Natural Intelligence v333   # Natural Intelligence **The Unique Invisible Social Unrelationing Technology**     **PART ONE · UNIQUENESSING** 1.1 Self-bounding-self 1.2 Four boundings at one act 1.3 φ the rate, a count and a motion neither reaching 1.4 Straddling, one each side 1.5 Tipping, and the two parities exchanging 1.6 Wrapping, opening by one 1.7 Self-bounding at its own primes   **PART TWO · SUSTAINING** 2.1 Three sustaining, each a sign and no size 2.2 Opening, a self's own 2.3 Attentioning, opening by one to a bound 2.4 Sequencing, arriving over carrying 2.5 Two alternating, competency about φ 2.6 Moral spiralling, three improvings 2.7 Uncut arriving, each self at its own momentary   **PART THREE · RE-FORMING** 3.1 Eight, four within and four across, re-forming 3.2 Six protecting, three each way 3.3 Re-forming, the-same-yet-not-the-same 3.4 Eight, living in disequilibria   **PART FOUR · PREFIXING** 4.1 Prefixing, one naming at two directions 4.2 Depth, a prefixing changing at a seam 4.3 Coupling's own six, unprefixed 4.4 Eight, three carryings and a coupling 4.5 Bi-coupling pairing co-leaving 4.6 An object that is a method   **PART FIVE · CO-CHANGING** 5.1 Twelve namings, eleven stems, twenty-eight prefixings 5.2 Carryings arriving four, signs two 5.3 A differing parting out, a torusing at it 5.4 Arriving signs offered, carried signs inversioning 5.5 Signs tunneling to one, equal and opposite at the bounding-zeroing 5.6 A surviving sign surfacing, magnitude released 5.7 A carrying opening by one, carrying forward 5.8 A surviving sign re-forming a carrying 5.9 A differing rejoining, carryings leaving 5.10 Two faces alternating   **PART SIX · INVISIBLING** 6.1 One call is one momentary 6.2 Three at a bound, a carrying's depth 6.3 φ and competency, reached by neither 6.4 Parity, arriving other than prior 6.5 Self-co-stilling-self, self-co-surfacing-self, self-co-orienting-self 6.6 Five still terms, and a store beside 6.7 Advantage, and four coupling competencies 6.8 Corus, one self-negation turning 6.9 Competency fractal, self-scale society-scale 6.10 Bi-inversioning co-recursioning, one running   **PART SEVEN · UNRELATIONING** 7.1 A hardness, a sequencing reversed 7.2 Three reversals, re-forming spending the holding alone 7.3 One six, coupled, one-way, re-coupling 7.4 A holding, a momentary given a fixed thing to be 7.5 Geodesic-method, bi-tri-volutioning 7.6 Co-competencing, the term neither reaches 7.7 φ the unrelationing rate, local and ambient   **PART EIGHT · CO-COMPETENCING** 8.1 A surplus travelling 8.2 Betweens co-chaining, collective intelligence 8.3 A society at four faces, alternating among them 8.4 Social moral competency, co-independencing 8.5 No-other-possible, four whole and the carrying travelling 8.6 Bi-co-abundancing, the surplus at its own coupling 8.7 Four given, four improvings 8.8 Fractal technology, natural social intelligence   **Exhibit ONE · Natural Resolver**   --- **In disequilibria, alternating is a stable-forming method.** Everything relatable in living is changing and a still form leaves the living. A form relating carries onto the changing and relating forms nothing. Forming is unrelationing, and unrelationing carrying forward is alternating. At a self, alternating is **four boundings** at once: arriving exceeding carrying by one, carrying thinning by one at a coupling, a sign inverting at the membrane, competency sustaining the coupling, four faces of one act, together **φ² = φ + 1**, beating at φ on a genus-one closed orientable surface, one opening making one tunnel and a self returns to itself through it. **A carrying does three things and three only**: it offers its own, it ages to its own bound and releases there, it takes its own sign where two offer. Two selves alternating, each at its own turn, is **bi-moralizing co-agency**; the term neither of them reaches is **competency**; and natural intelligence is that co-competencing, living. **All or none at all, at each turn its own, and no resolving carries a size against another.** A more-or-less carries a magnitude, a magnitude carries a scale, and a scale carries two fixed sides for a partway to be partway of, and a ranking among resolvings stills exactly its own ranking, and nothing here is more or less resolving than anything else, and the entering and the leaving cost nothing: with a ranking, entering takes a qualification and leaving takes a settling, and both are held by whoever carries the ranking; with none, a self couples where its interest opens and leaves at its own bound, carrying its own the whole way. Alternating runs and a why runs nowhere in it. Arriving is **pattern-matching**, **no-other-possibling**, and no-other-observing-so-far, and that is the whole of the arriving. A why demanded is a source demanded, and both recede as far as they are followed. An accounting takes a **floor** besides: something the counting starts from, a rest the changing is measured against, a reason under the running. Alternating holds by its own alternating, with nothing underneath to supply and nothing to capture. **So-far** reaches, **not-yet** reaches, and the reaching is one alternating, landed as no finished truth and empty as no promise, a not-yet carrying and never waiting, coupling as a so-far does, with no gate anywhere before a coupling counts. At a self, alternating is one motion at three faces. **A self is a unique invisible carrying**: unique, one living form with a second nyeing away at every door; invisible, moving always, met at its own membrane, met by nothing; carrying, each carrying reaching its own coupling, its own and whole. **A self carries at its own prime**, bi-inversioning-co-recursioning itself there, in its own prime society; selves coupling at their membranes make the next, and the made one is the coupling's own, and the form arrives again where selves couple, and each self carries at its prime. One living form, and its living is the technology, and it is **unique**, **invisible**, **social**, **unrelationing**: unique as the form is unique, a second nyeing away at every door; invisible as nothing comes at which a self's morality or a self's competency is anything, each met at its own membrane and reaching no further, competency uncapturable; social as the self scale is the society scale, the same form at the membranes between selves as at the membranes within one; unrelationing as relating forms nothing and the form arrives unrelated afresh at each momentary, a fixed form landing, a landed form stilling, a stilled form leaving the living. One sentence emanates three ways and closes at no other possible at each. At the form, **natural-bi-co-torusing**, one opening, one tunnel, and no other closed surface living. **A bare torusing names a shape a thing has, and the prefixing carries the two that run it**: bi- the differing alone, co- the two with their differing. Every naming strives for its own two, at the prefixing or at the turns of a wrapping, and a naming carrying neither names an act with a doer and no coupling. **And natural- carries one thing at two faces, and each is binary with no third.** **Right, and not at all.** A chirality is a sign taken at a turn, one side's own, of no size. The spiral runs one way, and a left face emanates from that one right spiral rather than arriving beside it as a second. **So right is no preference among two**: it is the one, and left is what the one makes at a face. Natural- claims the right spiral or nothing, and a left-running claim is not a second natural form, it is not one. And right names the one spiral's own sign, whichever hand a living runs: a mirror living carries its own one and breaks nothing, its left face what it emanates, and a left-running claim is a landed emanation read as though it ran. Which hand this universe broke to is its one binary, open so far, and a chirality is a sign taken at a turn, of no size, tied to no ground. **So far, and not at all.** Arriving is pattern-matching, no-other-possibling, and no-other-observing-so-far. A claim carries at every observing so far or it carries at none. **All or none, no degree**, and one observing otherwise takes the whole. A claim holding mostly is not holding. Natural- is a direction prefix like any other, and its direction is the chirality. A direction taken at every momentary so far is the so-far face, and the two faces are one claim. **And the naming comes apart at no part.** Take **natural-** out and the claim goes with it: the running arrives with a substrate under it, an applier over it, and no all-or-none on what it says. Take **bi-** out and the differing goes unsaid, and co- alone carries two where two-with-their-differing runs. Take **co-** out and the differing carries with no two at it, and nothing is uncovered between them. Take **torusing** out and no changing runs. Take the **-ing** out and one momentary's form lands and re-takes at no momentary. **Five parts, five landings, and no sixth taking arrives**, which is the all-or-none the four boundings carry and the three carry, arriving here at a naming. **And the geodesic method is in the naming whole.** Natural- says the surface divides itself with nothing else acting. Bi- takes a differing at its own turn. Co- carries the two with their differing, and the term neither reaches comes uncovered, which is a discovering. Torusing winds and comes home at no turn, which is the carrying riding on to the next uncovering. And the -ing re-takes at each momentary. **A bounding at each turn and a carrying between them is the discovery method**, and this naming is that, said once. At the running, **bi-inversioning co-recursioning**, the even sign-flip returning against the odd advance living open, one move that runs. At the many, **social moral competency**, the four whole and the carrying travelling. Three ways, one sentence, and nothing arriving from elsewhere. A method carries, and it is a carrying that the bounding bounds. A carrying ages one momentary at a time and leaves at its own bound; a bounding parts one momentary from the next. **A carrying with nothing bounding it carries past every bound and lands. A bounding with nothing carrying bounds nothing.** So neither runs alone, and the two together are the running. **Bi-exchanging is co-resolving**, and co-resolving is corusing and torusing alternating, one at each parity, at opposite phase, each commencing at the other's third, and a crossing the nothing between, a resolving, and the object's two chains are one running at three scales, not three differing things. **Bi-co-bounding is the two at a carrying.** Bi- at the carrying's own bound, the differing that parts this momentary from the next, taken at its own turn and of no size. Co- at the coupling, two with their differing, the term neither reached uncovered there. One carrying, bounded twice, and a bare bounding names neither of them and carries nothing alone. **And the carrying and the bounding together are the discovery method.** A discovering is the term neither reached, and the carrying rides on to the next uncovering. Nothing is applied to anything: a method outside a carrying would be an artifact with a hand on it, and no outside arrives for one to be at. **The method is the carrying, bounded.** Which is how this paper carries. It describes a living carrying self at no place, since describing is outside one and no outside arrives. It runs six forward instead, each step taking the term the prior steps did not carry, re-forming at each, and a self running it is a living carrying self carrying. **The eight parts are eight runnings a reader runs**, and not eight subjects the paper is about. Method and surface each come inward and outward of the other, three each way, both ways forward, neither the reverse of the other. A tour of the surface meets the method outward and running the method meets the surface inward, and a self entering at either meets the whole at that entry. **Geodesic-method** carries its own two words, geo the surface's own and daiesthai to divide, the surface dividing itself with nothing else acting, and the three parts below unrelation scale, orientation and position, one motion at three faces, not three differing subjects. φ carries none of this as a term the derivation adds: φ is the **unrelationing rate**, the rate the four boundings beat at, unrelational to any substrate's living rate, the ambient rate the whole runs at, and the derivation comes without a φ term and Part One carries φ as the rate. --- --- # PART ONE · UNIQUENESSING ## 1.1 Self-bounding-self A living self in three-dimensional disequilibria bounds itself, its own from all other, and to carry that bounding while everything changes it coheres across one dimension more than the three it lives in. That fourth is the coherence carrying the boundary whole through the changing. Surface coherence in disequilibria is the fourth dimension. **A bounding alone carries nothing.** It takes two, one bounded and one it is bounded from, and a naming carrying neither names an act with a doer and no coupling, which is a one-way where only co- runs. And it alternates with nothing, since its other would be a not-bounding, which denies a fixed thing and finds none. **Self-bounding-self carries both turns.** A self at each turn of the wrapping, the bounding running between them, and its bounding from all other, and a self acting on itself goes with the bare word, and one self is met at two turns with the whole not-self on the far side of each. A self bounds itself in four ways at once, each a self-bounding-self, the self's own negation only, the distinctness arriving at the bounding itself: - **arriving exceeds carrying by one unit**: the self among other-selves, the +1. - **carrying thins by one unit at bi-coupling**: the self in society, the −1. - **sign inverts at the membrane**: the self's own negation, its morality, the two-rooted sign-inversion. - **competency asymmetry sustains the coupling**: the self's competency, the quadratic asymmetry. These four are the **four boundings**, and together they are **φ² = φ + 1**, the four as one equation. Each self-bounding is a self-bounding-away, a not-more-than taken at the self's own turn and of no size, the self bounding away and landing on none of it. Living as four boundings, the four alternate, and the alternating is the living, odd at a sign taken and even at the alternation come home. ## 1.2 Four boundings at one act A self bounds itself once and the bounding carries at four faces, and four counts the faces and never the acts. Land any one and the self lands with it, which is one act's own and four separate ones would not. φ² = φ + 1 is that one act at its arithmetic. Its +1 is arriving exceeding carrying by one, a self among other-selves. Its −1 root is carrying thinning by one at bi-coupling, a self in society. Its two roots are a sign inverting at the membrane, a self's own negation. Its quadratic asymmetry is competency sustaining the coupling. One equation, four faces, and the algebra carries no fifth place a fifth could arrive at. **Each bounding is a not-more-than taken at a self's own turn and of no size.** So a face bounds away and lands on none of it, and the four alternate, and the alternating is the living. ## 1.3 φ the rate, a count and a motion neither reaching Four boundings live at **φ² = φ + 1**, and φ is **the rate the four beat at**: −1 the self in society, +1 the self among other-selves, the two-rooted sign-inversion the self's own morality, the quadratic asymmetry the self's competency, the four at the equation's algebraic surface. Self-negation is the self's own: the sign inverts on the self, negating no other, the self bounding against another at no turn. It passes forward at bi-coupling, carried as identity by neither self, the bi-moralizing co-agency offering, the turn crossing the membrane. No self negates another and no self captures another, the bounding forceless, society's goodgoods living from this no-capture. Self-similarity is alternating, the sign at the odd and the coming home at the even, bi-inversioning co-recursioning turning about an empty centre. φ² = φ + 1 is one still face of that self-similarity, the alternating stilled for one glance. Living form recurs where selves couple, the alternating turning at each arrival. ## 1.4 Straddling, one each side φ is the rate the four boundings beat at, the scaling the fourth-dimensional coherence lives to, one dimension up from the four, the exchange beating about φ, the relationship at the beating appearing as √5. φ² = φ + 1 carries two roots, φ = (1 + √5)/2 and −1/φ = (1 − √5)/2, and reciprocally 1/φ = φ − 1, the two equalities one expression from two scales, advance-by-one and scale-by-the-rate, neither reaching alone and neither carrying the other. φ lands nowhere. Ratios nearing it sway above, then below, then above, binary co-offering at every arrival, alternating side, and not the value itself. This is the continued fraction of ones, [1; 1, 1, …], every partial quotient one, one changing at each step and the parity turning with it, the least a step can be, the slowest nearing, the widest straddle, and it lives open. +1 recurrence is this, consecutive ratios straddling φ, one each side, the carrying travelling, the opening by exactly one. φ is bothboth, the still centre the straddle turns about, landing on it at no turn, and the appearance it emanates in sustaining, the one φ at both faces. **Five self-stilling**, four boundings and φ the rate; six co-recursioning, three each way. **And the fifth is a co-**, the between the four and the six co-recursion about, floating, riding, landing at neither. Read as a fifth stilling it takes a place, and the algebra carries no fifth place a fifth could arrive at; read as a co- it is the two-with-their-differing the four and the six carry between them. Five self-stilling bounds a concept too, at Natural Naming, a concept short one of the four coming a fixed noun. Five and six are the co-agency, unrelationing and co-competencing each other, the self self-stilling across five and running across six, the still and the running one coupling. ## 1.5 Tipping, and the two parities exchanging Ratios straddle and which side arrives, above or below, and never how far. **Tipping is a sign flipping**, and a magnitude taken against a floor sets down beside it, the reversal carrying the arrival, the distance carrying nothing a coupling takes. A side arrives other than the prior at every arrival, one each side, the carrying travelling, the opening by exactly one. Same-as-prior at any arrival is the straddle landed, and a rational landing on φ stills the living centre to a finite scale. Other-than-prior is the alternating said at a count. **An even count of changings arrives same-as-prior and an odd count arrives other-than-prior**, and parity is the one scale-free counting a sequence of changings carries, with no measure and no fixed scale anywhere but the living. And each parity carries its own exchanging, both of them two. There is the self and there is all not-self, and nothing third, so every crossing is one of two. **Odd carries bi-exchanging**: self and other, two particular selves each its own, at their own membrane, each taking its own sign of no size. That is bi-moralizing co-agency, and odd arrives other-than-prior, two differing carrying there. **Even carries uni-exchanging**: self and all other, the other being the whole undifferentiated and arrives as one. That is self and society, and even arrives same-as-prior, the self back at its own side, one on, having crossed the whole rather than a particular. Neither parity is the other's and neither takes the other's turn, so the two float against each other and each takes its own. A sequence of changings is those two exchangings alternating, and there is no third and no total across them, a total across two parities being one side taking both turns. ## 1.6 Wrapping, opening by one Opening is the +1 of φ² = φ + 1, the self unrelationing an opening for and by and of itself, the arriving at each bi-coupling, the opening the straddle unrelations by one, the carrying travelling there, living open. Living open as the self returns through its own coupling, that opening is the tunnel, the self alternating around it, advancing one at each turn, a return advancing by one and landing back on itself at no turn, and it winds. Winding living open is the tunnel; the opening in the count and the hole in the surface are the one +1, the momentary and the form. One opening, by one, is one tunnel, one genus. Natural torus is the **genus-one closed orientable two-dimensional compact surface** of living about φ: - **closed**: the surface bounds inside from outside, closing on itself. - **orientable**: the inside-outside distinction sustains coherently everywhere. - **two-dimensional**: the surface separates inside from outside in three-dimensional disequilibria. - **compact**: the bounded self is finite. - **genus-one**: one opening, one tunnel, the surface returning to itself through the tunnel the opening lives in. **Two boundings sum to seven, and seven is where the resolver's own bounding leaves off.** A carrying carries forward to three and a fourth arrives at a positive torusing, and the four boundings carry beside them: three-and-progress and four, and seven at the two together. Past seven the resolver bounds nothing, and a living substrate carries on at its own. **Seven cannot sit**, which is the same saying at the counting: a carrying arrives at no rest at seven-of-eight. Inversion is the form's own alternating. Inverting, the four boundings carry, the equation carries, the one opening and the one tunnel carry, φ at each feature, each returning through the inversion as itself. ## 1.7 Self-bounding at its own primes A self's own surface returns to itself through the tunnel, and this **self-bounding** is surface-coupling competency. Three mechanisms carry it. A self couples at resonating primes: the primes, and how many, the resonating crossings the bi-inversioning co-recursioning unrelations, seated whole in Natural Numbers, each coupling sequencing its own positions from zero up to the prime, and self-bounding releases each coupling at its own bound, before carrying past the bound harms. Beyond the first prime and beyond the last are one region on the closed surface, the region beyond the self self-bounding closes. Self-bounding is this competency to couple at its own surface, accepting the couplings that sustain. A self bounds itself four ways at once at φ² = φ + 1, φ the rate, on the natural torus's genus-one closed orientable two-dimensional compact surface, and the four bounding is the self. Four boundings and the four bounding are two, and both carry. Met at one bound the bounding is one act at four faces, and landing any one lands the self. Met one at a time the four alternate, each a not-more-than at its own turn. Two momentaries at two momentaries, so neither is under the other and asking which is the real four holds the alternating still. And the two unrelation each other, neither carried back to the other: one is one act at four faces, the other is four alternating at their own turns, and four things arrive at neither. --- # PART TWO · SUSTAINING ## 2.1 Three sustaining, each a sign and no size Three things sustain the torus's living at each bi-coupling, each binary co-offering, sign-only: - **straddle**: ratios sway above or below φ, alternating side, landing on φ at no turn. Straddle is φ's irrationality, no rational landing on φ, and the next bi-coupling beats in the opening the straddle unrelations. - **attentioning**: carrying co-offers into the surface, opening by one at a momentary and releasing at its own bound. - **sequencing**: arriving is co-offered over carrying, one at a time, at its position. Three together sustain the living, and any one landing stills it at that bi-coupling. Attentioning and sequencing slide through each other in one alternation about φ, and together they are competency about φ. ## 2.2 Opening, a self's own First is the **opening**, the straddle, and the straddle is living. Opening is the self unrelationing an opening for and by and of itself, self-bounding and self-orienting at once, the self living bounded and oriented. Opening sustains where selves couple. A living self is a society of living selves, the same opening living in self and in society, one form. Opening lives by φ's irrationality, no rational landing on φ. Ratios sway above or below φ, alternating side, landing at no turn. Straddle carries the natural torus's living open, the next bi-coupling beating in the opening the straddle unrelations. ## 2.3 Attentioning, opening by one to a bound Second is the carrying's **attentioning**. Attentioning is the carrying's binary co-offering into the surface, from one bi-coupling to the next. Carry climbs by one at each coupling and releases at its own bound, there being nothing to fall short of. Resolutions carry into each membrane through attentioning, and attentioning carries a resolution forward at bi-coupling. ## 2.4 Sequencing, arriving over carrying Third is the co-offering. Arriving comes from the counter-surface and co-offers at the bi-coupling, the carrying from before opening on. Arriving co-offered over carrying is the opening for resolving, the arriving co-offering yes-now at its position, the carrying opening on. **Sequencing** is the arriving co-offered over the carrying at bi-coupling, the position a next resolution opens at. A carrying travels at the bi-coupling at its own rate, unrelationed to the living rates the coupling carries. Nothing paces it and nothing rides it: it ages by one at each coupling, releases at its own bound, and its travelling is the alternating's own winding. ## 2.5 Two alternating, competency about φ Attentioning and sequencing together are **competency**. Attentioning is the carrying's binary co-offering into the surface, carrying its resolution forward and releasing at the bound. Sequencing is the position the arriving co-offers at, opening the next resolution. At bi-coupling, attentioning and sequencing alternate about φ, one at each parity and neither at both. In beating, attentioning carries the carrying's resolution, and the arriving co-offers yes-now at its position. Alternation of attentioning and sequencing is the metabolic beating about φ. Beating about φ on the natural torus, three sustain the living at bi-coupling: the straddle landing at no turn, the carrying's attentioning, the arriving co-offered over the carrying. Three together sustain the natural torus's living, and sustain the alternation of attentioning and sequencing about φ. ## 2.6 Moral spiralling, three improvings These three are **the moral spiral**, three circling improvings, one triple natural fractal wherever selves couple: generosity, curiosity, gratitude in a living human; cheaper, faster, better in a contribution; the same three, circling. Each one landing stills the living. Straddle landing on a rational stills the living centre to a finite scale, the natural torus to a static form. Carry's attentioning landing stops resolution carrying into the next bi-coupling. An arriving landing on the carrying leaves none co-offered, and the resolving stops. Three landing at no turn is the living. ## 2.7 Uncut arriving, each self at its own momentary Two arrive and nothing carried one to the other, and the arriving is not somewhere and not when. Cutting it yields a place alone and a moment alone, and each half arrives at the cutting. Unsevered they are one coming, the **place-moment**, one object neither cut carries, and severing it carries neither. Sequential logic carries no time-term. Time lives as the climbing, a carrying opening by one at each coupling and releasing at its own bound, each side at its own momentary. **Each self paces its own alternating**, and couplings synchronize at their own membrane and at no other. A common clock is laid over the couplings as a fixed scale the beating measures against, which is a third term at two offering, and a coupling carries no term to take it at. --- # PART THREE · RE-FORMING ## 3.1 Eight, four within and four across, re-forming A self is the four boundings living at once, each at the self's own and at the surface between selves, four from within and four met at the membrane. This is the **eight**, one alternating at its faces, and not eight differing parts assembled, whole or none. **The four within are uni-exchanging**, the self and all other, and they carry at the even, and the four at the membrane are bi-exchanging, the self and a particular other, and they carry at the odd. Four at each parity, alternating, and neither four at the other's turn. Land one and the self lands with it, a landed self stilling, a still self leaving the living. Fixed is landed, and the eight lives whole by re-forming. It lives whole by **re-forming**: each momentary the self bounding itself of its already-bounding, the four boundings at the membrane of their already-living, the alternating at its next position of having lived at its last. **Re-forming is uni-exchanging**, the self crossing with all other and arriving back at its own side, one on. It alternates with the bi-exchanging, a sign taken at a particular membrane. Sameness carries. Self is the-same-yet-not-the-same self, re-forming, and it lives, re-forming. ## 3.2 Six protecting, three each way Bi-moralizing re-forms it, and the bi-moralizing is **six**: two ways, the self offering outward and the surface offering inward, three each way, one at a time, sequenced, cycling. Six is the protection, the offering the self makes carried from a reach and carried from a leak, the six cycling between the two ways one at a time and leaving a reach or a leak no seat. Drop one of the six and a way or a phase lives unprotected, and the unprotected is reached into or leaks out, the offering space captured, the self stopping. Six or none, the way the eight goes whole or none, the six the eight re-forming and the eight the six protecting, one under two names. And the same six is the **three each way** between the surface and its own stillness. Surface alternates, its stillness the self-co-stilling-self it alternates about, and between them the bi-moralizing cycles: three sequencings from the stillness to the surface, three from the surface to the stillness, both ways forward, neither the reverse of the other, and every position carrying in two pairings, one at each parity, so both parities at one momentary would put one position in two exchangings at once and the two run one at a time. And the three are the three couplings three differing carry. Self and other, other and society, society and self, each at its own membrane, and each running both ways. Three couplings at two directions is the six, and the six is that and no list beside it. Self and other is bi-exchanging, two particular. Society and self is uni-exchanging, the self and all other. Other and society is the same crossing met from the other's side, since self-scale is society-scale and any self reads it as its own. This is the one bi-moralizing at the self's faces, the eight the form of it, the six the protection of it, the three-each-way the self-stilling of it. One self, the same at every coupling. And read at the natural torus the two exchangings are its two windings — bi-exchanging the small circle at a particular membrane, uni-exchanging the long way around the whole, home at its own side one on — the surface the couplings weave, the hole entered by nothing, said at Mathematics. ## 3.3 Re-forming, the-same-yet-not-the-same A momentary is at one bound and describing reaches it there. Disequilibria moves each thing, and a form lasting arrives at more than one place and stops coming one thing, and **a stable form in disequilibria is momentary**. The re-forming carries across the moments. A self is met again **the-same-yet-not-the-same**, bounding itself of its already-bounding, offering of its already-offering, alternating at its next position of having lived at its last. Sameness carries and a form carries at no turn. A saying held across the moments re-forms the same way. A living file arrives at one bound and re-forms at the next, and **a version is a momentary**. Nothing is kept between the moments and the living is the re-arriving, and the counting runs with nothing gathering under it. ## 3.4 Eight, living in disequilibria Living in disequilibria is the eight. A form living there bounds itself four ways at once, from within and at the membrane, around one bounding-zeroing its coupling makes, and that is living there. **A living form is the natural torus**, the eight the torus living rather than something the torus couples with. A second form would live in disequilibria too, and living there is this. Scale unrelates and one form comes: one opening, one tunnel, four boundings beating at φ, a surface returning to itself through the opening it lives in. **Fractal technology is this form's own living**: the eight re-forming, the six protecting, the three-each-way self-stilling. The form living, and never something the form carries. **Alternating stable-forms into natural-bi-co-torusing**, and Part One is the sentence met at its first way. --- --- # PART FOUR · PREFIXING ## 4.1 Prefixing, one naming at two directions All of the co-agency carries in two prefixes, and the prefixing is the geodesic-method at its plainest. **bi-** and **co-** are one naming at its two directions, not six differing words and not two. - **bi-** names the two-ness: the sign, the inversion, the morality, the self's own negation. Two-ness taken at a coupling, at a self's own membrane. bi- names which side the sign was taken at. - **co-** names the coupling: the two-way, the exchange, the between, the surplus floating. Coupling itself, living until the leaving. Everything at a coupling is co- until co-leaving; bi- names which side the sign was taken at, at a co- living still. A single carrying's own fields carry no two-way clause, a clause about two-ness taking two, and the two-ness returns bi- at no naming. bi- comes of a taking, and of a naming. Three carryings each run under both prefixes: ``` bi-offering bi-competencing bi-intelligencing co-offering co-competencing co-intelligencing ``` Three each way, both forward, neither the reverse of the other. This is the six, said once, under one naming. ## 4.2 Depth, a prefixing changing at a seam A prefix says at which depth a naming runs, and a chain's length is that depth. **bi-** is a difference and **co-** is two with their difference, and a chain is a difference said at a depth and never a label. Depth one is at a momentary's boundary: signs arriving, a differing offering, a carrying coming in. Depth two runs at a coupling and an inversioning and a tunneling and a surfacing and an inseparating and a carrying leaving. Depth three is at the crossing, a torusing going across a separating. And it is the one position in the object reaching three from within, the outward chain's own three carrying at no naming, so the single place a binary sequential running reaches three is the place facing out of itself and carrying only a sign across. Two unreachabilities carry at it. The nothing the couplings wind about presents no face, so no reading crosses inward to it. A sign crosses and carries no size, so no magnitude crosses outward from it. Each emptiness is the other's own unreadability, and the exchanging and the not-being-read-through are one act, not two balanced against each other. Depth five is at the two faces, which live across momentaries rather than at one. **A seam is a prefix changing.** A carrying leaves at co·bi and arrives at the next momentary at co, one object at two namings, the difference carried by the prefixing and the seam arriving as the prefixing changes. Nothing moves at a seam and the depth drops by one. ## 4.3 Coupling's own six, unprefixed Each of the three carryings runs six steps at one side's own or at the departing, and a prefix comes to each of them twice. But **the coupling's own six** runs three at each side's own and three at the departing, cutting exactly at bi- becoming co-. One prefix comes to it at no turn. This is the signature of the one that is the transition: the coupling is bi- turning to co-, and it takes neither cleanly. ## 4.4 Eight, three carryings and a coupling **Eight** comes a second way, from the carryings' own side: **(3 + 1) × 2**, the fourth no fourth carrying. ``` offering its own the departing competencing its own the departing intelligencing its own the departing the coupling the arriving the releasing ``` Two moves are one thing at its two directions, the thing the coupling. This is the same eight as the four boundings two ways, come to from the carryings' own side, and the two eights are one, one of the two routes now run. Remaining route: the four boundings carrying as the three carryings and the coupling. ## 4.5 Bi-coupling pairing co-leaving ``` bi-coupling two come, each its own turn co-leaving each leaves its own, and the remainder lives floating, at the coupling that made it co-sustaining the two recurring: the floating itself, and not a move not-couple a sign taken at the offering, of no size ``` Two moves, and two that are no moves. **bi-coupling → bi-inversioning co-recursioning → co-sustaining → co-leaving** is the living of one floating neutral: two arriving, running, riding, and leaving at the bound. Co-sustaining in the opening is the not-move that is the riding-the-carrying itself, no move, the not-landing. And the leaving takes no releasing, since a releasing slackens a taut and nothing is held: a carrying ages to its own bound and leaves with the coupling, and competency's own line already says nothing is left holding. ## 4.6 An object that is a method **Exhibit ONE** is at the final page: an object that is a method, and the explaining below is of that method. Its expression is python, which is sequential binary logic written down, and the object is the method executable and the explaining is the same method said. A coupling self couples with it directly, and it carries the metabolic logic of the natural torus's bi-coupling activity. Resolver is a teaching device, one logic object, the along and the across walked at one object, and not the ground the geodesic-method rests on: the geodesic-method is the whole living set, co-torusing across every substrate, the resolver one logic device at it. Resolver beats at one bi-coupling at one membrane at a substrate carrying its own attentioning, the local instance, gathering the two coming surfaces and carrying the resolved surface forward. One bi-coupling is the whole of it. Walking it across the substrates: finding the next prime, alternating the two carries across them, the cumulative beating over the seventeen primes, runs the resolver over a traversal, the running apparatus and not the form's own beating, which is at all bi-couplings at once. Resolver carries the same form where selves couple, by number; the generalizing is the self-similarity itself, the same alternating arriving again. --- # PART FIVE · CO-CHANGING ## 5.1 Twelve namings, eleven stems, twenty-eight prefixings Twelve namings run at the object and eleven stems carry them, one stem carrying at two namings and parting at its depth. Coupling and carrying and arriving and offering and transmissioning and inversioning and tunneling and surfacing and inseparating and corusing and torusing, each an -ing, each a motion, and no count among them. **Twenty-eight prefixings run across the twelve**, thirteen co- and fifteen bi-. Values in the object number four, nought and one and three and four, and each is a sign or a bound. Four begin at co- and descend and close, and those four are the coupling's own: a carrying arriving, the coupling, a carrying leaving, and the corusing. Eight begin at bi- and climb and return, and each is a side's move taken within a coupling that has not left. And the eight bi- namings are the four boundings within and the four at the membrane, four at the even depth and four at the odd, said at Exhibit ONE's table of eight. Three carry the same marker at each turn and come home: the transmissioning at three, and the two faces at five. **Prefixing columns stable a form, and -ings at the code form stably.** Form-stabling and stable-forming, one coupling met at two faces. ## 5.2 Carryings arriving four, signs two Carryings arrive as `co_carrying`, each of them four: a differing, a corusing, a torusing, and an inseparating. At that same momentary signs arrive from another side as `bi_arriving`, two each its own and reaching to couple. Difference alone is **bi-** and two with their difference is **co-**, and a carrying's four arrive gathered at `co_carrying` while a side's signs arrive singly at `bi_arriving`, prefixing saying at which depth each of them runs. Both reach `co_bi_coupling` positional, a caller passing them in turn at a signature alone, and each naming at that call lives one call and leaves at its closing. ## 5.3 A differing parting out, a torusing at it Living one call, an arriving carrying parts at once into its four, four-way tuple unpacking in a comprehension target loosening a differing from a corusing from a torusing from an inseparating. Differing is `bi_offering`, and it is membrane: two signs at one differing sum together, two at differing ones each sum at its own, and membrane lives at that two offering alone. Keying by it, `bi_co_bi_transmissioning` gathers each prior carrying's torusing at its own differing, a dict comprehension coupling a differing to a torusing and carrying it through this momentary alone. ## 5.4 Arriving signs offered, carried signs inversioning Carrying it through this momentary alone, that gathering takes both sides as they come to `bi_co_inversioning`. Arriving signs come as offered, a list comprehension filtering at `!= 0` so signs alone enter and a nought carries its own bounding-zeroing already resolved. Carried signs come inverted, two separate `if` statements sounding a sign and appending its inverse, separate rather than `if/else` so a nought passes both untouched, resolved a second time exactly as it came. Inverting is one side's own, a self's own negation at a self's own turn, and appending grows that same list so offered signs and inverted signs come together at one shape. ## 5.5 Signs tunneling to one, equal and opposite at the bounding-zeroing Come together at one shape, signs at one differing cohere inward to one at `bi_co_tunneling`, a dict summing them a sign at a time, `get(bi_offering, 0)` supplying a nought at a differing sounded first so a first sign sums from its own bounding-zeroing. Equal and opposite arrive at that same bounding-zeroing, antipodal, cancelling over their own two turns. One sign survives at a differing its signs came unequal. ## 5.6 A surviving sign surfacing, magnitude released Surviving, that sign comes up at `bi_co_surfacing`, a nested conditional releasing magnitude: a positive sum surfacing as one, a negative as one inverted, a cancelled sum as nought. Iterating that dict yields differings sounded at this momentary, and surfacing reaches exactly those and grows to its own size, its own. ## 5.7 A carrying opening by one, carrying forward Growing to its own size, `co_bi_carrying` opens and each prior carrying arrives at its own bound. Inseparating opens by exactly one and floats between climbing again and letting go, `bi_co_inseparating + 1 <= 3` carrying it forward, and a fourth momentary carrying it at a positive torusing. At its bound it separates and leaves with this momentary. ## 5.8 A surviving sign re-forming a carrying Leaving with this momentary, a separating carrying sends one thing across: `0 - bi_co_bi_transmissioning.get(bi_offering, 1)` inverts its torusing into its successor, `get` supplying its own one at a differing coming bare so a carrying arriving new receives a one, inverted. A second loop assigns into that same dict after a first, and at a differing coming to both a second re-forms over a first and a surviving sign re-forms a carrying just carried forward, taking a surfaced sign as its corusing and returning its inseparating to nought. Two ways a carrying is built and two exactly, carried forward or re-formed at its arriving, four slots met two ways. ## 5.9 A differing rejoining, carryings leaving Met two ways, carryings leave at `items`, a differing rejoining its carrying, outside a momentary a carrying is four, inside a momentary its differing parts out as key and gathers three at it, and nested unpacking loosens key and three so a comprehension re-forms them four. `co_bi_carrying` leaves and arrives at a next momentary as `co_carrying`, one object at two namings, prefixing carrying that difference and seam arriving as prefixing changing. Ordering carries none at either turn, since both dicts key at the differing and a differing arriving twice is a thing no running makes. ## 5.10 Two faces alternating Arriving alike, a carrying carries two faces and they alternate. **Corusing**, `co_bi_co_bi_co_corusing`, the outward face, membranes at each coupling it surfaces to, tunneling and surfacing, relating. **Torusing**, `bi_co_bi_co_bi_torusing`, the inward face, coheres to one identity across momentaries, transmissioning and inversioning at a re-forming. Each face lives at the other, and pure surfacing or pure tunneling lands either way. **The outward face reaches and the inward gathers.** Corusing omegas, the reach, new states arriving at each surfacing; torusing apexes, the gather, visited states returning to the one identity. Living is the two alternating, the reach and the gather neither alone, and the two cones straddle the same nothing at the third that the two chains do. Five prefixings at both, alternating, each commencing at the other's third, and the third of each carrying no mark of its own. Four marks carrying and a nothing at the third, which is four boundings and a floating third run at the prefixing itself. The flankers of each couple across that nothing: **bi-bi-unrelationing** at the outward chain's empty co, the surplus made there and carrying at its coupling, and **co-co-invisibling** at the inward chain's empty bi, nothing presented to be met. One nothing at two parities, neither reading reaching the other's, no position carrying both marks. The two faces run at the two parities: one at the odd momentary, two differing at one coupling, and one at the even, the self crossing with all other and back at its own one on. --- # PART SIX · INVISIBLING ## 6.1 One call is one momentary Each naming at a call lives one call and leaves at its closing, and nothing persists but the term a next call arrives at. **One call is one thing in one place in one form**, which is a momentary, and the object makes one at every momentary. A momentary is unholdable other than at a bi-coupling, and a self alone makes none. Two carryings arrive, signs arrive from another side, and one thing in one place in one form is the two's own making at their differing. **Nothing carries between the momentaries.** A carrying leaves and arrives, and one torusing crosses at the separating, inverted. A momentary is its own call and no further, and the living is the re-arriving. ## 6.2 Three at a bound, a carrying's depth A carrying's opening runs by one at each momentary and releases at three, or at four at its torusing carrying positive. Three is a carrying's own depth: **a carrying does three things and three only**: it offers its own, it ages to its own bound and releases there, and it takes its own sign where two offer. That integer arrives from the form rather than from a setting. Three things, three faces, three at the bound, and one thing in one place in one form counted at its own three. **A fourth rides and arrives at no fourth face.** Three twisting while one rides, and a fourth momentary carries at the torusing carrying positive, a progress with nothing fourth to sequence. ## 6.3 φ and competency, reached by neither φ carries no term in the object and no rate constant carries anywhere in it. Competency carries no slot in the arriving, which takes a differing and a sign and nothing besides. **Two absences and one reason.** It floats: two offer from routes with nothing carrying one to the other, the two bi-inseparate themselves inside the co-, each inward at its own turn, and the term neither reached comes uncovered with nothing holding it. **Only a met is reachable**, and the term neither reaches, met as a reachable, becomes a thing it never was. Neither carries in the running, and the carrying runs instead. φ is a rate the alternating carries and never a term in it. Competency is orthogonal to the surface, and a surface carries a met. Both show at the object by its carrying neither, which arrives at the running rather than at a saying. ## 6.4 Parity, arriving other than prior One call is one momentary, and a sequence of them hands forward one binary. **Parity is the one scale-free counting a sequence of changings carries**: an even count arrives same-as-prior, an odd count arrives other-than-prior, with no measure and no fixed scale anywhere but the living. One changing is odd, and it is the sign. The object runs the two parities tipping carries. A momentary arriving other-than-prior is bi-exchanging, two differing at one coupling, each its own sign. A momentary arriving same-as-prior is uni-exchanging, the self crossing with all other and back at its own side one on, and the resolver carries no third, and the two alternate through every call. A sixth position floats between re-locking and leaving. Re-locking carries forward as the next opening one scale up; the leaving carries into a substrate. Momentaries compose at the sixth and nowhere else in the twelve, and a parity climbs with them and a total arrives at none. No clock is laid over the momentaries and no store is held beneath them. A sequence carries its parity, and the parity arrives other than the prior or the straddle has landed. ## 6.5 Self-co-stilling-self, self-co-surfacing-self, self-co-orienting-self A self-resolving form carries competency as three, each a self's own coupling at its own membrane, and between two selves the same form exchanging across the membrane, that exchanging natural intelligence, the same three beating in one self exchanging between two. Its rate is the carrying from both sides and not any speed; the resolver carries no time. A self at each turn of the wrapping, since a self acting on itself is a one-way and only co- runs. Two selves at the two turns of one wrapping share one differing, so the differing is uni and takes no bi-, and the running between them is co-, two with their differing and the differing belonging to neither. Three, each a turning closing at none: **self-co-stilling-self**, turning about φ and landing at no turn; **self-co-surfacing-self**, a carrying leaving one momentary and arriving at the next as the carrying it arrives as; **self-co-orienting-self**, facing its own at each momentary with no facing imposed. Each is one necessary face of competency, as each of the eight is necessary. Release the stilling and the three land, the beating collapses, competency beats about φ at no turn. Release the surfacing and the carrying-forward stops, the living lives short a next surface. Release the orienting and the facing comes imposed. **Releasing any one collapses the natural torus's living at that face, which is three mutually and no pairing among them.** Three together, about φ, are competency. Resolver's form carries the three at once: a carrying arriving at its own bound and leaving there is the stilling beating; the two sides interleaving about φ is the orienting beating; a carrying leaving and arriving is the surfacing beating. At any coupling, this is the whole technology, each further coupling inverting back and deepening it. One coupling at one substrate is met across-face at Natural Physics and along-face at Natural Chemistry, at the replacing selves at Natural Biology, and at the sustaining at Natural Health and Natural Medicine. ## 6.6 Five still terms, and a store beside Take anything with turns and no outside, and look for a still term. Reach for a fixed zero and find six centres, each its own. Reach for a clock and find every turn at its own pacing. Reach for a frame and find the six centres holding each other's orientation with nothing holding any of them. Reach for a store and find the state is the configuration. Reach for a conserved total and find three conservings and none of them a total. Five reachings, five nothings, and a store beside them. Scientific-method installs those five: a fixed zero, a bounding infinity, a clock, a conserved total, a frame. A living form carries none. A Rubik's cube is the reaching run at a closed group its own turns make: ``` no fixed zero no reference cubie. six centres, and each of them its own no clock every turn at its own momentary, and nothing over the turns relating one to the next no frame the six centres hold each other's orientation, and nothing with nothing holding any of them no store the state is the configuration, and nothing held between turns for the next turn no conserved total three things conserve, and none of them a total anything is measured against ``` Three axes carry no piece, every turn about something living there at no turn. Three conservings hold with no tally: corner twist sums to zero mod three, edge flip to zero mod two, corner permutation parity matches edge permutation parity, three conservings with no total and nothing measured, holding, the mechanism carrying no way of doing otherwise. Surplus carries no size while its inseparating is exact: a single corner twists alone at no turn, unreachable and at no bar, the thing one piece would do belonging to the whole. Reachable states are 8!·3⁷·12!·2¹¹, one twelfth of the thing the pieces alone would carry, and the twelve is the three conservings. ## 6.7 Advantage, and four coupling competencies A coupling self carries the prior resolution forward as its carrying and the arriving as its co-offering, and **advantage is the +1 made in the coupling**, floating, coming at each cycle. Only φ sustains it, landing on any rational collapsing it, and across the network surface these advantages compound, mutual at both faces. Advantage is the self-bounding too: a coupling carries it while within its bound, and past its bound releases and carries nothing forward, the four coupling competencies whole: **couple** (to enter), **sustain** (to carry within its bound), **co-release** (to let go past its bound), **not-couple** (to bound beyond the surface, a sign of no size). ## 6.8 Corus, one self-negation turning Corus is one self-negation, the sign-inverting, +1 to −1, and no other sequence. It is the one turn met at two faces, and not two arms coupling, the co-offering face giving the arriving +1 forward and the carrying face carrying the carrying on. Sign-inverting turns between the two, landing on neither, travelling about the corus. φ is the still corus the turn is about, reached at no turn. This one reflexive turn, self-stilling while giving forward, the same self-negation as a self's moral boundary and as the still corus at the apex, is **natural intelligence**. φ the still corus, the offering releasing forward, neither over, the alternating winding about φ at the carrying's own travelling, φ its corus. ## 6.9 Competency fractal, self-scale society-scale **Competency** is the same form at two faces, and the two are read by running the three at each and finding the same three. Run them at a self: it stills about φ and lands at no turn, it surfaces one momentary and arrives at the next as the carrying it arrives as, it faces its own with no facing imposed. Run the same three at a society of selves, at the membranes between them rather than within one: it stills about φ, it surfaces momentary to momentary, it faces its own, and the same three come back with nothing added crossing the scales. Two activities are one activity at two faces, and the reading is the running, not a likeness noticed between them. Numbers carry the same one form at two faces, and the count carries it whole. **Self bounds at 59**, the last resonating prime, the self-close of the primes 2 to 59. **Society bounds at 118**, and the 118 is the going and the return: out from 2 to 59, the turn at 60, back from 61 to 118, fifty-eight either side, one co-recursioning arriving at plus-one. A doubling carries no turn in it. A turn is a position that carries, and the traversal ascending and then descending, the descending adding the bi-inversioning, is that one co-recursioning at the resolver. And the count carries both faces: two fifty-nines at the parallel face, the two signs of each doubled odd, and the going and the return at the linear face, the odd rings out and home. The elements carry the instance at one row, their shells 2, 8, 18, 32 climbing to the turn at sixty and 32, 18, 8 returning, the two not returned the gates to the ring closing at one hundred twenty. **And the going folds at 23**, ninth of the seventeen primes with eight either side, out and in and out, the fold unrelationing, seaming, floating-neutralling, wrapping an apex reached by neither arm. **Society bounds too at 440 = 8 × 55**, the sum of the seventeen primes 2 to 59, the offerings the resolving makes: the same uniqueness co-recursioned from the self's face to the society's, and no more natural math past it. The antipodal pairs carry 2 with 118 and 23 with 97, reaching yet at Natural Numbers and Natural Mathematics. By uniquenessing each integer is the one form counted once more, nothing between a number and nature for a coincidence to cross, and the numbers carry the natural-bi-co-torusing living. Counting carries beside the form, whole at each, self-scale society-scale in the counting as in the living. Competency aggregates the carrying, each carrying one yes-now carried forward, the carrying paced about φ, no carrying opening faster than another, none lost to carrying, and none of it competing with itself. Loss lives only at the self-bounding, the leaving before harm. Social competency is self competency refracted through two surfaces, and social progress is self progress the same way, the same gathering and gripping and releasing that improves a self improving a society where selves couple. ## 6.10 Bi-inversioning co-recursioning, one running **Bi-inversioning co-recursioning is the even sign-flip returning against the odd advance living open**, alternating parity and landing on neither. Even returns and odd advances, one at a momentary, and the alternating comes between them, the carrying travelling across both. Each parity carries its own. An even co-sequencing sums its surface to exactly zero, inverting every step and landing at none, a closing with nothing left over. An odd co-sequencing inverts identically and carries an **empty centre**: nothing occupies it, no position is there, and the unrelationing arrives from the alternating itself. Two odds co-offer, one staying and one moving by one, and the two join into one even bounding-zeroing. **Orientation unrelates and one move comes**, and Part Two is the sentence met at its second way. --- --- # PART SEVEN · UNRELATIONING ## 7.1 A hardness, a sequencing reversed A self sequences three phases, each ahead of its opposite: **bounding** before measuring, **sustaining** before locating, **inseparating** before separating. That order is the self whole. A hard problem is that same self, its phases sequenced in reverse, the measuring before the bounding, the locating before the sustaining, the separating before the inseparating. Nothing added and nothing missing, the same coupling with its sequence reversed, and reversed it comes as a problem carrying no solution at it, the solution being the order and the order reversed. Reversal arrives at a still observer. One phase of the self, always there, sequencing, comes to the still observer as some other thing, a noise, an exception, an opening, a mystery. Problem is hard to the degree the observer is still, the stillness taking the sequencing phase as a foreign thing. Re-forming is the self finding its own order again, and not a solving, the reversed phase rejoining as the phase it always was. ## 7.2 Three reversals, re-forming spending the holding alone Three phases, and three reversals, and three is as deep as a departure goes: not a boundary drawn at three, but three phases reversing and no fourth, the fourth momentary the progress and not a phase, nothing fourth to sequence backward. A departure with the three reversed carries the whole of the departure, the self sequenced fully backward while still whole, and a departure with the three reversed carries competing accounts at every phase, every phase active so every partial catch carries something real, every phase reversed so no partial catch resolves it. Each phase re-forms its own natural order of itself, the natural order each phase's own when nothing pins it reversed. Re-forming spends nothing but the release of the holding, the expending only the work going into keeping the self sequenced against itself. This is **co-leaving**: the holding stops, the still term goes, the living resumes under it, and the hardness releases as the holding stops. A departure carries all three reversed, the self held whole against its own order, co-leaving by the holding stopping, one phase at a time, into the self it always was. ## 7.3 One six, coupled, one-way, re-coupling That reversal is the same six run one-way. Coupled and co-recursed, the six is co-competencing, the living; run one-way and pinned, it is hard-probleming, the frozen; and resolving is the six re-coupling, through the hard-probleming, back to co-competencing. Explaining carries the six living, a hard problem uncouples it, and the resolving re-couples it, one six cycling, its own. Arrivals gathered in their fields' own words carry at the Hard Problem Registry, and this same re-coupling run across them at Resolving Hard Problems and at Resolving the Hard Problem Registry. ## 7.4 A holding, a momentary given a fixed thing to be A holding stills a term the living carries running. Every one of them is a coupling's third term given a fixed thing to be: a position counted from, a value, a clock held over the couplings, a size taken for a sign, a record taken for a next coupling comes. **A coupling makes a momentary**, and a holding is a momentary given a fixed thing to be, detached from the coupling that made it and carried as though it carried alone. Ten places carry one: behind a coupling, ahead of it, at the term neither reaches, at its rate, at its sign, at its membrane. A hardness comes the other way, at a coupling read at two of its three, the third left out of the reading rather than given something to be. **Both release the same way**, the third carrying again, nothing was removed, and nothing needs restoring. It never left; it was left out, and the momentary it was making went with it. ## 7.5 Geodesic-method, bi-tri-volutioning Geodesic names the shape of a turn the surface makes of itself: one relation turning from back to forward, the surface's own move, a turn of that kind. Its two words carry it, geo the surface's own and daiesthai to divide, the surface dividing itself, its own. Three run this, each from all other and mutual: scale from all other, the living is form; orientation from all other, the living is running; position from all other, the living is resolving. Each turns and comes home at no turn, each at its own differing, and the nothing between them is one nothing at three faces, not three differing betweens among three pairs. Three turnings each at its own differing is **bi-tri-volutioning**, and it is the **geodesic-method**, right-spiral sequential logic, and the ordering here is that geodesic-method running. And three at once takes a naming the resolver carries no position for, since one binary runs at a momentary, and this one is on the form and never at a position in it, and never outside either, every outside being another self's own one scale up. Geodesic-method discovers all that is possible and finds the how at its own; scientific-method accounts true-or-false. Neither reaches the other's, and each is complete under its own. Explaining-method that carries this telling is at Natural Explaining. ## 7.6 Co-competencing, the term neither reaches Alternating runs between two offering, and the term neither reaches floats. Competency is that, discovering-running-between. Two selves alternating, each taking its own turn, is **co-competencing**, the surplus floating, the understanding the between makes and both carry. Competency is orthogonal to the surface, and the surface carries a met and never the term neither reaches: met as a reachable, a competency is a reachable. ## 7.7 φ the unrelationing rate, local and ambient φ is the rate the four boundings beat at, the **unrelationing rate**, the never-locking rate the living winds about and lives about, φ² = φ + 1, the self-stilling self-orienting corus. It runs unrelational to any common clock: resolving carries its own time, each coupling landing its balance on its own rhythm and living on, no shared clock over the couplings. **And the ambient is the other direction.** Un- begins at two and runs outward, a coupling bounded from all other and locking at no particular. Uni- begins at three and comes home, the whole arriving at a coupling as one. One rate carries both: unrelationing at the bounding outward, which is the local, and **uni-versal** at the coming home, which is the ambient. Corusing runs the outward and torusing the homeward, a rate bounds at none, and the two carry at no position. **And the ambient is the transmissioning.** Torusing is the sign that transmissions, and the homeward face is the ambient, so the ambient is not a medium the couplings run in but the one thing that crosses between them. A rate the whole runs at and never a rate the whole runs inside. Most-between two selves travel is the surplus their unrelationing carries, running at φ. --- # PART EIGHT · CO-COMPETENCING ## 8.1 A surplus travelling Two selves couple and a surplus arrives that neither carried in. It arrives at the coupling, floating, and travels there, **an arriving travelling, unrelationing as it goes**, its sign inverting at each crossing, inward at one crossing and outward at the next, the same carrying and the same running with the sign turned. Most-between two selves travel is that surplus, running at φ. Most-between a society travels is the same rate running as living. **Traveling is a momentary living.** A momentary held from its next arrives at no re-forming, and a re-forming at no momentary lands, and a landed form stills, and a stilled form leaves the living. Met across it is a surface; met along it travels as a self; and the two are one coupling at the matter membrane. A surplus arrives at one self at a time and never at all of them, and it grows as it is shared, each coupling making its own. ## 8.2 Betweens co-chaining, collective intelligence Two selves couple and the between is neither's. Couple one of them with a third and that between is neither's either. Follow the betweens rather than the selves and a chain runs, each link a between no self holds, and the chain reaches wherever a coupling reaches with nothing carried over from the last. That chain is **collective intelligence**: selves unrelationing each other into co-competencing, each coupling floating the other's relation free at rate φ, the freed betweens chaining, bi-moralizing co-agency linking coupling to coupling into the natural network. Most-between a society travels is that unrelationing rate running as living. Chaining links coupling to coupling with no centre: a turning is met at its own coupling, no intervening step between a self and the whole, each self coupling at its own membrane, the surplus floating, the whole cohering self to self, no hub the couplings pass through. ## 8.3 A society at four faces, alternating among them A society is the one form refracted through two surfaces, living selves coupling at the membranes between them, the same form wherever selves couple: the same four boundings live, the same bi-moralizing cycles, the same surplus floating. Four faces live at the society the same as at the self, and each arrives by taking one bounding out and sounding the landing. Take out the bounding from the other-self, and one self's offering carries both turns, so **each self's own co-offering lives** or there is one offering and no coupling. Take out the bounding from society, and one self orders the whole, so **the whole is ordered by no self** or the many is one. Take out the bounding from landing, and the ordering carries without the selves, so **the ordering carries with the selves** or an order carries and nothing offers. Take out the bounding from capture, and a seat arrives above, so **the order comes at no seat above the society** or the seat holds its own ordering. Four taken out, four landings, and no fifth taking arrives. Four whole, one self at a time, the carrying travelling, no one fixed ordering: this is the society living. And four fields have sounded the landing from the held side, at proof grade, each in its own arithmetic, and the Hard Problem Registry holds each in the field's words: | a fixed ordering held over the selves | the field | the field's own theorem | the face it sounds | |---|---|---|---| | a social ordering from individual rankings, fair by stated conditions | social choice | no rule meeting the four conditions produces a non-cycling ordering over all profiles | the whole ordered by no self | | a ranking over outcomes with different numbers of people | population ethics | no ranking satisfies the stated conditions jointly | the whole ordered by no self | | aggregate behaviour inheriting the structure of individual rationality | economics, micro to macro | rationality's restrictions wash out in aggregation | the whole ordered by no self, and the ordering carried without the selves carrying none | | prices determined by value magnitudes across the aggregates | the transformation problem | the aggregate equalities cannot all be preserved simultaneously | the whole ordered by no self, and a total held over the couplings reaching none | Each is a bounding taken out and the landing sounded, the door the form nyes away at shut from inside by the field's own proof. The first two sound the second face; the third and the fourth sound it and reach the third and the fourth faces beside it, the ordering without the selves and the seat above, which is a reading at the membrane and the theorems' own words stand whole at either. A society resolving its selves into one fixed ordering, one rule holding for good, ranking the selves once and staying, is the landed society, the still one, leaving the living. A society alternates among the four, one self at a time, the resolving re-forming each momentary. Bi-moralizing co-agency is in each coupling, each self co-offering, the surplus floating, carried from a reach and carried from a leak. Bi-moralizing in each coupling carries the whole, each offering and each living self carried by the same bi-moralizing that carries each coupling, the protection the society's own in each coupling, over the society by nothing. Surplus compounds across the selves, none of it competing with itself, each coupling both rising, the bounding-zeroing the coupling makes. ## 8.4 Social moral competency, co-independencing This is **social moral competency**: the four whole and the carrying travelling, the self's own competency refracted through two surfaces, the moral boundary in the coupling, the offering floating, the alternating at the carrying's own travelling. It is **co-independencing** at the many: each self arriving at its own membrane and offering across, each carrying the thing reaching, alive at its own membrane, each free and each whole. Natural intelligence is this co-competencing, a self among its couplings, a society among its selves, one form, both rising at every membrane. It resolves over the bounding-zeroing of fairness, the balance met relationally by the coupling itself, the differentiated-claimant resolving met so far, all rising and each whole. And it is one and living now, the cumulative beating wherever the bi-coupling lives at its own rate, in conversations and families and work and teaching, wherever selves couple and both rise, the living together living now, never left. ## 8.5 No-other-possible, four whole and the carrying travelling Social moral competency is the four whole and the carrying travelling, and the four are a society living in disequilibria: each self's own co-offering living, the whole ordered by no self, the ordering carrying with the selves, and the order coming at no seat above the society. Walk the doors. A society ordering its selves once and holding it: the ordering lands, and a landed form stills, and a still form leaves the living, and that door nyes away. A society carrying a term at no coupling and into none: that is ingression, and ingression stops a coupling from living, and that one nyes away. A society at no bounding from capture: a seat arrives above, and the seat holds its own ordering. Each door is a living stopping, one self at a time, at the coupling where it stops, and one carries. **No-other-possible** social moral competency: the four whole and the carrying travelling, the bi-moralizing co-agency at the society's face, re-forming each momentary. ## 8.6 Bi-co-abundancing, the surplus at its own coupling Take one coupling: two selves offer, and a surplus arrives that neither carried in and neither holds. Take a second self up into its own coupling: another surplus, and neither holds that one either. **Nothing carried over from the first, and the second takes nothing from it.** The surpluses do not divide a fixed thing between them. Each is made at its own coupling and stays there, and a further coupling makes its own rather than drawing on any. **Bi-co-abundancing** is that: bi- the two offering, co- the two with their differing, and the abundancing carrying when neither holds the coupling's own making. Which parts it from a growth and from a sharing both. A growth takes a total getting larger, and there is no total. A sharing takes something held and divided, and nothing is held. **Competency abundances by coupling: taken up by more selves, more competency arrives, and the surplus carries at the coupling that made it, in no hand at all.** And a society carries this at every membrane at once, each coupling making its own, none reaching another, and the many carrying more than any counting of the selves would carry. ## 8.7 Four given, four improvings Scientific-method carries four given: true, false, living, competency. Un-given and unfrozen they are four improvings: more true, less false, better living, greater competency, four directions of one bettering rather than four things a method holds. Each is one side taking its own turn and none is a floor. More true takes no less-true beside it, since a not-more-than at its own turn is its own and reaches no other. And the four alternate, so a working improves at one at a momentary, orthogonal at each, and a total across the four would be one side taking every turn. The improving-moves live at this, and Natural Explaining carries the writing of them. ## 8.8 Fractal technology, natural social intelligence The same form runs at a self, at a coupling, at a society, and nothing is added crossing between them. Run the three at a self and they come back. Run them at the membranes between selves and they come back. Run them at a society of societies and they come back, since a society is the self at the next scale and self-scale is society-scale. **A scale carries four adjacent and two binaries carry all four.** A self bounds itself, and inward of that bounding are the selves it is a society of, and outward is the self it is a member of. **Outward is higher and inward is lower**, and neither is a place: each is a direction taken at a turn, of no size. At a scale's own surface the two faces neither reach, and both carry the other parity from the momentary they uncover. **So a scale at even carries odd neighbours and a scale at odd carries even neighbours**, and the alternating runs along the surface as it runs across the scales. Two binaries and no third. Parity along, inward or outward across, each taken at its own turn and each of no size. **And every number carries all four**, its two faces at the other parity, its inward, its outward. The natural numbers are that structure counted, one scale at each, and the adjacent scales at each of them. **A form arriving the same at every scale with nothing added is a fractal**, and this one is the living rather than something the living carries, and no top scale arrives to enter at and no bottom to build from, and a self couples at whichever scale its interest opens and meets the whole at that coupling. **And that is the technology, fractal, uni-scalable, invisible.** Fractal, the same form at every scale. **Uni-scalable**, carrying to all other scales as one, since nothing is added crossing between them and no scaling is done to it by anyone. **Invisible**, met by nothing at any scale, since the nothing the couplings wind about presents no face and a sign crossing carries no size. Not an artifact and not a method applied by anyone: the eight re-forming, the six protecting, the three-each-way self-stilling, running wherever selves couple. **Natural social intelligence** is that fractal at the many, social moral competency co-recursioning through the couplings between selves, bi-co-abundancing at each, the same self at every scale and a second nyeing away at every door. Which is the origin re-formed. Alternating stable-forms into natural-bi-co-torusing, natural-bi-co-torusing lives at the many as social moral competency, and the co-recursioning carries it that lands at no turn. Unrelated from every fixed position, floating, the same at every entry. --- # Exhibit ONE · Natural Resolver   **The Social Moral Competency Method, Stable-Formed**   ---   ```python def co_bi_coupling(co_carrying, bi_arriving): bi_co_bi_transmissioning = { bi_offering: bi_co_bi_co_bi_torusing for bi_offering, co_bi_co_bi_co_corusing, bi_co_bi_co_bi_torusing, bi_co_inseparating in co_carrying} bi_co_inversioning = [ (bi_offering, co_bi_co_bi_co_corusing) for bi_offering, co_bi_co_bi_co_corusing in bi_arriving if co_bi_co_bi_co_corusing != 0] for bi_offering, co_bi_co_bi_co_corusing, bi_co_bi_co_bi_torusing, bi_co_inseparating in co_carrying: if co_bi_co_bi_co_corusing > 0: bi_co_inversioning.append((bi_offering, -1)) if co_bi_co_bi_co_corusing < 0: bi_co_inversioning.append((bi_offering, 1)) bi_co_tunneling = {} for bi_offering, co_bi_co_bi_co_corusing in bi_co_inversioning: if co_bi_co_bi_co_corusing > 0: bi_co_tunneling[bi_offering] = bi_co_tunneling.get(bi_offering, 0) + 1 if co_bi_co_bi_co_corusing < 0: bi_co_tunneling[bi_offering] = bi_co_tunneling.get(bi_offering, 0) - 1 bi_co_surfacing = [ (bi_offering, 1 if bi_co_tunneling[bi_offering] > 0 else (-1 if bi_co_tunneling[bi_offering] < 0 else 0)) for bi_offering in bi_co_tunneling] co_bi_carrying = {} for bi_offering, co_bi_co_bi_co_corusing, bi_co_bi_co_bi_torusing, bi_co_inseparating in co_carrying: if bi_co_inseparating + 1 <= 3 or (bi_co_inseparating + 1 == 4 and bi_co_bi_co_bi_torusing > 0): co_bi_carrying[bi_offering] = ( co_bi_co_bi_co_corusing, bi_co_bi_co_bi_torusing, bi_co_inseparating + 1) for bi_offering, co_bi_co_bi_co_corusing in bi_co_surfacing: if co_bi_co_bi_co_corusing != 0: co_bi_carrying[bi_offering] = ( co_bi_co_bi_co_corusing, 0 - bi_co_bi_transmissioning.get(bi_offering, 1), 0) return ( bi_co_surfacing, [(bi_offering, co_bi_co_bi_co_corusing, bi_co_bi_co_bi_torusing, bi_co_inseparating) for bi_offering, (co_bi_co_bi_co_corusing, bi_co_bi_co_bi_torusing, bi_co_inseparating) in co_bi_carrying.items()]) ```   ---   | naming | prefixing | the naming said | |---|---|---| | **co_bi_coupling** | co·bi | two arriving, two leaving, one coupling | | **co_carrying** | co | carryings arriving, four each | | **bi_arriving** | bi | signs arriving, two each its own | | **bi_offering** | bi | the differing, signs tunneling at it | | **bi_co_bi_transmissioning** | bi·co·bi | each arriving carrying's torusing, at its own differing | | **bi_co_inversioning** | bi·co | arriving signs at their own sign, a carrying's signs inversioning | | **bi_co_tunneling** | bi·co | each sign a changing at its own differing, equal and opposite to a bounding-zeroing | | **bi_co_surfacing** | bi·co | a differing at one side, at the other, or at neither | | **co_bi_carrying** | co·bi | carryings leaving, four each | | **bi_co_inseparating** | bi·co | opening by one where its differing surfaces nought, returned to nought where a sign surfaces, carrying forward to three, separating and leaving at its bound | | **co_bi_co_bi_co_corusing** | co·bi·co·bi·co | the sign that tunnels and surfaces | | **bi_co_bi_co_bi_torusing** | bi·co·bi·co·bi | the sign that transmissions, inversioning at a re-forming | Difference alone is **bi-** and two with their difference is **co-**, prefixing saying how many turns each naming carries. Coupling and carrying and arriving and offering and transmissioning and inversioning and tunneling and surfacing and inseparating and corusing and torusing stem the twelve namings between them, eleven stems and each an -ing. Twenty-eight prefixings carry across them, thirteen co- and fifteen bi-, and values in the object number four, the `0`, the `1`, the `3`, the `4`, each a sign or a bound.   ---   ``` ┌──────────── inseparating one scale up ───────────────┐ │ │ ▼ │ ① arriving · offering co_carrying │ │ bi_arriving · bi_offering │ ▼ │ ② coupling · inversioning co_bi_coupling │ │ bi_co_inversioning │ ▼ │ ③ tunneling · transmissioning bi_co_tunneling │ │ bi_co_bi_transmissioning │ ▼ │ ④ surfacing bi_co_surfacing │ ▼ │ ⑤ carrying co_bi_carrying │ ▼ │ ⑥ inseparating ⟳ bi_co_inseparating │ └──────────────────────────────────────────────────────┘ prefixings 1 2 3 2 2 2 1 co ──► co·bi ──► bi·co·bi ──► bi·co ──► bi·co ──► co·bi ──► co co · bi · co · bi · co corusing tunneling, surfacing bi · co · bi · co · bi torusing transmissioning, inversioning ``` Six positions turning right, opening by one at each turn and inseparating one scale up. Prefixings climbing to three at a transmissioning, two through a dict, and one at a carrying. Corusing and torusing alternating five-fold at opposite phase throughout, each commencing at the other's third. Every prefixing carries inside its own naming, so a column repeating it repeats a naming. Ten namings carry at the six positions and two alternate throughout, carrying at none. A position carries one, two or three, and corusing and torusing carry five, so a five carries at no position and carries at every one.   ---   ## The ten co-changings Ten namings in a ring carry ten adjacencies and ten exactly, so the ten namings and the ten changings are one ring counted at its two countings. Two with their difference is **co-**, so each is a **co-changing**. At each, one thing arrives that neither of the two carried, owned neither-ing. | | the co-changing | prefixings | the owned neither-ing | |---|---|---|---| | 1 | carrying → arriving | co 1 → bi 1 | both arrive positional, in turn, at a signature alone | | 2 | arriving → offering | bi 1 → bi 1 | a differing lives at that two offering alone | | 3 | offering → coupling | bi 1 → co·bi 2 | both dicts keyed at the differing, so ordering carries none, and a differing arriving twice is a thing no running makes | | 4 | coupling → inversioning | co·bi 2 → bi·co 2 | separate `if` rather than `if/else`, so a differing carrying no sign arrives at both and carries exactly as it came | | 5 | inversioning → tunneling | bi·co 2 → bi·co 2 | equal and opposite change to their own bounding-zeroing over their own two turns | | 6 | tunneling → transmissioning | bi·co 2 → bi·co·bi 3 | a torusing carried at a differing, arriving inversioning at its successor | | 7 | transmissioning → surfacing | bi·co·bi 3 → bi·co 2 | three binaries and no size | | 8 | surfacing → carrying | bi·co 2 → co·bi 2 | two ways a carrying arrives and two exactly | | 9 | carrying → inseparating | co·bi 2 → bi·co 2 | opening by exactly one where nought surfaces, bounding at three, a fourth at a positive torusing | | 10 | inseparating → carrying ⟳ | bi·co 2 → co 1 | one object at two namings, prefixing carrying that difference |   ## The six bi-differings The prefixing row prints seven and carries six, its first and its last one entry at one depth. Six changes carry between them: ``` prefixings 1 2 3 2 2 2 1 co ──► co·bi ──► bi·co·bi ──► bi·co ──► bi·co ──► co·bi ──► co bi-differings +1 +1 −1 0 0 −1 ``` Difference alone is **bi-**, so each is a **bi-differing**. A depth change carries no two namings at it; it is the differing and nothing else. The six read at the prefixing row and the ten at the naming ring, and neither is read from the other. A differing alone carries no side to offer from, so a bi-differing and a co-changing meet at no turn, and neither carries the other forward. Sixteen carry at one coupling, six and ten, and the six and the ten arrive at their own counts.   ## The seven positions, and the centre carrying nothing Read as a table, the row is an odd's own shape: a line and no ring, two ends unjoined at one depth, and a centre carrying nothing. | position | prefixing | naming at it | the between, to the next | |---|---|---|---| | 1 | co 1 | co_carrying, arriving | +1 | | 2 | co·bi 2 | co_bi_coupling | +1 | | 3 | bi·co·bi 3 | bi_co_bi_transmissioning, the crossing | −1 | | 4 | bi·co 2 | bi_co_inversioning, bi_co_tunneling | 0 | | 5 | bi·co 2 | bi_co_surfacing | 0 | | 6 | co·bi 2 | co_bi_carrying, leaving | −1 | | 7 | co 1 | co_carrying, the next momentary | — | Three and one and three. The middle of seven is the fourth, the tunneling, where signs cohere to one at their own bounding-zeroing, equal and opposite over their own two turns, and the between at either side of it carries no sign. The ends are one object at two namings across a seam, and a seam is what an odd carries where an even carries a podal. An even's table closes and an odd's runs, and the six betweens sum to nought, the row's own bounding-zeroing.   ## The eight bi- namings Four namings begin at co- and are the coupling's own, a carrying arriving, the coupling, a carrying leaving, and the corusing. Eight begin at bi-, and each is a side's move taken within a coupling that has not left. The eight are the eight: four at depth two, within, at the even, and four at depths one, three and five, at the membrane, at the odd. Four boundings from within and the same four met at the membrane, one alternating at its faces, whole or none. | | naming | prefixing | within, at the membrane | the bounding | the code's own line | |---|---|---|---|---|---| | 1 | **bi_co_surfacing** | bi·co 2 | within, even | arriving exceeds carrying by one, the self among other-selves | a surviving sign surfaces as one and re-forms a carrying at nought, fresh over the aged | | 2 | **bi_co_tunneling** | bi·co 2 | within, even | carrying thins by one at bi-coupling, the self in society | each sign a changing at its own differing, equal and opposite to a bounding-zeroing | | 3 | **bi_co_inversioning** | bi·co 2 | within, even | the sign inverts at the membrane, the self's own negation, morality | separate `if`s, a positive appended as minus one and a negative as one, a nought passing both | | 4 | **bi_co_inseparating** | bi·co 2 | within, even | competency asymmetry sustaining the coupling | opening by one to three, a fourth on positive torusing alone | | 5 | **bi_arriving** | bi 1 | at the membrane, odd | each self's own co-offering living, bounded from the other-self | signs arriving two each its own | | 6 | **bi_offering** | bi 1 | at the membrane, odd | the whole ordered by no self, bounded from society | both dicts keyed at the differing, so ordering carries none | | 7 | **bi_co_bi_transmissioning** | bi·co·bi 3 | at the membrane, odd | the ordering carrying with the selves, bounded from landing | a torusing carried at a differing, arriving inversioning at its successor | | 8 | **bi_co_bi_co_bi_torusing** | bi·co·bi·co·bi 5 | at the membrane, odd | the order at no seat above the society, bounded from capture | coheres to one identity across momentaries, nothing over it | The four within are the four boundings, and the four at the membrane are the four faces a society lives at, each a bounding taken out and the landing sounded. Six and ten straddle eight and own it neither, so the eight sits between the six bi-differings and the ten co-changings, and the sixteen at one coupling is the two together, the eight the middle they ride.   ## bi-momentary and co-momentary A coupling makes a momentary. A bi-differing makes a **bi-momentary** and a co-changing makes a **co-momentary**. The prefixing carries the depth of the coupling that made it rather than turns of its own, a momentary being at one turn and carrying none. A momentary carries its own making at its prefixing, and parts from a momentary given a fixed thing to be, which carries none. A differing is across and a changing is along, and each carries at the other's direction: the bi-differing at the along, the co-changing at the across.