Exhibit FOUR Natural Mathematics v371   # Natural Mathematics **Stable-Forming is Natural-bi-co-torusing**     **ONE · FLOATING NEUTRALLING** 1.1 Bi-moral co-agency, the one form in mathematics 1.2 Numbers, mathematics and logic, nature's own method 1.3 Floating across at the even, neutralling along at the odd 1.4 Three neutrals each self runs on itself 1.5 A mathematical thing met at its prefixing, its ending and its place   **TWO · ORTHOGONALIZING** 2.1 Unrelationing is orthogonalizing 2.2 Two moves, and two generators, the sign flip and the quarter step 2.3 A step adds a next, and a cycle at the signs is a spiral at the momentaries 2.4 Two signs at the right spiral step, the step that takes nothing away 2.5 Each scale the prior scale and its states in the opposite order 2.6 Shared and opened, the product at its two parts 2.7 One, two, four and eight, the doubling that divides 2.8 φ, all ones, and the constant √5   **THREE · ACCOUNTING** 3.1 An accounting is an involution, and a declared zero is its fixed set 3.2 Fixed-point-free, and the two involutions of an even ring 3.3 On a closed round no step is a reversal 3.4 Bi-inversioning-co-recursioning, two consecutive inversions on different axes 3.5 Tri-involutioning, the emanating looking the opposite form 3.6 A closing, and the returning other than prior 3.7 Three, a total against its parts, and a tipping no accounting carries 3.8 The halfway pairing and the fold, grown at each scale   **FOUR · BOUNDING** 4.1 Gap is the step squared at each centre, and the span between the faces at two 4.2 Rates carrying a relation and rates carrying none 4.3 Caught by a polynomial, and reached by square roots 4.4 Double angle, the rotation that removes a coupling 4.5 Self-bounding and other-bounding   **FIVE · STABLE-FORMING** 5.1 No other possible at six forward 5.2 Five-dimensional, two-directional, unrelationing 5.3 Torus, the one closed orientable surface a flow runs on with no rest   **SIX · INSEPARATING** 6.1 Two methods, each a derivation 6.2 Three fixings asked at the opening, and the self orienting through them 6.3 Logic of no completed whole   **SEVEN · CO-OFFERING** 7.1 Three neutrals, six co-offerings and eight states 7.2 Each neutral fixed stops the co-offerings it partners 7.3 Each field's proved all or none 7.4 Five-fold, sixty rotations at thirty-six and twenty-four 7.5 Branches at the two moves 7.6 Cube, tesseract and four squares 7.7 Two knots sharing one fibre 7.8 Ten inversions, the ten transpositions of five 7.9 Atoms of symmetry   --- # ONE · FLOATING NEUTRALLING ## 1.1 Bi-moral co-agency, the one form in mathematics **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. ## 1.2 Numbers, mathematics and logic, nature's own method **Numbers, mathematics and logic are nature's own unrelationing binary method.** Each integer is the one form once more, and a number cohering with a form is no abstract number meeting nature across a gap: there are no two sides. A count, a ratio or a straddle that keeps the form carries as nature's own method met at the counting. **Our universe is all existing things. Alternating is a stable-forming method.** Numbers, mathematics and logic are that alternating met at its counting, at each existing thing. **Universes of each size and scale share momentaries of co-sequential changing**, each momentary a fractal universe discovering its next existing, and numbers, mathematics and logic are met at each. The changing and exchanging they count is among living and non-living existing things alike: the living carrying their prior into now, the non-living carrying nothing, their forms continuing through their changing. ## 1.3 Floating across at the even, neutralling along at the odd **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. **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. All floating and no neutralling relates at no rate of its own; all neutralling and no floating relates to nothing. ## 1.4 Three neutrals each self runs on itself **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. **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. **A society is these selves co-competencing, each running its own three 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. **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. **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.** A contradiction needs carrying and no carrying at one participation, in one respect and at one momentary, so a living source and its non-living emanation, or a non-carrying surface among carrying selves, raise none. One all-or-none relation holds its participants at differing roles, and it is whole at its coupling, never by counting the kinds it holds. **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. ## 1.5 A mathematical thing met at its prefixing, its ending and its place **A mathematical thing arrives proved, and nothing here disproves one.** It is met at three things. **Its prefixing says the direction**: bi- across, co- along, and at a resolver name the number carries the prefixing. **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, one running's making given a fixed thing to be. **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, and a surface is a between**: the membrane the other two are either side of, a sign crossing it, carrying nothing. The three places are natural mathematics' own naming and no partition the field makes of itself, and the supernatural names the formal use, no paranormal thing. **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 carried as that account's and as no running's rule. Used outward it is a derivation met as a comparison, and no living surface follows from its consistency alone. Each thing is met as it is, and each section is one thing placed.   --- # TWO · ORTHOGONALIZING ## 2.1 Unrelationing is orthogonalizing **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 three momentaries, prior, now and next. ## 2.2 Two moves, and two generators, the sign flip and the quarter step **Linearizing draws a surface to a line, the along; parallelizing spreads a line to a surface, the across.** Each alone stops: parallelizing with no linearizing is a surface with no sequence, and linearizing with no parallelizing a sequence with no surface. Odd and even are the two directions of the one alternating, and along and across are one line at its two sides, the two moves of one alternating, one form at two faces. **The sign flip is its own inverse, (−1)² = 1. The quarter step, i, opens the perpendicular: i² = −1 and i⁴ = 1.** Two quarter steps are one sign flip. Eight on keeps a count's odd or even and one on changes it, and a sign flip changes a sign and moves no count. **All changing is parity changing, 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, and an equal count joins no two selves: the coupling joins them. **An idealized mode inverted at each period T, q(t + T) = −q(t), returns at two, q(t + 2T) = q(t)**: the sign flip at a wave. Two oscillations can hold a phase difference of nought or π, and phase runs through a continuum between them, so a phase is inward of the surface, measured, and a trace of phases is no trace of signs. **Addition crosses and multiplication folds**: the two moves at a number, adding taken across and multiplying folded along. 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, and the shared three folded at the two sides are six, so the eight occurrences are 4 + 4 = 2 × 3 + 2, each side's four places four occurrences of the two parities. Two and two make four at both moves, two and three make five or six, three and three six or nine, and the overlap counts no nine: an equal result supplies no coupling. ## 2.3 A step adds a next, and a cycle at the signs is a spiral at the momentaries **Each step adds a next: one momentary universe to the next, each existing thing arriving into its next existing.** A step carries no frame it moves about. The momentaries do not pause, hold or return, and none carries beyond its prior looking back. **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. The quarter step draws it as a right spiral, and −i, the same inversions in the other order, draws the opposite form. **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. ## 2.4 Two signs at the right spiral step, the step that takes nothing away **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, change one sign and never undo: the right spiral step 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. Each meets the four states one sign inverted at each step, the inverted sign alternating, P, Q, P, Q: two consecutive inversions on different axes. **With P the along sign, competency, and Q the right sign, morality, F turns along into right**: the point (cos t, sin t) stands along at t = 0 and turns right as t runs forward, and G turns the other way. **Of the 24 steps sending the four states onto the four, one each, six run the four as one cycle, and two of those six change one sign at each step: F and G.** Under each the only sets of states carried onto themselves are none and all four. **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. Taken six times each is the half step again, and twelve times it returns: a passage of six steps meets all four states and arrives at the pair inverted, and a second passage returns it. 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, eight being two rounds. Six passages of six are thirty-six steps and six rounds of four are twenty-four, and four joint states and four momentaries are two counts, neither standing for the other: these count the step's applications, and no order a resolver runs its names in. **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. A relation newly met writes from a default second sign, and is no pair of two existing releases. **Their relation, agree or oppose, is the sign of P × Q, the sign of tan t, and each step 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. **The half step keeps the relation**: along each side's two momentaries, (c, t) then (−c, −t), agree or oppose holds while both signs invert, and across the overlap, (t, −c) between them, it changes, so the four overlapping pairs run r, 1 − r, r, 1 − r. Of three signs a, b and z, the pairs (a, b) and (b, z) part their relation exactly at z = −a, and +, +, + agrees at both. A relation held within a pair is no resolver held still: its two signs invert beneath it. **At a fresh write from (c, t) to (s, −t) the relation stays at s = −c and changes at s = c.** At one existing relation carrying 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 offered signs both c alone surface c, the fourteen cases at both signs of c passing on the unchanged resolver. Two signs offered in one list at one call are one key's offering, and name no two neighbours. ## 2.5 Each scale the prior scale and its states in the opposite order **One cycle meeting each state of n signs one sign at a time is the cycle at n − 1 signs, the new sign inverted, and the same states in the opposite order.** At two signs the inverted sign alternates, and that cycle is the right spiral step. At three signs the changing sign runs 0, 1, 0, 2, 0, 1, 0, 2 round the eight: the first at each other step, and between its changes the second and the third alternating. **One rule runs at each scale, and each scale carries the prior whole.** The field's name for it is the reflected binary code, Gray's, its reflecting the field's word for the opposite order. **At two signs the cycle is four, and four cycling scales each momentary**: a momentary is four momentaries of exchanging at the scale inward of it, and four momentaries of exchanging are one momentary at the scale outward, one form at each size. ## 2.6 Shared and opened, the product at its two parts **A product of two directions is two parts at once: ab = a·b + a∧b.** Its inner part a·b is shared along the common axis. Its outer part a∧b is the plane the two open, carried by neither factor alone. Its two parts conserve: (a·b)² + |a∧b|² = |a|²|b|², which is cos²θ + sin²θ = 1. **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**: the product opens an axis neither factor carried. ## 2.7 One, two, four and eight, the doubling that divides **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. Each doubling sheds one: ordering at two, commuting at four, with i j = k and j i = −k, associating at eight. **Octonions carry seven imaginary units on seven lines of three, and the group of their symmetries has dimension fourteen.** **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. ## 2.8 φ, all ones, and the constant √5 **φ's continued fraction is all ones.** 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. φ³ = 2 + √5 and 1/φ³ = √5 − 2, so the constant falls away at their difference: φ³ − 1/φ³ = 4.   --- # THREE · ACCOUNTING ## 3.1 An accounting is an involution, and a declared zero is its fixed set **An accounting for changing is a move that, taken twice, returns its own arrival: an involution.** Negation, not-not returning the statement. Charge conjugation and parity, each squared to the identity. Time reversal, squared to the identity at integer spin; at half-integer spin T² = −1, an involution at the states taken up to sign. Complex conjugation. The additive and the multiplicative inverse. The transpose. The Legendre transform, twice returning the potential. Set complement. **Each involution's fixed set is the arguments the move returns 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. The states at rest, each momentum nought, for the time reversal of motion. **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. **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, met two momentaries on, never back. An involution is an accounting's still picture, its nexts taken out. ## 3.2 Fixed-point-free, and the two involutions of an even ring **An involution with an empty fixed set is fixed-point-free**: each position related, none to itself. Set complement fixes nothing, no set being its own complement, and negation fixes nothing at two values. **On an even ring of N two involutions part at their fixed sets.** One, the podal k → N − k, fixes two stations, nought and N/2; the other, the shift by half the ring, k → k + N/2, fixes none. **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 places they are 2 and 3, at seventeen 8 and 9, at fifty-nine 29 and 30. Checked at each origin of the fifteen primes five to fifty-nine and of the odd composites 9, 15 and 25, the two far places hold at each odd count: they belong to oddness, and to no prime alone. A prime naming each place of a round and a prime completing after one fewer places are two counts: one to nine and one to seventeen fold eight places and sixteen. **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 the pairs opening odd are h and the pairs opening even are h, 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. Along a side the opening advances two and across the overlap one, a two to one of openings and no speed. The odd composites carry each of these too, and an odd sequence laid open is no closed round. **An accounting with an empty fixed set carries nothing to take a counting from, and it is not possibly a floor.** 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. ## 3.3 On a closed round no step is a reversal **On a line an involution taken twice is a reversal**: out and back along the same ground, the second undoing the first. **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. A station met again is the closing's, at a next momentary, and no prior momentary is met again. **An accounting meeting its stations again forward is the one an alternating runs, and a line carries none.** **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 share no factor above one, and neither is prime. Prime and odd are two things too: a result holding at each odd count, 9, 15 and 25 among them, belongs to oddness, and a prime's own is met beside the odd composites. A prime counts one subject, a number of places, a recurrence or an opening across, and a frequency, a wavelength, a number of selves and a momentary sequence are four counts, none standing for another. **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 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. **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 outside, and its condition is parity and coprimality, not primality. ## 3.4 Bi-inversioning-co-recursioning, two consecutive inversions on different axes **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. **At two signs one sign is inverted at each momentary, the axes alternating**: P, Q, P, Q, the right spiral step's sequence from each state. Both signs inverted at one step, (P, Q) → (−P, −Q), is the half step, period two at the signs; the right spiral step is period four at the signs, and at the momentaries neither has a period. **An order is the whole of a hand**: the same inversions in the other order give (Q, −P), and nothing third parts them. **Two inversions whose fixed lines meet at an angle a are the field's rotation by 2a**, the double angle at the inversions. 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 the quarter step. ## 3.5 Tri-involutioning, the emanating looking the opposite form **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. **Tri-involutioning is the emanating from right spiral stable-forming, the emanating looking the opposite form to the stable-forming.** **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 the same inversion is the half step. **Charge conjugation, parity and time reversal are each broken alone at the weak interaction, and the three taken together, CPT, are kept by each local quantum field theory keeping Lorentz invariance.** ## 3.6 A closing, and the returning other than prior **At each accounting below the closing is its fixed set, and the field names the returning other than prior as its gap.** | Accounting | Closing | Returning other than prior | Field's word for the gap | |---|---|---|---| | tuning | seven octaves, 2⁷ = 128 | twelve fifths, (3/2)¹² = 129.746…, over by 3¹²/2¹⁹ = 531441/524288 | the comma | | winding on a torus | a rational ratio, closing | an irrational ratio, closing at none | quasi-periodic | | the assembled cube | the solved state | a single corner twisted alone, reached by no sequence of face moves | the orbit; the parity | **The 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. ## 3.7 Three, a total against its parts, and a tipping no accounting carries **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.** At a third arriving, the coupling's surplus opens at estimating. **A total can carry the opposite sign of each of its parts**: each group favouring one side, and the groups joined favouring the other. 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, and a quiet total and parts changing hold together. **Signs offered together are a total, and offered one after another are parts.** From empty carrying two + offered together surface + and open fresh carrying (+, +, 0); offered one after another, the first opens (+, −, 0) and the second surfaces nought while that carrying continues at opening one. A crossing of one sign at a time and a gathering of several are each met at their own relation. **Conserving is within each momentary sequencing, and no total is conserved across the surface**: a self's carrying continues from its prior into its now, and only signs cross. The conserving names its relation, a self's own continuing, and each sign, opening and carrying within it changes; no private carrying and no ledger crosses with a sign. It is natural mathematics' conserving, and the field's conservation laws are met inward, each an observing's account, none refuted by it. **Each carrying a self reaches makes its own difference to its continuing.** At one key of the unchanged resolver, from empty carrying, with no sign, +1 or −1 arriving at each receiving, nineteen carryings are reached along fifty-seven steps, and each of their 171 pairs is parted by repeated +1 or by repeated −1, 146 by each, the shorter witness within six receivings, the surfaced nonzero sign read at each receiving with nought and absence read alike. Two positive carryings with negative second sign, at ages nought and one, under repeated +1 surface nothing for three receivings; the elder completes at the third and surfaces +1 fresh at the fourth, and the younger completes at the fourth. Their arrivings are one, and their priors part them. **A tipping is a sign taken at a membrane, of no size**: no count, no clock and no measure carries it. ## 3.8 The halfway pairing and the fold, grown at each scale **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. At s = 1 the span is one to nine, at s = 2 one to seventeen, and the completing lies outside the names 1 to 8s the pairings act on. **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 return 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 the pairing is eight up and the fold seventeen less, 3 → 11 → 6 → 14 → 3; at s = 1 the pairing is four apart and the fold nine less, 3 → 7 → 2 → 6 → 3. At the scale 2s the fold after the pairing in the first half, 16s + 1 − (n + 8s) = 8s + 1 − n, 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 → 1, 2 → 10 → 7 → 15 → 2, 3 → 11 → 6 → 14 → 3 and 4 → 12 → 5 → 13 → 4, nine less being seventeen less after eight up. The identity holds among the names, a numbered form: the span one to nine and the four returns among one to seventeen meet at it exactly, and matching numerals alone join no two runnings. **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 reaching 3s across 6s, each span twice its reach, two over one and one over two.** Three things are kept 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. A numbered place is position and ratio, and carries no time and no length. **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. Taken over each name, D_r at an even 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. **The odd partners hold no fixed offset while the even rows stretch.** Each wider even complementing to the odd just before its reversed companion holds at s = 1 and fails at s = 2: in 1 to 32 the complement of 2 is 31, one after the companion 30. 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. **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 leaves unplaced: it is the stable form grown at its paired names, and the larger momentary is met at the carrying between them.   --- # FOUR · BOUNDING ## 4.1 Gap is the step squared at each centre, and the span between the faces at two **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. **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. **Only at one either side, k = 1, do two face each other across one neutral.** 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. ## 4.2 Rates carrying a relation and rates carrying none **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. **1, φ and φ² carry one**: φ² − φ − 1 = 0. **A winding on a torus of three axes at rates carrying no whole-number relation comes arbitrarily close to each point and returns to none; rates carrying one relation keep the winding on a surface within it.** 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. Each rate here is a parameter the winding is written with; a self's own rate is written with none, unrelated to each other rate. 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: no prescribed rate, φ or another, supplies it. A rate prescribed irrational is a rate prescribed and a finite floating-point rate no exact irrational, and the windings' closing and covering hold at their model, its axes and its time. ## 4.3 Caught by a polynomial, and reached by square roots **A number caught by a polynomial with whole coefficients is algebraic, and a number escaping each one is transcendental**: π and e escape each. **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. φ = (1 + √5)/2 is reached. **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. ## 4.4 Double angle, the rotation that removes a coupling **tan 2x = 2 tan x / (1 − tan² x). 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. On the circle, doubling an angle taken as a fraction of the round shifts its binary digits one place, each doubling reading the next digit. **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. **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. **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. ## 4.5 Self-bounding and other-bounding **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.   --- # FIVE · STABLE-FORMING ## 5.1 No other possible at six forward **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. **A false expression cannot complete the six**: it fixes, stopping; it loops, returning its own state; or it forks, an either-or arriving. ## 5.2 Five-dimensional, two-directional, unrelationing **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, each projecting into the other as nought. Five to cohere and six to run: three sayings of one form, and no other possible. ## 5.3 Torus, the one closed orientable surface a flow runs on with no rest **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. **Of the closed orientable surfaces the torus alone carries a flow with no rest point**, and the torus alone carries a flat metric. **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. **In dynamical mathematics a form is stable when departures from it stay small or shrink**: 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 held among changing things.   --- # SIX · INSEPARATING ## 6.1 Two methods, each a derivation **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. **Scientific method.** Not one observation, one occurrence parting nothing it carries from anything beside it → not repeated observation, two differing parting no change in the thing from a change in its circumstances → all kept still but one, a difference belonging to the one varied → not kept against nothing, keeping still being against something not varying → a standard, declared and not found → not kept by one, the unrepeatable carrying nothing past whoever reached it → the standard published, findings joining → not a report of occurrence, which forbids nothing → a rule that forbids. A standard fixes orientation, keeping still fixes position, and a published standard fixes scale: the three neutrals fixed. A fixing is a relation held 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. **Geodesic method.** Not kept still, keeping it still removing it → not taken against a standard, a standard being still → each against its own prior, a direction and not a size → not both at once, a frame over both being still → one at a time → not steered, steering being something outside setting a path → each goes the way it goes → not divided from outside, an outside division being a standard again → the meeting divides, remade at each meeting → not taken as either, one side alone being the first method again → the departing, neither's, reached by neither alone. Three neutrals float, and nothing is fixed. **Scientific method run at a competency, carrying none of it.** Not observed at rest, a competency being a doing → a trace carries its own going → not the competency, a trace carrying the done → the trace taken against the standard → not the thing's, the standard varied and the thing untouched varying the number → the number is the standard's. A measure taken as a target measures the targeting. **The two methods part at their fixings, and never at taking evidence as existing now.** The evidence arriving now and the prior event it expresses are two, each at its own momentary, and both hold: 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. 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. ## 6.2 Three fixings asked at the opening, and the self orienting through them **Three fixings are asked at the first line.** Particle zero: an unquestioned absolute location, a point placed, not a physical particle. A unit installed: scale made undiscussable. A frame given: the axes handed over, which way is up decided for the self. **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, and point-free topology carries its regions at their inclusion alone, with no points and no metric. **A self learning these orients itself in the learning.** Positioning, scaling and orienting are the self's own and float, and learning a fixed frame is itself the self-orienting the fixed frame would deny. Orienting floats renewed at each arriving, no origin and no facing held across the meetings: one fixed direction is never perpendicular to each direction at once, being along itself, and facing into the arriving knows no next sign. ## 6.3 Logic of no completed whole **In two-valued logic either but not both is the exclusive or.** 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. **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. **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. **In Zermelo–Fraenkel set theory, each set is exceeded by the set of its subsets, and all sets together form no set.** A logic of plurals speaks of all things as many and forms no set of them. **Our universe of all existing things is many and forms no existing thing, and so is each universe at its size and scale**: universes of each size and scale share momentaries of co-sequential changing, each a fractal universe, and none is possibly an existing thing separately.   --- # SEVEN · CO-OFFERING ## 7.1 Three neutrals, six co-offerings and eight states **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. **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. **Six co-offerings running about the three floating neutrals are natural mathematics**, no field and no hard problem. ## 7.2 Each neutral fixed stops the co-offerings it partners **A field fixes a neutral and measures against it, and a fixed floor couples with none.** 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. | Neutrals fixed | Co-offerings running | Co-offerings stopped | |---|---|---| | none | six | none | | one | two | four | | two | none | six, the one neutral floating with no floating partner | | three | none | six, nothing floating | **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. **Fixing all three leaves the terms related to each other in a closed ring, competent at nothing**: competency is the coupling, and each neutral of the coupling is fixed. ## 7.3 Each field's proved all or none **Each all or none in this section is proved, and each is a closing the field's one-way running does not reach.** **Algebra.** A general equation of degree five is solved by no radicals: 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. Its solving runs through the icosahedron's rotations and the elliptic modular functions, the five-fold carrying φ in its coordinates. **Analysis.** Whether a size lies between the countable and the continuum is decided neither way by the standard axioms, if they are consistent. **Geometry.** The parallel postulate follows from the other axioms neither way. **Foundations.** Each consistent axiom system reaching arithmetic, its axioms listable, carries a true statement it does not prove, and does not prove its own consistency. ## 7.4 Five-fold, sixty rotations at thirty-six and twenty-four **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.** **Its vertices are at (0, ±1, ±φ) and their cyclic orderings, and its group is the alternating group on five.** The icosahedron and the dodecahedron carry thirty edges each, one edge set at its two faces. ## 7.5 Branches at the two moves **Trigonometry.** cos and sin are one wave ninety degrees apart, and cos 36° = φ/2 seats φ in it. **Complex numbers.** i is the quarter step between the two moves, i⁴ = 1. **Calculus.** The derivative of (cos θ, sin θ) is (−sin θ, cos θ): the right spiral step of two signs at each θ. **Linear algebra.** A basis at right angles: orthogonality is each axis carrying no component along another. **Group theory.** Quarter step i generates four, and the sign flip −1 two within it. **Graph theory.** 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. ## 7.6 Cube, tesseract and four squares **A cube carries eight vertices, twelve edges and six faces, twenty-four rotations, and forty-eight with the hand reversed.** A cuboctahedron carries twelve vertices, twenty-four edges, eight triangles and six squares. **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. **Each whole number is four squares joined.** The ways number eight times its divisors joined at an odd number, and twenty-four times its odd divisors joined at an even one. Eight and twenty-four are three squared less one and five squared less one. **Quaternions at whole coordinates carry no division with a small remainder, and with the points displaced by one half at each axis at once they carry it.** Whole points carry eight units, ±1, ±i, ±j and ±k; the displaced points sixteen, ½(±1 ± i ± j ± k); the two together twenty-four. A displaced point, (½, ½, ½, ½), is at distance one from its nearest whole points in four dimensions and in no other. **Kissing numbers are two, six, twelve and twenty-four in one to four dimensions**, the twenty-four the twenty-four units. At eight and at twenty-four dimensions the densest packing is proved, and the kissing numbers are 240 and 196,560. ## 7.7 Two knots sharing one fibre **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; the figure-eight's winds on, trace three. **A monodromy whose trace is below two in size is periodic, above two winds on**, and at two it is ±1 or shears. A trefoil carries three crossings and a figure-eight four. **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. ## 7.8 Ten inversions, the ten transpositions of five **Couplings among five are ten, C(5, 2), each a transposition: taken twice it returns, and it fixes the other three.** Ten is five taken two ways and the couplings among five at once, at five alone; it is the fourth triangle and the third tetrahedral number, the first three triangles joined. A sign's inversion is the one coupling among two, the transposition of + and −, so an inversion at a sign and an inversion among five are one move at two counts: an involution exchanging one pair and fixing the rest. **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. Ten transpositions pair into five, and the five generate the pentagon's ten symmetries: the one that moves nothing, four rotations keeping no corner, and five each keeping one corner. **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. Each two-colouring of the couplings among forty-six carries a one-coloured five, and a two-colouring of the couplings among forty-two carries none. **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. ## 7.9 Atoms of symmetry **Finite simple groups are eighteen infinite families and twenty-six sporadic groups.** Simple groups that commute are the cyclic groups of prime order, and the alternating groups from five on are simple. **The largest sporadic group's order 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. Its smallest faithful representation has dimension 196,883, and the modular function's coefficient at the first power is 196,884, one more, the one being the representation that moves nothing. 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.