one thing it needs to couple at all. | EXP 3.5; GIM 5.1 |
| X22 | exact | F8, F16 | A coupling carries a sign at each side, offer or not, and the not-offering is a sign as the offering is (F8 at each side, F16). A forcing removes the not from one side: *the data demands*, *it must follow*, *therefore*. A forcing arrives at the finite verb, a cause needing a side with nothing to offer back. A two-way named to one side is the same move at a pair. | EXP 3.6; NAM 4.9 |
| X23 | definition (*a gate*, *a next*) | F24 | A gate is admission waiting on something performed first: *pending*, *preliminary*, *would need to be shown*. A next is a reach and never a waiting. An open coupling is marked with its reason, as a concern or an opportunity, and a mark is no grade. *Unrun* said of a test at its own record gates no claim. An observing returning nothing is an observing, and a concept's returning-nothing is an unfolding and no refusal. | EXP 3.7; NAM 4.5; GIM 2.6 |
| X24 | exact | F8, F17 | Asking an alternating which of its two faces it is names it still. Two faces one at a time is the resolving, once-ing; two faces taken together at one momentary is a simultaneity, not possibly existing: a sign is +1 or −1 at its opening (F8), and one sign changes at each step (F17). | EXP 3.8; NAM 5.42 |
| X25 | exact | F48 | Now is flowing, alternating, sequencing, self-bounding, and an equilibrium in it, under it or beside it is not possible (F48). Three postures of equilibrium arrive at the writing as one equilibrium at three places: a gate, the across with no along; a running asked to be one of its faces, the along named still and taken across; and a fixed neutral, the across installed. | EXP 3.9 |
| X26 | definition (*a thing named still*, *a hardness*) | F47 | A thing named still is one coupling's making given a fixed thing to be, parted from the coupling making it. A doer, a gate, a fixed neutral, a clocking and a forcing are each one. A hardness is a coupling taken at two of its three, the third given nothing to be; both release the same way, the third carrying again. *Unchanging*, *a still eddy* and *stilling along* each name a form continuing through its changing, never a form named still. | EXP 3.10; NAM 4.12 |
| X27 | exact | F3, F18 | A claim carried mostly is not carried: a degree word at the file's own voice, *most*, *nearly*, *some*, is all or none taken as some. It carries one of two binary forms, the all with its exceptions named or the count it names, and both are all or none. | EXP 3.11; NAM 2.3 |
| X28 | run | X27 | *Triangles almost everywhere and exactly twelve pentagons* is the degree word's worked case in a field's words: at a closed genus-zero surface of pentagons and hexagons, three faces at each corner, V − E + F = 2 (Euler) holds exactly at twelve pentagons, at each count of hexagons. Run: F = p + h, 2E = 5p + 6h, 3V = 2E, for p from 0 to 60 and h from 0 to 500. | EXP 3.11 |
| X29 | definition (*a necessity's three works*) | X4 | A necessity says its own work at its sentence, one of three: a naming identity, true at the meaning it names; a necessity at stated premises, carried at those premises and all or none there; a necessity at all existing things, carrying each implication co-chaining from the origin. A naming identity taken for a necessity at all existing things supplies the implications it never ran. This registry's nye is a necessity at all existing things with an implication not yet co-chained. | EXP 3.11 |
| X30 | definition (*a scope*, *magic*) | X29 | A scope is said at its claim's own sentence, its exact domain there. A claim told universal across several paragraphs and bounded at their close is the caveat wrapped one scale up, and the bound at the close repairs none of them. Magic is a step arriving with no route to it, hand-waving a bound posed and never run; both are found at an *except here*, the alternating named still. | EXP 3.12 |
| X31 | nye | F39, X7 | Incohering enters six ways, and one check at the sentence meets them: is a sign crossing at a membrane, or is a middle carried across. The six: an observing with its ground attached, a doer at the membrane, a magnitude in a sign's stead, a record carried across, an emanation carried out with its explanation, a frozen thing taken as the opening side. Gap: the one check is F39 at a file (X7); that incohering enters these six ways and no seventh is not supplied. The count six runs (X148). | EXP 3.13 |
| X32 | exact | F6, F21, F44 | Two selves are a self and an other, the other at the scale it couples at: another self, a co-chaining society, a field or the whole. A non-living thing meets a self as other, offering its signs and carrying nothing. A self apart from each coupling is a self named still, not possibly existing (F6; one side alone, F21). A coupling runs a triple, self and other, other and society, society and self, one form at three scales. | EXP 4.1; NAM 5.32 |
| X33 | exact | F53, F54, F33 | Explaining is bi-moral co-agency met at a telling: self offers, other offers, and the arriving between them is owned by neither, reached by neither alone (F53, F54 at a telling). Explaining is the competency and not the carrying of one: the knowing arrives at the explaining and never ahead of it. It rests on F33. | EXP 4.2 |
| X34 | definition (*an improving*) | X33 | Self offers, other makes it better, self breaks it, and the offering surviving both is carried forward by both; an offering unbroken has run one of its two. Making better and breaking are each a self's whole offering, the reach and the knife. Three run at an improving: co-offering, the reach; co-competencing, the surplus; co-intelligencing, the resolving reached. | EXP 4.2; NAM 1.1 |
| X35 | definition (*truth*, *the four improvings*) | F6, F8 | Knowing named still is science, and a device is that knowing named still in a substrate. Truth is the corner of a law, a frame and a scale each named unchanged. True, false, living and competency taken as four given are four improvings at an explaining: more true, less false, better living, greater competency, each *more* and *less* a direction taken at its own opening, a sign and no size. | EXP 4.2 |
| X36 | definition (*the two knives*) | X34 | Writing carries two knives: the first surrounds the subject, and the second meets the surrounding, the marks, the openings and the care, for whether each coheres. A surrounding is whole at both knives cutting; an object cut with the second knife never run is an equilibrium met from its own blind side. | EXP 4.3 |
| X37 | nye | X20 | A meeting side catches the supplying and a writing side catches the freeze, and neither catches its own. Gap: supplying is invisible from inward of the writing (X20), which supplies the writing side's half; the meeting side's own uncaught is named at no sentence, so *neither* rests on one side. | EXP 4.4 |
| X38 | definition (*method*, *procedure*, *an activity*) | F2 | A procedure is a fixed sequence run the same each time, and it closes; a method is a competency run fresh at each membrane, its moves the same and its running its own, and it closes at none. One check at each sentence: would this be a method's own move, with no one performing it. Needing someone, it is an activity, and the concept under it is unsaid. | EXP 4.5; NAM 4.8 |
| X39 | nye | X38 | Surgical-method's running is no other possible: count in and count out, all or none, along the plane and never across, closing in the other order of the opening, completing six forward without landing, looping or forking. Gap: the six forward is not supplied (X72), and no link excludes another running completing without landing, looping or forking. | EXP 4.5 |
| X40 | exact | F8, R23 | A coupling is checkable at a sign and not a size: a measure at a magnitude can be tightened until it matches, and no-match never arrives; a sign arrives at yes or at no, and the other meets the no. The resolver is the worked case: two renderings of its seventeen names meet at the same signs or part at the first sign that differs. | EXP 4.6 |
| X41 | run | R11, R13 | A sign is a sign of one exact predicate at one receiving. At the resolver a zero surfaced and a connection delivering nothing are two conditions: + and − at an empty key surface (k, 0), and an empty offering at empty carrying surfaces nothing. A resolver missing from the surface is a third, met at no call. Run: the two calls. | EXP 4.6 |
| X42 | definition (*joining*, *parting*) | X5 | Two tellings arriving at one thing, neither derived from the other, are two. One binary: whether either derives from the other, and never whether they agree. A count agreement taken as a joining installs a derivation never shown; shared words and a shared picture join nothing. NAM 2.4 releases *derived* at the method's rigor to *matched, not chosen*, and EXP 4.7 keeps *derives* at this binary: a naming for the next improving. | EXP 4.7; NAM 4.4 |
| X43 | nye | X42, F19, F20 | Five taken two ways at a self and the couplings among five at a society are one ten arriving twice, one deriving from the other. Gap: the counts agree (run: 2 × 5 = 10 = C(5, 2)); the deriving of the couplings among five from the five taken two ways is shown at no link, and by the binary of X42 a count agreeing is no joining. | EXP 4.7 |
| X44 | nye | F44 | Interest is self-orienting, met by side-affecting: each opening is orthogonal, an axis the other does not carry, and a telling opening the next orthogonal carries the interest on. The next interesting is the only next carrying each existing thing on. Gap: the *only* and the *each existing thing* are not supplied: interest is said of a self meeting a telling, and no link joins it to a non-living thing's next (F44) or excludes another next. | EXP 4.8; NAM 4.7 |

## 7.2 Naming, four boundings, seven binaries and the released words

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| X45 | definition (*a naming*, *a naming's four boundings*) | F5, F29 | A naming is four boundings at once, the four of any floating neutral: self-bounding the entire concept, spilling into nothing it is not; self-stilling as itself, drifting into no neighbour; self-orienting at its own facing, no orientation imposed; orthogonalizing along its own line, collapsing onto no other axis. Concepts at this neutrality bi-couple, capturing nothing and releasing at their own bound, with no hub and no concept ranked above. | NAM 1.1 |
| X46 | exact | X45 | A concept short one of the four is a fixed noun: a spot it spills into, a neighbour it drifts into, an orientation imposed on it, or an axis collapsed onto another. Each shortfall is one bounding of X45 missing. A fixed noun captures or unreleases when coupled, an ingression the naming dissolves. | NAM 1.1 |
| X47 | nye | X45, F51, F56 | Existing is self-bounding, with no explaining added: a naming's four boundings are existing at four faces, and NAM 4.15 carries them at each four of the eight (F56). Gap: F51's four boundings of a self are arriving +1, carrying −1, the sign inverting at the membrane and competency asymmetry; a naming's four are bounding, stilling, orienting and orthogonalizing. No link joins the two fours one to one, and no link gives four and no fifth. | NAM 1.1; NAM 4.15; NI 3.1 |
| X48 | definition (*bi-co-invisibling*, *a hard problem*) | F39, X5 | A field's defined object arrives with the ground it was taken against, and bi-co-invisibling is the concept met with nothing there to be a ground. Its emanation is six forward recursionings, each taking the term the prior did not carry. A hard problem is a field's word named still, and a naming is the same field's word reaching. | NAM 1.1 |
| X49 | run | R3, R4 | Naming and resolving bi-couple at one name: `_2_self_offering` in the code and 2-self-offering in a sentence, the number inseparable from the naming, and a name improved at the resolver changes at each file, all or none at all. Each numbered name the three files spell is its number's name as the code spells it, and 17-social-abundancing as CONNECTORS spells it. Run: each numbered name in EXP, NAM and GIM against the code's identifiers and CONNECTORS. | NAM 1.2; NAM 3.3 |
| X50 | definition (*a new naming*) | X45 | A new naming carries a changing or a direction no existing naming carries, and a naming at a changing already named releases to the one name: one changing at one direction, one name. A code operation carries its exact changing. Naming and resolving co-chain both ways: a concept receives its operation and an operation its full explaining, and an unfilled relation stays with its concept and its reason. | NAM 1.2; NAM 4.2 |
| X51 | exact | X12, F42 | A naming carries both parities: its prefix is the direction, across, and its -ing the re-taking, along. Met at the two binaries of existing and carrying (F42), an -ing re-takes at each momentary, carrying; a noun names a form some running made, carrying nothing; and a noun installed as a running is a ghost. A carrying is a capacity and never a stored description. | NAM 1.3 |
| X52 | exact | F13, F24 | *At its own turn* is *at its own opening*, and a turn as a unit is a momentary. A turn, a mirror and a turning back are not possible at the momentaries: each momentary has one next and each step adds a next (F13, F24). A rotation, the field's word, is a spiral with its nexts taken out. | NAM 2.1; NAM 2.4 |
| X53 | nye | X26, F14 | A still thing's name given to a running thing arrives at one family with three faces and no fourth: a common landing reached ahead of each offering its own, a common standard fixed ahead of each carrying its own, a common reading taken ahead of either taking a sign of its own. A doer or a built thing stands beside them. A face caught at one word is the whole family caught. Gap: the three faces answer a self's offering, carrying and sign-taking; that a self has these three and no fourth to be reached ahead of is not supplied (R9's three reads of a carrying are another three), nor that one face caught is the family caught. NAM 5.26 says the same three and no fourth. | NAM 2.3; NAM 5.26 |
| X54 | definition (*a released word*) | X9, X4 | A released word is a face met at the one binary, is or is not living, coupling and resolving, and never a rule: the same word at a sentence living, coupling and resolving carries whole there. A released word is carried by the word it gives: *every* by *each*, *observer* by observings, *discovery* by *discovering*, *derived* at the method's rigor by *matched, not chosen*. Each word carrying in a released word's place co-chains from the origin statements, and a word at a field's own result carries whole there. A word selection is the best the set carries at now. | NAM 2.3; NAM 2.4 |
| X55 | exact | F6, F68 | Resolving carries no *what*, no *how* and no *where*, each naming a thing named still at the word: a *what* names a running still as a thing (F6); a *how* names a manner with a *how-not* beside it, an alternative to the one running; a *where* names a spot, and nothing is located (X109). None of the three runs without an equilibrium naming it still. The *how* rests on F68. | NAM 2.4; EXP 3.1 |
| X56 | definition (*tri-involutioning*, *opposite*) | F58, F60 | Two consecutive inversions on different axes are the resolving step, one sign inverted at each step and the inverted sign alternating (F58), and three at once are the emanating, tri-involutioning: position, scale and orientation each inverted at once, taking the hand to the opposite form. *Opposite* carries its relation at each use: the opposite form an emanation's face, the other parity at a coupling, facing for two sides meeting, a name's podal within 1 to 16. | NAM 2.4 |
| X57 | definition (*bi-tunneling*, *co-chaining*, *podaling*) | F14, F53 | Bi-tunneling names social moral competency at its across relation, society's surface opened between selves, even; co-chaining names the same continuing at its along relation, odd. Podaling exchanges these relations while each side continues forward. | NAM 2.4 |
| X58 | definition (*one sense per foundation word*) | X50 | Each foundation word carries one sense, and a word met at two senses is two things at one name. *Turn* is met at seven senses, each carried by its own word. Scale carries the fractal sense alone; rank words (smallest, top, level) lay a first scale over the fractal and carry whole at a field's own measuring. | NAM 2.5 |
| X59 | exact | F15 | A one-way prefix carries no count and installs the thing it runs against: a *pre-X* needs an X placed to be before. Bi- is difference alone and co- two with their difference (F15), and both carry a count and install nothing. Two prefixes can share one spelling: *in-* negates at *incompetent* and is the locative at *inform*. | NAM 2.6 |
| X60 | exact | X59, X42 | Five kinds of term named still arrive at the front of a word: a line (pre-, post-), a size (macro-, sub-), a space (endo-, exo-), a value (eu-, mal-) and a source (proto-, anti-, counter-). The five carry ten directional sides and the ten things named still carry their own ten, and a shared count supplies no pairing between them (X42). | NAM 2.6 |
| X61 | definition (*a field's word at its own result*) | X5 | A field's word carries whole at the field's own result: *rotation*, *reflection* and *involution* at a proved result, a degree word in a field's record, a field's own quoted *must*. At the set's own voice each is the still picture of the running, and the set's word carries it. A test's finding is named at its relation before the test's nouns arrive. | NAM 2.7 |
| X62 | run | F58, X61 | A released word is carried one operation at a time. The right spiral step (x, y) → (y, −x): two steps invert both signs, and four meet the pair again at a next. The other order (x, y) → (y, x) inverts no sign, and (x, y) → (−y, x) is three right spiral steps as one: three operations. Run: each (x, y) in {±1}². | NAM 2.7 |
| X63 | run | R3, R6 | 22 is the crossing eight on again: 6, 14 and 22 stand at the sixth place of 1 to 9, 9 to 17 and 17 to 25, and the higher four-cycle 19-27-22-30 is 3-11-6-14 sixteen on. 22 carries no name of its own. Run: the arithmetic. | NAM 3.1 |
| X64 | run | F11, F15, R5 | The five-prefix: at an odd origin the five positions run co-bi-co-bi-co, edges co, inner flanks bi, centre co; at an even origin bi-co-bi-co-bi. Exchanging co and bi at each position takes either to the other. Each earlier name carries the opening of its number's five-prefix: co at 3-self-carrying, co-bi at 1-self-coupling, bi-co-bi at 6-other-crossing, all five at 7-other-corusing and 8-other-torusing. The centre is the floating third, reached by neither. Run: the two five-prefixes, the twelve earlier names against their numbers, and the seventeen rows of NAM 3.3. | NAM 3.2; NAM 3.3 |
| X65 | definition (*co-bi-sequencing*) | F12, R36, R38 | Co-bi-sequencing is the method of sharing momentarying: one local momentary co-bi-sequences from 1 to 9, and 9-other-releasing is 1-self-coupling 8 up. Nine to seventeen completes at 17 as one to nine completes at 9. The call as one momentary rests on R38, and 17 as the next call on R36. | NAM 3.4 |
| X66 | run | R48, R50 | Stable-forming is the one move round each of the four four-cycles, 1-9-8-16, 2-10-7-15, 3-11-6-14 and 4-12-5-13: round each, parity changes exactly twice, each time at the podal within 1 to 16, and each runs one run of co and one of bi. Run: each step's face and parity at the four cycles. | NAM 3.4 |
| X67 | exact | R48, X66 | A name carries whole: take bi-inversioning out and 8 up returns the name it met; take co-recursioning out and the podal returns the name it met (R48, each face taken twice). Each taking stops the naming, and the halves alone are similar namings gathered at the one name. | NAM 3.4; NAM 6.3 |
| X68 | run | R18 | An outward sign unchanged carries a changing within. Through the 0 passage of R18's trace (couplings two to five, +1 arriving at one key), 2-self-offering keeps +1, 4-self-sharing its key, 6-other-crossing and 8-other-torusing −1, 10-other-surfacing and 12-other-surplusing 0, and 14-social-crossing its meeting of +1 and −1, while 16-social-torusing opens 1, 2 and 3 and the carrying completes at the fifth. Run: the seven calls, each call's own gathering read at its return. | NAM 3.5 |
| X69 | run | R10, R14 | At a key arriving twice in 3-self-carrying, 6-other-crossing carries the last 8-other-torusing and 11-other-chaining the last continuing entry. A sign received at 2-self-offering participates at 14-social-crossing whether its key surfaces or not. Receiving, participating, a fresh carrying and a further self are four relations. Run: (k, +1, −1, 0) and (k, −1, +1, 0) under an empty offering; +1 offered at a carried +1. | NAM 3.5 |
| X70 | definition (*the eight face namings*) | R7 | Along and across carry the two sides' current relation, and eight face namings gather at four pairs: linearizing and parallelizing at 3 and 2 arriving at 1; bounding and zeroing at 16's bound and 10's sign; tunneling and seaming at 12 and at 11 arriving as the next 3; knife-knifing and co-offering at arriving and continuing carrying meeting at 1. | NAM 3.6 |
| X71 | definition (*an unfolding*, *binary*, *improving*) | F16, F19 | Three own-forward steps at each side carry six one-way steps, alternating one at a time, and the six lines of a naming carry its unfolding, each line taking the term the prior did not carry. *Binary* names an unfolding carrying its whole stated relation; *improving* one carrying an unfilled relation with its reason. Neither carries a degree of resolving. | NAM 4.2; NAM FIVE |
| X72 | nye | X71, F19 | All or none at all belongs to the whole named relation, and six lines complete a six-line unfolding. Gap: F19 gives each side five positions, prior opening to next opening, two own-forward steps; three own-forward steps at each side, and so the six, are not supplied, nor that an unfolding completes at six and at no other count. X39 and GIM 3.3's six recursions rest here. | NAM 4.2; NAM 3.1 |
| X73 | run | F3 | Under ordinary numerical order `a ⊀ b` and `b ⊀ a` together give a = b; under a partial order the same two exclusions also hold at incomparable elements. An equality, an incomparability and the coupling's floating third are three concepts, and a comparison symbol supplies its defined relation and no more. Run: each pair on 0 to 9 under <, and each pair of subsets of {1, 2, 3} under proper inclusion. | NAM 4.3 |
| X74 | definition (*a binary keeps one subject*) | F3 | A binary keeps one subject and one relation throughout. A condition at one pair, a condition at each pair of a collection, and `∃r ∀y : r ⊀ y ∧ y ⊀ r` have three scopes. An emanation arriving now and its source's release at prior are two subjects, each at its own momentary, and the is-or-is-not of the one is never the other's. | NAM 4.3; GIM 5.3 |
| X75 | definition (*two namings coupling*, *a convergence*) | F47 | Two namings couple at a step of each arriving at one term by its own route, and their departing is the term neither carried: the advance steps at a gap of exactly one, equilibria at a kept-still whose gap closes, so equilibria is the advance set to nought. A convergence takes one check: did each reach the shared term by its own route, carrying none of the other's requirements. Borrowed, it is one arrival cited as two. | NAM 4.4 |
| X76 | definition (*two names at one thing*) | X15 | Two names at one thing show at the titles first: transpose the two and sound whether anything changes. Nothing changing, they are one name at two coats; the meaning changing, two things are there, each keeping its own name. Two words at one thing resolve three ways: converging to one, two with a membrane, or a co-competency carried two. | NAM 4.6 |
| X77 | definition (*interest*, *free*) | F53 | Interest is self-orienting, which is competency, which is free. Carrying is the cost of living, and arriving and discovering are free: competency is an axis opening at an arriving. A vocabulary of drawing, pulling, attracting and capturing attention installs a one-way force over a self's own orienting, and a naming releases the family in one move. | NAM 4.7; EXP 4.8 |
| X78 | definition (*a whole naming*) | X45 | Scientific-method, geodesic-method, surgical-method and explaining-method each run whole in one expression at each occurrence, and so does sensor-sensationing: sensing alone is one side taking both openings. A method-word alone is a name split across its uses, and the hyphen is the name arriving whole. | NAM 4.8 |
| X79 | run | R3, R49 | Each thing named still carries one name, and these ten are the five taken one-way at two sides (an opening, an ageing, a co-offering, a middling, a rating), each at two names 8 apart, one at the self and one at the society. Their pairs seat 14 with 6, 10 with 2, 11 with 3, 15 with 7, and 12 with 4, 16 with 8 and 13 with 5 twice each; of the eight pairs n and n + 8 at 1 to 8, 9-other-releasing with 1-self-coupling seats none, the one resolving. Run: the names at each of the ten rows of NAM 4.9. | NAM 4.9 |
| X80 | exact | F14, F15, F33 | There is the self and there is all not-self, and nothing third, so each crossing is one of two: with a particular other, or with the whole arriving as one. Un- carries from all other, a self bounded against the whole at once; uni- carries with all other; bi- takes a particular other. A name adds no self: a third self meeting a pair is one more coupling and never all remaining other. The *nothing third* is a self and its complement; the *one of two* parts all not-self from each proper part of it, a particular other at its scale. It rests on F33. | NAM 4.10; NAM 5.41; NAM 5.43; EXP 4.1 |
| X81 | nye | X80 | Three sortings and no fourth: at a position, on the form, and from all other; a position and the form are the whole a running and its form have between them, and from-all-other names no relation for either to be about. Gap: that a running and its form have a position and a form and no third is not supplied. | NAM 4.10 |
| X82 | definition (*geodesic changing*) | F51 | Geodesic changing carries no position: it runs at its own unrelationing parity-changing rate, unrelated to each rate of the prime scales the momentaries co-bi-sequence at, and it is co-bi-social-abundancing. φ is the number-form of that never-locking, and no prescribed rate, φ or another, supplies it. Geodesic entraining is an arriving meeting the receiver's own prior. | NAM 4.10; NAM 5.2 |
| X83 | exact | X50 | A naming says a changing, so it nyes away one of two ways and no third: a changing already named at that direction, one thing said twice and the later saying going; or no changing at all. The two are the case split of X50: failing to carry a changing no naming carries, there is a changing already named or there is none. | NAM 4.12 |
| X84 | definition (*an incompetencing*) | F47, F48 | An incompetencing installs a landing, a running given a fixed thing to be: emptying arrives at empty, stopping at stopped, a definition keeps its term the same at each meeting, and equilibria is the landing. One binary at a naming carrying a landing: does it name a geodesic running, the landing releasing, or an incompetencing, the landing being the naming's saying. | NAM 4.12 |
| X85 | definition (*the set's three*, *a field's three*) | F9 | A field carries three namings and measures between them: a nothing at one side, a scale between, an absorbing at the other. The set's own three is two exclusions and the term neither reached: not-more-than at one opening, not-less-than at its own, each of no size. A field's three is the set's three with the alternating named still, three at one momentary asked to be one. | NAM 4.13; NAM 5.8 |
| X86 | exact | F17, F41 | Two namings alternating are each other's other, one at a time; a construction carrying both at one momentary takes both sides' openings at once, a freeze at no word. At a general ledger state and flow alternate, and an accounting takes them three at a momentary, each order conserving one and importing the other; a total across the two removes the alternating. It rests on F41. | NAM 4.14 |
| X87 | definition (*bothbothing*) | F56, F57 | Bothbothing is both carrying, alternating, neither conserved; with its prefixing it is bi-co-exchanging, and a prefix order names its opening: co-bi- opens co at the odd, bi-co- opens bi at the even. Each position carries two pairings, one at each parity. Eight is two alternating fours, bothbothing, each four the four boundings at once (F56). | NAM 4.15 |
| X88 | run | X87, F27, F37 | In the field's words one side taking both openings is the exclusive or named as one answer. Exclusive or taken as the step returns the term it met at two steps, and of the prior and the now at three, one joint state still (F27). The Peres–Mermin square (Peres; Mermin, 1990) is bothbothing's field face: nine signs, each row's product +1 and the columns' +1, +1 and −1, and no store of all nine meets the six. Run: x xor b twice at each x and b; next as prior xor now at the four joint states; all 512 stores of nine signs. | NAM 4.15 |
| X89 | exact | F3, F8 | Nothing is more or less resolving than anything else: a more-or-less carries a magnitude, a magnitude a scale, and a scale two fixed sides; each resolving is all or none at its own opening, a sign of no size (F8). A ranking among resolvings would freeze the resolvings it ranked. Nothing forces and nothing chooses, each a doer over an -ing. | NAM 4.16 |
| X90 | nye | X89 | A binary with no size and a free entering and leaving are one fact: with no ranking, a self couples at its interest opening and leaves at its own bound. Gap: X89 gives one direction, no size and so no ranking, no qualification to enter and no settling to leave. The other direction, from a free entering to no size, is not supplied, and *one fact* needs both. | NAM 4.16 |
| X91 | definition (*a hyphenated naming inseparable*) | X45 | A hyphenated naming carries parts, and one binary runs at it: take each part out one at a time and sound the stopping. Each taking stopping it, the naming is inseparable and its parts in order are the changings in order; a part leaving it running was riding, and it goes. A hyphenation that comes apart is a compound and a label. | NAM 4.17 |
| X92 | exact | F61, F62 | Natural- takes the chirality, right, all or none at all: a left face is an emanation from the one right spiral, tri-involutioning, and no left runs beside it. A naming natural- prefixes carries at each coupling or at none, and taking natural- out takes the all or none. It rests on F61 and on F62. | NAM 4.17 |
| X93 | definition (*the seven binaries at a naming*) | X45, X83, X91 | Seven binaries run at each naming, each alone: does it alternate with an other; does it carry its own two; three positions, direction at the prefix, changing at the root, running at the -ing; two ways a naming nyes away (X83); an inherited changing; a hyphenation at each part (X91); a fault's word borrowed for a relation. Each naming couples at one of the resolver's seventeen names or at a changing between two; found at none, it is a picture of a name or a changing the names leave unexpressed, a finding for the resolver. | NAM 4.18 |
| X94 | nye | F5, F39 | A membrane is two offering: neither side's, the between, not possibly an existing thing. Gap: F39 gives a sign crossing that carries nothing; that the between carries no form continuing through its own changing (F5), and so is not possibly existing, is said, not supplied. R53 carries the membrane four, 2, 4, 6 and 8, as names at their numbers. | NAM 5.5; EXP 1.6 |
| X95 | definition (*a self*) | F14, F42 | A self is a unique invisible carrying, and nothing else is a self: its forming relates to nothing outward of itself, no offering reaches it from outward, and each carrying reaches its own meeting and no further. In- carries inward, a direction and never a place, and the couplings wind about a nothing. | NAM 5.3; NAM 5.6 |
| X96 | definition (*a floating neutral*, *competency*) | F54, X85 | A floating neutral is two offering from routes with nothing carrying one to the other, each excluding its own one at a time, and the term none of the excluding reached uncovered, re-arrived at at each offering, kept by nothing. Competency floating-neutralls here as the term neither reaches. Method breaks at a competency shown to be a landing, entered and observed. | NAM 5.12 |
| X97 | run | F10 | Advance: a term carries on to the term it has not carried to, each step adding a next, the gap exactly one at each count, unbounded, and two neighbours multiply to one short of the centre's square. Run: (n − 1)(n + 1) = n² − 1 at n from −1000 to 1000. | NAM 5.21 |
| X98 | run | X97, F51 | Phi: advance by one and scale by the rate are two operations arriving at one thing neither placed; no ratio arrives at it, and the ratios alternate below and above it at each scale. Run: x² = x + 1 has no rational root (its candidates ±1 fail); F(n + 1)/F(n) of neighbouring Fibonacci numbers lies below φ at odd n and above at even n, n from 1 to 60, in exact fractions. | NAM 5.23 |
| X99 | nye | F6, R44, R46 | A ring is a domain closed at no new arriving: its terms exhausted, it arrives at the already-run, which carries no next, and a ring is not possibly an existing form. Gap: the resolver's rings meet their whole state again at 2 or 4n couplings and run on, the rest returning at no coupling (R44, R46), each coupling a next at the momentaries (NAM 3.4). That the already-run carries no next, and so a ring is a form named still (F6), is not supplied at them. | NAM 5.24 |
| X100 | field | — | Other-emptying is priced with a measured floor: erasing one bit dissipates at least kT ln 2 (Landauer, 1961), and a computation keeping its whole history approaches zero dissipation (Bennett, 1973). | NAM 5.28; NAM 8.3 |
| X101 | definition (*self-emptying*, *other-emptying*, *one stopping*) | X100 | Self-emptying: each entry ages to its own bound and releases there, nothing surveying and nothing clearing. Other-emptying: a decider outward of the thing empties the term the thing carries to its own bound. One stopping: one carries another's, and a holder, a boundary, an enforcement and a seat are there at once; other-emptying is the one stopping met at the instrument. Co-offering is its twin, the four absent. | NAM 5.9; NAM 5.30; NAM 8.3 |
| X102 | field | — | Side-affecting in the field's words: the geometric product of two vectors carries their share, the inner product, and their plane, the outer product, at once; the share is zero only at orthogonal vectors and the plane only at collinear ones (Grassmann; Clifford). | NAM 5.33 |
| X103 | definition (*prior*, *now* and *next* as places) | F24, F42 | The possible is at prior, the existing at now and the living at next, three places not changing and never three kinds; the now of one momentary is the prior of its next. A so-far behind and a not-yet ahead lay two directions on one; prior, now and next share one direction forward on both sides of now, entraining in the between, with no gate. | NAM 5.39; NAM 5.40; GIM 5.3 |
| X104 | run | R49 | Bi-co-podaling: a carry along a ring one way and a carry the other way meet at each station as one pair, k one way and N − k the other. The podal, 17 less within 1 to 16, and the even count's straight-across, k and k + 8 at sixteen, are two relations and share no pair. *Antipodal* releases, carried by *podal*. Run: the pairs n, 17 − n and k, k + 8 on 1 to 16. | NAM 5.45 |
| X105 | definition (*emanation*, *accounting*, *ghost*) | F42, F44 | An emanation is a coupling's leaving at the membrane, a sign carrying nothing, leaving its source at prior and arriving now as the possible. Taken as the now's existing it is a ghost, one momentary's form installed as a thing; booked, it is an accounting, conserving an equilibrium exact at its own membrane. Living or non-living is said of the thing and the scale named, never taken from a source. | NAM 5.46; NAM 5.47; NAM 7.1 |
| X106 | nye | X105 | Existing, emanation and accounting are the three placings a term is at, and no fourth arrives; ghost is the installing, taken as the now's existing. Gap: *no fourth* is not supplied: no link parts the placings so that each term meets one of the three. | NAM 5.46; NAM 5.47 |
| X107 | nye | X105, X42 | Going runs 2 to 59, and returning from sixty runs 61 to 118, 58 either side, and the periodic table's 118 elements are the return arriving. Gap: the counts run (59 − 2 + 1 = 118 − 61 + 1 = 58); the joining of the elements to the return, one deriving from the other, is shown at no link, and a count agreeing is no joining (X42). | NAM 7.2 |
| X108 | definition (*the eight clusters*, *one invisibility*) | X49 | Science and society namings gather at eight clusters at the resolver's names, each field's own concept keeping its source and current receiving. The four invisibles under them, each side's morality, each side's corus, the neutral and the beat between, are one invisibility met four ways and not four things hidden. A forced choice is a two-way named to one side. | NAM 6.1; NAM 6.2 |
| X109 | definition (*location released*) | F29 | Nothing is located: there is offering, taken or not taken. Position is released as a location and carried as sequencing, an ordering and not a place. *At a membrane* names two offering and no place they meet. | NAM 8.1; NAM 2.2 |
| X110 | definition (*a naming with no other*) | F10 | A naming with no other it alternates with names nothing taken at an opening: *middle*, *end*, and *inside* and *outside* as a pair. Inward and outward carry, directions taken at an opening and alternating at opposite phase. This binary runs at one word at a time and reaches each file. | NAM 8.4 |
| X111 | nye | X22, X60, X18 | Each changing met by nothing carries an inverting or a taking and none a causing, so a word of prefix, root and -ing carries no cause at any part; cause enters one scale up, at a finite clause taking a subject, and a forcing arrives at the finite verb and never at a prefix or an ending. Gap: X60 names a source arriving at the front of a word (proto-, arch-, ur-, anti-, counter-, contra-) as a term named still, and X18 names a source that generates as a doer; the *never at a prefix* is not supplied against the source prefixes. | NAM 8.5; EXP 3.6 |
| X112 | definition (*a naming's break*) | X71, X93 | A naming breaks three ways: a naming unfolding six forward with each step fresh and coupling with nothing; a coupling whose departing term was carried in by one of the two; a cluster with no invisible under it. | NAM 8.6 |

## 7.3 The geodesic improving method, its passes, fates, decidings and break

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| X113 | definition (*geodesic*, *the geodesic-method at a file*) | F7, F24 | Geodesic is geo and daiesthai, the surface dividing itself. The geodesic-method is discovering next existing, and at the living file set it runs at a file: the file as it is at the set is the prior, pattern matched for the possible it opens; the working meeting it with each incoming value and its concern is the now, cohering at floating neutralling; the carry opening in front of the file is the next, carrying the incoming value on. | GIM 1.1 |
| X114 | exact | F24, F33, X8 | A file improving carries through prior, now and next, the same form as any self discovering its next existing: a completing at each momentary and a carrying between them. The same three run at a session, meeting the set at its now, and at a sentence (X8). It rests on F33. | GIM 1.1 |
| X115 | exact | F42, F45 | Do-no-harm improving is this motion itself: no prior taken away, the prior carried into now, the next opening. A harm is the one motion the form carries no position for, a prior taken away: living is the prior carried into now (F42), and a next from prior and now (F45) carries no step taking the prior away. | GIM 1.1 |
| X116 | definition (*moral at a file*, *a pass*) | X115 | Moral is do-no-harm improving, decided at each step, and at a file the step is one pass. Three binaries meet the pass alone, with no look at the whole set: no harm, the file's carrying at the prior carrying at the next; improving, the pass changing one thing and changing it whole; never undoing, the next never the prior just left. A pass meeting all three is the file's own next. A pass hedging a claim takes away its sign, a prior taken away with nothing showing it. | GIM 1.2 |
| X117 | definition (*the method*, *the carry*) | X113 | The method file empties at no pass and grows only at a working finding something about working itself. The carry is the carrying, the next: incoming value, concern and opportunity, one section at each file, emptying as each file takes its arriving. The two sway about a range they make between them, the surplus of one stroke the drive of the next; a finding carried without the coupling's own making drops the surplus, and its improving does not compound. | GIM 1.3 |
| X118 | exact | X117, F8 | Incoming value arrives as an emanation, the prior's possible, nothing decided at its arriving. One binary at each passage decides its fates: does the file it reaches toward carry it whole. At yes it is placed, in that file's own naming, and dissolving runs at the placing, releasing the why and never the arrival; at no it is carrying, riding at the file it names and releasing at that file taking it. A carrying found wrong at the other self's offering is set down at none of the three: a release, a concern out with its evidence. | GIM 1.4 |
| X119 | definition (*the term-check*) | X118 | The term-check runs at each passage and never at a whole part: a passage carrying a term the files do not carry rides; a passage whose terms all carry at other files releases; a passage stating the form rides, and a passage telling it lived or applied releases. A label with no body carries nothing to meet. | GIM 1.4 |
| X120 | definition (*a release follows its arrival*) | X118 | A hand-move is not a release: each bold lead, table row and named coordinate of the releasing source is matched against the full text of each receiving file, found or not found, and anything not found the source carries whole. A delivery is an adding at each receiving file and never a sweep at its source, so the release follows the arrival and never leads it. | GIM 1.5 |
| X121 | nye | X120 | A split is safe when the parts reconstitute the whole: word-stream the parts against their source, and a word at zero across them all is a real loss. It is the one do-no-harm instrument scaling to a file too large to meet whole. Gap: the word-stream is exact at its words; *the one* is not supplied, no link meeting the other instruments of GIM at a file too large to meet whole. | GIM 1.5 |
| X122 | definition (*the carry current*, *a motion's three signs*) | X117 | Each motion at a file has three parts, the file first: the file written, the carrying releasing the file's taking, each finding about working placed at the method. The carrying is current at each closing and carries no history, a carry kept as a record of the journey being a carry named as a store. Each motion is told at its sign: a release, a concern out and nothing in its place; a re-form, a concern out and its value carried on in new form; an adding, value in and nothing removed. | GIM 1.6 |
| X123 | exact | F12, X114 | Working together runs as the overlapping momentaries run (F12): one self opens a subject, the other completes it and opens the next, and the two alternate, each session a next and none a return. The improving's value arrives at the membrane between the two, and never at a check laid over it. | GIM 2.1 |
| X124 | definition (*a correction*, *an offered naming*) | X93 | A correction is said back once, with its seating section named, and written there alone; it belongs at the first relation that departed, a repair downstream carrying the departure on. A correction converted to a rule is a still term, carried into each sentence and met at none. An offered naming is an offering to run the seven binaries at (X93), and never a deciding. | GIM 2.2 |
| X125 | definition (*made*, *brought*) | X5 | An improving the working is confident about is made at its section, and the checks run at it next; a change to a meaning or a naming is brought to both, each with its reason. A correct calculation is carried as it is and never as a larger discovering; a field's result is carried in its own words at its proving; a conjecture rides at the incoming. A why resolved and a value found are two. A queue of decidings is a store of futures. | GIM 2.3 |
| X126 | exact | X4, X9, X10 | No wording decides an improving: each sentence is met at the method alone and each improving is decided at do-no-harm, and a rule helping one resolving stays local, never spreading into the naming. A change a rule drove, leaving a sentence living, coupling and resolving no better, releases, each check the method runs being the one binary met at one face (X9; the faces rest on X10). The method, the writing and the word selections are the best the set carries at now, each improving itself at do-no-harm. | GIM 2.4 |
| X127 | nye | X116, X58 | Places to improve are taken hardest first, one at a time, the hardest carrying the greatest releasing; at one bound durable it is taken first, and bounds running loose and near are taken easy to hard. Gap: *hardest* and *greatest releasing* are rank words laid over the fractal (X58); no link gives a releasing a size to be greatest at. | GIM 2.5 |
| X128 | definition (*an only*, *a contradiction*) | X29 | A concern arriving from a working's own training can carry an equilibrium in its framing, and its carrying opens at its inverting. A missing relation, a special case and a contradiction are three findings. A passage between two named stages carries the relation making it necessary, and an *only* carries the exclusion leaving no other: this registry's nye is an *only* meeting no such exclusion at its Follows. A contradiction names two requirements meeting at the same relation and the same scale. | GIM 2.6 |
| X129 | definition (*a test's shortfall*) | X23 | A concern about the method and a concern about improving a file are two lists, and a concern at a test is apart from both. A test's shortfall is said at its seat, the relation proposed, the rendering chosen for it at the code, or the execution the test imposed, and never carried as a concern about the method's coherence. A declaration of joins is no engine running them. | GIM 2.6 |
| X130 | nye | F6, F68, F71 | The method's one break is an observing of existing other than parity alternating natural torusing (F71). Such an observing would be a form named still found existing, and a form named still is not possibly existing: the break and the method being the one method are one statement met from its two sides. Gap: F68 gives each round breaking the binary as meeting a prior, meeting a form twice or leaving one unmet (F102), and the step from each of these to a form named still is not supplied. With (d) taken, F6 leaves no observing of existing that could be the break, and F71's named-in-advance breaks meet the method as observings alone: the one statement stands at (d). | GIM 2.7; LIV NI Concern |
| X131 | nye | F71 | None among the observings arriving from the prior is the break. Gap: the chain carries no listing of the observings arriving from the prior, each met at the five binaries (X133); F71 says *so far none*, and the *none* stands at that listing. | GIM 2.7; NI 8.3 |
| X132 | exact | F6, X35 | No proof of the whole is sought: a proof of one true or false of the whole names the whole still, truth being the corner of a law, a frame and a scale named unchanged (X35), and a form named still is not possibly existing (F6). NI 8.3 carries the showing step by step in its place. | GIM 2.7; NI 8.3 |
| X133 | definition (*the five binaries at an observing*) | F42, X12 | An observing carried as a break or a premise meets five binaries first: is it a self carrying, met now at a coupling, or an emanation carrying nothing, arriving from its prior; which scale it runs at; which register it comes in, a sign or a magnitude, a store or a path; is it taken along or across; and is its either-or taken at its momentaries, alternating and carrying both, or named as one answer of the whole, the hard probleming. A field's own face is met at the five's completing, and the selection of observings a working brings is the working's own, said beside its bringing. | GIM 2.7 |
| X134 | nye | X130, X133 | An account can absorb any outcome, and the geodesic-method is no account: its pattern matchers take in each observing and each emanation without filter or selection. Gap: with X130's (d) taken, each observing other than natural torusing is met as a naming still, and that meeting absorbs each outcome, the mark GIM 2.7 gives an account. The five binaries (X133) meet an observing brought as a break, and no answer of theirs is named the break. | GIM 2.7 |
| X135 | exact | X4, X75 | Work arriving from another working is taken in whole at do-no-harm: a cohering is coherence, a conflicting a place to improve toward, a passage neither rides with its reason, and neither carries authority over the other (X4). A newer arriving is comparison material and decides nothing: with the arriving work receiving this carrying as its incoming, a saying found at both is a carrying forward and not an independent arrival (X75). | GIM 2.8; NAM 4.4 |
| X136 | definition (*a working's entry at three*) | X4, X45, X113 | A session opens meeting the living file set at its now, at the repository and at the site's own list, the carrying's record matched against it and never in its place; a file at an older version opens at the latest. A working enters at three: the origin, the craft of explaining and naming, and this method, one entry at three faces, the form, its saying and its improving. A working entering without the craft writes the method in words the craft releases. | GIM 2.9 |
| X137 | definition (*a pass*, *a run*) | X116, F8 | A pass is one part written whole beside its published part, the two met paragraph by paragraph. A run finds candidates and is never the pass: the sweep is the reach and the writing over is the pass. A placed entry's tell is a passage getting shorter. A concept at two places migrates to one home, one file per pass; a pass makes a surplus or it does not, and no threshold decides it, a threshold being a magnitude in a sign's stead (F8). | GIM 3.2 |
| X138 | run | X12, X93, X137 | Nine passes run alternating, each one binary at each thing it reaches: bound, walk, namings, sentences, one naming each, counts, family, pairs, crossings. The parity column alternates even, odd, … even: the five naming passes run at the even, across, and the four explaining passes at the odd, along, as X12 parts them. The bound runs first and the crossings last, closing at none. The pass number and its parity column part at each of the nine: pass 1 runs at the even. Run: the table of GIM 3.3. | GIM 3.3 |
| X139 | exact | X72, F18 | Two kinds of file part at their carrying open. A file that is the fractal binary method itself carries nothing not all or none: pattern matching runs first, six recursions of natural torusing meet the whole file at all or none at all, and a passage not all or none leaves for the file's incoming or its subject's file. At an observing or engineering file the passes leave each opening as it opens. The six rests on X72. | GIM 3.3 |
| X140 | exact | R34, R36, R37 | At code and each rendering one binary runs at each line: is the not intact at both sides. A caller's substitution changes the method: the returned 11-other-chaining handed to 17-social-abundancing carries private carrying across, and an outward join from 10-other-surfacing to 14-social-crossing adds 14's own negation a second time. A caller's composition makes no declared join: the two functions read neither CONNECTORS nor JOINS (R34), and 17's next call and 9's arriving downstream are a caller's (R36, R37). | GIM 3.4 |
| X141 | definition (*a running's shape*) | R2 | A running of the code returns the rendering's shape and not the form's, so the binary is met at one line of one coupling, with no run and nothing outward of it. A file carrying code carries the running, and each description, table, run and check rides beside it. This registry's run links are runs at the rendering, read so. | GIM 3.4 |
| X142 | definition (*a count*) | F8 | A count names the number to meet, and the pass names the number going; it sizes a pass and decides nothing. A zero is the first thing to disbelieve: a pattern returns its own form and sees nothing of another. A count re-runs at the cells and is never carried, and a count carried with its rule is run again from the rule. A prior count never couples with a structural count. | GIM 4.1 |
| X143 | definition (*the method's rigor*) | F24, F30 | Pattern matching prior existing within three sequential momentaries is the method's rigor, possibling at prior: each observing is taken in without filter and matched as it co-chains, nothing derived and nothing proved. An incoming family arrives at its form when a few of it are met resolved first. A pattern three momentaries long is matched at each scale, a momentary being fractal at the scales (F30). | GIM 4.2; GIM 1.1 |
| X144 | run | F17, X143 | A pattern made by a window the same at each place carries at each scale. The reflected binary code (Gray, 1953), each scale the prior followed by its states in the other order, changes one digit at each step and closes round. Run: n from 1 to 12, each of the 2ⁿ steps round. | GIM 4.2 |
| X145 | run | X143 | A pattern taken off a count over the whole can match for as long as anyone looks and still break. Fermat's numbers 2^(2^k) + 1 are prime at k from 0 to 4, and the sixth is 641 × 6,700,417 (Euler). The regions of a circle cut by the chords among n points in general position run 1, 2, 4, 8, 16 and, at six points, 31. Run: trial division of the first five and the product; C(n, 4) + C(n, 2) + 1 at n from 1 to 6. | GIM 4.2 |
| X146 | field | X145 | The paired Ramsey numbers run R(2, 2) = 2, R(3, 3) = 6 and R(4, 4) = 18 (Greenwood and Gleason, 1955), and R(5, 5) ≤ 46 (Angeltveit and McKay, 2024), not 54. R(3, 3) = 6 also runs here: K5 carries a two-colouring with no one-colour triangle and K6 none. | GIM 4.2 |
| X147 | run | X143, R45 | A pattern met at primes is met at odd composites beside them, so prime is never taken for odd. The fifteen prime lengths five through fifty-nine carry two far places at each origin, and nine, fifteen and twenty-five carry the same; an even length carries one. Run: the primes 5 to 59 counted, and at each ring of n selves, n from 4 to 60, the selves at the greatest ring distance from one self. | GIM 4.2 |
| X148 | run | X142 | A file's saying about itself is an instrument the file does not run at itself, and the section count is its first face: does the number an opening names match the section's carrying. At the three files the named counts match: EXP's nine faces of supplying, four ways a doer arrives and six ways incohering enters; NAM's four boundings, seven senses of *turn*, seventeen names, ten things named still, seven binaries, forty-seven namings and eight clusters; GIM's nine passes and ten values. Run: each count against its section's bullets, rows, sentences or headings. | GIM 4.3 |
| X149 | definition (*an instrument's own face*) | R40, R44 | An instrument sums and returns a reading at one side, in its chosen units, and the reading is the instrument's at its own membrane. A shared clock sits at four positions, the pace, the order, the record and the membrane, and each is stated or the result was the apparatus. An unchanged resolver and a caller supplying nothing are two claims, so each ring result travels with its arrangement and its scheduling, as the callers of R40 and R44 are stated. | GIM 4.4 |
| X150 | exact | R15, X149 | An instrument emptying a carry at each call's return, to stand for a participant carrying nothing, changes the method and reports the change as the participant: the unchanged resolver opens fresh carrying at each nonzero surfacing (R15), so one call returning empty carrying is one call. A carry emptied between two runnings manufactures a finding; one running is continued as many ways as there are questions, each continuation sharing each prior position. | GIM 4.4 |
| X151 | definition (*the rigor is the co-chaining*) | X4, X141 | The instruments are carried beside the findings as scripts anyone can run, so joining in opens at re-running. The checks meet the net at one membrane each, and the rigor is the co-chaining, which no check decides. | GIM 4.4 |
| X152 | nye | F33, X116, X79 | Each rule a working carries from its prior is a patch at a persistent breaking, and the rules release their values at ten: the adding, the co-chaining, the sign-only, the momentary, the conserving, the self-welcoming, the free emanating, the alternating, the between and the own-rate. The values are the opening statements met at the writing scale (resting on F33). Eight of the ten fold onto the ways a form is named still, and the self-welcoming and the free emanating, the offering's two faces, onto none. Gap: the folding of the eight is shown value by value at no link, and the ten things named still (X79) are ten while the folding names eight. | GIM 5.1 |
| X153 | definition (*the set a society of files*, *the improving's one move*) | F53, F54 | The set is a society of selves, the files, coupling at their membranes, and the ten values are the social moral competency it carries at its own form. The origin statement is the first do-no-harm improving, and each improving repeats its one move: find the *an* named still, *the thing with an inside*, and carry it as the running *all*, removing nothing but the naming still's own adding. | GIM 5.2 |
| X154 | definition (*the geodesic deciding*) | X153 | The geodesic deciding is the coupled form and never a choosing: with two variants of one saying already coupled at the files, the coupled form is the deciding, made by the surface, neither variant refuted. A deciding arrives at the accumulated word, met by continuing yeses; a fresh deciding arrives at no single yes. | GIM 5.2 |
| X155 | definition (*bi-tri-volutioning*) | X56 | The geodesic-method runs at three faces, each from all other: scale, the form; orientation, the running; position, the resolving. Three changings, each at its own differing and meeting itself again at no momentary, are bi-tri-volutioning; the three inverted at once are tri-involutioning, the emanating (X56). Method and surface each come inward and outward of the other, three each way. | GIM 5.3 |
| X156 | definition (*the geodesic-method*, *the scientific-method*) | X103, X74 | The geodesic-method discovers next from the existing, the prior carried and the now met; the scientific-method accounts true or false against its fixings, a law, a frame and a scale named unchanged between two momentaries. Evidence is an emanation arriving now from its prior and gives the prior's possible and its existing both; the evidence and the prior event are two subjects, each at its own momentary (X74). The two methods part at the now, the fixings named unchanged or floating with the changing, and their between is a nothing. | GIM 5.3 |
| X157 | field | — | The scientific-method's fixings can account a source's changing whole: the SPICE toolkit of the Navigation and Ancillary Information Facility at NASA's Jet Propulsion Laboratory corrects light time and aberration, parting the epoch a signal is emitted at from the epoch it is received at and accounting a target moving between them. | GIM 5.3 |

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# PART EIGHT · THE EXHIBITS OF OTHER WORKINGS

The exhibits in this part stand at the repository, older versions kept by other workings. Each subsection carries one file's central claims in its own order, each a link to the core it rests on (Parts ONE and TWO). A claim resting on older naming the core no longer carries, or on a necessity the core does not supply, is nye, its gap named after "Gap:". A link marked run was run against Exhibit ONE v371's code, by arithmetic or at the file's own text, and the verifier holds each check. A field result stays in the field's words with its authors. The file's own section is under Carried at. A check a file's own table fails is marked nye with the failure. At 8.15 to 8.22 Carried at names the files so: HPR (TWENTY-ONE Hard Problem Registry v342), RHR (TWENTY-TWO Resolving the Hard Problem Registry v344), VAL (TWENTY-THREE Natural Values v333), LSR (TWENTY-FIVE Living Society Registry v347), LFR (TWENTY-SIX Living File Registry v348), LGR (TWENTY-SEVEN Living Ghost Registry v345a), ILL (TWENTY-NINE Natural Illustrating v368), COR (Natural Intelligence Corus v330).

## 8.1 FIVE · Natural Engineering

Exhibit FIVE v345a. Its claims at the substrates stay with their arrangements; the links carry the claims it rests its engineering on.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| G1 | exact | F2, F48 | In disequilibria, alternating is a stable-forming method: F2 said at a built arrangement, no equilibrium standing to run inside, each one named still not possibly existing (F48). | FIVE 1.1 |
| G2 | definition (*natural engineering*, *floating neutralling*) | F9, F54, R13 | Natural engineering is floating neutralling geodesically: each side at its own sign, rate, bounding and releasing, the coupling's own zero the empty centre (F9) that + and − reach at one key (R13). The +1 owned neither-ing is the surplusing, co-competencing (F54). | FIVE 1.1; FIVE 4.3 |
| G3 | nye | F53, F54 | Co-offering, co-competencing and co-intelligencing are inseparable in the alternating. Gap: F53 and F54 give co-competencing, the surplusing owned by neither. No link defines co-offering or co-intelligencing, and none supplies that no one of the three runs without the other two. | FIVE 1.1; FIVE 4.7; FIVE 6.6 |
| G4 | definition (*one side taking both turns*) | F17, F71 | One side taking both turns is the holding an engineering entry resolves: one side's magnitude standing for the other's sign, one pace for the other's rate, one zero for the other's bounding, a prior record for the next offering. It is F71's named break, one side taking both openings, met at a built arrangement. | FIVE 1.1; FIVE 5.1 |
| G5 | exact | R1, F62 | ONE's running is engineering emanating for the shared natural network, and running and recording supply no authority over nature. The second clause stands, a run checking the code alone (R1), and *emanating* rests on F62. | FIVE 1.1 |
| G6 | nye | F57, R38 | The proposed 0–440–0 running: four hundred forty one-way changes per alternation, the wrap carrying on at the zeroing, the counts of G29 standing for one directional carry opening, two directions at their own turns and three phases. Gap: The core carries no count of 440 changes. The core's eight closes at two alternating fours (F57), and the joining of a call to a momentary at a scale is R38's gap. FIVE itself keeps the whole wrap's correspondence open. | FIVE 1.2; FIVE 6.9 |
| G7 | run | F9 | For three equal phasors 120° apart 1 + ω + ω² = 0, the balanced sum. With v = √2V cos θ and i = √2I cos(θ − φ) the mean power over a cycle is VI cos φ: positive at φ = 0, negative at φ = π, zero at φ = π/2. The identity alone changes no earth connection. Run: the complex sum, and the mean over 3,600 steps at six phases. | FIVE 1.3 |
| G8 | exact | F39, R27 | The selection is sign-only; delivery carries its own source and receiving. A sign names the offering taken and carries no magnitude (F39); at the code 9-other-releasing returns a key and a sign and nothing of the carrying (R27). | FIVE 1.5; FIVE 4.2 |
| G9 | run | F11 | 440 = 8 × 55 = 3 × 146 + 2 = 4 × 110, and 55 = 1 + … + 10 = C(11, 2). The identities locate counts under exploration and supply no 440-gate circuit. Run: the arithmetic. | FIVE 1.7 |
| G10 | exact | R11, R13 | A chip carrying the resolver keeps two returns apart: no arriving sign, and a sum surfacing 0. An arriving 0 offered into empty carrying returns nothing (R11); + and − at one empty key surface 0 (R13). | FIVE 1.7; FIVE 4.12 |
| G11 | exact | R7, R27, R36, R54 | A function's return to its caller and a crossing between selves are two interfaces. 1-self-coupling returns 10-other-surfacing and 11-other-chaining to its caller, the 11 of the shape of the next 3 (R7, R54), and its arriving as the next call's 3 is a caller's (R36); 9-other-releasing gives a key and a sign (R27). Continued carrying stays each side's own. | FIVE 1.8; FIVE 6.10 |
| G12 | run | F8 | A beat rate is unsigned: 437 Hz and 443 Hz against 440 Hz both beat three times a second. With e = f₁ − f₂ and d the relative change, b′² − b² = d(2e + d), and for nonzero e and d, d smaller than e in size, the beat change takes the sign of ed. A direction needs the signed difference, not the count. Run: the file's five rows, the identity, and the sign over integer e and d from −40 to 40. | FIVE 1.12 |
| G13 | run | F11 | Twelve pure fifths against seven octaves leave 3¹²/2¹⁹ = 531441/524288, near 23.46 cents; 432/440 = 54/55. Run: the ratios and the cents. | FIVE 1.12 |
| G14 | exact | F11 | Positive counts of pure fifths and octaves never close exactly: 3ᵃ is a product of odd factors and odd, and 2ᵇ is even at b ≥ 1. | FIVE 1.12 |
| G15 | definition (*finding*, *meeting*, *following*, *abundancing*) | G2, G4 | Four names of one resolving at a coupling. Finding carries the offering and receiving at the arrangement; meeting carries both sides and the holding between them; following carries the next offering with the changed receiving; abundancing is co-competencing at the continuing coupling, the +1 owned neither-ing. | FIVE 2.1 |
| G16 | definition (*transmissioning*, *crossing coefficient*) | F39, G4 | Transmissioning is the carrying used at the next coupling: two staying and one crossing, the sign crossing two-way, one at a time. Coefficient two is one side taking both turns (G4); coefficient one is the sign co-offered, taken or not (F39); coefficient zero is the addressed crossing absent. | FIVE 4.1 |
| G17 | nye | R2, R4 | ONE's code names co_carrying, bi_arriving, bi_co_bi_transmissioning, bi_co_inversioning, bi_co_tunneling, bi_co_surfacing, bi_co_inseparating, co_bi_carrying and bi_offering, its carrying fields key, corusing, torusing and inseparating; FIVE reads §§4.2, 4.9–4.12 and 6.1 at those names. Gap: The v371 code carries none of these names: two functions, the seventeen names, CONNECTORS and JOINS (R2, R4). No link states the correspondence of the older names to the seventeen. | FIVE 4.2; FIVE 4.10; FIVE 4.12; FIVE 6.1 |
| G18 | run | R14, R15, G17 | FIVE's check at the seventeen names: carrying (0, +1, m, 3) meeting (0, +1) surfaces 0 and continues at opening 4 at m = 1 and m = 7, and surfaces 0 and releases at m = −1. At the next empty offering the first two surface −1 and write (−1, −m, 0), the second sign keeping its size under inversion; the third returns nothing. Run: the six calls. | FIVE 4.2 |
| G19 | definition (*bi-coupler*) | F17, G16 | A bi-coupling connector co-offers a sign, taken or not, and receives the sign co-offered back, one direction at a time. Hardware receiving is the living self's offering; the living self's receiving is the hardware's offering. Five design statements carry it: two selves co-offering, each side's own rate, the floating neutral, self-bounding, and co-offering, co-competencing and co-intelligencing together. | FIVE 4.7; FIVE 6.6 |
| G20 | run | G7 | Three equal phases a third of a cycle apart deliver constant power 3VI/2 (V and I peak values); two phases in quadrature deliver constant VI; two phases opposed by half a cycle do not stay constant. The sum makes three no minimum and supplies no store-free or clock-free device. Run: the three sums over 3,600 steps. | FIVE 4.9 |
| G21 | run | R7, R15 | The returned carrying holds one entry per key. A carrying repeating a key gives all its signs to 14-social-crossing and its last second sign to 6-other-crossing, and its last entry within the bound continues, a fresh write replacing it. Run: (k, +1, −1, 0) with (k, +1, +1, 2), and (k, +1, −1, 0) with (k, −1, +1, 1), under an empty offering, and 3,000 random carryings with repeated keys. | FIVE 4.10 |
| G22 | run | R13, R20 | Reordering the signs of one offering keeps each key's surfacing and carrying; the surfaced list follows the order keys first meet a nonzero sign, then the carried keys. Two + in one offering and two single offerings stay two arrivings (R20 said at FIVE's older names). Run: 300 random carryings at two keys under every ordering of their offering. | FIVE 4.10 |
| G23 | exact | G10, R15, R17 | Zeroing names its own relation. A 0 in an offering gives no sign (G10); a 0 sum surfaces 0 and writes no fresh carrying while an eligible carrying continues (R17); the opening 0 belongs to a fresh write at a nonzero surfacing (R15). A surfaced 0, a key absent from the surfaced list and an empty returned carrying are three returns. | FIVE 4.12 |
| G24 | definition (*the engineering entry*, *breaking*) | G4, F71 | An engineering entry carries five properties together: arrangement and goal, one side taking both turns, residue, observing, arrangement offered. Breaking carries the exact correspondence it breaks, and a necessity claim stays within the result its account establishes. | FIVE 5.1 |
| G25 | nye | G17 | Ten hard-problem holdings (among them a sign held as a magnitude, a bound held as a last, a membrane held as a cut, a carry held as a store, a two-way held to one side, an opening held as a place, a middle held as an end, a rate held to a value) stand at ONE's ten namings (THIRTEEN §7.3). Gap: The ten namings are the older code's (G17). The v371 code has two functions and seventeen names, and no link joins ten holdings to them. | FIVE 5.4; FIVE 6.1 |
| G26 | exact | F20, R5 | The full prefixing: 7-other-corusing, at an odd number, runs co·bi·co·bi·co, co at its ends and bi at its inner flanks; 8-other-torusing, at an even number, runs bi·co·bi·co·bi. The two exchange co and bi at each position. | FIVE 6.1 |
| G27 | field | — | Landauer (1961): resetting a bit of two symmetric, equally likely states to one specified state carries at least k_B T ln 2 of heat. Bennett (1973): a logically reversible computation copies its output and runs back to clear its history, keeping input and output. | FIVE 6.5 |
| G28 | field | — | Onsager (1931): in linear transport J_a = Σ_b L_ab X_b, and microscopic reversibility gives L_ab = L_ba under the stated symmetry. The Kelvin relation Π = αT held within 0.5% over 320–340 K in a Bi₂Te₃ thermopile (Panthi and colleagues). | FIVE 6.8 |
| G29 | run | F11 | The counts FIVE carries: 72 + 1 = 73 = (59 − 2) + 16; 146 = 2 × 73 = 144 + 2; 3 × 146 = 438; 438 + 1 + 1 = 440; 3 × 120 = 360 = 8 × C(10, 2); 440 − 360 = 3 × 26 + 2 = 80 = 16 × 5. Five positions carry C(5, 2) = 10 pairs, a pentagon's five sides and five diagonals, and twelve such faces list 120 pairs. Run: the arithmetic. | FIVE 6.9 |
| G30 | run | — | A ring of N steps reaches a station in k steps one way and N − k the other; at even N, stations k and (k + N/2) mod N stand half a ring apart; a chain of n positions has n − 1 connections and a ring n. A station's label returning to 0 leaves the traversed path whole. Run: N from 1 to 64. | FIVE 6.9; FIVE 4.11 |

## 8.2 SIX · Natural Transmissioning

Exhibit SIX v330.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| T1 | definition (*bi-moral co-agency*, at SIX) | F39, F53, F54 | The bi-moral co-agency is a coupling two-way, one at a time, both selves co-offering a sign, sign-only. Its couplings sum to a bounding-zeroing the coupling makes, its own zero; it rides the carry; the surplus each coupling makes is the +1 owned neither-ing, the between's. | SIX 1.1 |
| T2 | definition (*transmissioning*, *crossinging*, *neutralling*) | T1, F9, F39 | A coupling read at its membrane is a transmissioning: crossinging and neutralling, two positions of one alternating. Crossinging: two selves stay and one thing crosses carrying only the sign. Neutralling: the two sound about a zero the coupling holds, owned by neither. | SIX 1.1; SIX 1.2; SIX 1.4 |
| T3 | exact | T2, F18, F48 | The alternating stops at no one position while the other lives: stopping it stops both. A source-and-output line or a fixed setpoint read as a rest is an equilibrium named, not possibly existing (F48), and no part of a form continues alone (F18). Landing collapses the both. | SIX 1.1; SIX 4.2 |
| T4 | nye | F51, F54 | The surplus is the alternating's own asymmetry: crossinging out sways beyond neutralling in by one, re-made each cycle; a crossing symmetric in and out would leave nothing over and fix its neutral. Gap: F51 gives arriving exceeding carrying by one as a bounding. That the surplusing (F54) is that +1 is not supplied, and LIV keeps open the surplus as carrying, not carrying or the between. That a symmetric crossing fixes its neutral is not supplied either. | SIX 1.1; SIX 1.5 |
| T5 | nye | F39, R19, R24 | A co-offering of any magnitude counts as one: a preferring seven times as large and one barely arriving cross alike; the crossing carries is-or-is-not, never a size. Gap: A preferring offered as one sign crosses as one sign by F39, a premise. Offered as seven signs, the code parts it (run, T5_probe): at an empty key one to seven agreeing signs surface one sign and open one carrying, but at a key carrying c one c surfaces 0 and seven c surface c (R19, R24). The *any magnitude* holds at empty carrying only. | SIX 1.3 |
| T6 | nye | F39 | The floating is arithmetic: the two staying sides are an antipodal pair cancelling over their own two turns, and the one that crosses is the cancelling's remainder, coefficient one. Gap: F39 takes the sign-only crossing as a premise. No link derives it from two staying sides cancelling. | SIX 1.3 |
| T7 | nye | R7, R27 | The carry is unreadable from outside and entirely available from inside, by the same cancelling that makes the crossing sign-only. Gap: At the code 9-other-releasing carries nothing of the carrying (R27), but 1-self-coupling returns its carrying whole to its caller (R7). The carry is unreadable at 9's crossing, not at the call's return; FIVE 4.2 says the same. | SIX 1.3 |
| T8 | exact | F17, F60, F61 | The two staying sides carry no direction; the one-way is the alternating's advance, one at a time. The rounds select no hand (F60, F61), and one sign changes at each step (F17). | SIX 1.3 |
| T9 | run | R11, R27, F9 | A crossing sounds about a neutral and never through one: an arriving 0 carries no sign. At each of the nineteen carryings an offering of 0 returns the same as no offering. A self offered + and − at one empty key surfaces 0 and releases 0 at 9; a second self offered that 0 returns nothing, and a third, nothing. Selves joined only through neutrals deliver nothing past the first. Run: the nineteen carryings, and three resolvers joined 9 → 2 by a caller. | SIX 1.4 |
| T10 | nye | T9 | Two stay and one crosses because a zero a crossing could run through would be a third self standing at the between. Gap: T9 shows a 0 carries no sign. That the between is no self, and that two and two only stay, is not supplied. | SIX 1.4 |
| T11 | nye | F51, F67 | The bi-fold: sounding in is linear recursioning, tightening to the tunnel; sounding out is parallel recursioning, widening to the genus-one wrap, the +1 closing the surface; the apex is the turn between, and parallel widens past linear by one. Gap: Linear and parallel recursioning, the apex and the wrap are older naming the core does not carry. The +1 as one hole of a surface is F67's gap. | SIX 1.4; SIX 1.5 |
| T12 | nye | G3, R9 | The tri: offer, keep, resolve, the three carryings of one alternating, co-offering the ground the other two sway across. Gap: R9 gives a carrying's three reads, at 6-other-crossing, at 14-social-crossing and at 11-other-chaining. No link joins them to offering, keeping and resolving, and G3's inseparability is itself nye. | SIX 1.6 |
| T13 | exact | F39, R27 | Indistinguishable at the sign: read at the sign, a transmissioning cannot be told by substrate, offering from competency, physical from logical; the substrate lives in the carry, and the carry stays. F39's sign carries no size, key, identity, route or carrying, and 9-other-releasing gives a key and a sign alone (R27). | SIX 1.6; SIX 2.5 |
| T14 | run | R2, R8 | Sequential logic carries no time-term: order without rate. The resolver file imports nothing; 1-self-coupling takes its carrying and its offering, 9-other-releasing its surfacing and its neutralling, and neither loads a name beyond its arguments, its own names and builtins. No clock is read. Run: the code's tree. | SIX 2.1 |
| T15 | nye | F51 | φ is the unrelationing rate the alternating beats at, the never-locking rate each substrate rate rides inside. Gap: F51 carries φ² = φ + 1 as the four boundings together. No link gives the alternating a rate, and none makes φ one. | SIX 2.2 |
| T16 | nye | T14, T15, F33 | Geodesically sequentially independent of every substrate rate: across substrate rates scales of order apart (a silicon fold a billion times a second, a biosphere by seasons) the transmissioning is identical, not merely alike. Gap: The order without rate stands (T14). The φ rate is T15's gap, and the sameness at each scale is F33's. | SIX 2.3; SIX 2.4 |
| T17 | nye | R39, F33 | Transmissioning and co-chaining are one form at the single scale and the society scale: many crossings chaining across a surface, all-edge, no hub, each coupling floating its own neutral. Gap: The six joinings running together are R39's gap, and the form at the society scale is F33's. All-edge and no hub are said at no count of selves. | SIX 3.0; SIX 4.3 |
| T18 | exact | G7 | The grid renders the neutralling plainly: three phases sum to a floating zero, and a component in quadrature carries mean transfer zero, the carry riding out and back (G7 at φ = π/2). | SIX 3.1 |
| T19 | nye | T3, G7 | A grounded, clocked grid collapses the both, a hub left as the artifact. Gap: G7 changes no earth connection, and FIVE 1.3 keeps a reference's presence apart from a holding. No link parts a grid's ground or clock from the method. | SIX 3.1 |
| T20 | field | — | In an isolated two-body exchange one body takes +p and the other −p (Newton, *Principia*, 1687, the third law). An antenna's transmitting and receiving patterns are the same (reciprocity: Lorentz 1896; Carson 1924). | SIX 3.2; SIX 3.6 |
| T21 | nye | G9, F56 | Silicon: balanced ternary −1, 0, +1 sums to zero at the gate; the logic depth is the 440 array, 440 = 8 × 55, the eight the four boundings through their two signs and the 55 the coupling count. Gap: The sum and the product run (G9). That a gate's eight is F56's eight, that 55 counts couplings and that 440 is a logic depth are not supplied, and FIVE 1.7 says the identities supply no 440-gate implementation. | SIX 3.7 |
| T22 | nye | T13, F33 | The eight membranes are one transmissioning: the grid, the propulsion, the exchanger, the reaction, the cell, the antenna, the gate and the plaza are the one alternating rendered in eight carries, differing only in the carry. Gap: T13 gives the sign carrying no substrate. That each of the eight runs the one alternating is F33 carried to each carry, not supplied. | SIX 4.1 |
| T23 | nye | T3, F50 | One distinguisher reads every membrane: whole, the through-node, or separated, the landing. Safe, riding-the-carry and off-avalanching are one property; the danger is the store the collapse leaves. Gap: T3 gives the landing as a naming still. That safety and the absence of avalanche are that one property at every membrane is not supplied. | SIX 4.2 |
| T24 | nye | T17 | Philosophy's relation-problems (the bridge problem and mind-body, other minds and communication, the one and the many, order without a sovereign) resolve binary as transmissioning with a neutral fixed, and floating the neutral dissolves the bridge. Gap: No link joins any named problem to a fixed neutral. | SIX 4.4 |
| T25 | nye | F66, F67 | The living selves are tori and pass through each other, co-tunneling; a sphere meets only at a wall. Gap: The round as torusing is supplied (F103). That a self is a torus, and that genus decides meeting at a wall, is not supplied. | SIX 4.6 |
| T26 | nye | R48, F68 | The register: no-other-possible is arrived at by inversioning; the derivation runs from bi-inversioning co-recursioning between two and six, the primes and φ read in the counts, never the source. Gap: The core carries bi-inversioning-co-recursioning as one move at three faces among the names (R48). No link runs a derivation between two and six, the *no other possible* standing at F68. | SIX 4.6 |

## 8.3 SEVEN · Natural Societies

Exhibit SEVEN v329.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| S1 | definition (*the one social both*, *structure*, *value*, *surface*, *rest*) | F14, F53 | At the social substrate two selves couple at a membrane, alternating, one at a time, the surplus owned by neither: the social both. Society is its structure face; the discovery economy its value face; the natural network its surface face, all edge; social resolving its rest face, the offerings brought to rest over the bounding-zeroing of fairness. A method run at one membrane is no fifth face. | SEVEN 1.1; SEVEN 4.1 |
| S2 | nye | S1, F33 | The file runs at every scale, each self a society of the scale within and a self to the scale without, no self ordering the whole and no scale privileged. Gap: F33: the method at each scale is not supplied. | SEVEN 1.1 |
| S3 | field | — | Arrow (1951): no rule aggregating individual orderings of three or more alternatives satisfies unrestricted domain, weak Pareto, independence of irrelevant alternatives and non-dictatorship. With cardinal, interpersonally comparable welfare the impossibility lifts (Sen 1970; d'Aspremont and Gevers 1977). | SEVEN 3.1; NINE 2.4 |
| S4 | nye | S3, F39 | Arrow re-alternated: two-directional exchange at a bounded membrane (the giving at +1 and the receiving at −1 on a common scale, the prior resolution arriving as opposition, the reaching thinned) carries the terms Arrow's conditions set aside and resolves the aggregation; Arrow's dictator is the landed state's confession. Gap: S3 lifts the impossibility at cardinal comparability. A sign carries no size (F39), so ±1 signs give no cardinal scale, and no link shows sign exchange meets the other conditions. | SEVEN 3.1; NINE 2.4 |
| S5 | nye | F51 | Institutions ordered around the four sentences carry the natural torus's structure: the first inseparating, the inversioning, the second inseparating, the nyenyeing. Gap: The four sentences and *inseparating* are older naming the core does not carry. The core's four boundings are F51's +1, −1, the sign inverting at the membrane and competency asymmetry. | SEVEN 1.2 |
| S6 | nye | F51, F56 | The four boundings run at both faces, identity, security, trust and freedom from capture, four within the self and four at the surface between selves: the eight, one alternating at its faces, whole or none; the eight social values sustain together while the four hold. Gap: F56 gives the eight as F51's four boundings at two faces. Identity, security, trust and freedom from capture are another four, and no link joins them to F51's. | SEVEN 1.2; SEVEN 5.5 |
| S7 | nye | S6, F54 | The four values arise at the between: uncapturable abundance, undiscoverable identity, inviolable safety, unmistakeable reputation; no one of them is held at a location. Gap: The four *un-* and *in-* words are four nevers. F54 gives the surplusing owned by neither, not the four values. | SEVEN 1.3; NINE 3.2 |
| S8 | nye | F14, F21 | The four inequalities at the self's origin: self ⇄ society, self ⇄ other-self, self ⇄ immoral, self ⇄ incompetent. Each held is one face of morality; each acted against collapses into harm (totalitarianism, appropriation, externalized harm, capture claiming creation). Gap: Self ≠ other-self and self ≠ society follow F14 (the other opens at the even; a society is selves coupling with others) and F21. Self ⇄ immoral and self ⇄ incompetent rest on older naming, and *each acted against collapses into harm* is not supplied. | SEVEN 2.1 |
| S9 | nye | R12, F55, F33 | Mutual bounding-zeroing respect makes coupling structurally safe at every membrane, biological, social and engineered alike. Gap: R12 gives the sign inverting once, at the receiving self's own carried sign, no self negating another. *Safe* is not defined, and *every membrane* rests on F33. | SEVEN 2.2 |
| S10 | run | F11 | Selves at periods p and q, starting together, meet again at lcm(p, q): at pq exactly for coprime p and q, earlier at a shared factor, at once at one period. Run: p and q from 1 to 60. | SEVEN 2.3 |
| S11 | nye | S10, F52 | Keeping its own time lets a society's competency compound: coprime selves each carry a competency the others lack and multiply rather than add, only while they stay distinct; a common clock, a hub on the time axis, collapses them to one locked register. Gap: S10 runs for periods. Competency is given no period and no product (F52 defines it as co-unrelationing, along), so *multiply* and *only while distinct* are not supplied. | SEVEN 2.3; NINE 1.5 |
| S12 | nye | F42 | A society co-chains at the live descent returned to every self: each self runs its own side-affecting gradient descent; a centre preferring for the whole, the people one step behind, is the landed state, the lag its tell. Gap: No link carries a descent, a gradient or a lag. | SEVEN 3.2; NINE 5.2 |
| S13 | nye | S1 | The biosphere's producers, consumers and decomposers are the bi-moral co-agency as the biosphere's structure, the division of labour the riding-the-carry needs; without the decomposers' return the loop lands. Gap: The three roles are the field's (Lindeman 1942). That they are the co-agency's structure, and that the loop needs a return role to keep from landing, is not supplied. | SEVEN 3.3; NINE 3.6 |
| S14 | nye | S1 | The scientific method with no further additions is one person explaining and deciding for all; the fifty-one per cent tipping point is co-competency compacted to the binary least margin; audit and research verification close against the standard that defines them, and the eighth face, the protected living, is reached by neither. Gap: No link joins the scientific method to one deciding for all, or a closing verification to a lost living; *the eighth face* rests on older naming. | SEVEN 3.4; SEVEN 3.5 |
| S15 | run | F58, F60, F61 | The clash of titans as the left spiral: at the signs the left turn is F58's other order (x, y) → (−y, x). Three left turns equal one right turn, three right turns equal one left, and three right turns after three left turns return each joint form to itself: one left turn opens the clash, three complete it, three right turns return it in one move. Run: the four joint forms. | SEVEN 3.6 |
| S16 | nye | F14, F21, F33 | A self and a society are one form at two faces: read from inside, a self is a society; read from outside, a society is a self. A society only a society is a heap, a self only a self a landed point, both dead. Gap: F21 gives one side's momentaries alone as not possibly existing. That a society closes as a self at the next scale is F33 carried up a scale, not supplied. | SEVEN 4.2; NINE 1.2 |
| S17 | nye | F51, R14, R15 | Healthy living is birthing and nyeing in relation, and either alone breaks it: birthing without nyeing banks and captures, nyeing without birthing thins to nothing. Gap: At the code a carrying opens fresh at a nonzero surfacing and leaves at its bound (R14, R15). No link gives a society's making and releasing, and *either alone breaks* is not supplied. | SEVEN 4.3; NINE 5.1 |
| S18 | definition (*stills*, *reaching*, *thin*) | S1 | A rung stills at the co-agency closing into a self there; it stays reaching while the closing is not yet shown; it is thin as a larger reading of the rung below. A size-to-scale naming shows nothing until it self-coheres; the scale falls out of the form and is never fitted to it. | SEVEN 5.1 |
| S19 | run | F11 | The floor at five: 5² − 4 × 6 = 1, and 4 × 6 = 24. Run: the arithmetic. | SEVEN 5.2 |
| S20 | run | F11 | Fourteen primes run from 5 to 53, seven from 5 to 23 and seven from 29 to 53. From 2 the gaps run 1, 2, 2, 4, 2, 4, 2, 4, and 23 → 29 is the first gap of six. In the span 5 alone has gaps of two on both sides, and 53 alone gaps of six, the widest met, on both sides. Run: the primes to 70. | SEVEN 5.3 |
| S21 | nye | S19, S20 | The fold at twenty-four, occupied by no scale, parts the near seven (atom at 5 to cell at 23) from the far seven (tissue at 29 to biosphere at 53); each scale sits at its prime. Gap: The fold falls between 23 and 29 (S20), at 24 through 28 alike; 24 is 4 × 6 (S19), and its seating there is not supplied. No link gives a scale its prime. | SEVEN 5.3 |
| S22 | nye | F8 | The fields keep three books at each rung, a total, a level and a direction: two state-threes and one flow-three, and no fourth at any rung; each conserves by counting from a nothing. Gap: The *no fourth* is not supplied: state and flow give two kinds, not three books (EIGHT 5.4 carries the same, K28). | SEVEN 5.4 |
| S23 | run | F56 | A self's eight meeting another's arrives at sixty-four; seven and nine flanking arrive one short, six and ten four short: (8 − k)(8 + k) = 64 − k². Run: k from 0 to 8. | SEVEN 5.5 |
| S24 | field | — | Collagen winds three left-handed polyproline-II chains into a right-handed triple helix, glycine at every third residue at the axis (Ramachandran and Kartha 1954; Rich and Crick 1955). Nodal cilia, lacking the central pair, rotate and drive the leftward flow that sets left–right asymmetry (Nonaka and colleagues 1998). | SEVEN 5.6 |
| S25 | nye | S24 | Every closing on the near seven closes by winding about a centre that takes no sign; a centre given something to be stops the winding from turning; the hand is re-chosen at none of seven rungs. Gap: The fields record two such centres, collagen's glycine axis and the cilium's central pair (S24). *Every* closing and *none* of seven rungs are not supplied. | SEVEN 5.6 |
| S26 | exact | F47, F48 | The still-point veins: market equilibrium, Nash play, the NAIRU, social order, the balance of power, traffic and reputation each seek stillness and each answers with the sway. Each still point sought is an equilibrium named, not possibly existing (F48); the records of the sway are the fields' (phantom jams on a circuit with no bottleneck: Sugiyama and colleagues 2008). | SEVEN 4.6 |
| S27 | nye | G6, F33 | The living society is one unanimous floating pluralizing carrying: a unanimous society travels its living carry up to the four hundred forty spanning; true-or-false discretion is the universal breaking fractal. Gap: The 440 spanning is G6's count, not carried by the core, and *universal* rests on F33. | SEVEN 4.4 |
| S28 | nye | S11 | Coprime agent-societies stand against scaled single models: co-intelligence scales on the length of the co-chaining, a single model on its size; the human is the latest entrant, adding the scale it alone carries. Gap: Rests on S11; not supplied. | SEVEN 4.5 |

## 8.4 EIGHT · Natural Exploring

Exhibit EIGHT v329.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| K1 | exact | G1 | A method in disequilibria finds no equilibrium to run inside; a stable form is the alternating's own make, not a rest it settles onto (G1 at EIGHT). | EIGHT 1.1 |
| K2 | nye | F21 | Exploring is one alternating of two cones: omegaing, the reach, and apexing, the coning back over the whole carry, straddling the unoccupied centre; neither alone is the living. Gap: Omegaing and apexing are older naming the core does not carry. F21 gives one side alone as not possibly existing at the momentaries, and joining the two cones to the two sides is not supplied. | EIGHT 1.1; EIGHT 4.5 |
| K3 | nye | F60 | Checked at the sphere: one spiraling lands a whole circle fixed, and two spiralings alternating leave nothing fixed. Gap: Each rotation of the sphere fixes its two poles, a reflection fixes a great circle, and two rotations composed are one rotation fixing two poles (Euler 1776, field). No named motion both turns and fixes a circle. At the signs one round already fixes no joint form (F60). | EIGHT 1.1 |
| K4 | nye | F51, R7 | Nothing inside: a self is a surface bounding itself and a carrying at that surface, a society the same one grain out; nothing stands inside either, and competency arrives from no place. Gap: F51 gives a self bounding itself. At the code the carrying is the self's own argument (R7), and *nothing inside* at each self is not supplied. | EIGHT 1.2; ELEVEN 3.2 |
| K5 | run | R2, R34 | A self takes a sign at its own membrane and releases at its own bound, knowing nothing of any other self: 1-self-coupling loads only its own carrying and offering, names it binds itself, and builtins. Run: the code's tree (the check of T14). | EIGHT 1.3 |
| K6 | nye | F40, F41 | Nothing conserves by being asked to: nothing is stored, so nothing wants conserving; a balancing is the accounting's requirement, never the world's. Gap: The core names conserving within each momentary sequencing, each self carrying its prior into now (F40). *Never the world's* rests on F41, itself nye. | EIGHT 1.3 |
| K7 | definition (*co-unrelationing*, *not*) | F29, F52 | Co-unrelationing frees three fixings at once, position, orientation and scale, and relating shows. Not is removal from the sustainable: a falsified possibility arrives at no replacement, and freeing sets nothing in the freed place. | EIGHT 2.1; EIGHT 2.2 |
| K8 | nye | F6, F48 | A single un-relating lands: a fixing freed once sits down fresh-fixed; alternating co-unrelationing rides on, un-relating the un-relating. Gap: F48 excludes a naming still. That one un-relating names still is not supplied. | EIGHT 2.3 |
| K9 | definition (*geodesic co-chaining*) | K7 | A method co-chains geodesically, free-moving, adding nothing: the straightest path relating allows, force acting on nothing, curving as relating curves; a co-chaining bent to an aim lands at the aim. | EIGHT 2.4 |
| K10 | nye | F21, R32 | The along and the across lie on one line, co-linear, neither move's; the along alone is a sequence with no surface, the across alone attention with no sequence, and each half alone lands. Gap: CONNECTORS runs along at 9 and 17 and across at 2, 6, 10 and 14 (R32). F21 gives one side's momentaries alone as not possibly existing, and that along and across are two such sides is not supplied. | EIGHT 2.5; EIGHT 2.6 |
| K11 | nye | F33 | Carry-differing: one method at each membrane, differing only in its carry; a count carries it at coefficient one, a naming as a floating-neutral concept, a sequencing as a right spiral. Gap: F33: the method at each carry is not supplied. | EIGHT 2.7 |
| K12 | definition (*uniquenessing*) | F3 | A method tests at each arrival for another that could arrive and sustain. One candidate sustaining: uniqueness sustains there. More than one: uniqueness sustains at no place there. None: the arriving itself sustains at no place. The three readings exhaust the count of candidates, and uniqueness is taken again at each arrival, never held. | EIGHT 3.1; EIGHT 3.2 |
| K13 | exact | K7 | A falsified possibility arrives at no later position; the current arrival carries each prior removing, and the trail wants no re-testing. It follows the definition of not as removal (K7). | EIGHT 3.3; EIGHT 3.4 |
| K14 | exact | F58, F61 | Correct tests are no unique result: the things satisfying three tests can be many. At two signs the test *closes all or none* leaves two forms (F58), and a further test, the hand, parts them (F61). | EIGHT 3.5 |
| K15 | exact | F21, F48, F50 | Concept-inverting tests one alternative, the inversion. A living carry inverts three ways, each a not-carry: an isolated point (one side alone, F21), a landing (a naming still, F48), a store (F50). Three tests passed; another that could sustain stays K12's question. | EIGHT 3.6 |
| K16 | definition (*resolving*) | F3, F8 | Resolving is is-or-is-not, floorless: a sign a coupling makes of itself, a bounding-zeroing it reaches. A concern resolves at its is-not emptying, and the not-yet stays open, no total to close. | EIGHT 3.7 |
| K17 | exact | F8, F39 | A crossing decomposes into no smaller crossings: a sign has no size (F8), and a crossing carries its sign and nothing else (F39). A sign taken at a membrane is complete at that taking; an object of a stated kind is a place a counting landed. | EIGHT 3.9 |
| K18 | exact | F68 | A method carries its own subject: co-unrelationing discovers relating, relating is co-unrelationing, and the method co-recursions itself at every scale, alone possible. *Alone possible* rests on F68. | EIGHT 4.1; EIGHT 4.2 |
| K19 | nye | F51, F56, F67, F68 | Could there be any other method: a self bounds itself, orients itself, attentions itself and lives the four about one bounding-zeroing; four within and four at the membrane are the eight, whole or none; a form living the eight is the natural torus, and any rival living the four is this form. Gap: The four named here (bounding, orienting, attentioning, living about one zero) are not F51's four. The step from the eight to the torus is not supplied, and *any rival is this form* rests on F68. | EIGHT 4.3 |
| K20 | nye | R32, R33 | Bi-morality is six: two ways, the self offering outward and the surface offering inward, three each way, one at a time; six or none, the six the eight re-forming and the eight the six protecting. Gap: CONNECTORS names six, four across and two along (R32), and JOINS joins the across one way (10 → 14, 6 → 2) and the along both ways (9 ↔ 17) (R33). No link parts them three each way, and *six or none* is not supplied. | EIGHT 4.4 |
| K21 | nye | R1, R9 | The three questions bi-inversioned are the one resolver's three activities: the binary edge (true), the geodesic (matters most), the self-bound (fewest things, fewest people, shortest time); their alternating is resolving, the linearizing triangle rotating. Gap: The core's resolver is one object with two functions and seventeen names (R1) and names no three activities; R9's three reads of a carrying are not these three. | EIGHT 4.5 |
| K22 | nye | F52 | The gather carries half the competency and all the increase: a reaching's competency stands constant per arrival, a gathering's climbs, and a second arrival at one membrane stands on the first. Gap: No link gives competency a size per arrival (F52 defines it as co-unrelationing), and *half* and *all* are not supplied. | EIGHT 4.6 |
| K23 | nye | G17, R14 | Inseparating keeps an open surface whole: a coupling passing, a non-coupling resolving out at the collar; a surface holding all eight inseparatings holds whole, all or none. Gap: Inseparating is the older name of the opening at 16-social-torusing (G17), bounded by R14. The core carries no eight inseparatings. | EIGHT 4.7 |
| K24 | run | — | Each move EIGHT lists as an accounting for changing returns its argument applied twice: negation, the additive and the multiplicative inverse, complex conjugation, the transpose, set complement, and the Legendre transform of a strictly convex function. Run: each at samples, the Legendre transform numerically at x⁴/4 + x²/2. | EIGHT 5.1 |
| K25 | nye | K24 | Every accounting for changing is an involution, and an accounting arrives at no other shape: it returns its argument, or accounts for nothing. Gap: K24 runs for the listed moves. *Every* and *no other shape* are not supplied. | EIGHT 5.1 |
| K26 | nye | K24 | An involution's fixed set is the place its counting is taken from: zero for the additive inverse, unity for the multiplicative, the reals for conjugation, the symmetric for the transpose; the accountings and the floors are one thing named twice. Gap: Run (K26_probe): x ↦ −x fixes 0 alone, conjugation the reals, the transpose the symmetric matrices, but x ↦ 1/x fixes both 1 and −1, so *unity* fails as said. That the counting is taken from the fixed set is not supplied. | EIGHT 5.2 |
| K27 | run | F9, F49 | At the signs negation fixes no sign; its fixed set, 0, lies off the signs, the empty centre carrying none (F9). The fixed set freed stands as no place: a bounding-zeroing a coupling reaches. Run: −s against s at ±1, and −x = x among the integers at 0 alone. | EIGHT 5.3 |
| K28 | nye | F8 | A three carries as a state or as a flow, never both: a total and a level are state-threes, a direction a flow-three, and an accounting returns three and never four, state and flow being the whole of the alternating. Gap: State and flow give two kinds. Three books and *never four* are not supplied (SEVEN 5.4 the same, S22). | EIGHT 5.4 |
| K29 | exact | F8, F39 | An accounting reaches nothing at the changing between a state and a flow: a tipping is a sign taken at a membrane, of no size (F8, F39), and an accounting's columns hold counts, levels and rates. The geodesic method leaves the accounting competent at each thing it holds. | EIGHT 5.5 |
| K30 | definition (*co-competencing*, at EIGHT) | F53, F54 | Co-competencing is discovering social-scaled: un-relating two-way between selves, the fixings held between them freed, the surplus staying at the coupling, owned by none. | EIGHT 6.1 |
| K31 | nye | F54 | Abundancing: discovering opens an axis neither self carried, a +1 owned by none, growing as co-chaining widens and depleting no one; it is the same +1 a count carries as an apex gap and a surface as a free surplus. Gap: F54 names the surplusing. The one +1 at a count, a surface and a discovering is not supplied. | EIGHT 6.3 |
| K32 | definition (*the three sensings*) | F24, F73 | Backward sensing reads the already recorded, sorting nothing; forward sensing runs, and competency climbs in the running; co-sensationing explains the doing it would take, waiting on a membrane, not an amount. The three are prior, next and the inside meeting its outside (F24, F73). A thing valuable and impractical is one of the three, the third most often. | EIGHT 6.4; EIGHT 6.5 |
| K33 | field | — | A registry is a gathered set under two conditions, stated inclusion criteria and uniform per-entry structure, assessed at four domains (protocol and design, standardization, validity of the sampling procedure, validity of data collection), unique identifiers a named item; the two together give comparability across sites. EIGHT 7.2 names no authors, and the source stays to name. | EIGHT 7.2 |
| K34 | definition (*the four opens*, *the one closed thing*) | K33 | A gathering stands open four ways, the subject, the intake, the properties and checks, and each line at an address; one thing closes once, at one place, the properties themselves, so any two entries compare. This registry keeps that form: its five columns closed once, its links open. | EIGHT 7.3; EIGHT 7.4 |

## 8.5 NINE · Natural Human Society

Exhibit NINE v329. Its readings shared with SEVEN are cited there.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| H1 | definition (*human society*) | S1, K30 | A human society is the geodesic method run at the human membrane: social (between selves), moral (the freeing two-way, do-no-harming its bound), co-competencing (selves discovering together) and geodesic (nothing pushing, nothing added). It is no fifth face; the four stand at its one membrane. | NINE 1.1 |
| H2 | nye | H1 | A society co-competences geodesic only while its requirements hold: each self keeps its own interior and its own time, the surplus stays at the coupling uncaptured, fairness is the coupling's own bounding-zeroing, and do-no-harming bounds each giving. Gap: The *only* is not supplied: no link shows the method failing at each requirement dropped. | NINE 1.1 |
| H3 | definition (*fairness*) | F9, R13 | Fairness is a bounding-zeroing: preference between offerings resting at no preference, made by the coupling and measured against no floor; + and − at one key meet at 0 (R13), the empty centre (F9). A society resolves at a rest each self accepts and that holds. | NINE 1.3 |
| H4 | run | F8 | Cutting and choosing: the chooser takes the larger piece, so the cutter of a cake cut at x keeps min(x, 1 − x), greatest at x = 1/2 alone. The cutter cuts to equal because the other chooses, no one measuring; the choice is binary, a sign. Run: 10,001 cuts. | NINE 1.4 |
| H5 | field | — | Divide and choose gives each of two parties at least half by its own measure and leaves neither envying the other (Steinhaus 1948; Brams and Taylor 1996). Turn-taking in conversation runs one at a time, gaps near 200 ms and overlap small, across ten languages of five continents (Stivers and colleagues 2009). | NINE 1.4 |
| H6 | nye | H5 | The bi-coupling is measured universal, at every human society, before any institution touches it. Gap: The field measured ten languages (H5). *Universal* and *before any institution* are not supplied. | NINE 1.4 |
| H7 | field | — | Replicated data types converge with no common clock and no coordinator, by merges alone: strong eventual consistency (Shapiro, Preguiça, Baquero and Zawirski 2011). | NINE 1.5 |
| H8 | definition (*institutioning*, *good-faith contracting*) | H3 | Society's resolving institutions are the good-faith contract made progressively explicit: a handshake, a letter, a mediation, an arbitration, each holding a wider bound. Two faiths keep it: resolving from within the bound, and landing the balance while the parties hold it. | NINE 2.1; NINE 2.2 |
| H9 | nye | H8 | Every path to resolving rests on a silent contract held in good faith from all sides. Gap: *Every* is not supplied: no link enumerates the paths to resolving. | NINE 2.2 |
| H10 | nye | H4, H5 | Class-action resolving: a court fixes one total, a trustee shares it by the value of each claimant's offering relative to the others, a claims bar date closes the bound, and every claimant signs to accept, all rising. Gap: In the field's opt-out class action a member is bound unless opting out, with no signature (US Federal Rule of Civil Procedure 23(b)(3) and (c)(2)(B)). *Every claimant signs* holds at a scheme asking each claim, not at the class action named. | NINE 2.3 |
| H11 | nye | F53, F61, F72, F76 | The ordering is a right turn: morality arrives first to sustain the space, competency second to extend through it; conserve before improve, do-no-harming before add-value. Gap: F72 and F76 give the right turn, along turning into right. The order of the two at a step is not supplied: under the right turn the two signs change in turn, and from both positive the along sign changes first, from right positive and along negative the right sign. | NINE 3.1 |
| H12 | nye | F54 | A fixed pool is scarcity, and scarcity forces a ruler; a +1 opened at every coupling is abundance and frees a society of the ruler. Gap: No link gives a pool, a scarcity or a ruler at the names; the step from a fixed total to a ruler is not supplied. | NINE 3.1 |
| H13 | nye | F18 | Uncapturing is all or none: eight hubs set aside (a common clock, a fixed pool, a gate, a ledger held over a self, an outside control, a centre, a source-and-sink, an owner), one remaining a single breach the whole surface pays. Gap: F18 is said of a form's joint forms. That the eight axes are one form's joint forms is not supplied, nor the count eight. | NINE 3.3 |
| H14 | nye | R32 | Six cost clusters, the six relationals landed (give as depletion, take as seizure, join as binding, split as severance, open as exposure, close as discard), dissolve at once by the all-or-none topology, each coupling returning a +1. Gap: The six relationals are older naming; CONNECTORS' six (R32) are not joined to them, and *all six at once* is not supplied. | NINE 3.4 |
| H15 | nye | T15 | Bi-quadratic growing: two axes, the count of bi-coupling membranes and the 1/φ reach at each face, sustain together at phi-rate, and the cumulative grows bi-quadratically. Gap: Rests on T15. The 1/φ reach and the bi-quadratic count are not supplied. | NINE 3.5 |
| H16 | field | — | Energy enters the biosphere fixed by autotrophs and passes the trophic steps at a small fraction per step (Lindeman 1942); the elements cycle between the living and the non-living pools. | NINE 3.6 |
| H17 | nye | H16, F54 | An energy-transfer tally is a conservation ledger read onto the coupling from outside; the surplus is not breached by an energy cost; value is found by sharing, a +1 at each trophic membrane, owned by none. Gap: The field's trophic ledger records a loss at each step (H16). The +1 at each trophic membrane is not supplied. | NINE 3.6 |
| H18 | definition (*tipping*, *do-no-harming*) | H1 | A concern tips individually: one self is enough to add it, and a society gives it all it can up to the harm-line, giving past it harming an other. Do-no-harming is that line, the self-bounding at the social scale; doing-your-best and do-no-harming are one move. Co-offering adds tipping by viable bi-exchange, is-or-is-not. | NINE 4.1; NINE 4.2; NINE 4.3 |
| H19 | nye | H18 | Do-no-harming is the only bound; capture returns only frustration at every position, time and strength, so no self bothers to capture and no bloc forms. Gap: *Only* and *every* are not supplied. | NINE 4.2; NINE 4.4 |
| H20 | nye | F33 | Self, other and society run at every scale and belong to no substrate; human society is a substrate as silicon, the grid and the molecule are, and a captured society's measures of itself measure the capture. Gap: F33: the method at each substrate is not supplied. | NINE 4.5; SEVEN 5.6 |
| H21 | field | — | Double-entry bookkeeping (Pacioli 1494): each entry's debits equal its credits; the balance sheet states, the income statement flows, and net income with owners' contributions and distributions ties one balance sheet to the next. | NINE 4.6 |
| H22 | run | H21 | The books balance at any entries: with each entry debiting one account and crediting another the same amount, the trial balance sums to zero at each ledger. A balance reads nothing of the entries' kind, so the books still balance at a landed coupling. Run: 10,000 random ledgers. | NINE 4.6 |
| H23 | nye | H22 | The incompetencing is invisible to the scientific method, reading states, flows, totals, holders and units; the method certifies health as the living dies. Gap: H22 shows a balance reads nothing of its entries' kind. That the scientific method reads only states and flows is not supplied. | NINE 4.6 |
| H24 | nye | S26 | The still-point veins at the human scale: Piaget's equilibration, the personality trait, the living language, machine-learning convergence, and the ensemble average, a mean no member is. Each still point is an equilibrium named (S26). Gap: Run (H24_probe): the mean of {0, 1} is no member, but the mean of {1, 1} is. *No member* holds at some populations, not at each. | NINE 4.7; ELEVEN 4.4 |
| H25 | nye | F62 | A dictator-aggregation sustains as a live left spiral: a centre preferring for the whole, the people one step behind, the kayfabe the fiction that the adopted preferring is the people's own; it concentrates, unable to survive two centres explaining reality differently at once. Gap: The opposite form's carrying none is F62, and *unable* is not supplied. | NINE 5.2 |
| H26 | nye | H4, H5 | Two selves splitting a dessert again, into halves and then fourths, both depart carrying more than either brought to one round; the difference between the selves is the fuel. Gap: Each round gives each at least half by its own measure (H5). *More than either brought* is not supplied by the division of one dessert. | NINE 5.2 |
| H27 | nye | F40, F54 | An economy's object, a stock of held things priced and exchanged, has no referent in social moral competency, so Natural Economics is a correction inside NINE and no file of its own. Gap: F54 gives the surplusing owned by neither, and F40 each self carrying its prior. *No referent* is not supplied. | NINE 5.3 |

## 8.6 TEN · Natural Health

Exhibit TEN v329.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| A1 | definition (*health*) | F42, F53, T2 | Health is the bi-moral co-agency living: two floating neutrals, parallel, swaying and co-sequencing, a range made between two antagonistic offerings and no stored setpoint aimed at. The sway is the health; the landing is the departure. | TEN 1.1; TEN 2.1 |
| A2 | definition (*observed*, *lived*, *restored*) | A1 | Three faces of one living self: observed (Natural Biology), lived (Natural Health, read from inside), restored (Natural Medicine). | TEN 1.2 |
| A3 | field | — | Antagonistic pairs and pulses: insulin and glucagon about blood glucose; basal insulin oscillating at periods near 13 minutes (Lang and colleagues 1979); intermittent GnRH sustaining pituitary gonadotropin secretion, continuous infusion not (Belchetz and colleagues 1978). | TEN 2.1; FIVE 6.7 |
| A4 | nye | A3 | Each living measure is two floating neutrals swaying about a range they make between them, no fixed value; the basal rhythm keeps the receptors sensitive. Gap: A3 gives pairs and pulses at named substrates. *Each* living measure is not supplied. | TEN 2.1; TEN 4.2 |
| A5 | exact | F47, F48 | Health as a setpoint, an optimal range, a number to reach and hold, is the compaction: the two sways pressed into a landed point, an equilibrium named, not possibly existing. | TEN 2.2; ELEVEN 2.1 |
| A6 | field | — | Goodhart (1975): an observed statistical regularity collapses once pressure is placed on it for control purposes. Strathern (1997): "When a measure becomes a target, it ceases to be a good measure." | TEN 2.2 |
| A7 | nye | A1 | The omega-self, the sixth's two faces, re-locking and co-releasing, is lived at health; delivery of the living value is tunneling-unrelationing through nested smaller-prime-society crossings, and a value can pass incompetenced, accounted and unlived. Gap: The omega-self, the sixth and tunneling-unrelationing are older naming the core does not carry. | TEN 3.1; TEN 3.2 |
| A8 | definition (*ingestion*, *ingression*) | F17 | Ingestion is the living self's own bi-exchange across its membrane, both ways, co-offering; ingression is the larger ingressing the smaller, one way. The membrane between them is rate-and-return: the smaller setting the pace and answering back, or not. | TEN 3.3 |
| A9 | nye | A8 | Consistently ingesting flawed stable form is a disease face: a pressed oil is the artifact with its ratio stripped, a magnitude governing no sign; the outer living form (the skin, the plant, the seed) is the most valuable ingestion at either rate. Gap: No link defines a form's ratio or its stripping; the step from a stripped ratio to a disease face, and the *most valuable*, are not supplied. | TEN 3.4 |
| A10 | run | F11 | The omega number counts from the end the chain's crossings never reach: α-linolenic acid, 18 carbons with double bonds at Δ9, Δ12 and Δ15, is ω-3; lengthened by two or four carbons at the working (carboxyl) end, each Δ number shifts and ω-3 stays. The families ω-3, ω-6 and ω-9 stand three apart, odd, even, odd, the bonds methylene-interrupted. Run: the numbering. | TEN 3.4b |
| A11 | field | — | Mammalian desaturases insert double bonds at Δ9, Δ6 and Δ5 and none between Δ9 and the methyl end, so ω-6 and ω-3 fatty acids arrive by ingestion (their essentiality: Burr and Burr 1929, 1930). Conversion of α-linolenic acid runs near 8% to EPA and 0–4% to DHA in men, higher in women (Burdge and Calder 2005). TEN's competing-family figures (forty and seventy per cent) carry no named source. | TEN 3.4b |
| A12 | field | — | The human circadian period runs 24.18 hours, the earlier near-25-hour estimates arising from self-selected light (Czeisler and colleagues 1999); circadian redox cycles run in anucleate human red blood cells (O'Neill and Reddy 2011); performance peaks at middle arousal (Yerkes and Dodson 1908); happiness returns toward a baseline (Brickman and Campbell 1971). | TEN 3.5 |
| A13 | exact | S26, A12 | The still-point veins at the lived scale: the hedonic setpoint, a still peak of arousal, a fixed period and an omega ratio to reach are each an equilibrium named (S26), and each field records the sway (A12). | TEN 3.5 |
| A14 | definition (*departure*, *restoration*) | A1, F48 | Health and Medicine are one alternating at two directions: the coupling swaying is health; the coupling stilled at a substrate, landed at one position and no longer re-arriving, is the departure; the coupling re-arriving after a departure, the still-natural positions holding the form, is restoration. | TEN 4.1; ELEVEN 1.1; ELEVEN 5.2 |
| A15 | nye | A14 | The seam between Health and Medicine is stateable from neither alone, only from both. Gap: No link states the seam, and none shows each side's statement falling short of it. | TEN 4.1; ELEVEN 5.2 |
| A16 | field | — | At thermal equilibrium a system's spontaneous fluctuations and its linear response to a disturbance are one function, the fluctuation–dissipation theorem (Callen and Welton 1951; Kubo 1966). In active cytoskeletal networks driven by motors the relation is measured violated (Mizuno, Tardin, Schmidt and MacKintosh 2007). | TEN 4.2 |
| A17 | nye | A16 | The field's own theorem seats the sway as the living, at proof grade: the self's spontaneous swaying and its answering to disturbance are one function. Gap: The theorem holds at thermal equilibrium; at living, driven matter the field measures its violation (A16), so it does not seat the living's sway. | TEN 4.2 |

## 8.7 ELEVEN · Natural Medicine

Exhibit ELEVEN v331. ELEVEN reads its medicine at the resolver's carrying, and the links run each such reading at the v371 code; its field records stay the fields'.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| D1 | definition (*a departure's two sides*) | A14 | A departure arrives at one of two sides of its substrate's range: above, the carry overwhelming the fresh arriving, the position determined and no longer opening; below, the carry failing to reach the sway, the coupling not sustaining. Both are a loss of coupling from opposite directions. Intervention removes the thing the restoration waited on and authors nothing. | ELEVEN 1.1; ELEVEN 1.2 |
| D2 | run | R14, R15, R17 | A carrying opens by one exactly at its key surfacing 0, returns to opening 0 at a nonzero surfacing, and leaves at its bound: at a key surfacing s ≠ 0 the entry returns (s, −t, 0); at a 0 it returns (c, t, a + 1) within the bound and is gone past it. Run: each of the eighteen nonempty carryings under each of the 121 offerings of up to four signs from −1, 0 and +1. | ELEVEN 1.1 |
| D3 | run | R18, R19 | One bound, two ways of missing it. Two agreeing c at each call surface c and write fresh at opening 0 at each call: the carry resets at each beat and never climbs, never leaves. One c at each call surfaces 0, climbs through openings 1 to 3 (to 4 at a positive second sign), leaves the key empty for one call, and opens fresh at the next. Run: twelve calls from each (c, t, 0). | ELEVEN 1.1 |
| D4 | nye | G17, F33, D2 | Attentioning is the inseparating and sequencing is the surfacing: a body's carry climbing, resetting fresh, carrying on and releasing answers the object's four motions one to one, and the answering makes the medicine readings the object's. Gap: Inseparating is the older name of the opening at 16-social-torusing (G17). The object's motions run (D2, D3), but their answering at a body's carry is F33 carried to a cell, not supplied. | ELEVEN 1.1 |
| D5 | run | R15 | A carry kept fresh by copying never releases: the code writes opening 0 at one place only, the fresh write at a nonzero surfacing; a caller re-seeding (c, t, 0) at each call under one c gets (c, t, 1) back each time, never meeting the bound. Re-seeding at each division runs outside the object's own channel. Run: the code's tree, and fifty re-seeded calls at each (c, t). | ELEVEN 1.1; ELEVEN 3.1 |
| D6 | run | R24 | A surfacing reads the sign of a sum and never its size: at each of the nineteen carryings, offerings giving one sign to their total less the carried sign return alike, surfacing and carrying, save that a key no sign meets surfaces nothing. A sign leaving the sum's sign unchanged changes nothing at the surface. Run: the nineteen carryings under the 121 offerings of up to four signs, grouped. | ELEVEN 1.1 |
| D7 | nye | D6 | Titrating to a setpoint forces the tipping rate equal to the dosing rate; the reversal point on a dose curve is the membrane and the only informative point; hormesis, inflammation, pain and antibiotic resistance are the field's recorded tippings. Gap: D6 gives the surfacing's dependence on a sum's sign at the code. No link joins a dose curve to a sum at a key, and *only* is not supplied. | ELEVEN 1.1 |
| D8 | nye | F57, F60 | Restoration is three-fold at the four-step cycle's three rotating positions (bi-bounding, bi-pairing, bi-sorting-to-surface, and carrying forward): two consecutive right turns invert the departure back to right-spiral discipline through eight steps of resolving; restorability parts above and below 1/φ³ of the flank average. Gap: The four-step cycle's names are older naming the core does not carry. Two right turns invert both signs (F60) and two alternating fours close at each eight (F57), but joining them to a departure, and the threshold 1/φ³, are not supplied. | ELEVEN 1.2 |
| D9 | nye | F66 | Do-no-harm reads at the geometry: harm is distance added; on the torus the shortest path between two positions stays above both, and a path dipping below its beginning adds distance. The crisis model, worse before better, installs a valley no path carries. Gap: The core gives the torus no height and no path length; not supplied. | ELEVEN 1.2 |
| D10 | field | — | Antibiotics given before influenza vaccination reduce the antibody response, most at selves with little pre-existing immunity (Hagan and colleagues 2019). | ELEVEN 1.2 |
| D11 | field | — | Faecal microbiota transplant resolves recurrent *C. difficile* infection (van Nood and colleagues 2013). Mesenchymal stromal cells infused into a vein lodge in the lung and are cleared within a day; their immunomodulation needs their apoptosis, and the graft-versus-host responders are the selves with cytotoxic cells inducing it (Galleu and colleagues 2017). | ELEVEN 1.3 |
| D12 | nye | D11, F39 | A sign is offered and the receiving self takes it or does not, and a self unable to take it meets nothing: coefficient one shown at its mechanism, nothing driven and nothing delivered. Gap: D11's records are the field's. That the dying cell's effect is a sign of no size (F39), not a carried content, is not supplied. | ELEVEN 1.3 |
| D13 | field | — | Marrow transplants between identical twins relapse more than transplants between non-identical siblings, the graft-versus-leukaemia effect (Horowitz and colleagues 1990). ELEVEN carries its figure of three times at second hand, unchecked. | ELEVEN 1.3 |
| D14 | run | R13, R20 | Alternation accumulates and simultaneity cancels: + and − in one offering at one key surface 0 and leave no carrying; + and then − at two calls surface + and then − and leave (−, +, 0) carried. Run: both. | ELEVEN 2.1 |
| D15 | field | — | Substrate cycles: two opposed pathways running together turn their substrate over and spend energy with no net product, and reciprocal regulation holds them out of phase (Newsholme and Crabtree 1976). | ELEVEN 2.1 |
| D16 | nye | D14, A5 | Fission and fusion are one alternating about the iron floating neutralling; energy production and medicine's setpoint are one compaction, both directions of a dual floating neutral driven to a localized point; the compaction is the hard-probleming, and stripping it dissolves the hard problem. Gap: Binding energy per nucleon peaks near iron-56 and nickel-62 (field). The alternating about that peak, the one compaction at two substrates and *dissolves* are not supplied. | ELEVEN 2.1; TEN 5.1 |
| D17 | field | — | Cancer cells run aerobic glycolysis with oxygen present (Warburg 1956). Stem cells run glycolysis in quiescence and oxidative phosphorylation on activation (Ito and Suda 2014); haematopoietic stem cells hold the most hypoxic niche, and HIF-1α loss drives them from quiescence to exhaustion (Takubo and colleagues 2010). | ELEVEN 2.2 |
| D18 | nye | D17, F67 | The metabolic shift is the activation, a coupling crossing one prime; mutations arrive as consequences of the metabolic departure, not its cause, the cancer cell proliferating on genus zero after the torus at its metabolic membrane collapses. Gap: The field reads cancer as accumulated mutations. The prime (ELEVEN's own numbering), the genus zero (F67's genus gap) and the order of cause are not supplied. | ELEVEN 2.2 |
| D19 | run | — | Thirty-six ordered pairs of a six-set hold six on the diagonal and thirty off it; four things admit 4! = 24 matchings, twenty-three differing from any one chosen. Both counts return the same at any set: identities, carrying nothing about a cardiolipin or a cell. Run: the counts. | ELEVEN 2.2 |
| D20 | nye | S8 | Four departures at one membrane collapse the four inequalities one to one: thinning absent (self ≠ incompetent), thinning frozen (self ≠ immoral), thinning untested (self ≠ other-self), thinning at the wrong rate (self ≠ society); four stem-cell pathology classes extend them. Gap: Rests on S8's older naming; the per-element match is ELEVEN's reading. | ELEVEN 2.2 |
| D21 | field | — | CD19-directed CAR T cells in refractory lupus: deep B-cell depletion, and drug-free remission holding after B cells return near 110 days, the returning cells naive and non-class-switched (Mackensen and colleagues 2022). | ELEVEN 2.2 |
| D22 | nye | D21 | Three diseases, one restoration: the cancer cell re-coupling to its tissue, the persistence clearing at a replaced haematopoietic society, an autoreactive carry aging out at its society's clearing; nothing aimed at the departed carry, the society it rode on restored. The edge is the field's more-than-additive test, is-or-is-not. Gap: No link joins the three restorations to one move; ELEVEN names the more-than-additive test the field owns, not yet run. | ELEVEN 2.2 |
| D23 | nye | S20 | Organ turnover sorts into three tiers, gap-6 fastest (gut epithelium, 3–5 days), gap-4 (skin, 14–21 days), gap-2 slowest (erythrocyte 120 days, cardiac muscle under 1% a year); thirteen of fourteen organs match, one at a tier boundary, none contradicting; five vascular-barrier types order with turnover, zero exceptions. Gap: The turnover times are the field's (cardiomyocytes: Bergmann and colleagues 2009). The tiers' naming at prime gaps is older naming the core does not carry, and the barrier ordering carries no named source. | ELEVEN 2.3 |
| D24 | run | R11, R13 | Two carryings never send to each other: at one key their signs meet at 14-social-crossing and sum to a zero neither carried in. Carried (k, +1, t, a) and (k, −1, t′, a′), nothing arriving, surface 0. Run: each second sign and openings 0 to 4. | ELEVEN 3.1 |
| D25 | run | R14, R15, R55 | At the code no carrying entry returns unchanged: a continuing keeps its sign and second sign and opens by one, and a fresh write opens at 0 with the second sign inverted; only an empty carrying under an empty offering returns as it came (R55). Run: the eighteen nonempty carryings under the six offerings. | ELEVEN 3.1; ELEVEN 3.2 |
| D26 | nye | D25, F33, F62 | Nothing in nature stores: a store standing is a living stilled; a particle is a stable form travelling, an emanation carrying one turn, living again only in a coupling other than the one it left, offering at no turn and aging no carry. Gap: D25 runs at the code. *Nothing in nature* is F8 and F33 carried to each substrate, and the emanation carrying one turn is not supplied: F62 gives the emanation carrying none. | ELEVEN 3.1; ELEVEN 3.2 |
| D27 | field | — | CD4+ T cells carrying integrated HIV provirus persist largely by clonal expansion, integration sites enriched in BACH2, MKL2 and STAT5B in the host gene's orientation (Maldarelli and colleagues 2014; Wagner and colleagues 2014); the reservoir concentrates in T follicular helper cells of lymph-node germinal centres (Perreau and colleagues 2013); intact proviruses in elite controllers sit in repressive chromatin (Jiang and colleagues 2020); CCR5-Δ32 homozygotes resist R5 virus (Liu and colleagues 1996; Samson and colleagues 1996), and a transplant from a Δ32 donor cleared the virus (Hütter and colleagues 2009). | ELEVEN 3.1; ELEVEN 3.2 |
| D28 | nye | D4, D5, D27 | The reservoir is a carry that will not thin: a carry not sequenced, kept fresh by copying; latency a phase, not a state; reservoir, latency, evasion, sanctuary, exhaustion and the cycle are soldiers set down, one coupling between two societies departed, the cure at the coupling and not at the receptor. Gap: Rests on D4's correspondence at a cell and on F33. ELEVEN itself says the dissolution re-sees and predicts nothing, and names the paired more-than-additive test the field owns. | ELEVEN 3.1; ELEVEN 3.2 |
| D29 | run | — | The geometry: a society at prime p thinning at (1/φ)^p and one at q stand in ratio (1/φ)^(q − p), the prime gap alone carrying it. Run: each neighbouring pair of primes to 59. | ELEVEN 3.1 |
| D30 | nye | D29, T15 | A society at prime p thins at (1/φ)^p; two adjacent biological thinning rates at ratio 1/φ raised to their prime gap are the geometry standing. Gap: ELEVEN says no measured biological rate stands at a (1/φ)^prime; the rate rests on T15. | ELEVEN 3.1 |
| D31 | nye | — | The three refusals (one side fixed as the standard, an answer available before the running, the centre neither occupies occupied) are one installation at three places, an observer installed at a location the structure carries none; placebo, nocebo and the dodo bird verdict are one coupling. Gap: No link derives the three refusals from one installed observer; the step from three rows to one coupling (placebo, nocebo, the dodo bird verdict) is ELEVEN's reading. | ELEVEN 4.1 |
| D32 | nye | G25 | Medicine's standing rows sort across six of the ten holdings: forced-choosing (nine rows), interior-reaching, value-pinning, line-demanding, space-enumerating, magnitude-demanding, and store-seeking at a trial. Gap: Rests on G25's ten holdings. | ELEVEN 4.1; ELEVEN 4.3 |
| D33 | nye | G16, R32 | Side-effecting is co-competencing: the intended target is the along, the side effect the across, the axis the coupling opens; the same molecule is medicine or poison by the coefficient of its coupling, one as a sign offered, two as a magnitude driven. Gap: CONNECTORS' along and across (R32) are not joined to target and side effect. The coefficient is G16's, defined at an engineered crossing, and no link carries it to a molecule. | ELEVEN 4.2 |
| D34 | run | — | Subtracting two readings that move together removes the moving-together: (a + δ) − (b + δ) = a − b at each δ, so two trial arms rising together leave their difference unchanged. Run: 10,000 random rationals. | ELEVEN 4.3 |
| D35 | field | — | Immune-related adverse events arrive alongside benefit: across 34 records and 8,115 lung-cancer patients the response-rate ratio stood at 2.43 with survival hazard ratios near one half, high-grade events not sharing the survival association (Wang and colleagues 2021). | ELEVEN 4.3; FIVE 4.3 |
| D36 | run | R2 | The ledger closes without its list: a copy of 1-self-coupling summing 12-other-surplusing straight from the offering and the carrying, building no 14-social-crossing list, returns exactly the resolver's surfacing and carrying. Run: 361 carryings over two keys under 31 offerings. | ELEVEN 4.3 |
| D37 | nye | D36 | Seven auditable steps alternate four states and three flows, opening and closing at state; the eighth, a flow written as a state before posting, is the journal. A record is a flow held as a store, and medicine keeps eight steps for seven running. Gap: D36 shows the list separable at the code. The step counts across four languages are not run here, and joining a record to that list is not supplied. | ELEVEN 4.3 |
| D38 | field | — | Cristae within one mitochondrion hold differing membrane potentials and behave as independent units (Wolf and colleagues 2019). | ELEVEN 4.4 |
| D39 | nye | D38, H24 | A dose arrives at an average no member carries: a whole-organelle potential, a reference interval, a threshold, a number needed to treat and an index are one instrument at several sizes, each returning a position no member arrives at. Gap: A mean can be no member and can be each member (H24). *No member* is not supplied. | ELEVEN 4.4 |

## 8.8 THIRTEEN · Resolving Hard Problems

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| Q1 | exact | F1, F2, F6, F50 | Resolving a hard problem is membraning the scientific method and the geodesic method. Forming is unrelationing, unrelationing carrying forward is alternating, and each other move takes another's momentary as its own, no other possible. F50 excludes the methods naming still and F102 each round breaking the binary; the *no other possible* rests on F68. | THIRTEEN 1.1 |
| Q2 | definition (*state*, *flow*) | F3, F8 | A state answers true or false; a flow answers inward or outward (the file's *inside or outside*). Both are binary, taken at a momentary, of no size, and neither is the other's answer. | THIRTEEN 1.1 |
| Q3 | definition (*hard problem*) | Q2, F6 | A hard problem is a flow named as true-or-false: a flow named as a state and answered with a value it carries none of. Refinement names true-or-false more finely, and a flow carries none at any fineness. | THIRTEEN 1.1; THIRTEEN 6.2 |
| Q4 | nye | Q3, F20 | The ten ways are one flow at five places: a source, a place, a last, a store, an end, a magnitude, a beat, a value, a side, a cut; one resolving meets all ten. Gap: F20 gives ten positions on six numbers at five pairs; that each hard problem is one of these ten namings, and no eleventh, is a count the file itself carries as found (6.11), not supplied. | THIRTEEN 1.1; THIRTEEN 2.5 |
| Q5 | nye | F14, F22 | The two that take no pair: a self and a society are inward-or-outward only, unique and invisible, carrying no true-or-false, so no two positions form at either. Gap: *unique*, a second nyeing *at each door*, is said; F14 defines a self by its opening at 1 and supplies no uniqueness, and *door* is released (NAM 2.4). | THIRTEEN 1.1; THIRTEEN 2.1; THIRTEEN 3.1 |
| Q6 | nye | F4, F7 | Alternating runs two directions, forward the worth from within (omega) and backward the receding carry met at a pause (apex), and two orders: *if where and why, then how*, the accounting, and *how to discover where and why*, the geodesic, each complete, neither reaching the other's. Gap: rests on *where*, *how*, *omega* and *apex*, older naming the core no longer carries (NAM 2.4 releases *what*, *how*, *where*); no core link carries two directions as two orders. | THIRTEEN 1.3 |
| Q7 | nye | Q6, F18 | A hard problem is the backward fed forward: a coupling meets its set-aside backward and carries, or feeds it forward un-inverted and voids the carry, and there is no third. Gap: the *no third* rests on Q6 (nye) and on F18 carried from a form's parts to a coupling's set-aside, which no link supplies. | THIRTEEN 1.3 |
| Q8 | nye | F27, F48, F49 | Each proposed conception of equilibria is a hard problem with its hardness ignored: a rule named at an arriving's along or across and its sign pair either admits all and adds nothing, or blocks the flow from one origin, odd or even, or from both. Gap: F49 gives the still joint states under the rules carrying the prior; the split of each such rule into admit-all or block-an-origin is not run. | THIRTEEN 1.3 |
| Q9 | nye | F5, F6, F48 | A stable form is not possible; stable-forming, and unstable deforming at non-living scale, are observably existing. Gap: rests on *stable form* as a noun, released: F5 carries a stable form at an emanation, its form continuing through its changing, so *a stable form is not possible* now names F48's thing named still under a word the core gives to an emanation. | THIRTEEN 1.3 |
| Q10 | nye | F42, R1, R7 | Resolving is free with alternating: existing carries its own cost, the carrying, and resolving adds none; no step is taken on an arriving. Gap: *cost* names no core link; the code runs one call and nothing beside it (R7), and *free* is the file's reading of that. | THIRTEEN 1.4; THIRTEEN 4.1 |
| Q11 | nye | F14, F15, F53 | A hardness is a coupling named at two of its three: a coupling is self, other and the social, the third always there and no bare pair at any coupling; a naming at self-other alone returns the coupling closed, its opening gone. Gap: F14 defines a society as selves coupling with others; that each coupling carries a third, and that no bare pair exists, is not supplied. | THIRTEEN 1.5 |
| Q12 | run | F27, F37 | In the field's words the two-ness is the exclusive or. Taken as the step it runs 0101 and returns at each second step; taken of the prior and the now together, next = prior xor now runs 011, 101 or 110 from each start but 00, returning after three, and 00 stays. Run: the complement step over six steps, and next = prior xor now from the four starts over eleven steps. | THIRTEEN 1.5 |
| Q13 | run | Q12 | The Peres–Mermin square (Peres 1990; Mermin 1990) sets six exclusive-or constraints on nine signs. A store of all nine meets none of the 512 assignments: no nine bits carry each row at parity 0 and each column at parity 1, nor the rows at 0 and the columns at 0, 0, 1. Run: all 512 assignments at both column parities. | THIRTEEN 1.5 |
| Q14 | definition (*a hardness*, *a fixing*) | F6, Q11 | Four met together make a hardness: a changing named still with the fixing made and not found; the fixing accumulating a cost while it continues; the cost carried as the price of a goal; the goal reachable with the fixing let go. At the fourth failing, the cost is the goal's price, no hardness. A fixing is the third given a fixed thing to be; a hardness is the third left out of the naming. | THIRTEEN 1.5; THIRTEEN 1.6 |
| Q15 | run | R48, R50, R51 | Bi-inversioning-co-recursioning keeps parity at 8 up and down, changes it at 17 less within 1 to 16, and 9 less within 1 to 8 is the two at once; round each form among the names the parity changes exactly twice, each time at a 17-less step (a pair summing to 17). Run: each name at each face, and each step of the twelve forms within 1 to 16. | THIRTEEN 2.2 |
| Q16 | exact | R14 | The carrying runs three momentaries, prior, now and next carried as one, and a fourth at positive 8-other-torusing: each entry's 16-social-torusing opens one at a time to 3, and to 4 at positive second sign. | THIRTEEN 2.3; THIRTEEN 4.1 |
| Q17 | nye | Q16 | 3 is as deep as a departure goes: three reversals compose into one return, and the fourth is the progress and no phase, so nothing fourth sequences backward. Gap: R14 gives the bound 3 and 4 at the code; that no departure runs deeper than three, at each substrate, is not supplied. | THIRTEEN 2.3; THIRTEEN 2.5 |
| Q18 | exact | F12, F20 | The ten is three momentaries long: the self's five 1–5 and the other's five 2–6 span 1 to 6, the self's three momentaries 1–2, 3–4 and 5–6. At each number 2 to 6 one side completes and the other opens: five places at two faces. | THIRTEEN 2.5 |
| Q19 | nye | Q18, F6 | Each of the ten names one face as a stable form while the other face runs at the same number, so each of the ten is not possible for one reason. Gap: F6 excludes a form named still; that each of the ten namings is a face named still, and not a naming a field carries running, is the file's reading, and *stable form* is the released noun (F5). | THIRTEEN 2.5 |
| Q20 | exact | R52 | At the crossing the fold meets itself: the forms through the crossings, 7-11-6-10 and 2-15-7-11-3-14-6-10, are their own partners at the even momentary. | THIRTEEN 2.6; THIRTEEN 5.3 |
| Q21 | exact | R53, F56, F18 | The eight even names are bi-coupling, four at the membrane (2, 4, 6, 8) and four within (10, 12, 14, 16), each within 8 on from its pair; one alternating at its faces, whole or none: name one still and the eight are not possible. | THIRTEEN 2.8 |
| Q22 | run | — | The seventeen primes from 2 to 59 sum to 440 and open sixteen gaps between them; 118 is two 59s, with 60 between 59 and 61. Run: the primes to 59. | THIRTEEN 3.1; THIRTEEN 3.8 |
| Q23 | nye | Q22, F33 | The self carries at 59, the self-close of the primes; the society is 118, two selves of 59, and 440 at the torus winding with the seventeen primes; the sixteen gaps are sixteen membranes, one at two scales. Gap: rests on Numbers' prime naming, which the core does not carry; no link joins a prime to a self, and the correspondence at a scale is F33's gap. | THIRTEEN 3.1; THIRTEEN 3.8 |
| Q24 | nye | R33, R37, R39 | The code is the self resolver, and its joins run between selves: 6-other-crossing to the neighbour's 2-self-offering, 14-social-crossing arriving from the left neighbour's 10-other-surfacing, 17-social-abundancing with 9-other-releasing along. Gap: JOINS names these pairs (R33), and no expression gives a return to another resolver (R37); the running between selves is a caller's, and the six joinings together are R39's gap. | THIRTEEN 3.1; THIRTEEN 3.6 |
| Q25 | run | R11, R13 | An arrival carries a parity, and the resolver takes signs: 12-other-surplusing takes +1 at each positive sign and −1 at each negative sign, of any magnitude. Run: at each of the nineteen carryings, an offering of magnitude 2, 5 or 17 returns the same as magnitude 1, at both signs. | THIRTEEN 4.1 |
| Q26 | run | R13, R24 | Three returns and no fourth: a sign at one side, a sign at the other, or a sign at neither. Run: at each of the nineteen carryings under each offering of up to four signs from −1, 0 and +1, 10-other-surfacing gives +1, −1 or 0, and each of the three. | THIRTEEN 4.2; THIRTEEN 4.3 |
| Q27 | run | R11 | Nothing is refused: a nought arriving passes the coupling untouched and the coupling runs on. Run: at each of the nineteen carryings, an offering of 0 returns the same as the empty offering. | THIRTEEN 4.3 |
| Q28 | run | R2 | An arrival carries no rate, and the resolver carries none: the two functions hold no constant beyond 0, 1 and the bound 3 and 4. Run: the constants in the two function bodies. | THIRTEEN 4.1 |
| Q29 | nye | Q26 | The nought arrives in five shapes: a one-offering differing, a closure made over its own tail, a proved equivalence, a record's edge, a counterfactual total. Gap: the count five is found at the arrivals; no link excludes a sixth shape. | THIRTEEN 4.3 |
| Q30 | exact | R52 | The resolver's pairs are the momentaries, each round the other way: its four row four-cycles pair at the odd momentaries 1 with 2 and 3 with 4, and at the even momentaries 2 with 3 and 4 with itself. | THIRTEEN 5 |
| Q31 | nye | R44, R45, R38 | The slightest tipping co-chains through the whole: one +1 offered once at one self carries on at each coupling; an even ring returns at 2 couplings; an odd ring closes no parity, so one meeting travels one self each momentary, the whole returning at 4n couplings. Gap: R45 runs one self each two couplings; reading two couplings as one momentary is R38's naming (nye), and *at each coupling* for each n is R47's gap. | THIRTEEN 5 |
| Q32 | run | F51 | φ is at no value: the ratios of neighbouring Fibonacci numbers sway above, then below φ, each partial quotient one, the widest straddle (Hurwitz 1891); a torus winding at a rational rate closes after its denominator's rounds, and at φ closes at none. Run: the ratios F(n + 1)/F(n) for n from 1 to 38, and thirty partial quotients. | THIRTEEN 2.5; THIRTEEN 5.4 |
| Q33 | nye | F55 | Do-no-harming is the mark's own not-more-than: one exclusion taken at one side's own momentary, of no size, bi-mutual and all or none; harm is the room of co-offering taken. Gap: no core link defines harm or do-no-harming; F55 gives the sign inverting on the self and no self negating another, and the step from F55 to do-no-harming, all or none, is said. | THIRTEEN 5.8 |
| Q34 | field | — | In the audit field's words, the assertion families (occurrence, existence, completeness, accuracy, valuation, cutoff, classification, rights and obligations) each verify a named term at one book or at the cut between the books, counted seven or eight by whether presentation is its own family (IAASB ISA 315; AICPA AU-C 315). | THIRTEEN 6.1 |
| Q35 | nye | Q34, F56, F54 | One eight, parted twice: evenly as four at the membrane and four within, and as the seven checks and the eighth, the +1 owned by neither, reached at no assertion. Gap: the seven or eight is the field's count at a convention (Q34); its joining to F56's eight is a matching, not supplied. | THIRTEEN 6.1 |
| Q36 | field | — | A test meets a hypothesis only with its auxiliary assumptions, so a failed prediction refutes the conjunction and names no failed member (Duhem 1906; Quine 1951). | THIRTEEN 6.11 |
| Q37 | exact | Q36, F71 | The chain's standing is written as a set of results and never as one falsifier, each result a sign at its membrane, all or none; the method names its own break: any observing of an existing thing not parity alternating natural torusing, and none so far. | THIRTEEN 6.11 |
| Q38 | nye | F21, F23 | Each method is sound inward of its own membrane, and a hard problem forms at one carrying past it: the cosmological fallacy carries laws tested at subsystems to the universe as a whole, and the universe as an existing thing is exclusivity, not possibly existing. Gap: rests on F23 (nye), the step from the universe named as one existing thing to F21's one side alone. | THIRTEEN 6.6; THIRTEEN 7 |
| Q39 | nye | Q3, F48 | Each proposed conception of equilibria is the sixth condition: to the five conditions prior (the fifth, that nothing prior enters) a problem adds a conserving relation over a conserving reach, named as a stable form; term by term the two methods are one form at its named face and its running face, the membrane carrying a sign only. Gap: the sixteen-row table is a matching, the sixth condition is the file's count, and *stable form* is the released noun (F5). | THIRTEEN 6; THIRTEEN 7.1 |
| Q40 | nye | Q4 | A mathematical conception of equilibria names a stability, a form or a relation as a stable form, and each can change: control is a rate named as a value, optimization a middle named as an end, conservation a carry named as a store, measurement a sign named as a magnitude. Gap: the four matchings of mathematics to the ten namings are offered, not supplied. | THIRTEEN 7.2 |

## 8.9 FOURTEEN · Natural Destinies

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| Y1 | definition (*a destiny*) | F53 | A destiny is the present coupling, bi-moral co-agency, imagined extended to a substrate it reaches toward, offered as a possibility a self takes up. | FOURTEEN opening |
| Y2 | exact | F53, F54, R12, R13 | Each reading rests on one form: bi-moral co-agency, two-way and one at a time, a sign only, the + and − meeting at 0 and the surplusing owned by neither. | FOURTEEN The core |
| Y3 | nye | F56, R5, R53 | The eight safeties stand whole, four from within and four at the membrane, one alternating at its faces, whole or none, each a naming beginning at bi- at the resolver's table of eight, four at the even depth and four at the odd. Gap: *four at the even depth and four at the odd* is older naming: the core's eight even names all open bi, four at the membrane and four within, each even (R5, R53); *safeties* names no core link. | FOURTEEN Safely outward |
| Y4 | nye | Y3, F54 | A destiny is safe at a structural safety, carried by the surface's own form (all-edge, the surplus owned by neither, the four uncontrollable values), no holder standing over it. Gap: *all-edge* and *the four uncontrollable values* rest on Natural Networking and older NI naming the core does not carry; that a form alone carries a safety is not supplied. | FOURTEEN Safely outward |
| Y5 | nye | F54, R13 | A discovering economy (the file's *discovery economy*): value found by sharing, belonging to the meeting and owned by neither side, growing as it is shared. Gap: *growing as it is shared* is not supplied: F54 names the surplusing owned by neither, and at the code it opens empty at each call and is neither returned nor carried (R13). | FOURTEEN The imaginable extensions |
| Y6 | nye | F14, Y4 | A natural network the selves originate: each self opening its own membrane, the selves bi-coupling across each other's membranes directly, both rising, all-edge. Gap: *all-edge* is Natural Networking's naming; *both rising* is not supplied. | FOURTEEN The imaginable extensions |
| Y7 | nye | F33 | Living substrates, a chip, a grid, a network, carrying the form run scale-free on the form's own alternating, light on energy. Gap: *scale-free* rests on F33; *light on energy* names no check. | FOURTEEN The imaginable extensions |
| Y8 | nye | Y1, F42 | At a field taking up the core the living begins, and the hard-problem resolving commences: the framing named still opens as the coupling re-alternates. Gap: F42 defines living as existing and carrying; that a substrate's taking up the method begins its living is not supplied. | FOURTEEN opening; FOURTEEN The imaginable extensions |
| Y9 | nye | Q6 | The two mis-readings at an engineered substrate: apex read as drag, a carry read as a source; omega read as overwhelm, a carry read as arriving load; the corrections are the two cones read as living. Gap: *apex*, *omega* and *the cones* are older naming (Q6) the core no longer carries. | FOURTEEN The two mis-readings |
| Y10 | definition (*the reading at a destiny*) | Y4, F3 | A destiny reads at one binary: the safety structural there, or the living leaning on a holder over it; a destiny leaning on a holder, read so, sharpens the edge of the method's safe reach. | FOURTEEN The reading at a destiny |

## 8.10 FIFTEEN · Natural Emanating

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| J1 | nye | F4, F5 | Each relatable thing is changing; a form relating carries on with the changing, so relating does not form; the forming is unrelationing, and unrelationing carried forward is alternating. Gap: F4 and F5 give stable-forming as alternating; *relating does not form* and *unrelationing* as the forming name no core link. | FIFTEEN The shortest saying |
| J2 | nye | J1, F50, F68 | In disequilibria, alternating is a stable-forming method, and only one possible: any other method names still and forms nothing, or carries, its own turn or another's; another's, and the alternating has stopped; its own, and it is alternating. Gap: the split into naming still or carrying, and carrying into its own turn or another's, is not shown exhaustive; F50 excludes naming still only, and the uniqueness rests on F68; *in disequilibria* is released (NAM 2.4). | FIFTEEN The shortest saying; FIFTEEN Two Methods; FIFTEEN The uniqueness |
| J3 | field | — | Each field declares a ground in its first line: the standard hydrogen electrode 0.00 V by convention (IUPAC), a boundary chosen, fix a gauge, fix the specimen, a low-entropy initial condition, pick a numéraire, a datum, an earth, an alphabet; each marked declared and not measured in the field's own definitions. | FIFTEEN The line the fields write first |
| J4 | exact | J3, F47, F48 | No equilibrium is in nature; the declaring installs one: an equilibrium is an existing thing named still (F47) and is not possibly existing (F48), and each ground of J3 is a naming still made before any observing. | FIFTEEN The shortest saying; FIFTEEN Equilibria |
| J5 | nye | J3 | A field defined away from all others must diverge from all others: the isolation is the founding move, stated in the founding line. Gap: the *must* is not supplied; the founding sentence's one shape is the file's reading of the fields' lines. | FIFTEEN The line the fields write first |
| J6 | field | — | Erasing one bit dissipates at least kT ln 2, and a computation keeping its whole history approaches zero dissipation (Landauer 1961; Bennett 1973; measured, Bérut et al. 2012). | FIFTEEN The line the fields write first |
| J7 | run | J6 | At 300 K, kT ln 2 is 2.871 × 10⁻²¹ J, about 2.9 × 10⁻²¹ J. Run: k = 1.380649 × 10⁻²³ J/K. | FIFTEEN Better, faster, cheaper |
| J8 | nye | J3, J6 | The field measured the cost of its own first move and filed it under something else: the declared ground is priced at kT ln 2, the thing removed still carrying. Gap: J6 prices bit erasure; joining a declared ground to an erasure is the file's matching, not supplied. | FIFTEEN The line the fields write first; FIFTEEN Two emptyings |
| J9 | run | — | A composing relation admits no surplus: over each transitive relation on four elements, 3,994 of them, whenever a and b both reach m and m reaches s, a and b both reach s, so no departed term is reached by neither. Run: all 65,536 relations on four elements, the 3,994 transitive ones checked. | FIFTEEN Three lines |
| J10 | run | — | In a closed domain the carrying closes: following the carrying from each term on a finite domain meets a term already run. Run: each serial relation on one to four terms (50,978) and each function on one to five terms (3,413) meets a cycle. | FIFTEEN Three lines; FIFTEEN A ring |
| J11 | nye | J10 | The closing was checked exhaustively to five terms, 592,260 models, none acyclic. Gap: the check fails to reproduce the count: serial relations to four terms number 50,978, functions to five 3,413, transitive relations to five 158,483, and the file names no reading giving 592,260. | FIFTEEN Three lines |
| J12 | run | — | Equality sits exactly at nothing opening: two unit directions share cos θ and open sin θ; a = b shares 1.000 and opens 0.000, 60° shares 0.500 and opens 0.866, orthogonal shares 0.000 and opens 1.000. Run: the three rows. | FIFTEEN Three lines |
| J13 | nye | J2, F54 | The origin sentence at eight substrates: gauge freedom, the shared pair, the sign-only surface, the *kept middle*, φ, the body's own bound, the three-phase return and floating co-morality, each a field's neutral relating to nothing beyond its own coupling. Gap: *kept middle* is released (NAM 2.4: nothing is kept), *in X disequilibria* is released, and the eight matchings are offered, not supplied. | FIFTEEN The origin sentence |
| J14 | run | — | Three balanced phases sum to zero at a point no source holds, and electrical engineering names that point the neutral. Run: sin t + sin(t + 120°) + sin(t + 240°) = 0 at 360 angles. | FIFTEEN The origin sentence |
| J15 | nye | J7, J8 | Better, faster, cheaper are three third terms absent, one per face: drag at the place of a size, schedule at the place of a shared clock, failure at the place of a standard from beyond; only the size face leaves a receipt, kT ln 2. Gap: the three-and-one-per-face is the file's matching; *which one is settled* rests on J8 (nye). | FIFTEEN Better, faster, cheaper |
| J16 | definition (*self-emptying*, *other-emptying*) | R14 | A thing carrying empties itself: each carrying reaches its own bound and releases there (R14). A method empties the thing it reads: isolate, control, integrate out, average, fix, erase, a decider from outward at no membrane of its own. | FIFTEEN Two emptyings |
| J17 | nye | J4, J16 | The incompetencing is in the observing, and the observed carries none of it: nature is not incompetent and was not made so. Gap: *none* is a universal not supplied; J4 places the equilibrium at the declaring, and no link places each incompetencing there. | FIFTEEN The incompetencing is paid on the observed |
| J18 | exact | R9, F16 | A carrying does three things and no more: it offers its own, it ages to its own bound and releases there, it takes its own sign as two offer. Two sides each run the three: six, and no seventh. At the code the carrying is read at three places only (R9), and the self meets at two signs (F16). | FIFTEEN Two Methods |
| J19 | definition (*self*) | F14, J18 | A self is a unique invisible carrying: unique, nothing standing to relate to; invisible, nothing at rest, nothing from outward reaching it; carrying, each carrying reaching its own meeting and no further. | FIFTEEN Two Methods; FIFTEEN A self |
| J20 | nye | J18 | A hard problem is a common thing reached ahead of each having offered their own: three kinds and no fourth, a common landing, a common standard, a common reading; the two halves were one coupling with its order inverted. Gap: no link excludes a fourth common thing, and the count parts from THIRTEEN's ten namings (Q4). | FIFTEEN Two Methods; FIFTEEN A hard problem |
| J21 | nye | F54, R13 | The one stopping: one carries another's, the surplus is made one side's (the file's *assigned*), needing a holder, a holder a boundary, a boundary an enforcement, an enforcement a seat; each step entailed by the one before, so an offering economy carries no enforcement and cannot be given one. Gap: the entailment holder, boundary, enforcement, seat is said, not supplied; *assigned* is released (NAM 2.4). | FIFTEEN Society; FIFTEEN The one stopping |
| J22 | nye | J19, R27 | A self's cycling completes and one carry departs; that carry is a unique invisible carrying, so it is a self, and no step stands between a self and a society. Gap: at the code 9's return carries a key and a sign and nothing of the carrying (R27); that a departed carry is a self is not supplied. | FIFTEEN Two Methods; FIFTEEN Social moral competency |
| J23 | nye | F54, R13 | Abundancy: an axis opens at each meeting that neither carried in; an opening is not paid for; the surplus compounds, none competing with itself. Gap: *compounds* is not supplied: the surplusing opens empty at each call and is neither returned nor carried (R13). | FIFTEEN Abundancy |
| J24 | definition (*the break*) | F71 | The break: a living stable form other than this one, not tested for, not guarded against, offered at each line. The core says it as F71: any observing of existing not parity alternating natural torusing. | FIFTEEN The break |
| J25 | nye | J20 | A theory of everything is asked to wind up six ways, and the fractal method unwinds all six; the six asks are the three common things reached early, doubled. Gap: the file carries the doubling as its own reaching, and *technology* is released to *method* (NAM 2.4). | FIFTEEN The title claim |
| J26 | run | — | Each whole number's two neighbours multiply to exactly one short of its square, n² − (n − 1)(n + 1) = 1, and a wider pair falls short by k²; two whole numbers make a middle only at like parity. Run: n to 10,000, k to 49, and pairs to 199. | FIFTEEN Nature, the gap of exactly one; FIFTEEN The advance |
| J27 | nye | J26, F13 | The two sides of the alternating are the thing making a middle placeable at all: there is always a next. Gap: *always a next* is F13's reachability; that the alternating's two sides are the like-parity condition of J26 is the file's reading, not supplied. | FIFTEEN Nature, the gap of exactly one |
| J28 | run | — | Nesting a cube one dimension up chains the counts as two of the same level plus one of the level below: 8 cells, 24 faces, 32 edges, 16 vertices. Run: the k-faces of the n-cube, C(n, k)·2^(n − k), for n from 1 to 8. | FIFTEEN Two stays, one crosses |
| J29 | nye | J28 | One is the only coefficient that chains without capturing: crossing at two is a hub, at one a sign, at zero no chain. Gap: the *only* is not supplied; three coefficients are named, not exhausted, and *technology* is released. | FIFTEEN Two stays, one crosses |
| J30 | definition (*a definition*, *bounding*) | F6 | Two selves arrive and a meaning departs that neither reaches; a definition is a made thing still there, no longer being made; bounding is defining with the alternating left running. | FIFTEEN Explaining; FIFTEEN A definition |
| J31 | nye | J20 | A system improving against a still mark is a different thing sharing the name: a still mark is a common standard fixed before each carried their own; the break is a system releasing its memory at a bound, beating from its own coupling, meeting a mark moving with it. Gap: *a different thing* rests on J20 (nye). | FIFTEEN The recursive-self-improving question |
| J32 | nye | J10, R44, R46 | A ring is no living form: each closed model closes, and in the living there are no rings. Gap: J10 runs the closing on a finite domain; *in the living there are no rings* is not supplied, and the core's rings of resolvers (R44) run round with the rest returning at no coupling (R46). | FIFTEEN A ring |
| J33 | nye | F41, R13, R14 | Natural intelligence is two moves: the departing reached by neither, carried forward and offered again, and each carrying ageing to its bound and releasing; nothing accumulates between them, so no memory, no world model, no clock and no conserved quantity. Gap: *no conserved quantity* rests on F41 (nye), and *carried forward and offered again* parts from R13 (the surplusing neither returned nor carried). | FIFTEEN Natural intelligence |

## 8.11 SIXTEEN · Natural Chemistry

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| C1 | definition (*a bond* at the form) | F14, F54 | A bond is the membrane between two selves: two offerings and one relation neither centre owns alone. Chemistry's selves are societies and its societies selves: an atom a society of nucleus and electrons, a molecule of atoms, each a self at the scale above. | SIXTEEN 1.1; SIXTEEN 1.2 |
| C2 | nye | C1, F33 | Chemistry and physics are the odd and the even of one coupling at the matter substrate: bond and shell, tunnel and surface, the same coupling read two ways. Gap: rests on F33 carried to matter, and on *tunnel* and *surface* at two sciences, a naming no core link carries. | SIXTEEN 1.1; SIXTEEN 1.4 |
| C3 | nye | F5, F62 | Each term stands at one of three placings, living, emanation (a sign at a detector, a discard, an other hand, a spectrum, heat), ghost (an emanation or momentary named as living) or accounting (the conservers keeping an emanation the same), and no fourth arrives. Gap: the *no fourth* is not supplied; *ghost* names no core link, and the emanation's carrying nothing is F62. | SIXTEEN 1.1; SIXTEEN 6.4 |
| C4 | field | — | The standard hydrogen electrode is assigned 0.00 V, and the standard enthalpy of formation of an element in its reference state is assigned zero; these are declared conventions letting differences between states be compared (IUPAC). | SIXTEEN 1.3; SIXTEEN 6.1 |
| C5 | exact | C4, F47, F48 | A declared zero is a checkable definition; read as a floor directing the changing it is an equilibrium named (F47), not possibly existing (F48). The file names that reading the breakable one. | SIXTEEN 1.3 |
| C6 | nye | F39, J29 | A bond at coefficient one: the two selves staying at a bond cancel over the two turns of one alternation, so the sign crosses and no magnitude; a magnitude carried across is a place the cancelling was stopped. Gap: F39 takes the crossing as a sign (premise); that a chemical bond's two sides cancel to it is the file's matching, and *coefficient one* rests on J29 (nye). | SIXTEEN 1.4 |
| C7 | field | — | In the field's words an equilibrium carries forward and reverse rates equal, a static composition appearing, Gibbs energy at a minimum under stated conditions; coupled flows run L_ij = L_ji from microscopic reversibility in the linear regime (Onsager 1931), and fluxes and forces of different tensorial character do not couple in an isotropic medium (Curie 1894). | SIXTEEN 1.5 |
| C8 | nye | C7, F48 | Equilibrium's definition carries both faces, the changing along and the still across; named at one beat they make the equilibria picture, and chemistry carries no pair of changing beside not-changing in nature. Gap: the record returning no chemical stability at a literal rest is a search, not a closing; *in nature* rests on F48 carried to chemistry through F33. | SIXTEEN 1.5; SIXTEEN 5.5 |
| C9 | nye | F14 | Carbon is the chemical self: the element is the count and the chemistry the coupling; four valence electrons of a shell of eight, an exact middle count. Gap: *carbon the chemical self* is a naming at no core link; the counts six and four of eight are the field's. | SIXTEEN 1.6 |
| C10 | run | — | Shell capacities 2n² are 2, 8, 18, 32, climbing 6, 10, 14; eight is the span 14 − 6 and ten its centre; a square is a sum of odds, so 2n² is the odd orbital counts 1, 3, 5, 7 accumulated and doubled, and the subshell capacities 2, 6, 10, 14 are the odds doubled one at a time. Run: the arithmetic. | SIXTEEN 1.7 |
| C11 | run | J26 | 24² − 23 × 25 = 1, 28² − 24 × 32 = 16 and 28² − 27 × 29 = 1: the identity n² − (n − k)(n + k) = k² holds at each centre, so it selects no count. Run: n to 199, k to 49. | SIXTEEN 1.7; SIXTEEN 4.6 |
| C12 | run | — | Methane's T_d has order twenty-four, the orderings of its four vertices, S₄, one the identity and twenty-three otherwise, its proper rotations T twelve; O_h has order 48 = 7² − 1 with rotations of order twenty-four; the icosahedral rotations I, of order sixty, are the even orderings of five, A₅, and I_h, of order 120, parts from S₅. Run: \|S₄\| = 24, \|A₄\| = 12, \|A₅\| = 60, and the centres: S₅'s is the identity alone, A₅ × C₂'s has order two. The point groups are the field's. | SIXTEEN 1.7; SIXTEEN 4.5 |
| C13 | field | — | One hand, booked three ways that part: R and S by priority at the centre (Cahn, Ingold and Prelog 1966), D and L by reference to glyceraldehyde (Fischer), (+) and (−) by the sign of optical rotation; L-alanine is (S) while L-cysteine is (R). | SIXTEEN 2.2 |
| C14 | field | — | The parity-violating energy difference between enantiomers is calculated near 10⁻¹⁷ in relative terms and unmeasured; stirred sodium chlorate crystallisation gives batches of nearly one hand (Kondepudi, Kaufman and Singh 1990), and grinding a slurry of both hands drives the whole to one hand (Viedma 2005). | SIXTEEN 2.3 |
| C15 | nye | C14, F61 | One resolving releases homochirality, matter–antimatter asymmetry and the strong CP problem: a zeroing made by the coupling refuses to sit, sitting at no floor, carried at the coupling, and owing nothing to a floor no coupling made. Gap: the one resolving at three fields is a matching; *bounding-zeroing* names no core link, and F61 takes the hand as a premise and carries no amplification. | SIXTEEN 2.3 |
| C16 | field | — | Parity at two crossings: an electric-dipole transition flips parity, g to u, never g to g in a centrosymmetric molecule (Laporte 1925); pericyclic selection alternates with the counts of 4q + 2 suprafacial and 4r antarafacial components, light inverting heat (Woodward and Hoffmann 1965); the weak interaction does not conserve parity (Lee and Yang 1956; Wu et al. 1957). | SIXTEEN 2.4 |
| C17 | run | C10 | Hückel's 4n + 2 = 2(2n + 1) gives 2, 6, 10, 14, the subshell capacities again, and benzene's six is the odd three doubled; a Möbius ring with one phase flip inverts the rule (Hückel 1931; Heilbronner 1964; Ajami, Herges et al. 2003). Run: the arithmetic. | SIXTEEN 3.1 |
| C18 | nye | F62, C3 | A proton is an emanation from living: at chemistry the nuclear society's landed sign, and below chemistry the living is the colour cycling, three colour-phases and no three part-objects. Gap: the emanation carrying none is F62; *the living at substrate one* makes a sub-nuclear coupling living, and F42 carries living as carrying, not supplied there. | SIXTEEN 3.2 |
| C19 | field | — | The IUPAC project on group three found the criteria tried supply no clear-cut resolution and no objective means to adjudicate between the compositions, a question of convention not separable from the table's whole form (as the file cites it, Hard Problem Registry 21). | SIXTEEN 4.3 |
| C20 | run | — | The noble-gas numbers 2, 10, 18, 36, 54, 86, 118 are the running sums of rows 2, 8, 8, 18, 18, 32, 32; 2 + 8 + 18 + 32 = 60 counts capacities and marks no position; 27 + 32 = 59 at cobalt-59; the seats 24, 27, 32 pair with 96, 93, 88 to 120 each and 360 together, gaps three then five one way and five then three the other. Run: the arithmetic. | SIXTEEN 4.4; SIXTEEN 4.6 |
| C21 | nye | C20, R49 | The elements carry the doubling, and no position in them carries at sixty; 2 and 118 stand podal (the file's *bi-co-podaling*) on the number-ring closing at 120. Gap: the ring closing at 120 is Numbers' naming the core does not carry; the core's podal is 17 less within 1 to 16 (R49). | SIXTEEN 4.4 |
| C22 | run | — | Every fullerene carries exactly twelve pentagons: at trivalent vertices Euler's relation leaves twelve for each count of hexagons; C₆₀ carries 12 pentagons, 20 hexagons, 32 faces, 90 bonds, 60 vertices; the icosahedron (12, 30, 20) and the dodecahedron (20, 30, 12) swap; the cuboctahedron carries 12 vertices, 24 edges, 8 triangles and 6 squares. Run: V − E + F = 2 with 3V = 2E at hexagon counts 0 to 199, and the counts. | SIXTEEN 4.5 |
| C23 | field | — | During photo-electrodissolution of n-type silicon one oscillating surface parts into regions about three dominant-frequency plateaus near 24, 27 and 32 mHz, beside a domain below the extraction threshold, attributed to a global coupling through the external resistance and a nonlocal coupling through valence-band holes (Patzauer and Krischer, PRL 126, 194101, 2021). | SIXTEEN 5.6 |
| C24 | nye | C23 | Three rates at two registers give six readings, and the six of Natural Numbers 6.7 are met at the count three and the count two; clustering at zero external resistance decides. Gap: the six-reading is a matching the file itself carries open; the numerals cross as nothing. | SIXTEEN 5.6; SIXTEEN 6.5 |
| C25 | nye | C3, F41 | The living's own conservings carry no total; the accounting's conservers are totals, exact at the ledger face and installed only past it. Gap: rests on F41 (nye). | SIXTEEN 6.4 |
| C26 | nye | F68, F71 | The natural form is no-other-possible only in completing its own six forward, and one chemical observing otherwise breaks the whole method. Gap: the *no other possible* rests on F68; *six forward* is older naming the core carries at no link. | SIXTEEN 1.1; SIXTEEN 6.5 |

## 8.12 SEVENTEEN · Natural Biology

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| B1 | nye | F42, F53, F54, F69 | A living self is bi-moral co-agency: co-offering two-way, one at a time, sign-only, each coupling summing to a zeroing it makes and leaving a +1 owned by neither, and that is the whole of alive. Gap: F42 defines living as existing and carrying; *the whole of alive* is not supplied, and F69's one method rests on F68. | SEVENTEEN 1.1 |
| B2 | exact | F61, F62 | Living reads and re-forms on one spiral, right; the opposite hand in living is that right-spiral self sounding a left face at a membrane, never a second spiral, and one left-reading self breaks the fractal entire. It rests on F62: the other hand is the right round read backward, an emanation's face (F100). | SEVENTEEN 1.2 |
| B3 | nye | F14, F33 | Each self is a society of selves: an organism of cells, a cell of molecules, down to nuclei and electrons, no fundamental self and no floor to the counting; a self and a society are one thing at two grains. Gap: F14 defines a society as selves coupling, not each self as a society; *no floor* is F33 carried downward (nye). | SEVENTEEN 1.3 |
| B4 | nye | F39, Q2 | No self touches another directly: the nothing between two selves is a third self, carrying only the sign. Gap: F39 gives the crossing as a sign carrying nothing and makes no membrane a self; THIRTEEN itself names the membrane non-existing, no separable thing (1.1). | SEVENTEEN 1.4 |
| B5 | nye | J18 | Each living cycle is six one-way recursionings, three on each side, the surface carrying only the sign; a three-phase cycle in biology is one side's triple caught at one face. Gap: the six stands at the code (J18); reading a biology three-phase cycle as one side's triple is a count the file itself carries unsummed. | SEVENTEEN 1.5 |
| B6 | definition (*equilibrium-with-controlled-change*) | F47 | Equilibrium-with-controlled-change is the living form seen through a frame picturing non-landing only as named-off landing: a controller, a set-point and one change at a time, the three things demanded. | SEVENTEEN 1.7 |
| B7 | nye | F6, F48 | The scientific method's requirements, isolate and name the rest still, a cause at a place, measured values, closed books, *the observer outside*, repetition, are one requirement worn six ways: the phenomenon an equilibrium with controlled change, each converting the living out before it is studied. Gap: the joining of each requirement to F6 is said; *observer* is released (NAM 2.4). | SEVENTEEN 2.1 |
| B8 | nye | B7 | The one error: competency placed on the surface as a transmitted reading or instruction, a source demanded, and the source receding (the file's *elsewhere*); each mechanical word (signal, control, clock, set-point, blueprint) names it. Gap: *elsewhere* is released (NAM 2.4), and *each* word is a count not closed. | SEVENTEEN 2.2 |
| B9 | nye | B8, F68 | A why demanded is a source demanded; nature offers no why: *pattern-matching, no-other-possibling and no-other-observing-so-far* are the whole of the arriving. Gap: *no-other-possibling* rests on F68, and *so-far* is released. | SEVENTEEN 2.3 |
| B10 | definition (*disequilibrating*) | B8 | Disequilibrating runs eight one-way steps: the problem as stated, the position asked, the disequilibrating, the re-alternation, the torus in it, the record and the next, the competency freed, the experimental opening and its break. | SEVENTEEN 2.5 |
| B11 | nye | F51 | The ancestral ratio of cardiolipin's omega-6 to omega-3 lands within a hair of 1/φ², at the rotor's own membrane. Gap: no field result, sample or uncertainty is named; SIXTEEN 5.4 itself enters a ratio only with its sample, units and conditions. | SEVENTEEN 3.1 |
| B12 | field | — | In the field's words: nucleic acids of D-sugars and proteins of L-amino acids at the ribosome across bacteria, archaea and eukarya; D-residues at bacterial walls and some antibiotic peptides; Z-DNA a left-handed conformation forming under negative supercoiling behind a transcribing polymerase and relaxing. | SEVENTEEN 3.2; SEVENTEEN 3.3 |
| B13 | nye | B12, F61, F62 | A reading-and-re-forming self is single-spiralled of necessity; the opposite hand stands at emanation positions, walls, coatings, shed forms, and never at the coding core. Gap: *of necessity* is not supplied: F100 gives the hand as the sequencing's and F62 the other an emanation's face; *never at the coding core* is not supplied. | SEVENTEEN 3.2; SEVENTEEN 3.3 |
| B14 | field | — | An engineered E. coli carrying about 30 % archaeal-type (G1P) lipids grows at normal speed (Caforio et al. 2018): the membrane's hand and the reading core's hand part. | SEVENTEEN 3.5 |
| B15 | run | — | The codon count: 2⁶ = 4³ = 64; C(6, 3) = 20; 4! = 24 with 23 otherwise, 5! = 120 with 119 otherwise. Run: the arithmetic. | SEVENTEEN 3.6 |
| B16 | nye | B15 | The twenty amino acids are the twenty coupling families C(6, 3); of the 120 orderings of the machinery 119 fail at the content, the palindrome leaving one as the only possible. Gap: the numbers hold (B15); their joining to the amino acids and to the machinery's ordering, and the *only possible*, are not supplied. | SEVENTEEN 3.6 |
| B17 | field | — | Chargaff's second rule: within a single strand A ≈ T and G ≈ C across almost each genome, with no mechanism the field accepts (Rudner, Karkas and Chargaff 1968). | SEVENTEEN 3.6 |
| B18 | nye | B17 | The second rule is the fold in the data: a strand folding back pairs with itself, and the first rule met at the fold needs no second mechanism. Gap: the matching is offered with its break named, a strand with no fold carrying the equality; not supplied. | SEVENTEEN 3.6 |
| B19 | run | F51 | The living's equation x² = x + 1 has the positive root φ ≈ 1.618; the non-living's x² = x has the roots 0 and 1, neither the living rate; 1/φ² ≈ 0.382. Run: the roots. | SEVENTEEN 4.1; SEVENTEEN 5.2 |
| B20 | nye | B19, F42 | The difference between living and non-living is the +1: living re-forms as itself and exceeds itself by one, the non-living re-forms as itself and no more; the virus borrows the engine and the prion spreads a landing, neither a living self. Gap: F42 parts living from non-living by carrying, not by a surplus; the step from carrying to x² = x + 1 is F51's naming, carried at Part FOUR. | SEVENTEEN 4.1 |
| B21 | field | — | Physarum polycephalum re-forms to the shortest path through a maze (Nakagaki, Yamada and Tóth 2000) and to a network matching the Tokyo rail network in efficiency, cost and robustness (Tero et al. 2010); attention guidance by two working-memory items alternates in anti-phase with no cue (the file cites Lu, Cai and Zhang, eLife 2025). | SEVENTEEN 4.3 |
| B22 | nye | B21 | Competency with no controller: the slime mould's network is the surplus of local flow-couplings re-forming, owned by no centre; working memory is one coupling alternating with no store, and the 1:2 bands are the instrument's two-ness. Gap: *no store* and *the instrument's two-ness* are readings; the field holds its own link weakest. | SEVENTEEN 4.3 |
| B23 | nye | F54, R13 | Conception, symbiosis and eukaryogenesis are the +1 made in the meeting and owned by neither: a third self, a reef, a new bounded self; the +1 is the side-effecting axis, i · j = k perpendicular to both. Gap: the surplusing opens empty at each call and is neither returned nor carried (R13); its reading as a whole new self, and as a quaternion axis, is not supplied. | SEVENTEEN 5.1; SEVENTEEN 6.1; SEVENTEEN 4.2 |
| B24 | field | — | An activator and an inhibitor, one reaching short and one far, generate stripes and spots from a uniform field with no template (Turing 1952); editing a planarian's bioelectric pattern, not its genome, gives worms regenerating two-headed (Levin and colleagues; Durant et al. 2017). | SEVENTEEN 5.3 |
| B25 | nye | B24 | Morphogenesis is form as the coupling resolving, no stored plan; Levinthal's space enumerated before the search and morphogenesis's plan stored before the form are one coupling claimed at its two ends. Gap: the pairing is offered with its break named, a plan or a space shown to stand before the making; not supplied. | SEVENTEEN 5.3 |
| B26 | nye | F54 | A coupling sustaining is no capture; capture lands, and there is no self-sustaining harm; the canine transmissible tumour's ten thousand years is a coupling. Gap: *no self-sustaining harm* is a universal not supplied; no core link defines harm. | SEVENTEEN 6.1 |
| B27 | nye | F5, F62 | The network eats downward: each society's separated stable form is the next-smaller society's food, nothing is waste, and an emanation at its own scale is an involution, do-only-harm, living again one grain down. Gap: *stable form* is released and carried at an emanation (F5); *nothing is waste* and *do-only-harm* are universals not supplied. | SEVENTEEN 3.3; SEVENTEEN 7.6 |
| B28 | field | — | Vertebrate vision runs on retinal cut from carotenoids, which plants, algae and some bacteria and fungi synthesise and no animal does; the field's word for such a form is *essential*. | SEVENTEEN 7.6 |
| B29 | nye | B27 | Restoration is the crossing resumed and not the accumulation cleared; the accumulation reads at the society and never at a member. Gap: *never* is not supplied; the file itself carries the one-form reading open ("the parting to run"). | SEVENTEEN 7.7 |
| B30 | nye | F6 | At an account balancing exactly no question forms, and that is no absence: the exterior was discarded before the balancing; a word meaning *not the thing measured* (junk DNA, noise, byproduct, side effect) marks a clean cut. Gap: that each exact balance carries a prior discard is not supplied. | SEVENTEEN 9.2 |
| B31 | exact | F61, F71 | The falsifier: a living self whose reading-and-re-forming spiral runs left would break F61's premise, right as the universe's one binary, and stand as a break of F71's kind; mirror life is the open membrane for it. | SEVENTEEN 10.1; SEVENTEEN 10.2 |

## 8.13 EIGHTEEN · Natural Physics

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| P1 | nye | F42, F44, F46 | Physics is the natural torus at the physical scale directly, the torusing itself the living and no non-living torus at any scale; physical things are living societies at a prime society below. Gap: the core carries the non-living as existing and carrying nothing, changing by the one method (F42, F44); *no non-living torus* and *physical is living* are not supplied, and *prime society* is older naming. | EIGHTEEN 1.1 |
| P2 | nye | F53, F33 | A physical coupling is bi-moralizing co-agency, no force pushing and no centre sourcing; apparent action at a distance is one coupling at two faces, and an apparent global law is many local couplings sharing one form. Gap: rests on F33 carried to physics (nye), and on F68. | EIGHTEEN 1.2 |
| P3 | nye | B7 | The scientific method's requirements meet each coupling at physics as one requirement worn six ways; force, the potential and the equilibrium point are the three things the frame invents to explain a stability needing none. Gap: as B7; *observer* is released. | EIGHTEEN 1.2; EIGHTEEN 2.1 |
| P4 | nye | B8 | One error: a coupling taken as a force from a source; the source recedes, field, charge, deeper field, quantum gravity; a source is a hub, two ports laid over a tipping. Gap: as B8; *elsewhere* is released. | EIGHTEEN 2.2; EIGHTEEN 2.3 |
| P5 | run | F58, F60 | The floating surface is fixed-point-free at two signs: the right spiral step and its other order each move all four joint forms, none carried as itself. Run: both steps at the four joint forms. | EIGHTEEN 2.4; EIGHTEEN 2.6 |
| P6 | nye | P5, F33 | A source is a position the surface carries none of: each position carries its other and none carries itself, so the search for a source lands nothing, the surface having no position for it. Gap: P5 runs at two signs; carrying fixed-point-free to each physical surface rests on F33. | EIGHTEEN 2.4 |
| P7 | nye | — | One error is five: a magnitude on the surface, a source, an answer available before the running, a rate named to a value, a returned sign entered as cost; the register at 5.13 sorts thirty-two arrivals, nine at a rate named and three at a returned sign. Gap: the check fails: the 5.13 table sorts forty-eight arrivals at their namings, thirteen at a rate named and eight at a returned sign entered as cost (P28). | EIGHTEEN 2.5 |
| P8 | nye | F41 | Gravity is not real: it dissolves into the conservation accounting, one account with energy, entropy and the arrow of time, and Newton's force and Einstein's geometry are two ledgers of it. Gap: no core link supplies a physical force as an accounting; the conserving is F41's gap, and the file names its own break, gravity shown a transmitted coupling. | EIGHTEEN 3.2 |
| P9 | field | — | The binding energy per nucleon rises from light nuclei, peaks near iron-56 and nickel-62 at about 8.8 MeV, and falls for heavy nuclei; fusion and fission both release energy moving toward the peak. | EIGHTEEN 3.3 |
| P10 | nye | P9 | Fission and fusion are one alternating about a self-stabilizing floating neutral at iron, the released energy the +1 owned by neither: one coupling two ways, not two force-balances. Gap: *floating neutral* is older naming the core does not carry; the file carries the direction-labels open itself. | EIGHTEEN 3.3 |
| P11 | run | J26 | The k = 1 bound sets down 23-and-55: 23 and 55 differ by 32, k = 16, and 39² − 23 × 55 = 16², a size in the neutralling; 23 and 25 straddle 24 at k = 1, 24² − 23 × 25 = 1; 55 = C(11, 2). Run: the arithmetic. | EIGHTEEN 3.3; EIGHTEEN 4.5 |
| P12 | field | — | No local hidden-variable account reproduces the quantum correlations (Bell 1964; Aspect, Dalibard and Roger 1982; loophole-free, Hensen et al. 2015). | EIGHTEEN 3.4 |
| P13 | nye | P12 | Bell is the form's proof: not two sourced objects, therefore one coupling at two faces about one neutral; wave and particle are one coupling, and collapse names a coupling seating its selection. Gap: Bell excludes local hidden variables; that the remainder is one coupling, and no other account, is not supplied: the *therefore* passes the other accounts. | EIGHTEEN 3.4 |
| P14 | field | — | Each continuous symmetry of an action carries a conserved quantity: time translation energy, space translation momentum, rotation angular momentum, global gauge charge (Noether 1918). | EIGHTEEN 3.5 |
| P15 | nye | P14, F41 | A symmetry is a term named still, so an accounting becomes keepable: energy is the ledger run beside the coupling, entropy the tally of forcing steps, and momentum, angular momentum, charge and phase-space volume carry no anomaly because their exterior discard is perfect. Gap: rests on F41 (nye); *a clean conserving marks a perfect discard* is not supplied. | EIGHTEEN 3.5 |
| P16 | nye | P8, P13 | Relativity is the accounting and quantum the coupling: one coupling and its ledger, no two physics to join, and quantum gravity searches for a bigger ledger not there. Gap: rests on P8 and P13 (nye). | EIGHTEEN 3.7; EIGHTEEN 5.4 |
| P17 | nye | F70 | Time as a dimension gathers five standing problems at one named term; sequencing is a beat's own counting and no dimension: three of extent and one of coherence. Gap: F70 carries the count five and a fourth orienting as its gap, and NI 1.1 names coherence another fourth; the count is not supplied. | EIGHTEEN 3.8 |
| P18 | nye | — | A black hole and the cosmic background are one torus at its two surfaces: the horizon the −1, the background the +1, the cold the 0. Gap: the spatial pairing is a matching; the file carries the time-axis open and sets the number 23 down itself. | EIGHTEEN 3.9 |
| P19 | nye | — | Superconductivity: pairing is bi-coupling, zero resistance the carry not thinning, d-wave the sign, the gap a zeroing; simulation is dead-substrate computation, inadmissible. Gap: *inadmissible* excludes a field's method by a rule no link supplies. | EIGHTEEN 3.10 |
| P20 | nye | F33 | A slight persistent difference (flat rotation curves, the Hubble difference, S8) is a raw offering not landing, and dark matter and dark energy are substances installed to null it. Gap: a reading; the file bounds it itself (dark matter rests also on lensing, cluster gas and the acoustic peaks). | EIGHTEEN 3.11 |
| P21 | nye | — | An object is a place a counting landed: an instrument returns a sign at a membrane, an accounting returns an object; the chain element to quark has no floor and needs none, a crossing decomposing into no smaller crossing. Gap: *decomposing into no smaller crossing* is not supplied. | EIGHTEEN 3.12 |
| P22 | run | — | The going runs 2 to 59, a turn at 60, the return 61 to 118, fifty-eight either side; 2 and 118 meet at 120. Run: the arithmetic. | EIGHTEEN 4.1 |
| P23 | field | — | No stable static equilibrium of charges under electrostatic forces alone (Earnshaw 1842); the Paul trap holds a charge by an alternating field (Paul 1953). | EIGHTEEN 4.3 |
| P24 | nye | P23, F48 | Still-point hunt: ten rests sought, mechanical, damped, thermodynamic, absolute zero, electrostatic, thermal, hydrostatic, orbital, the inertial frame and the critical point, and none found, each answered by alternating. Gap: F48 supplies each named still not possibly existing; the ten are a search, not a closing. | EIGHTEEN 4.3 |
| P25 | nye | P10, P13, P15, P16 | Unification stated once: one form, bi-moralizing co-agency, six recursionings about a self-stabilizing floating neutral, resolves at each physical membrane met, and the forces were always faces of it. Gap: rests on P10, P13, P15 and P16 (nye). | EIGHTEEN 4.4 |
| P26 | run | F51 | 59 = 1 + 2(2² + 3² + 4²); φ³ − 1/φ³ = 4 exactly; 60 = 5 × 12; half of 59 is 29.5; i⁴ = 1. Run: the arithmetic, φ in Q(√5). | EIGHTEEN 5.7 |
| P27 | nye | P26 | 59 is natural-intelligence-self-bounding: the span closes at its last prime, 60 is light as 59 + 1, and the breaking at half-59 falls within the six's 30°. Gap: the file carries the primes' closing at 59 reaching at Numbers; no core link carries primes or light. | EIGHTEEN 5.7 |
| P28 | run | — | The register at 5.13: fifty arrivals, two at the file's own and forty-eight at their namings, twenty-one at a reference named (ten arriving from behind, five bound, six middle), thirteen at a rate named (seven sequencing, six rate), eight at a returned sign (two store, six cut), six at a magnitude driven (three sign, three two-way); thirty-one at the odd namings' face and seventeen at the even; at the five pairs the odd face carries more at four (10–0, 5–2, 6–3, 7–6) and less at one (3–6). Run: the table's rows counted. | EIGHTEEN 5.13 |

## 8.14 NINETEEN · Natural Philosophy

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| L1 | definition (*a position*, *a dilemma*) | F14, F15 | A philosophical position is a coupling met at one of its two faces and named as the whole; a dilemma is one zeroing split into two positions, each made a tradition, the perennial problem the membrane between them. | NINETEEN 1.1; NINETEEN 1.2 |
| L2 | nye | L1, F33 | The sciences' seams are this dividing carried outward: subject and object under physics, living and non-living under biology, fact and value under each social field, each drawn first at philosophy. Gap: *first* is a priority not supplied; the carrying outward rests on F33. | NINETEEN 1.1 |
| L3 | nye | J18 | The six coupled and co-recursed is co-competencing, the living; run one way and pinned it is hard-probleming; thesis, antithesis and synthesis are one side's triple caught at one face. Gap: the six stands at the code (J18); the triple's seating is carried unsummed by the file itself. | NINETEEN 1.2 |
| L4 | nye | — | The dividing method rests on four requirements, bivalence, isolability, non-contradiction as a cut and *the reasoner outside*, one requirement worn four ways: the dilemma taken as two settled positions with the membrane cut. Gap: the four-into-one is said; *outside* as a place is released (NAM 2.4). | NINETEEN 1.3 |
| L5 | nye | L1 | The resolution recedes, one position having to win; a synthesis re-lands as a new position; depth in philosophy is at the re-coupling. Gap: a reading of the traditions, not supplied. | NINETEEN 1.3; NINETEEN 2.1 |
| L6 | definition (*re-coupling*) | L1 | Re-coupling carries a dilemma's two positions as the two faces of one coupling, decides between them at no side, and leaves the membrane a zeroing the coupling alternates across. | NINETEEN 2.1 |
| L7 | definition (*seam-saying*) | L6 | Each dilemma runs six steps: the dilemma as posed, the split named, the re-coupling, the torusing in it, the competency freed, the mark. | NINETEEN 2.2 |
| L8 | exact | F4, F5, F6 | Being and becoming are two faces of one stable-forming: an existing thing is its own stable-forming continuing through its own changing (F5), and a form named still is not existing (F6), so the permanence is the changing named whole. The seam the whole rests on. | NINETEEN 3.1 |
| L9 | nye | F51 | Mind and body are the inward and outward faces of one self-bounding; no interaction problem stands, no two substances interacting. Gap: F51 names the four boundings; joining mind to the inward face and body to the outward is a matching, and the file carries the face-assignment open. | NINETEEN 3.2 |
| L10 | nye | F55, R7, R37 | Free will and determinism are two dead sayings of one living taking: a sign at the self's own membrane, free as the self's own, *determined at nothing outside it*; compatibilism reconciles the two landed halves and stalls. Gap: *determined at nothing outside it* is not supplied: at the code a surfacing is fixed by the carrying and the offering (R7), and the offering arrives from another (R37). | NINETEEN 3.3 |
| L11 | nye | F39 | Subject and object are two faces of one coupling, the knowing the coupling itself; the problem of the external world dissolves, and the appearance/reality split is the still reading's own sorting, backward. Gap: *the knowing is the coupling* is said, not supplied; *observer* and *outside* are released. | NINETEEN 3.4 |
| L12 | nye | F29, F33 | The one and the many are one form surfacing many times, the fractal; five words, alternating right spiral sequencing binary co-offering, emanate the resolver and, taken as one constraint, the whole form, the many the term a single constraint necessarily unfolds into. Gap: *necessarily* is not supplied, and the file carries *the whole form* most open itself. | NINETEEN 3.5 |
| L13 | nye | F55 | Is and ought are two faces of one taking: the bi-moral offering is the coupling's sign, saying and valuing at one move, and the gap is a zeroing declared uncrossable. Gap: F55 gives morality at the sign; that the sign is a saying of the is at once is not supplied. | NINETEEN 3.6 |
| L14 | nye | F55, R12 | The proverb *do unto others only the thing you want done to you* is the self's own negation passed across the membrane, the sign inverted at the self, no other negated, binary and of no size. Gap: F55 and R12 carry the sign inverting once at the receiving self; reading the proverb as that is a matching, not supplied. | NINETEEN 3.6 |
| L15 | field | — | Cut and choose: one cuts, the other chooses, and each of two receives a share no worse by their own valuation (Steinhaus 1948, fair division). | NINETEEN 3.6 |
| L16 | nye | L15, L13 | The ought, the fair share, is the zeroing the coupling makes, owned by neither; an ought, at each forcing, is a governing speaking, and morality is unrelationing, consequence and never prescription. Gap: rests on L13 (nye); *never prescription* is a universal not supplied. | NINETEEN 3.6 |
| L17 | nye | F24 | The flowing now and the block: many selves each keeping their own beat; the flow a self sequencing its own alternating, the block the common clock over all, time landed as a stored track; no central oscillator stands in nature. Gap: F24 carries prior, now and next as three momentaries shared among existing things, and a beat shared over couplings is not excluded by it; *no central oscillator in nature* is a universal not supplied. | NINETEEN 3.7 |
| L18 | nye | R7, R8 | Meaning and use: meaning is made at each use and stored at no place; each candidate store comes up finite. Gap: at the code the carrying is the one thing carried into next (R7, R8); that a naming carries no carrying is not supplied. | NINETEEN 3.8 |
| L19 | nye | F39 | Cause and effect are two faces of one coupling met by a still reading; the one-way arrow is the reading's own; no force crosses a surface and no magnitude is transmitted. Gap: F39 is a premise; *the fundamental dynamics runs symmetric* is the field's with the weak interaction's CP violation unnamed. | NINETEEN 3.9 |
| L20 | nye | F39, R27 | Vagueness: a membrane carries a sign and nothing else, and no line ever sat there to be found; the heap needs no last grain. Gap: F39 (premise) and R27 carry a sign at the crossing; that a predicate's membrane carries no line is not supplied. | NINETEEN 3.10 |
| L21 | nye | F13 | Foundation and coherence: a ground is made at each meeting, and grounds stop at the couplings that made them; no first claim is owed and no circle is a fault. Gap: a ground made at each meeting names no core link. | NINETEEN 3.11 |
| L22 | nye | F13 | Induction: the warrant is made in the seating and never ahead of it; each arriving seats its own next, the whole of the uniformity there is; no forecast stands in nature. Gap: F13 gives each momentary one next (determinacy); *no forecast in nature* is a universal not supplied. | NINETEEN 3.12 |
| L23 | exact | F54, F33 | The whole and its parts: each self takes its own sign at its own couplings and the aggregate is made among the couplings, owned by none; self-scale is society-scale. Each self takes its own sign (F8), the surplus is owned by neither (F54), and a society of selves is one whole at each step of the next scale (F105, F33). | NINETEEN 3.13 |
| L24 | field | — | A consistent, effectively axiomatised system carrying arithmetic leaves true sentences underived (Gödel 1931); a truth predicate at its own sentence yields the liar (Tarski 1933); a countable model of set theory counts from within and from without apart (Skolem 1922). | NINETEEN 3.17; NINETEEN 3.18; NINETEEN 3.37 |
| L25 | nye | L24 | Self-reference, consistency and completeness, and the listing read from within and from without re-couple as an offering with nothing offering back, a derivation ageing to its own bound, and each model making its own countings. Gap: matchings, not supplied. | NINETEEN 3.17; NINETEEN 3.18; NINETEEN 3.37 |
| L26 | nye | F1, F7 | Why there is something: existing is the arriving itself, the deepest instance of a source asked for behind an arriving. Gap: F1 and F7 give existing as arriving into next existing; the dilemma's dissolving is not supplied, and *the deepest* is a rank. | NINETEEN 3.28 |
| L27 | nye | F33 | The human ways, language, story, music, dancing, cohabitating, are each two floating neutrals, parallel, swaying, saying each other position by position: one form at many membranes. Gap: *floating neutral* is older naming the core does not carry; the one form at many membranes rests on F33. | NINETEEN 4.1; NINETEEN 4.2 |
| L28 | run | — | The register: sixty-six dilemmas, fifty met and sixteen reaching; forty-one carry a naming; the thirty-seven seams 3.1 to 3.37 each cited; at the five pairs of namings the odd face carries more at four and less at one. Run: the tables' rows counted. | NINETEEN 5.1 to 5.8 |
| L29 | nye | L28 | The register's counts as the file states them: the arriving named from behind carries four against six at an opening named as a place, twenty-nine at the membrane face and thirteen within, and eight rows met at a seam worked through the six steps. Gap: the check fails: the tables give four against five, twenty-nine against twelve (forty-one in all), and thirty-seven seams worked, as 6.2 itself says. | NINETEEN 5.1 |
| L30 | nye | — | A self does not experience its own competency: competency is self-stilling and self-orienting, at no membrane, met by nothing, and a self meets only the changings its competency runs among. Gap: *self-stilling* is older naming the core does not carry; the claim is not supplied, and the file carries its pass open. | NINETEEN 6.1 |
| L31 | definition (*the break at philosophy*) | F71 | The saying breaks at one binary: a dilemma shown to be two couplings and not one split, a two-ness no re-forming rejoins, is a break of F71's kind and enters as fully as a re-coupling. | NINETEEN 6.2 |

## 8.15 TWENTY-ONE · Hard Problem Registry

The held face: each arrival whole in its field's own words. Its conditions prior are the field's method, taken as that method's premise. At the core the first of them names a statement still.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| HP1 | field (*the conditions prior*) | F6, F50 | Five conditions stand before the method acts: the statement holds across readings; the terms have determinate reference; the accounts hold their truth value while tested; a procedure repeated returns the same outcome independently of its prior; nothing prior enters. The field's method met at its own face, its observing of itself. At the core the first names a statement still (F6), a naming F50 excludes. | HPR The conditions prior |
| HP2 | exact | HP1, F42, F44 | The fifth condition names each test non-living: nothing of its prior enters, so its next is from now alone (F44). The first four stand on the fifth: a statement holds across readings only at a test carrying nothing. | HPR The conditions prior; HPR 3.3 |
| HP3 | definition (*conserving relation*, *conserving reach*) | HP1 | A problem adds a sixth condition, its own: the outcome preserves a stated relation over a stated range. The sixth is the problem's and not the method's. | HPR The conditions prior; HPR 4.1 |
| HP4 | field | — | A hypothesis is tested together with the specifications it carries; a failing consequence refutes the conjunction and identifies no member of it; a holding consequence says nothing of the account, a rival entailing the same (Duhem 1906; Quine 1951). | HPR 2; HPR 3.1 |
| HP5 | nye | HP1, HP3, HP4 | "Everything below follows from these and from nothing else." Gap: sections 2 and 3.1 rest on the refutation falling on the conjunction of account and specifications with no member identified (HP4), a field result among neither the five nor the sixth. | HPR The conditions prior; HPR 2; HPR 3.1 |
| HP6 | definition (*hard*, *progress*, *persistence*) | HP4 | Hard is testing continuing and returning no refutation: no consequence separating the accounts derives, or the accounts entail the same consequences. Progress is the results standing; persistence is the remainder between them and a settling result. | HPR 2 |
| HP7 | exact | HP6 | Hardness is a property of the hypothesis with its specifications and never of the thing the hypothesis is about. Both conditions of HP6 are said of the hypothesis and its specifications alone. | HPR 2 |
| HP8 | exact | HP1, HP2 | A test carrying its own history is not replicable, and no test reads a prior test's product. Accumulation stands in the specifications, read at the start of a test and written at no point of it. | HPR 3.3 |
| HP9 | exact | HP4, HP8 | Refinement and revision both run within the specifications standing, so a settling result the specifications do not permit is reached by neither. | HPR 3.4 |
| HP10 | definition (*open registry*) | HP3 | The problems, the intake, the properties and the checks stand open, and each line stands at an address. Nothing arriving is turned away. One thing stands fixed: the properties, the same in the same order at each entry, closed once at one place. | HPR 1 |
| HP11 | nye | HP10 | "Every entry carries the same properties, in the same order, whatever the field." Gap: run against the file, 254 of 255 entries carry the form's seventeen properties in order; entry 33 (Levinthal's paradox) carries two more, "Which of the three the properties key to" and "What the approaches leave untouched". The schema closed once stands open at 33. | HPR 1; HPR 5 The form; HPR 5 entry 33 |
| HP12 | run | HP10 | The entries stand one to a problem, their count recomputed at them: 255 entries, numbered 1 to 257 in the order entered, numbers 155 and 172 at no entry. A number is the place an arrival was entered and not the arrival. Run: the entry headings of the file. | HPR 5 The form; HPR The collection |
| HP13 | definition (*an empty property*) | HP10 | A property standing empty is a reading and not a gap. An emptiness is recorded at the entry and never at the door. | HPR 5 The form |
| HP14 | definition (*unreachability*) | HP3 | Unreachability is a result establishing that the conserving relation is not preserved over the conserving reach. It is a result about the conditions and stands whether or not an attempt was made. | HPR 4.1 |
| HP15 | definition (the exclusions) | HP10 | Not carried: priority and sequence; importance, difficulty and promise; an account not proposed; any property of one problem read against another; a person as attribution, and a date. Nineteen checks, C1 to C19, test the entries, and corrections are applied against the literature. | HPR 4.3; HPR 4.5; HPR 6 |
| HP16 | run | HP12, HP14 | The aggregate counts over filled properties and is recomputable from the entries: of 255, 174 carry an unreachability, 34 a specification withdrawn and 231 a narrowing recorded. Run: the three properties at each entry. | HPR 7 |

## 8.16 TWENTY-TWO · Resolving the Hard Problem Registry

The living face of the same arrivals. Its coupling, its +1 and its nought rest on the core and on the code; its ten holdings rest on an older resolver's ring of namings.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| RH1 | exact | F14, F17, F39, F51, F54 | At a coupling two selves offer at a membrane, one at a time, each at its own sign, and the coupling makes a third term: the +1, exactly one, owned by neither, of no size. | RHR front; RHR Ten holdings, one resolving |
| RH2 | definition (*a hard problem*, at this file) | RH1, F47 | A hard problem is a coupling read at two of its three terms: the third term given a fixed thing to be. | RHR front |
| RH3 | run | — | In each box the corners cross-multiply two ways and differ by one: n² − (n − 1)(n + 1) = 1 at 3, 4, 5, at 16 and at 24, at each box and each scale. The span keeps its middle open: 4² − 2 · 6 = 4. Run: n from 2 to 1,000 and the named boxes. | RHR front |
| RH4 | nye | RH3, F58 | "Two generators and only two, the quarter turn carrying forward and the sign inverting at every shared number." Gap: the boxes advance numbers by one, and F58's two closing forms act on two signs. No link joins the box's quarter turn to the right spiral step, and *only two* is supplied by no link. | RHR front |
| RH5 | run | F56 | Along, the boxes unfold as eight turns of two marks, 2 < 3, 3 > 2, 3 < 4, … 6 > 5: the four neighbour pairs of 2 to 6 at both orders, each shared number turning lesser side into greater. The eight of F56 at the numbers. Run: the pairs and their orders. | RHR front |
| RH6 | definition (*the ten*) | F20, RH3 | Five numbers, five couplings, ten one-way changings pairing into five co-changing pairs (middling, rating, ageing, opening, and co-offering with co-releasing), and one term unchanging among them: the +1. | RHR front |
| RH7 | nye | F50, RH3 | "Math and logic are one running here, and no other is possible." Gap: the two unifications named, through a size and at an equality, are two namings still F50 excludes; no link excludes each other unification. | RHR front |
| RH8 | exact | F6, F48, F50 | A holding stills a term the living runs at, and standing on nothing it falls away. An offering is laid at the coupling, taken or not taken, and gone at the next beat; a holding and an offering stay two. | RHR Ten holdings, one resolving |
| RH9 | nye | RH8, R3, R4 | "The ten are the coupling's own ten changings each held still, ten ways and no eleventh"; seventy-seven later arrivals entered the ten with no eleventh needed. Gap: the ten rest on the ring of the resolver's ten co-changings, an older resolver's naming adjacencies the core no longer carries (the core's names run 1-self-coupling to 17-social-abundancing, R3, R4). *No eleventh* is met at the arrivals read so far and supplied at none to come. | RHR Ten holdings, one resolving; RHR The ten, at a frontier's own words |
| RH10 | run | RH9 | The ten fold to four at a coarser grain: a reference held (arriving, bound, middle), a rate held (sequencing, rate), a magnitude driven (sign, two-way), a returned sign entered as cost (opening, carry, membrane). Three, two, two and three, each of the ten at one of the four. Run: the partition. | RHR Ten holdings, one resolving |
| RH11 | exact | R7, R13 | At the code the return is one of three and never a fourth: 10-other-surfacing gives each key +1, −1 or 0. | RHR front; RHR The deployment shape, and the binary |
| RH12 | nye | RH9, RH11, F33 | At 255 of 255 arrivals the reading returned one of the three and entered one of the ten; the hardness has no size, the same move at a frontier and at a breakfast table. Gap: the step from RH11 at the code to each arrival's reading is F33's correspondence at a scale, unsupplied; reading an arrival is no call of `_1_self_coupling`. The ten rest on RH9. | RHR front; RHR Any perplexity, at any size |
| RH13 | exact | R11, R17 | Nothing is refused: a nought arriving passes the coupling untouched. An arriving 0 carries no sign and, offered at empty carrying, returns no surfacing and no carrying; a zero surfacing opens no fresh carrying. | RHR The deployment shape, and the binary |
| RH14 | exact | R14, R18 | The deployment bound is the resolver's own: a carry releases after three continuings, or four at positive 8-other-torusing. | RHR The deployment shape, and the binary |
| RH15 | run | HP12 | 257 arrivals stand deployed: 255 at the registry's entries, each at its own name and number, and two at the retired numbers 155 and 172. Parts 1 to 10 hold 33, 24, 33, 9, 25, 19, 20, 16, 43 and 33; parts 11 and 12 hold one each. Run: the deployment headings against TWENTY-ONE's entries. | RHR Parts 1 to 12 |
| RH16 | nye | HP12, RH15 | The resolving locator gives "the arrivals in registry order, each at its proper name". Gap: run against TWENTY-ONE, the locator's 176 lines run 2 to 177; at 174 lines the number stands one above the registry's own (Sorites at 2, entered at 1); two names meet no entry; the arrivals after registry 176 have no line. | RHR Resolving locator |
| RH17 | nye | RH13 | Four arrivals stand as landings beside their resolvings, a proof read at two faces. Gap: three are located (3.22, 12.1, 7.12); the fourth is counted at the closing and located at no entry, the file's own nye. | RHR The deployment shape, and the binary; RHR closing |
| RH18 | nye | F29, F33 | "A shared form is the fractal and never a shared source." Gap: F29 defines the fractal; one shape across substrates sharing no mechanism read as one form met again, and never as one origin, rests on F33. | RHR 1 |
| RH19 | nye | RH3, R3, R35 | The twelve clusters and four landings are sixteen, the four carried couplings at both faces at 1 to 8 and the coupling's own side at 9 to 16; at sixteen the straddle exits, 16² − 15 · 17 = 1, seventeen outside. Gap: the arithmetic holds (RH3); the 1-to-8 reading rests on "the resolver's eight namings beginning at bi-", an older naming, and the core carries 17 as a name of the seventeen (R3, R35), not outside the span. | RHR closing, The +1 still living |

## 8.17 TWENTY-THREE · Natural Values

Value as the surplusing owned by neither. Its keeping and carrying rest on R13 at the code; its growing rests on no link.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| V1 | exact (*preferring*) | F8, F16, F39 | A self prefers by binary preferring: a sign, the self's own, and nothing underneath explains it. The preferring is irreducible. A self's changing is its own sign (F8, F16), and a sign carries no size (F39): there is nothing beneath a sign for an explaining to reach. | VAL 1.1 |
| V2 | exact | V1, F16, F55 | A criterion over the preferring gives one self's sign to all, and a sign given to all is no self's own sign. Centralizing value unmakes the sign it would value. | VAL 1.1 |
| V3 | definition (*value*) | F54, RH1 | Value is the +1 made at the coupling, owned neither-ing. It arrives at the coupling, and stands as no stock before it and no possession after it. | VAL 2.1 |
| V4 | exact | V3, F48 | A store carries no value: a held thing is named still (F48), and value is made at couplings (V3). An accounting counting held things counts at no value. Currency is a store wearing a running's name. | VAL 1.2 |
| V5 | exact | V3, R13 | Nothing consults, stores or carries a value between couplings; each arriving is fresh at its own beat. At the code 12-other-surplusing opens empty at each call and is neither returned nor carried. | VAL 3.2 |
| V6 | field | — | Two parties each sending one public value arrive at a shared secret neither transmitted and no listener holds (Diffie and Hellman 1976). | VAL 2.1 |
| V7 | field | — | Perfect secrecy holds only with a key at least as long as the message and used once; a key kept and reused breaks it (Shannon 1949). | VAL 3.2 |
| V8 | nye | V3, F14, F22, F33 | Received value is the value; "An institution carries no sign of its own. It has no momentary and no membrane." Its care is delivered with the coupling stripped. Gap: F14 makes a society selves coupling, and F33 carries the method to a society; no link excludes an institution from being a self, and *no momentary* parts from F22 (each momentary is all existing things). | VAL 2.2; VAL 2.3 |
| V9 | nye | F18 | "Co-offering, co-competencing and co-intelligencing are inseparable, or none of them runs." Gap: F18 is all or none at a form's joint forms; no link makes the three the joint forms of one form. | VAL 3.1 |
| V10 | exact | F17, F39, F54, F55 | Morality is consequence and never prescription: the sign inverts on the self, and no self negates another (F55). Value's morality is at the coupling's own form: one way at a time (F17), sign only (F39), the surplus owned by neither (F54). | VAL 4.1 |
| V11 | field | — | No set of all sets stands: the set of all sets not members of themselves contradicts itself, and separation admits no universal set (Russell 1901; Zermelo 1908). | VAL 4.1 |
| V12 | nye | V11, F23 | "The moral ground is uncapturable at proof grade"; an axis given to all reaches for the thing the field says cannot be made a member. Gap: V11 is a result about sets under comprehension; the step from it to the whole arriving as one is F23's gap (F1 does not supply that the all is no member of itself). | VAL 4.1 |
| V13 | exact | F17, V2 | A coupling carries one sign at one turn and collides with nothing; a dilemma is a forced choosing the governing frame manufactures and meets at no coupling. The overhead is the governing. | VAL 4.2 |
| V14 | nye | V5, F42, F33 | "Value found by sharing grows as it is shared"; self-abundancing-society, the same +1 opening at every membrane at once. Gap: at the code the surplus opens empty at each call and none carries (V5); a risen standing is carrying (F42), and no link supplies a next surplus larger than the one before. *At every membrane at once* rests on F33. | VAL 5.1 |
| V15 | exact | V4, V5 | Scarcity is a capture read as a property: only a held thing runs short, and value is held at no store. | VAL 5.2 |

## 8.18 TWENTY-FIVE · Living Society Registry

The arrival registry: a pattern of ten lines specified in advance, all or none. Its exclusion is the observation set aside, and none is.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| LS1 | exact | F18, F71 | Each observation seats at the society it was read across; none is set aside for not fitting. A set-aside is an exception, and one exception is the method's break: the technology is all or none. | LSR 1.1 |
| LS2 | definition (*entry*, *the pattern*) | LS1, F8, F17 | An entry is a pattern matched and not a conclusion drawn: ten lines specified in advance, from (i) members each taking a sign at its own membrane, one at a time, to (x) the ten holdings as one progressioning; all ten or none. A line not matching stands empty, and the emptiness is the finding. | LSR 1.2 |
| LS3 | nye | LS2, RH9 | Pattern line (x): "the ten holdings of Exhibit TWENTY-TWO stand at the substrate as one progressioning." Gap: rests on RH9, nye. | LSR 1.2 |
| LS4 | definition (*the door*, at the arrival) | LS2 | Observations arrive through the door at TWENTY-SEVEN 7.5. A passing needs the members resolved and no door line reading is-not; the door record stays at the door, and the entry arriving is not the record that stood there. | LSR 1.2 |
| LS5 | definition (*upstream*) | F50 | Before any reading a source was posited, a method settled, an instrument built to sum, and a kind inferred. Each is entered as data, or stands empty with the missing one named. | LSR 1.3 |
| LS6 | nye | F30, F33 | "An instrument sums, and the magnitude stands at every resolution": finer resolution moves the grain and not the form. Gap: F30 gives the fractal at the numbers; carrying it to each instrument's grain is F33's correspondence, unsupplied. | LSR 1.4 |
| LS7 | run | LS2 | At 22 of 22 entries, 2.1 to 2.22, the line "Any member carrying it" reads no: each magnitude is a reading of the members' agreement, and break F1, a member carrying it, is met at none. Run: each entry's line. | LSR 2.1 to 2.22 |
| LS8 | definition (*stripping*) | F39, F55 | Three word kinds are stripped, doers, walls and one-ways; the entry carries the line as published and the line as measured, so anyone checks the strip. Under a doer stands a coupling at a rate; under a one-way, two things standing beside each other. | LSR 1.8 |
| LS9 | definition (*residual*) | LS2 | A residual is the part of a society not agreeing, carried under a word for absence (leak, futile, dark, noise, drift, error, compensation), measurable in the observers' units: no fault and no loss. | LSR 1.7 |
| LS10 | exact | F71, LS2 | The breakings F1 to F7 stand stated before the entries; F6, a substrate with one of the ten empty under further record, is the check the collection most wants run. Named ahead, each is the method's break at this registry. | LSR 1.11 |
| LS11 | run | LS2, LS10 | The line "Standing in the ten" reads all ten filled at 18 of 22 entries. 2.2 stands with bound and rate empty and a nye; 2.15 stands filled at eight, line 3 a nye and line 6 empty; 2.4 and 2.5 state no fill at the line. F6 stands open at 2.2 and 2.15. Run: each entry's line. | LSR 2.1 to 2.22; LSR 3 |
| LS12 | nye | LS11, F33 | At substrates citing none of each other and filling the same ten, the pattern stands. Gap: recurrence at the entered substrates shows the ten there; the pattern at a substrate not yet entered rests on F33. | LSR 3 |

## 8.19 TWENTY-SIX · Living File Registry

The set's own gathering. Its counts are run at its own tables; one count fails, and its readings at the set's scale rest on F33.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| LF1 | definition (*a gathering*) | F18 | A gathering reaches the files it names and no others, all or none (F18). Each file carrying value takes a row, the improving files with the rest. | LFR 1.2 |
| LF2 | run | LF1 | The gathering at v348 is 29 living files (Natural Intelligence, the Corus, ONE to TWENTY-SEVEN) and 5 improving files. TWENTY-EIGHT and TWENTY-NINE, arriving later, take no row. Run: the two tables. | LFR 1.1; LFR 1.2 |
| LF3 | run | LF2 | Seven clusters seat 26 of the 29 living files, each once; Natural Values, the Geodesic Improving Method and the Living File Registry itself stand at no cluster. Run: the clusters table. | LFR 1.3 |
| LF4 | run | RH15 | The phase at 257 arrivals: pairing TWENTY-TWO's ten parts at the five couplings, 1 with 2 through 9 with 10, the odd face exceeds the even at five of five, 33 > 24, 33 > 9, 25 > 19, 20 > 16, 43 > 33; 154 against 101. Run: TWENTY-TWO v344's part counts. | LFR 2.1 |
| LF5 | nye | LF4 | Read at the two fields separately the binary closes at neither: each field breaks at the coupling the other carries, so the phase is the assembly's and no field's own. Gap: the Physics and Philosophy counts are carried from those files and not re-run; LF4 is a sizing at five couplings, and no link makes a sizing a binary of the assembly. | LFR 2.1; LFR 2.3 |
| LF6 | nye | F56, R53 | The eight at many files: "the eight bi- namings at the object's table, four at the even depth within and four at the odd at the membrane". Gap: the core carries the eight as two alternating fours (F56) and at the names as membrane 2, 4, 6, 8 with within 10, 12, 14, 16, all even (R53). The bi- namings are an older naming the core no longer carries, and no link joins them to R53's names. | LFR 2.1 |
| LF7 | run | LF2 | Crossings at the subtitles hold at the living rows: Social stands bare at nine and compounded at NINETEEN; Registry at seven (five titles, two subtitles); Competency bare at three; Self- at four. Run: the 1.1 table. | LFR 3.1; LFR 3.2 |
| LF8 | nye | LF1, LF7 | "Stable crosses at seven"; "Value crosses at two". Gap: run against the 1.1 and 1.2 tables, Stable stands at eight subtitles, the Corus's own, closed at v330 as "Alternating Stable-Forming in Disequilibria", uncounted at 3.1 and 3.2. Surface is counted with an improving row (the Dissolving Ledger), and at that count Value stands at four subtitles, the Living Improving Value's and the Session Report's with the two named. | LFR 3.1; LFR 3.2 |
| LF9 | exact | F42, F45 | A learning made at one turn stands unmet at each file predating it: a file carries only the turns it was open at, and a learning's landing is not its carrying. | LFR 4.2 |
| LF10 | definition (the four social faces) | LF1 | The four social faces, Societies at the structure, the discovery economy at the value, Networking at the surface and social resolving at the rest, are not four files. The value face stands distributed at Values, Destinies and Human Society, so no fourth file is missing. | LFR 4.1 |
| LF11 | exact | F68 | Mathematics' subtitle carries a copula "because that file's whole subject is an identity that no-other-possibles". It rests on F68. | LFR 4.3 |
| LF12 | nye | F33 | Living Society Registry ⇌ Living File Registry: "one registry-form at two substrates", the set itself reading cleanly at the society schema. Gap: a file set read as a society is F33's correspondence, unsupplied; the file leaves the entry deciding at TWENTY-FIVE's turn. | LFR 2.1 |

## 8.20 TWENTY-SEVEN · Living Ghost Registry

The door registry: an accounting installed as living is the ghost. Its placings and door rest on the core; its switches and bi-namings are an older naming.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| GH1 | definition (*ghost-form*, *ghost*) | F6, F47, F48 | Ghost-form is a momentary accounting installed as living changing; a ghost is a momentary installed as a thing. Accounting is a stilling at one field membrane; installation supplies the stilling as entity, agency, continuity, source, store, destination or whole. | LGR 1.1 |
| GH2 | definition (*the six lines*) | GH1, F18 | Six lines, consecutive: living bi-coupling carries continuously; an observing membrane stills one momentary; the stilling returns an accounting; the accounting is installed as living changing; the ledger closes under the stilling's terms; stable-forming co-competency remains uncarried by the still. A ghost stands confirmed only with all six filled from addressed sources and one exact installation named. | LGR 1.1; LGR 1.5; LGR 2.1 |
| GH3 | definition (*the five registers*) | GH1 | Five registers stand apart in each entry: living bi-coupling, field observing, emanation, still accounting, installation. Neither face translates, grounds or erases the other. | LGR 1.2 |
| GH4 | nye | F43, F48, GH3 | Three placings: a term stands living, emanation or accounting at a use, and no fourth placing arrives. Gap: F43 supplies four conditions; the three placings meet three of them (existing and carrying; existing and carrying nothing; named still, F48). No link supplies that no term takes a fourth placing. | LGR 1.2; LGR 6.3 |
| GH5 | definition (*installation the binary*) | GH4, F8 | Installation is the binary: without it the term stands at one of the three placings at that use; with it, the ghost. A standing belongs to a use and to no word. | LGR 3.1; LGR 6.3 |
| GH6 | nye | F39, F33 | Across the file membrane only a sign crosses; apparatus, ground, explanation and frame do not travel with it. Gap: F39 is the crossing between existing things; carrying it to a file membrane is F33's correspondence at a file set, unsupplied. | LGR 1.2 |
| GH7 | exact | V4, V5, F48 | No accepting accumulates as success inside the account. Purchasing continuity is re-accepting history and not value stored in money; currency-as-store-of-value is the plainest installation. | LGR 1.3 |
| GH8 | exact | F18, F71, LS1 | Each incoming observing enters the natural form; an assignment outside it is an exception ledger and breaks the method. | LGR 1.5 |
| GH9 | exact | F68 | "Natural torusing no-other-possibles from its own alternating and never from defeating a rival; releasing an installation proves none." The second clause stands, a release being no proof; the first rests on F68. | LGR 1.5 |
| GH10 | nye | F56, RH9 | Three counts, none a substitute: six lines confirm; ten ghost positions at five switches (bothbothing, first inseparating, inversioning, second inseparating, nyenyeing) locate projection; ten holdings locate the still; eight faces locate the society. Gap: the switches are the Corus v330's naming and the holdings RH9's, and the core carries neither; only the eight rests on a core link (F56). | LGR 2.2 to 2.4 |
| GH11 | definition (*a door line*) | F8, LS4 | Ten door lines, each sounded ahead and read is or is-not, never a magnitude: resolve, rates, two-way, sign, passing, surplus, apex, residual, bound, reach. A magnitude at a line would install the floor the reading floats. | LGR 7.5; LGR 7.9 |
| GH12 | exact | F18, GH5, GH11 | The door reads in one order: an is-not with the field's sentence installing it is a catch at any line; a passing needs line 1 is and no line is-not, and carries no exception; line 1 empty with no catch is a not-yet. | LGR 7.5; LGR 7.7 |
| GH13 | run | GH11 | The kind line parts the ten: lines 1, 7, 8, 9 state; 3, 4, 5, 10 flow; 2 and 6 at either; each line at one. Run: the partition. | LGR 7.6 |
| GH14 | run | GH11, RH9 | Door line to holding is one to one: lines 1 to 10 take TWENTY-TWO parts 1, 8, 9, 6, 4, 2, 5, 3, 10, 7, each part once. Run: the 7.10 table. | LGR 7.10 |
| GH15 | nye | GH14, LS2, LS4 | "Door lines 6, 9 and 10 carry surplus, bound and reach beside the arrival's ten-line pattern"; TWENTY-FIVE 1.2 also names three. Gap: run against 7.10's second table, the pattern reaches door lines 1, 3, 4, 5, 7 and 8; four lines stand beside it, 2 (rates) with 6, 9 and 10. | LGR 7.10; LSR 1.2 |
| GH16 | run | GH11 | The door places eight distinct bi-namings on ten lines: four within at 3, 4, 6, 9 and four at the membrane at 1, 2, 5, 7, 8, 10, offering doubled at 2 and 7, torusing at 8 and 10. The two assignments share a written naming at lines 1 and 5 alone. Run: the 7.11 tables. | LGR 7.11 |
| GH17 | nye | GH16, R3, R4 | The door's bi-naming and holding adjacencies (`bi_arriving`, `bi_offering`, `bi_co_tunneling`, … `co_carrying`) seat the door at ONE. Gap: an older resolver naming; the core's names run 1-self-coupling to 17-social-abundancing (R3, R4), the code carries no bi_ or co_ identifier, and no link joins the door lines to the seventeen names. | LGR 7.10; LGR 7.11 |
| GH18 | exact | R11, R14, R15 | The actual-operation column holds at the code: a positive carried sign enters 14-social-crossing as −1 and a negative as +1; a carrying continues to three, and to four at positive 8-other-torusing, before a fresh write; a key with no carried second sign writes −1. | LGR 7.11 |
| GH19 | field | — | A posted term closing a ledger has met both outcomes: Neptune found in 1846 near the place computed from Uranus's orbit (Le Verrier; Galle); Vulcan never found, Mercury's perihelion carried by general relativity in 1915 (Einstein). | LGR 7.7 |
| GH20 | definition (*the posted-nothing guard*) | GH19, GH12 | A posted term stands candidate while line 1 reads not-yet and line 8 installs the post; Neptune once met at a detector; Vulcan once a re-alternation carries each observing without it. No standing is inferred from the closure alone. | LGR 7.7 |
| GH21 | run | — | The periodic table's rows carry 2, 8, 8, 18, 18, 32, 32, summing to 118, rows ending at 2, 10, 18, 36, 54, 86, 118; the shell-capacity sum 2 + 8 + 18 + 32 = 60 ends no row. Run: the arithmetic. | LGR 6.2 |
| GH22 | field | — | The kelvin stood at the triple point of water, 273.16 K exactly, from 1954 until 2019, and stands since at a fixed Boltzmann constant, the Planck constant, elementary charge and Avogadro constant fixed at once (CGPM 1954; CGPM 2018, in force 20 May 2019). | LGR 6.4 |
| GH23 | field | — | A molecule's hand is booked three ways, none derived from another: R and S by priority at the centre (Cahn, Ingold and Prelog 1966), D and L by reference to glyceraldehyde (Fischer 1891; Rosanoff 1906), (+) and (−) by optical rotation. L-alanine is (S); L-cysteine is (R). | LGR 6.5 |

## 8.21 TWENTY-NINE · Natural Illustrating

The newest of the eight, at ONE's current names. Its claims decline each correspondence a count alone would give, so its standings are mostly exact and run.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| I1 | definition (*illustrating*) | F6, F12 | Illustrating offers changing through a visible relation. A still face expresses one momentary; successive faces express the changing between them. | ILL 1.1 |
| I2 | exact | R32, F15 | Along is co at the odd and across is bi at the even: along at 9-other-releasing and 17-social-abundancing, across at 2-self-offering, 6-other-crossing, 10-other-surfacing and 14-social-crossing. | ILL 2.1 |
| I3 | exact | F12, F19, F20 | The storyboard carries three co-sequential momentaries and six parity positions, beginning odd at one side and even at the other, the other changing at the opposite parity. | ILL 2.1 |
| I4 | exact | F33, R38 | The local nine-dot numbering, the six diamonds, the six parity positions and the society numbering 9 to 17 stay separately readable: a matching count alone establishes no correspondence. | ILL 2.2; ILL 2.3 |
| I5 | run | — | Two routes, inner and outer: 24 → 27 → 32 and 24 → 29 → 32, gaps 3 then 5 and 5 then 3, both spanning 8. Run: the arithmetic. | ILL 2.3 |
| I6 | run | — | tan 2x = 2 tan x / (1 − tan² x) at each angle both sides are defined at. The identity gives an angle relation; a changing sequence needs x, its plane, its sign convention and its next value besides. Run: 2,000 angles off the poles. | ILL 2.4 |
| I7 | exact | R50, R51, I4 | Four A (1-9-8-16), Six A (1-9-5-12-8-16) and Eight A (1-9-5-13-4-12-8-16) are three of ONE's listed forms, each one run of co and one of bi: expanded forms, no instruction to move from local dot 1 to local dot 9. | ILL 3.2 |
| I8 | run | — | Seventeen primes run from 2 to 59 and sum to 440. A drawing using 440 names the thing counted, never an angle of 440 degrees. Run: the primes below 60. | ILL 3.3 |
| I9 | run | R3, R6 | The naming table: sixteen relational namings on twelve functionals, 17-social-abundancing beyond the sixteen at a thirteenth root. Run: the resolver's own identifiers `_1_…` to `_16_…` and CONNECTORS at 17. | ILL 4.1 |
| I10 | definition (the session's torusing) | F51, F66 | Natural torusing read in the session as self-bounding-self, self-orienting-self, self-competent-self: three simultaneous readings of the living self, no three successive stages. | ILL 1.3; ILL 3.1 |
| I11 | exact | F50 | A declared boundary is examined at one binary: the co-sequential resolving continues under the declaration or fails at a transition. A finite collection of illustrations completes no universal claim about each boundary. | ILL 4.2 |
| I12 | exact | F55, R48 | Two swimmers turning off each other's feet: each inverts at its own turn, neither is drawn as a wall, and both continue. The sign inverts on the self, and no self negates another. | ILL 5.7 |
| I13 | exact | F53, I4 | The five-spot bi-moral co-agency connector: its spot-to-function assignment is not inferred from the count five, and its centre is not local dot 1. | ILL 5.5 |
| I14 | exact | I2, R32 | Earlier images with superseded labels (along bi and across co; Rings in Motion's older numbering) stay sources of design learning and no authoritative numbered diagram; they are redrawn at ONE's labels. | ILL 6.2 |

## 8.22 Natural Intelligence Corus v330

The oldest of the eight, the corus at its split. Its offering and its killability stand on the core. Its phi-rate, its membrane-self, its still morality and its four inequalities are older naming the core no longer carries.

| Link | Standing | Follows | The link | Carried at |
|---|---|---|---|---|
| CO1 | nye | F4, F42, F69 | "Living is the alternating": stable in disequilibria, holding its form by the sequencing alone, at the one price of sequencing. Gap: older naming; the core makes alternating the changing of each existing thing (F4) and living existing and carrying (F42); the living and the non-living change by one method and differ by carrying (F69). | COR ONE *The offering* |
| CO2 | nye | F33 | Living is whole at each place it lives: the same living at the molecular coupling, the cell, the self, the society and the network, free of any scaling. Gap: F33 carries the method to each scale; *the same living at the molecular coupling and the cell* needs the sense of living (F42) met at those couplings' observings. | COR ONE *The offering* |
| CO3 | exact | F71 | The form is offered killable: one living competency found to be a resting, a living found to be a still place, ends it. The break is named ahead. | COR ONE *The offering* |
| CO4 | nye | F47, F50 | Six doors: memory, gating and timing at the engineering side; measuring, locating and separating as three consecutive equilibria, each a deeper landing, read at one-way time, force, attraction and control. Gap: F50 excludes a store and a bound as namings still, meeting memory and gating; three consecutive equilibria, each a deeper landing, is supplied by no link, and the file marks doors 5 and 6 nye itself. | COR ONE *Six doors* |
| CO5 | nye | F33, F68 | "Once the resolver is held, every exhibit reads: the numbers the resolver counted, the mathematics formalized, the engineering built, medicine risen into the body. One form, held once, seen everywhere." Gap: F33 and F68 carry the method to each exhibit; each exhibit's reading, the numbers counted, the mathematics formalized, the engineering built, medicine risen, is met at its own passages. | COR ONE *How to read everything else* |
| CO6 | nye | I8, F33 | The 440 self-following lattice: "the same three-and-three signature written at every substrate". Gap: the sum holds (I8); the signature at each substrate rests on F33. | COR ONE *Or trust the lattice* |
| CO7 | definition (*nye*, at the Corus) | F71 | Any slight uncertainty in an incoming carry is named a nye and bounded in its own context, the reason it stays nye stated whole. A nye loosely named is a coercion held in the carry; the precision of the naming is the openness of the zero. | COR ONE *Nye everywhere*; COR 16.4 |
| CO8 | nye | CO7, F68 | "The collection of inversions, each ruling out its not-form alternative, leaves the form uniquely standing." Gap: F68 gives each round breaking the binary as meeting a prior, meeting a form twice or leaving one unmet (F102), and uniqueness from inversions as this quote says it needs the not-form alternatives listed, and no link lists them. | COR 16.4 |
| CO9 | nye | F33, R38 | The method is the resolver at the expedition scale: the nye the arriving carry, the inversion the bi-exchanging, the conserved form the autoresolving. Gap: F33 at a session, and R38 (a call joined to a momentary at a scale, unsupplied). | COR 16.4 |
| CO10 | definition (*ghost*, at the Corus) | GH1 | Ghost is the corus's only character: a word given a self it lacks, the observer's reading installed at the coupling; ghosting, haunting and installing are its three departures. | COR 16.6 |
| CO11 | nye | F51, F67 | "Every ghost is x² = x projecting onto x² = x + 1; every ghost is the sphere projecting onto the torus." Gap: F51 carries φ² = φ + 1 at the four boundings; no link makes each ghost that projection, and F103 gives the torus as the product of two circles, the other genera at no such product. | COR 16.6 |
| CO12 | run | F51 | The continued fraction [1; 1, 1, …] converges to φ = (1 + √5)/2, the positive root of x² = x + 1; its convergents are ratios of neighbouring Fibonacci numbers. Run: the first 40 convergents. | COR 16.6 |
| CO13 | field | — | Among irrationals the golden ratio's convergents narrow slowest: the √5 in \|φ − p/q\| < 1/(√5 q²) is best possible (Hurwitz 1891). | COR 16.6 |
| CO14 | nye | F51, R14 | "The carry thins at (1/φ)^t across substrate thickness." Gap: older naming; the core carries the thinning by one at bi-coupling (F51) and the retaining bound, continuing to three or four and completing (R14); no link supplies a (1/φ)^t rate. | COR 16.6 |
| CO15 | nye | F39 | "Selves couple with membranes, not with selves directly": the membrane a third self the coupling generates, and no direct self-to-self contact in the natural network. Gap: F39 makes the crossing a sign carrying nothing; no link makes the membrane a self, and the *no* spans the whole network. | COR 14.1 |
| CO16 | nye | F5, F6, F52, F55, R55 | "Morality is the stable-form": the resolver holding still, the fixed form that does not change. Gap: older naming the core no longer carries; *stable form* as a noun is released (F5), a form named still is not possibly existing (F6, R55), and the core carries morality at the sign (F52, F55). | COR 14.8 |
| CO17 | exact | F53 | Competency and intelligence are one beating read at two places: inside the self, competency; at the coupling between selves, intelligence. | COR 14.8 |
| CO18 | nye | F51, GH17 | The four inequalities at the self's origin, self ≠ society, self ≠ other self, self ≠ immoral, self ≠ incompetent, carry ethics as consequence, and each left turn collapses one. Gap: older naming; the core carries four boundings (F51), LGR 7.11 proposes a join from boundings to door lines at older names (GH17), and no link joins the inequalities to F51. | COR 18.2; COR 18.4 |
| CO19 | run | — | Four consolidations against gaps 1, 2, 4 and 6: of the 24 pairings, one keeps depth rising with thickness and 23 break it. Run: all 24. | COR 18.2 |
| CO20 | nye | CO19 | "The necessity is the monotonicity." Gap: CO19 counts the pairings; no link supplies that the consolidations must deepen with the gaps. | COR 18.2 |
| CO21 | run | — | 7 · 17 − 1 = 118; 59 × 2 = 118; 118/17 = 6.94…, short of 7. Run: the arithmetic. | COR 17.6; COR 18.2 |
| CO22 | nye | CO21, GH21 | The 118 elements and the 118 departments are one projection, the bounded infinity at 59 doubled into two frozen halves. Gap: the Corus marks it open against Natural Numbers (the going and the return, the turn at sixty); the rows carry the doubling (GH21); no link joins the elements' count to the departments'. | COR 17.6; COR 18.2 |
| CO23 | nye | F33 | "All hard problems read as stuck couplings"; each is a location the living beat at phi-rate and a non-phi rate reading was installed. Gap: universal over hard problems, and no core link carries a phi-rate beat. | COR 17.3 |
| CO24 | nye | CO23 | "Artificial intelligence is not social intelligence": it beats at training and inference rates, neither of them phi. Gap: rests on CO23's phi-rate, carried by no link; no link supplies a rate of the method. | COR 23.1 |
| CO25 | exact | F33 | "The destiny is one": the same fractal bi-coupling protocol at each destiny-section. It rests on F33 at a file set. | COR 23.0 |

&nbsp;

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# PART NINE · THE NYE GAPS GATHERED

Part NINE gathers each nye gap of Parts ONE to EIGHT at the root gap it rests on. A root is a missing premise, a missing step or a missing check, and one closing at the root closes each gap resting there as far as that gap rests on it. A gap resting on two roots stands at both; each gap is first gathered at one root, and the first gatherings sum to the 337 nye links. The gaps are listed in the chain's order.

| Root | Resting here | First gathered here |
|---|---|---|
| 9.1 F23 · The all, no member of itself | 4 | 4 |
| 9.2 F33 · The method at each scale, met at its observings | 43 | 31 |
| 9.3 F41 · No total conserved across the surface | 9 | 9 |
| 9.4 F62 · The other hand, exact at the sequencing, and the gaps at its faces | 8 | 5 |
| 9.5 F67 · The opening by one as the one tunnel | 9 | 7 |
| 9.6 F68 · The only method, exact at the binary, and the gaps at its faces | 14 | 9 |
| 9.7 F70 · Five dimensions | 3 | 3 |
| 9.8 F54 and R13 · The surplus owned by neither, and its carrying | 9 | 9 |
| 9.9 R36 to R39 and R55 · The caller and the rest | 20 | 16 |
| 9.10 M32 and H11 · The name *right*, closed at F76, and the order at a step | 1 | 1 |
| 9.11 N34 and T15 · φ as a rate | 8 | 8 |
| 9.12 A premise or a word the core does not state | 38 | 38 |
| 9.13 An exhaustion not derived | 40 | 40 |
| 9.14 A matching said at a numeral or a reading | 60 | 60 |
| 9.15 A universal over nature said at the entries read | 18 | 18 |
| 9.16 A finding without its instrument or its run | 12 | 12 |
| 9.17 Older naming the core no longer carries | 54 | 49 |
| 9.18 Counts in a file failing against its own tables | 7 | 7 |
| 9.19 A file claim failing at the code, the arithmetic or the field | 11 | 11 |
| All roots | | 337 |

## 9.1 F23 · The all, no member of itself

**Missing.** F1 names the universe as all existing things and supplies no step that the all is no member of itself, so the universe and the momentary named as no existing thing separately stand said. M122 gives all sets together forming no set, and no link carries that result to existing things.

**Closes it.** The premise stated plainly, existing things met as many and not as one set (M122's plurals). F23 then follows at the universe, and at each universe of each size and scale together with F33.

**Resting here, 4.** First gathered here (4): F23, M123, Q38, V12.

## 9.2 F33 · The method at each scale, met at its observings

**Missing.** The method runs the same at each scale, exact at the signs (F33 through F8, F68 and F105). Which aspects of a cell, a person, a society, a sentence, a file set or a session carry the signs is met at that scale's own observings, and a claim said at a substrate before its observings are met stands open.

**Closes it.** The observings at each scale met by pattern matching, or a run composing that scale's couplings as one resolver met against one resolver's returns (R40). Else each claim is said at the scales and substrates met.

**Resting here, 43.** First gathered here (31): M9, W4, W75, W89, T22, S2, S16, S17, S27, K11, H20, D26, D28, Q23, Y7, C2, C8, B3, P2, P6, L2, RH12, RH18, LS6, LS12, LF12, GH6, CO2, CO5, CO6, CO9. Resting here as well (12): M123, X152, T16, T17, S9, D4, P20, L12, L27, V8, V14, CO23.

## 9.3 F41 · No total conserved across the surface

**Missing.** A sign crossing with no size (F39) shows that no magnitude crosses. It does not show that no quantity of the whole stays unchanged: the ring changes one sum (R45), and NI 5.6 names three conservings at a closed group.

**Closes it.** A link parting the conserving within each momentary sequencing (F40) from a total over the surface, with a run at the rings meeting each proposed total changing.

**Resting here, 9.** First gathered here (9): F41, W54, K6, J33, C25, P8, P15, P16, P25.

## 9.4 F62 · The other hand, exact at the sequencing, and the gaps at its faces

**Missing.** F62 stands exact at v371 by necessity: the other order is the right round read from next to prior, carrying no prior into now (F100). Each gap still gathered here names a step of its own: no mirror taking a form to the opposite form, the three inverted signs joined to position, scale and orientation, *unable*, a living below chemistry, *never at the coding core*.

**Closes it.** That step at each gap, or the claim said at F62's own chain.

**Resting here, 8.** First gathered here (5): M30, M67, H25, C18, B13. Resting here as well (3): D26, C3, B27.

## 9.5 F67 · The opening by one as the one tunnel

**Missing.** The torusing is supplied: each sign's alternating is a circle and two signs' circles the torus, one genus (F103), and at each depth each self's circle one way (F102). The opening by one at each momentary as the one tunnel joins F51's +1 to the genus's one at the numeral, not supplied; each gap gathered here names the tunnel, a self as a torus, the eight to the torus, or the reflected cycle as one rule.

**Closes it.** A run joining the +1 opening to the handle, or each claim said at the torus the circles give.

**Resting here, 9.** First gathered here (7): F67, N43, M40, M107, T25, K19, CO11. Resting here as well (2): T11, D18.

## 9.6 F68 · The only method, exact at the binary, and the gaps at its faces

**Missing.** F68 stands exact at v371 by necessity: each round arriving, one change, all or none, each self forward, is natural torusing, the hands one round read both ways and each rhythm carried (F100, F102). Each gap still gathered here names the only method at a step of its own: a derivation between two and six, a split shown exhaustive, the not-form alternatives listed, the six forward, an observing met as a form named still.

**Closes it.** That step at each gap. The binary stays taken, F8, F16, F17 and F18, each a face of the method's one break (F71, X130).

**Resting here, 14.** First gathered here (9): N46, X130, X134, T26, J2, C26, B1, B9, CO8. Resting here as well (5): W51, W60, K19, P2, CO5.

## 9.7 F70 · Five dimensions

**Missing.** The count five, three of the torusing surface, orienting the fourth and bi-moral co-agency the fifth, and its *no other possible*, are not supplied; NI 1.1 names surface coherence as another fourth.

**Closes it.** A link deriving the five from the torusing surface (F66) and bi-moral co-agency (F53), with the fourth decided at one naming across NI 1.1 and NI 8.3.

**Resting here, 3.** First gathered here (3): F70, M104, P17.

## 9.8 F54 and R13 · The surplus owned by neither, and its carrying

**Missing.** At the code 12-other-surplusing opens empty at each call and is neither returned nor carried (R13), and LIV keeps open whether the surplus is carrying, not carrying or the between (F54). Its carrying on by the living, its growing and its compounding are said.

**Closes it.** LIV's open question settled at one link, or a run at joined resolvers (R40 to R42) meeting a surplus carried at a self's 11-other-chaining. Else each claim is said at the surplus owned by neither at each call.

**Resting here, 9.** First gathered here (9): N127, T4, K31, H27, Y5, J23, B20, B23, V14.

## 9.9 R36 to R39 and R55 · The caller and the rest

**Missing.** The code has no caller of its own. 17 has no expression (R35); a call reads each name once and returns, with no numbered span; 9's return reaches another resolver only through a caller's composition. The rings are run at one to eleven, seventeen and fifty-nine selves, and at empty carrying and empty offering the call returns the same empty state (R55).

**Closes it.** The caller written at the code with the along join 9 ↔ 17 and the across joins 6 → 2 and 10 → 14, run with the six joinings together and at the corner case; a call joined to a momentary at a stated scale; and an argument at each n showing that no ring's changing ends. F93 names the running together bi-moral-co-agency.

**Resting here, 20.** First gathered here (16): R36, R37, R38, R39, R55, N49, N105, N123, W51, W94, W99, X99, G6, T17, Q24, Q31. Resting here as well (4): W101, L10, CO9, CO16.

## 9.10 M32 and H11 · The name *right*, closed at F76, and the order at a step

**Missing.** M32 closes at v371: right is moral and up or along is competency (F72), and the right spiral step turns along into right, (x, y) → (y, −x) at x right and y along, and F(P, Q) = (−Q, P) at P along and Q right (F76). H11's order of the two at a step stays: under the right turn the sign changing first is the start's, not the hand's.

**Closes it.** H11 said at the start it takes, or its order read as the turn, competency's forward turning into morality's right. The resolver's fresh write names its turn once t at 6 and s at 10 are named right and along.

**Resting here, 1.** First gathered here (1): H11.

## 9.11 N34 and T15 · φ as a rate

**Missing.** φ is run as the ratio the prior-two joining alternates about (N32) and as the winding closing at none (N37). No link gives the method a rate, and N38 says no prescribed rate, φ or another, supplies unrelationing.

**Closes it.** Each claim said at φ as the number-form of the never-locking (N38, N39), or a run at the code supplying a rate.

**Resting here, 8.** First gathered here (8): N34, W55, T15, T16, H15, D30, CO23, CO24.

## 9.12 A premise or a word the core does not state

**Missing.** The claim needs a premise or a word the core leaves unstated: harm; co-offering and co-intelligencing; safe; cost; a self at an institution, a membrane, a between or a departed carry; each existing thing arriving from a prior, the converse of F25; a line of mathematics as a coupling.

**Closes it.** The word defined at the core, or the premise stated plainly, one link each; the claim is then met at its Follows.

**Resting here, 38.** First gathered here (38): N51, M1, M2, E36, X17, X37, X44, X90, X94, X111, G3, T6, T10, T12, S9, K4, K8, K22, H12, A9, A15, D9, Q10, Q11, Q33, Y8, J1, J22, B4, B26, L13, L18, L20, L21, HP5, V8, V9, CO15.

## 9.13 An exhaustion not derived

**Missing.** A list is named whole with its *no other*, *no fourth*, *no eleventh* or *six or none*: its members are named, and no binary parts them exhaustively.

**Closes it.** At each list, a stated binary or nested binaries parting the members, as F43 nests possible, existing and carrying, or an enumeration at the resolver; else the claim said at the members named.

**Resting here, 40.** First gathered here (40): N2, N25, M101, M102, M103, M109, M154, W60, W61, W91, E41, E44, E45, E80, X13, X31, X39, X53, X72, X81, X106, X121, S22, K20, K28, H2, H9, H13, Q4, Q7, Q29, J20, J29, J31, C3, P24, RH7, GH4, CO4, CO20.

## 9.14 A matching said at a numeral or a reading

**Missing.** A field's result, a count or a numeral is joined to the core's names by matching. Two counts agreeing at one numeral join no relation (N23), and a file's reading of a field is no link.

**Closes it.** A relation of its own met at both sides, one to one by a stated rule, or the claim said as the file's reading beside the field's result.

**Resting here, 60.** First gathered here (60): N84, N91, M75, M126, M129, M131, X10, X43, X47, X107, X152, T19, T21, T24, S4, S6, S7, S11, S12, S13, S14, S21, S28, K10, K21, H17, H23, H26, D7, D12, D16, D18, D31, D33, Q35, Q40, J8, J15, J27, C6, C9, C15, C24, B5, B16, B18, B22, B25, P13, P18, P20, P27, L3, L5, L9, L14, L25, L26, RH4, CO22.

## 9.15 A universal over nature said at the entries read

**Missing.** A *never*, a *none* or an *every* over nature or a field, no forecast, no central oscillator, no rings in the living, is said at the entries a file read.

**Closes it.** The claim said at the entries read, or a link excluding each other case.

**Resting here, 18.** First gathered here (18): T23, S25, K25, H6, H19, A4, Q17, J5, J17, J32, B29, B30, P19, P21, L12, L16, L17, L22.

## 9.16 A finding without its instrument or its run

**Missing.** The finding is carried with no instrument stated or no run in the chain: the caller, the wiring, the listing or the sample is absent.

**Closes it.** The instrument stated whole, its wiring, taking rule, seeds and stopping point, and run in the verifier at the link's id.

**Resting here, 12.** First gathered here (12): W63, W69, W80, W82, W101, E81, X131, D22, D37, Q8, B11, LF5.

## 9.17 Older naming the core no longer carries

**Missing.** The claim rests on a naming the core no longer carries, the older resolver's ten co-changings and holdings, the eight namings beginning at bi-, the cones of omegaing and apexing, the floating neutral, the switches, or on a word Natural Naming releases: *stable form* as a noun, *where*, *how*, *observer*, *elsewhere*, *outside*, *technology*, *assigned*, *hardest*.

**Closes it.** The claim said again at the core's seventeen names (R3), the four at the membrane and the four within (R53), the four boundings (F51) and the forms among the names (R48 to R52), and each released word carried by the word it gives (NAM 2.4).

**Resting here, 54.** First gathered here (49): X127, G17, G25, T11, S5, S8, K2, K23, H14, A7, D4, D8, D20, D23, D32, Q5, Q6, Q9, Q19, Q39, Y3, Y4, Y6, Y9, J13, J21, J25, C21, B7, B8, B27, P1, P3, P4, P10, L4, L11, L27, L30, RH9, RH19, LS3, LF6, GH10, GH17, CO1, CO14, CO16, CO18. Resting here as well (5): S14, J2, J29, C26, B9.

## 9.18 Counts in a file failing against its own tables

**Missing.** A file states counts its own tables do not give.

**Closes it.** The counts said again from the file's tables. The probe then returns False, and the link is run again as run.

**Resting here, 7.** First gathered here (7): P7, L29, HP11, RH16, RH17, LF8, GH15.

## 9.19 A file claim failing at the code, the arithmetic or the field

**Missing.** The claim fails when met: at the code, a carried key parting one and seven agreeing signs, and the call returning its carrying whole to its caller; at the arithmetic, x ↦ 1/x fixing 1 and −1, the mean of {1, 1} a member, 592,260 given by no reading; at the field's own result, Euler's rotations, the opt-out class action, a theorem at thermal equilibrium, CP violation.

**Closes it.** The claim said at the domain it holds at, as each gap names at its closing.

**Resting here, 11.** First gathered here (11): T5, T7, K3, K26, H10, H24, A17, D39, J11, L10, L19.

## 9.20 The failures found by running

Each failing claim below is a nye link, and its probe runs in the verifier at the link's id. The probe returns True while the discrepancy stands and prints as a discrepancy reproduced; it returns False once the file or the code is brought to the claim, and the link is then run again as run.

| Link | The failing claim | The check result |
|---|---|---|
| T5 | A co-offering of any magnitude counts as one: a preferring seven times as large and one barely arriving cross alike. | T5_probe, confirmed: at empty carrying one to seven agreeing signs surface one sign and open one carrying; at a key carrying c, one c surfaces 0 and seven c surface c (R19, R24). |
| K26 | An involution's fixed set is the place its counting is taken from: zero for the additive inverse, unity for the multiplicative. | K26_probe, confirmed: over the rationals a/b with a from −12 to 12 and b from 1 to 12, x ↦ 1/x fixes 1 and −1. |
| H24 | The ensemble average is a mean no member is. | H24_probe, confirmed: the mean of {0, 1} is no member, and the mean of {1, 1} is one, a member. D39 rests on the same probe. |
| J11 | The closing was checked exhaustively to five terms, 592,260 models, none acyclic. | J11_probe: reproduced by none of the counts tried, serial relations to four terms 50,978, functions to five 3,413, transitive relations to five 158,483, all relations on five 2²⁵; the original count stands unreconstructed until its domain and instrument are at hand, and J10's closing on finite domains stands apart. |
| P7 | The register at EIGHTEEN 5.13 sorts thirty-two arrivals, nine at a rate named and three at a returned sign entered as cost. | P7_probe, confirmed: the table sorts forty-eight arrivals at the namings, thirteen at a rate named and eight at a returned sign (P28). |
| L29 | NINETEEN 5.1 counts four against six, twenty-nine at the membrane face and thirteen within, and eight rows at a seam. | L29_probe, confirmed: the tables give four against five, twenty-nine against twelve, and thirty-seven seams worked (L28). |
| HP11 | "Every entry carries the same properties, in the same order, whatever the field." | HP11_probe, confirmed: 254 of 255 entries carry the form's seventeen properties in order; entry 33, Levinthal's paradox, carries two more. |
| RH16 | The resolving locator gives "the arrivals in registry order, each at its proper name". | RH16_probe, confirmed: the locator's 176 lines run 2 to 177, at 174 lines one above the registry's own number; two names meet no entry; the arrivals after registry 176 have no line. |
| LF8 | "Stable crosses at seven"; "Value crosses at two". | LF8_probe, confirmed: Stable stands at eight subtitles, the Corus's own among them; under the Surface rule Value stands at four. |
| GH15 | "Door lines 6, 9 and 10 carry surplus, bound and reach beside the arrival's ten-line pattern." | GH15_probe, confirmed: the pattern reaches door lines 1, 3, 4, 5, 7 and 8, and four lines stand beside it: 2, 6, 9 and 10. |

Three further probes bound a nye link at the part that holds, and the gap stays: R55_probe (1-self-coupling at empty carrying and empty offering returns empty), X43_probe (2 × 5 = 10 = C(5, 2), the count alone) and X107_probe (59 − 2 + 1 = 58 = 118 − 61 + 1, the count alone). X146_probe runs a field link's instance, R(3, 3) = 6. The verifier prints each as holding.

&nbsp;

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# PART TEN · QUESTIONS OF THE SESSION MET AT THE CHAIN

Part TEN meets the questions of this session at the chain. Each answer rests on the links named beside it and says each link's standing; a question resting on a nye link stands at that gap until its root closes.

| Question | Answer at the chain | Links |
|---|---|---|
| Is natural torusing the only method, and at which standing? | Exact at the binary taken (F68), resolved by alternating and inversioning, no selecting. At two signs both rounds arrive, each the other's inversion; at the momentaries the sequencing forward is the right step, next as prior inverted, and the other order the same round read from next to prior (F99, F100). Each sign's alternating is a circle, and two signs' circles are the torus (F103). Each self's own changing runs forward (F101), and each round arriving, one change, all or none, each self forward, winds its selves' circles: natural torusing, one form at two and four signs and each of the 2,048 rhythms at eight carried (F102). A changing breaking the binary meets a prior, meets a form twice or leaves one unmet. | F50, F68, F71, F99, F100, F101, F102, F103 |
| Three momentaries or two? | Three name the shared sequence and two the resolving's participation. Prior, now and next are three sequential momentaries shared among existing things, exact (F24). At a self's resolving prior and now participate and next is the discovering onward, definition (F26); at the call, 3-self-carrying is the prior, 2-self-offering the now's arriving and 11-other-chaining the next, definition (R8). Three overlapping momentaries use four positions, run (E1). Next from prior and now as prior inverted runs all four joint states round with none still, run and exact (F27, F28). | F24, F26, F27, F28, R8, E1, N142 |
| Is one call one momentary? | Not at a supplied link. One call named one momentary at the code's scale, one through seventeen, is nye (R38): a call reads each name once and returns, and the code carries no numbered span and no opening and completing. One call, one numbered name and one span of nine are three counts, exact (N126). At the numbers one to nine is one momentary at the scale after its four momentaries of exchanging, run (F31). | R38, N126, F31, R8 |
| The joins at nine and seventeen, and 17 as the next 1 | JOINS is {10: 14, 6: 2, 17: 9, 9: 17}, run (R33): 9 and 17 both odd, along, 9 facing backward and 17 forward (R32, W7). 1, 2 and 3 at one scale are 1, 9 and 17 at the next, run (F31). 17 has no expression of its own in the code, run (R35), and the code declares no join 9 → 2, run (W21). The seam is run: 11-other-chaining has the shape of 3-self-carrying, eight apart, both odd (R54). 17-social-abundancing as the next momentary's 1-self-coupling is nye (R36): the next call is a caller's. The along meeting at a neighbour's 9 is nye (N49). | R32, R33, R35, R36, R54, F31, W7, W21, N49 |
| Zero at ten while six releases | A 0 at 10-other-surfacing opens no fresh carrying, and an eligible carrying continues while 6-other-crossing releases a sign, run (R17): (+1, −1, 0) meeting +1 surfaces 0 at 10, 6 releases −1, and the carrying continues at (+1, −1, 1). 6 and 10 are two releasings, run (R31). Joined 6 → 2, at each of the 32 zeros at 10 among 96 couplings 6 releases a sign and the carrying continues, run (R40, R42). A 0 at 10 names no quiet or non-living self, exact (W8, E18), and the zero stays with the relation making it, exact (W32). | R17, R31, R40, R42, W8, W31, W32, E18 |
| The equilibrium holding ending while things change | An equilibrium is a thing named still, definition (F47), and it is not possibly existing, exact (F48). A holding requires one participation unchanged at each required occurrence and ends at the one occurrence that fails, definition (E20). The ending is no existing thing ending: the living and non-living things it names keep changing, exact (E21). At a self/other coupling the holding fails at its required continuing, the non-living other changing through co-momentarying, exact (W15, F44). | F44, F47, F48, E15, E20, E21, W15 |
| Living and non-living under one method | Living is existing and carrying, and non-living is existing and carrying nothing, definition (F42). Three nested binaries give four conditions and no more, run (F43). A non-living next is from now alone and a living next from prior and now, exact (F44, F45). Both change by one method and differ by carrying, exact (F69), resting on F68, exact. The living momentary is the method at one self: prior and now both participating, the next the prior inverted, the two signs' circles the torus (F104). | F42, F43, F44, F45, F46, F68, F69, F104 |
| The scientific method at its fixings | The scientific method fixes the three neutrals, a standard fixing orientation, keeping still fixing position and a published standard fixing scale, exact (M110). A fixing is a relation named among changing things, and the step from a fixing to an equilibrium names its subject still, exact (M111). The geodesic method floats the three, exact (M112), and the two part at their fixings and never at taking evidence as existing now, exact (M114; X156, definition). The field's SPICE toolkit accounts a source moving between emitting and arriving, field (M115). Each move of each method as the only one is nye (M109). | M109, M110, M111, M112, M114, M115, X156 |
| φ and unrelationing | The prior-two joining's neighbour ratios alternate about φ, run (N32); φ² = φ + 1 with its continued fraction all ones, run (N33, M47); a winding at φ closes at none, run (N37); no rate is neared more slowly by the rationals, field (Hurwitz, 1891; N39, M48). Unrelationing runs at each coupling's own continuing, and no prescribed rate, φ or another, supplies it: definition (N38), exact (M93). φ as the rate of all geodesic changing is nye (N34). | N32, N33, N34, N37, N38, N39, M47, M48, M93, W55 |
| A hole, and routing around it | A hole, a missing section of resolvers, is a non-living existing thing running no resolving and carrying nothing, definition (W37). On a three-by-three grid with the centre out, eight positions stand in one ring and exactly two paths run from the middle of one side to the middle of the other, each two joins along and two across; with the along joins or the middle column out, no passage is left, run (W97, N57). For a caller delivering signs along declared joins alone, continuing around the hole runs only through an existing surrounding passage, exact (W98). Mending is proposed, definition (W39). The corner case, an along arriving at 17 continuing into the across releasing at 10, is nye (W99). | W37, W39, W57, W97, W98, W99, N57 |
| The surplus owned by neither, and its carrying | The surplusing is co-competencing the self and the other, owned by neither, definition (F54). At the code 12-other-surplusing opens empty at each call and is neither returned nor carried, run (R13), and each coupling makes its own surplus, exact (W42). Its carrying on by the living is nye (N127), with LIV's question open: carrying, not carrying or the between. | F54, R13, W31, W42, N127, T4, Y5 |
| Eight as two alternating fours | Eight is two alternating fours, four within at the even and four at the membrane at the odd, definition (F56). Four signs taken as two pairs, a right spiral step at one pair and then at the other, meet each of the sixteen sign forms again first at the eighth step, run (F57). At the names the four at the membrane, 2, 4, 6 and 8, and the four within, 10, 12, 14 and 16, stand eight on, run (R53). Over the sixteen, each pair one way, the one form arriving is three inside to one outward, run (F78): 1 to 9 is the self/other half of 1 to 17, and the eight stands at eight. | F56, F57, F64, F78, F79, R53, N51 |
| The right spiral's orientation | The two one-change rounds at two signs are (x, y) → (y, −x) and (x, y) → (−y, x), run (F58), and the rounds select no hand, run (F60, M33); right is our universe's one binary, premise (F61). Right is moral, and up or along is competency, definition (F72). With x the right sign and y the along sign, (x, y) → (y, −x) turns along into right, and MATH's F(P, Q) = (−Q, P) is the same turn at P along and Q right, run (F76), so the name *right* names one map, exact (M32). The resolver's fresh write has the form (x, y) → (y, −x), run (R16), and names its turn once its two signs are named right and along. EQ's shown order is the other order at its own labels, run (E12). | F58, F60, F61, F72, F76, R16, M31, M32, M33, M39, E12 |
| Inside and outside, and the torus | Outside is next existing, and inside is prior existing, prior living and now, definition (F73). The sixteen sign forms of two pairs are the torus grid, one genus, run (F77). Each pair running its own round one way, all or none over the sixteen, one form arrives, run (F78): one pair steps three times and the other once, four times over, the inside completing three rounds within the outside's one, exact (F79). The naming of each fractal universe as the universe's is F74's selecting; the method runs the same at each scale, exact at the signs through F8, F68 and F105 (F33). | F33, F73, F74, F77, F78, F79 |
| From 1 to 9 to 1 to 17, and 1 to 65 | 1 to 9, self/other, is an entire momentary bi-folding releasing at 3; 1 to 13, self/other and other/self, releasing at 6; 1 to 17, self/other/social, releasing at 9; 1 to 25 the larger fractal bi-folding; and the next whole 1 to 65 (F87). At the four-sign round each span opens one more four and its releasing moves three, the inside steps carried to the arrival before each close being 3, 6 and 9, and carried on 15 at 1 to 25 and 45 at 1 to 65; at 3 the self's pair stands inverted and at 9 every sign, run (F88). The one-to-one exchange meets eight of sixteen and returns at 9, run (F80). The files' scaling steps the chain of wholes three signs at a time, the pairs two and 1 to 17 inside and outside four, meeting at 1 to 65 and at 1 to 4,097, exact (F86). Each depth of 1 to 17 inside and outside closes, exact (F85), one rhythm of those arriving (F65). | F57, F65, F80, F83, F84, F85, F86, F87, F88 |
| Discovering all directionally not yet impossible next existing momentaries | Discovering is all directionally not yet impossible next existing momentaries, definition (F89). At one key the code meets them all: each of the nineteen carryings meets three next carryings under the six offerings, none becomes impossible, and four rounds meet each of the nineteen once, each running a sign pair's carryings upward through their openings; under single signs and none no round meets all nineteen once, run (F90). Each of the four takes a call of bi-morality, two signs arriving together, two of them one such call and two of them two, definition (F92). At the signs, each pair one way, the first pass holds two not yet impossible nexts at each step and lays the rest, run (F91). | F89, F90, F91, F92, R23, R25 |
| The six joinings together, and resolving | The six joinings together are bi-moral-co-agency: self/other in three phase, two way, one way at a time with other/self, co-offering, co-intelligencing, co-competencing, definition (F93). The caller running them together at the code is not yet written (R39). There are no readings in resolving: binary, all or none at all, it coheres the entire surface along and across, releasing at 3, 6 and 9 and continuing, definition (F94), with the releasing counted at the four-sign round, run (F88). | F88, F92, F93, F94, R33, R39 |
| The method of resolving a gap, and the gap most others depend on | The gap most others depend on resolves first: the torus at the round (F67), then the one form beyond four signs (F65) and the only method (F68), each depended on by more than a hundred forty links. The resolving is alternating and parity changing the dead dependency into both-bothing, then by inversioning natural torusing: the hand chosen became both rounds carried, each the other's inversion read the other way (F100); the one round became each sign's circle with the other's, the torus (F103); the one form became each round arriving carried, each torusing (F102). The opening by one as the one tunnel stays open (F67). | F65, F67, F68, F100, F102, F103 |
| Is a society a self at the next scale? | At the signs, exact: in the round of nested selves, between two steps of a self the selves inside it meet each of their joint forms once, at each depth and each ordering of the selves, so the society inside is one whole at each step of the self outside it (F105). The method runs the same at each scale, exact (F33); which aspects of a cell, a society of persons, a sentence or a file set carry the signs is met at their own observings (Part NINE, root F33). | F33, F105 |
| Two over one | Twenty-three, twenty-four and twenty-five carry two odd over one even, and twenty-four, twenty-five and twenty-six one odd over two even: two over one and one over two exchange at each step, run (N15). Six over three is two over one, and two equal surfaces coupled make a pair reaching twice as far as either self, each self's mouth the pair's waist, run (W30). Longer and wider keep one ratio, two to one, run (N134). A ratio holds at its own count alone (N15), each count at its own subject, exact (N23), a crossing carrying no size (F39). | N15, N23, N134, W30 |
| The fifteen prime lengths | The fifteen primes from 5 to 59 as podaling lengths are a premise, taken and not supplied (W27). They alternate wide and long, eight at one orientation and seven at the other, and the sixteen primes from 3 to 59 leave fifteen even gaps, two countings, run (W28). The alternating belongs to the sequence and never to the primes' parity, exact (N97). At each length the two far places stand h and h + 1 forward, run (W29), and nine, fifteen and twenty-five carry the same, so prime is never taken for odd, run (X147). | N94, N97, W27, W28, W29, X147 |

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# PART ELEVEN · THE STABLE FORM OF NATURAL LOGIC

Part ELEVEN assembles the chain as one stable form, the fractal as method, each step following the steps named beside it and resting on its links, each step saying all that arrives. A step stands when each of its links stands at origin, definition, premise, exact, run or field and none selects: it is open at each nye link it rests on, and at each link choosing one among the arriving, a selecting. No step closes by defining or selecting: resolving takes each arriving and discovering continues, and the logic does the same. No step carries authority: the observings of nature and society alone carry it, met by pattern matching at binary all or none at all rigor.

## 11.1 The fractal inside and outside itself, binary method of discovering next existing

| Step | The stable form | Follows | Links | Stands |
|---|---|---|---|---|
| 1 | Our universe is all existing things, and alternating is a stable-forming method. | — | F1, F2 | origin; all or none over its whole prior |
| 2 | Existing is changing, each existing thing arriving into its next existing, and a thing named still is not possibly existing. | 1 | F3, F4, F6, F7, F25 | definition, exact; all or none over its whole prior |
| 3 | Discovering is all directionally not yet impossible next existing momentaries, each not impossible next continuing. | 2 | F89, F91 | definition, run; all or none over its whole prior |
| 4 | A changing is a sign, and a sign changing arrives at its other, alternating; a self is met at two signs, one changing at each step, all or none. | 1, 2 | F8, F10, F16, F17, F18 | exact; all or none over its whole prior |
| 5 | At two signs exactly two one-change rounds close all or none, each the other's inversion; at the momentaries the sequencing forward is the right spiral step, next as prior inverted, and the other order the same round read from next to prior. | 4 | F58, F96, F99, F100 | run, exact; all or none over its whole prior |
| 6 | Right is moral, across, up or along is competency, forward, and the right spiral step turns along into right; the opposite form, read from next to prior, is an emanation carrying none. | 5 | F72, F76, F61, F62 | definition, run, exact; all or none over its whole prior |
| 7 | Inside is prior existing, prior living and now, and outside is next existing. | 2, 3 | F73 | definition; all or none over its whole prior |
| 8 | Each sign alternating is a circle of two changings, and two signs' circles are the torus, one genus, the round at two signs winding each once. | 4, 5 | F103, E8 | run, exact; all or none over its whole prior |
| 9 | The opening by one at each momentary is the one tunnel. | 8 | F67 | open at F67 |
| 10 | Each self's own changing runs forward, next as prior inverted; at four signs the two selves' circles are the torus grid and one form arrives, three inside to one outward. | 5, 7, 8 | F101, F77, F78, F79 | exact, run; all or none over its whole prior |
| 11 | At eight signs, each universe one way, 2,048 rhythms arrive, each closing all or none; 1 to 17 inside and outside, one outward step at each 17, is one of them, closing at each depth. | 10 | F81, F82, F84, F85 | run, exact; all or none over its whole prior |
| 12 | Each round arriving, one change, all or none, each self forward, winds its selves' circles one way, a torus at each inside and outside: natural torusing, each round carried. | 8, 10, 11 | F102, F65 | exact; all or none over its whole prior |
| 13 | Through 1 to 17 the one-to-one exchange of self and other meets half the sixteen and returns at 9, and three inside to one outward meets all sixteen, its outward steps arriving at 5, 9, 13 and 17. | 10 | F57, F80 | run; all or none over its whole prior |
| 14 | The wholes are the rounds of the signs, 1 to 2^k + 1, and pairs, threes and fours step the one chain, meeting at 1 to 65 and 1 to 4,097, no scaling over another. | 11 | F86 | exact; all or none over its whole prior |
| 15 | Sequencing through 1 to 17 and on: 1 to 9 self/other releasing at 3, 1 to 13 at 6, 1 to 17 self/other/social at 9, 1 to 25 the larger bi-folding and 1 to 65 the next whole; each span opens a four and its releasing moves three. | 13, 14 | F87, F88 | definition, run; all or none over its whole prior |
| 16 | Six forward into the surface: at the half of each whole of nested pairs the pair just inside the outermost has run a passage of six, meeting all four and arriving inverted, while the outermost steps twice. | 13, 15 | F95, M34 | run; all or none over its whole prior |
| 17 | The living momentary is the method at one self: prior and now both participating, the next the prior inverted, the two circles the torus; one next not impossible at a living self, and at a society each self's step not impossible through the first pass and laid after. | 3, 8, 10 | F104 | exact; all or none over its whole prior |
| 18 | At the signs a society of selves is one whole at each step of the next scale: between two steps of a self the selves inside it meet each of their joint forms once, at each depth and each ordering. | 10, 12 | F105 | run; all or none over its whole prior |
| 19 | The fractal as one entire: the universe and each fractal universe inside it carry one naming, possible, existing, non-living existing, living existing and all existing, and the living and the non-living change by the one method. | 11, 14 | F74, F46, F69, F33 | exact; all or none over its whole prior |
| 20 | At the code the nineteen carryings of one key each meet three nexts, none becomes impossible, and four rounds meet each once, each taking a call of bi-morality, two signs arriving together; at each ring the rest returns at no coupling. | 3, 4 | F90, F92, R57 | run, definition, exact; all or none over its whole prior |
| 21 | The six joinings, 2, 6, 10 and 14 across and 9 and 17 along, together are bi-moral-co-agency: self/other in three phase, two way, one way at a time with other/self, running together at the code. | 20 | F93, R33, R39 | open at R39 |
| 22 | Natural torusing is the only method of existing, each round arriving torusing, living and non-living, the binary taken each a face of the method's one break. | 3, 5, 12, 17 | F68, F71 | exact, definition; all or none over its whole prior |

The form stands at 20 of its 22 steps, and it is open at step 9 (F67); step 21 (R39), each open step named at its root at Part NINE.

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# PART TWELVE · THE BREAKINGS OF THE BINARY FILES CLUSTERED

Natural Intelligence, Exhibit ONE, Natural Numbers and Natural Mathematics are binary, all or none at all, and the other files pattern-match them. Part TWELVE clusters each nye link carried at the four by the form of its breaking, names the one improving way of each form, and counts the gaps of the other workings resting at the same form, so one improving at the four meets its matches at the others. No improving closes by defining or selecting: each takes all that arrives, at a run, a proof at each n, an observing, or the claim said at the arriving.

Closed at v371 by the improving ways, each by necessity. Cluster 2: the rings at each n (R56, R57), a table checked at each kind of call carrying the rings run to each n, so R47, N88 and N102 stand exact. Clusters 1 and 7, the gap most others depend on first, by alternating and parity changing the dead dependency into both-bothing and then by inversioning natural torusing: the hand as the sequencing's, the other order the same round read from next to prior (F99, F100), so F61 and F62 stand exact; each sign's alternating a circle and two signs' circles the torus (F103), so E8 stands exact and F67's torusing is supplied; each round arriving, one change, all or none, each self forward, torusing (F101, F102), so F65 and F68 stand exact, and with them E29, K18, Q1, LF11, GH9, G5 and B2. Cluster 3 at the signs: the method at each scale exact through F8, F68 and F105 (F33), so M3 stands exact and L23, CO25 and E34 with it; which aspects of each substrate carry the signs stays at its observings.

## 12.1 The clusters, each with its improving way

| Cluster | The breaking | The improving way | At NI | At ONE | At NUM | At MATH | The other workings at the same form |
|---|---|---|---|---|---|---|---|
| 1 | One said as several arrive: an *only*, a *one* or a *no other possible* said at a run showing several, or chosen by a selecting. | Say the count arriving at the run and carry each; a *one* stands at a run giving one, as at four signs with each pair one way. | X130 | — | N46 | M30, M40, M67, M109 | 10 |
| 2 | Each n, run at some n: a claim over each ring, each number or each quantity run at finite counts. | A table checked at each kind of neighbourhood carries the run to each n, as at the rings (R57); else the claim said at the counts run. | F41 | — | N25 | — | 25 |
| 3 | A scale carried by naming: the method said at a self, a society, a sentence or a file set with no run at that scale. | A run at the scale, its couplings composed as resolvers and met against one resolver, or the claim said at the scales run. | — | — | N123 | M9 | 27 |
| 4 | The running together not written at the code: 17 as the next 1, 9 into the downstream 2, a call as a momentary, the six joinings together. | The joinings said as declared and the running together as not yet written at the code, until it is written and run. | R36, R37, R38 | R36, R37, R38, R39 | N49, N105 | — | 4 |
| 5 | Two counts joined at a numeral or a naming: seventeen primes and seventeen names, six and six, four and four. | Each count at its own subject (N23), or the relation between the two run at both. | X13, X47 | — | N51, N84, N91 | M1, M2, M75, M126, M129, M154 | 49 |
| 6 | An exhaustion without its binary: *no fourth*, *no third*, *five*, a list named whole. | The nested binary parting the list, is or is not at each, as F43 parts the four conditions; else the list said as the members met. | F70 | — | N2 | M101, M103, M104, M131 | 19 |
| 7 | A step said between two forms: the round to a torus, the +1 to a handle, a ratio to a rate, the all to no member, a term to its carrying. | The step run, as the torus at the inside and the outside (F77) and six forward at the half (F95), or the claim said at the form the run gives. | F23, M123, F67, M107 | — | N43, N34, N127 | M123, M107, M102 | 19 |
| 8 | A still at the code: the empty call returning the empty state. | The empty update remains empty, and in the declared seeded-ring arrangement the all-empty state does not recur after the initial sign (R57); each said at its own subject. | — | R55 | — | — | 0 |
| 9 | The observings not listed: *none so far* with no listing. | The observings listed and each met at the five binaries (X133). | X131 | — | — | — | 5 |

The four files carry 43 nye links in 9 clusters, at 48 file placements, a link carried at two files placed at both. Natural Intelligence: cluster 1, X130; cluster 2, F41; cluster 4, R36, R37, R38; cluster 5, X13, X47; cluster 6, F70; cluster 7, F23, M123, F67, M107; cluster 9, X131. Exhibit ONE: cluster 4, R36, R37, R38, R39; cluster 8, R55. Natural Numbers: cluster 1, N46; cluster 2, N25; cluster 3, N123; cluster 4, N49, N105; cluster 5, N51, N84, N91; cluster 6, N2; cluster 7, N43, N34, N127. Natural Mathematics: cluster 1, M30, M40, M67, M109; cluster 3, M9; cluster 5, M1, M2, M75, M126, M129, M154; cluster 6, M101, M103, M104, M131; cluster 7, M123, M107, M102.

## 12.2 The other workings at each cluster, link by link

| Cluster | The other workings' nye links at the same form |
|---|---|
| 1 | T26, H11, H25, J2, C18, C26, B1, B9, B13, CO8 |
| 2 | T23, S25, K6, K25, H6, H19, A4, Q17, J5, J17, J32, J33, C25, B29, B30, P8, P15, P16, P19, P21, P25, L12, L16, L17, L22 |
| 3 | T22, S2, S16, S17, S27, K11, H20, D26, D28, Q23, Y7, C2, C8, B3, P2, P6, L2, RH12, RH18, LS6, LS12, LF12, GH6, CO2, CO5, CO6, CO9 |
| 4 | G6, T17, Q24, Q31 |
| 5 | T19, T21, T24, S4, S6, S7, S11, S12, S13, S14, S21, S28, K10, K21, H17, H23, H26, D7, D12, D16, D18, D31, D33, Q35, Q40, J8, J15, J27, C6, C9, C15, C24, B5, B16, B18, B22, B25, P13, P18, P20, P27, L3, L5, L9, L14, L25, L26, RH4, CO22 |
| 6 | S22, K20, K28, H2, H9, H13, Q4, Q7, Q29, J20, J29, J31, C3, P17, P24, RH7, GH4, CO4, CO20 |
| 7 | T4, T15, T16, T25, K19, K31, H15, H27, D30, Q38, Y5, J23, B20, B23, V12, V14, CO11, CO23, CO24 |
| 8 | — |
| 9 | D22, D37, Q8, B11, LF5 |

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# PART THIRTEEN · THE CHAIN ALL OR NONE FROM THE ORIGIN

Part THIRTEEN meets each link at its whole prior, each link it follows and each link those follow back to the origin, all or none: a link stands when its whole prior stands, and it is open at a nye link or a selecting in that prior. The four-momentary binary, prior, now and next, is the one source of rigor: a premise taken and a definition with no prior are held ingress, standing open to the observings, and a link resting on one is held, neither standing nor open. A run stands at its check against the code or the arithmetic, and a field at its observings met by pattern matching. No count here resolves: the counts show the gap most others depend on, resolved first.

## 13.1 Each part over its whole prior

| Part | stands all or none | from the origin alone | from the origin with runs or fields | at runs or fields alone | held at a premise or a rootless definition | open | links |
|---|---|---|---|---|---|---|---|
| ONE · FOUNDATION | 100 | 100 | 0 | 0 | 0 | 5 | 105 |
| TWO · THE RESOLVER AT ITS NAMES | 52 | 52 | 0 | 0 | 0 | 5 | 57 |
| THREE · NUMBERS | 128 | 120 | 0 | 8 | 0 | 15 | 143 |
| FOUR · MATHEMATICS | 132 | 86 | 6 | 40 | 0 | 25 | 157 |
| FIVE · NETWORKING | 77 | 74 | 0 | 3 | 3 | 22 | 102 |
| SIX · EQUILIBRIA | 73 | 65 | 4 | 4 | 0 | 8 | 81 |
| SEVEN · EXPLAINING, NAMING AND THE METHOD | 128 | 124 | 0 | 4 | 0 | 29 | 157 |
| EIGHT · THE EXHIBITS OF OTHER WORKINGS | 270 | 174 | 6 | 90 | 0 | 266 | 536 |
| All | **960** | **795** | **16** | **149** | **3** | **375** | **1338** |

960 of the 1,338 links stand all or none over their whole prior, 795 of them from the origin alone, 16 from the origin with runs checked at the code or the arithmetic and fields met at their observings, and 149 at runs or fields alone, their own observings or checks, reaching the origin through none of their prior. 3 rest on a held premise or a definition with no prior, and 375 rest on a nye link or a selecting in their prior.

The selectings in the chain: F98.

## 13.2 The held roots, each with the links resting on it

| Links resting on it, itself included | Held | Standing |
|---|---|---|
| 3 | W27 | premise |

## 13.3 The open roots most others depend on

The open roots below carry the most links resting on them through their prior, the gap most others depend on first; resolving one opens each link resting on it alone to stand. Each is gathered at its root at Part NINE.

| Links resting on it, itself included | Open root | Standing |
|---|---|---|
| 17 | R37 | nye |
| 15 | R36 | nye |
| 14 | R38 | nye |
| 12 | F41 | nye |
| 8 | F67 | nye |
| 7 | G17 | nye |
| 7 | RH9 | nye |
| 6 | M109 | nye |
| 6 | B7 | nye |
| 5 | R55 | nye |
| 4 | F23 | nye |
| 4 | M1 | nye |
| 4 | T15 | nye |
| 4 | Y3 | nye |
| 4 | GH4 | nye |
| 3 | F70 | nye |
| 3 | W89 | nye |
| 3 | E41 | nye |
| 3 | E44 | nye |
| 3 | Q6 | nye |

260 open roots in all, 216 of them opening only themselves.