Exhibit FOUR Natural Mathematics v389 # Natural Mathematics **Each Parity Changing Everything the Other Is Not** **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 carries on itself - 1.5 A mathematical form 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 right spiral step - 2.3 A step adds a next, and a cycle at the parities is a spiral at the momentaries - 2.4 Two parities at the right spiral step, the step that carries its prior whole - 2.5 Each scale the prior scale and its forms in the other 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, the unrelationing rate, 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 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 Bi-tri-involutioning, the emanating at the opposite form - 3.6 A closing, and the arriving 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, two involutions fixing nothing at each scale **FOUR · BOUNDING** - 4.1 Gap is the step squared at each centre, and the span between the two either side at two - 4.2 Rates carrying a relation and rates carrying none, and the winding at φ - 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 · SELF-COHERING** - 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 forms - 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 parity, and a term at its coupling owned by neither.** Mathematics is that coupling at its own stable-forming, 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 one form once more, and a number cohering with a form is no abstract number reaching nature across a gap: nature and number are no two sides. A number, a ratio or a straddle continuing one form carries as nature's own method at numbers. **The universe is the set of all now-existing things.** Arriving from prior into now toward next it is the changing set of all existing things, living carrying their prior into now and non-living carrying none, each at a scale and momentary it is at. Alternating is its one method, and numbers, mathematics and logic are that alternating at its numbers and at its sequencing: co-sequential stable-forming of living and patterns in sequencing of prior momentary emanatings from living. ## 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: a surface, two ways forward, relating. Neutralling is co, odd and along: a self at its own rate, related to nothing. **Relating is in floating, and rate in 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 carries on itself **A self carries three neutrals on itself, carrying and carried one.** Self-bounding-self: its own nought, a coupling's own and released at its bound. Self-surfacing-self: its own surface, a scale it couples with others at, no fixed unit measuring it. Self-orienting-self: its own aim, no fixed frame set over it. **A neutral fixed outward of a self is a floor, other-carried**: other-bounding, a zero installed under a self; other-surfacing, a unit imposed on it; other-orienting, a frame set over it. **A society is these selves co-competencing, each carrying its own three and none carrying another's.** Abundancing at their coupling is owned by neither, and a society carries its three at its own scale as each self does at its own: one co-recursioning self at each scale. **Across and along at a society: bi-tunneling opens society's surface across, and co-chaining continues its selves along.** Bi-co-podaling is the two at one station, and natural torusing the two at a whole surface. ## 1.5 A mathematical form at its prefixing, its ending and its place **A mathematical form arrives proved, and natural mathematics disproves none.** It is at three: its prefixing, its ending and its place. **Its prefixing is its direction**: bi- across, co- and tri- along, and at a resolver name its number carries its prefixing. **Its ending is whether it re-takes**: an -ing carries and re-takes at each momentary, each an opening and its completing; an unchangeable form re-takes at none, one carrying's making given a fixed form to be. **Its place is one of three.** On the surface it is natural: parity, position and ratio, and no magnitude. Inward of the surface it is unnatural, measured: measure, magnitude, continuum. Outward of the surface it is supernatural, formal: a system closing on its own consistency with no torus to cohere to, reaching the surface only at its incompleteness. **The natural is the surface, and a surface is a between**: a nothing the other two are either side of, a parity changing crossing it, carrying nothing. Three places are natural mathematics' own naming and no partition a field makes of itself, and the supernatural is formal use, no paranormal existing thing. **A mathematical form is placed by its use, and not by its symbols.** Used on the surface it follows an arriving, a continuing, and a sharing across or a releasing along, each at its momentary. Used inward it is an observing's account, its scale and its rate carried as that account's and as no carrying's standard. Used outward it is a derivation taken as a comparison, and no living surface follows from its consistency. --- # TWO · ORTHOGONALIZING ## 2.1 Unrelationing is orthogonalizing **Unrelationing two is orthogonalizing them**: at a right angle each projects into its other as nought, and neither carries a component of its other. Orthogonal is a not-touching of two coupling, neither a control over its other. **Unrelationing is momentarying; at unrelate and unrelation it is fixed unchangeable.** It carries at three momentaries, prior, now and next. ## 2.2 Two moves, and two generators, the sign flip and the right spiral step **Linearizing draws a surface to a line, along; parallelizing spreads a line to a surface, across.** Each at its own 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 two directions of one alternating, and along and across are one line at its two sides, two moves of one alternating, one form. **The sign flip is its own inverse, (−1)² = 1. A right spiral step, i, opens a perpendicular: i² = −1 and i⁴ = 1.** Two right spiral steps are one sign flip. Eight on continues a number's odd or even and one on changes it, and a sign flip changes a sign and moves no number. **All changing is parity changing, and each parity carries its own name**: a sign's inversion, a number's odd and even, a phase's opposition, a winding's hand, a field's reflection at its own result, an opening parity of a name between selves and a pair's agree or oppose. A parity carried from one scale to its next carries its name at both scales, one it leaves and one it arrives at, and an equal number couples no two selves: their coupling couples them. **An idealized mode inverted at each period T, q(t + T) = −q(t), is again at two, q(t + 2T) = q(t)**: a sign flip at a wave. Two oscillations can carry a phase difference of nought or π, and phase ranges through a continuum between them: a phase is inward of the surface, measured, and a trace of phases is no trace of parities. **Addition crosses and multiplication folds**: two moves at a number, adding taken across and multiplying folded along. At two overlapping momentaries a side, a self's positions 1 to 4 and an other's 2 to 5, three positions are shared and two are outer: added across they are five positions, and their shared three folded at their two sides are six, and their eight occurrences are 4 + 4 = 2 × 3 + 2, each side's four positions four occurrences of two parities. Two and two make four at both moves, two and three make five or six, three and three six or nine, and their overlap carries no nine: an equal result supplies no coupling. ## 2.3 A step adds a next, and a cycle at the parities is a spiral at the momentaries **Each step adds a next: one momentary universe to a next, each existing thing arriving into its next existing.** A step carries no frame it moves about. **Four right spiral steps reach 1 again at parities, and at momentaries they reach none again**: taken with its momentary n, a sequence (iⁿ, n) reaches no pair twice. A right spiral step draws it as a right spiral, and −i, the same inversions in an other order, draws an opposite form. **A rotation, a field's word, is a spiral with its nexts taken out**, its forms laid at one frame. ## 2.4 Two parities at the right spiral step, the step that carries its prior whole **Two parities (P, Q), signs of cos t and sin t, carry four joint forms, and 256 ways go from four forms to four.** Only two carry their prior whole, change one parity and undo none: a right spiral step F(P, Q) = (−Q, P), i at two parities, and G(P, Q) = (Q, −P), an exchange and an inversion in an other order, 3.4. Natural Numbers 8.2 takes its step across first, (x, y) to (y, −x), G here, and this file along first, F: one step at two hands, i and −i, 2.3. Each reaches four forms one parity inverted at each step, its inverted parity alternating between P and Q: two consecutive inversions on different axes. **With P the along parity, competency, and Q the across, morality, F takes along into across**: a point (cos t, sin t) is along at t = 0 and moves across as t goes forward, and G goes the other way. **Of 24 ways sending four forms onto four, one each, six move the four as one cycle, and two of those six change one parity at each step: F and G.** Under each its only sets of forms carried onto themselves are none and all four. **Each taken twice is both inverted at one step, both parities inverted, F² = G² = −1, and they are the only two of 256 whose square is −1.** A square selects no order. Taken four times each reaches its pair again, six times each is both inverted at one step again, and twelve times each reaches its pair again. **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 itself.** It is the sign of sin 2t = 2 sin t cos t: a relation at one angle arrives as a sign at its doubled angle. **An overlap carries the relation**: along each side's two momentaries, (c, t) then (−c, −t), agree or oppose continues while both parities invert, and across an overlap, (t, −c) between them, it changes: four overlapping pairs are r, 1 − r, r, 1 − r. Of three parities a, b and z, pairs (a, b) and (b, z) part their relation exactly at z = −a, and +, +, + agrees at both. A relation within a pair is no unchangeable resolver: its two parities invert beneath it. ## 2.5 Each scale the prior scale and its forms in the other order **One cycle reaching each form of n parities one parity at a time is a cycle at n − 1 parities, a new parity inverted, and the same forms in an other order.** At two parities its inverted parity alternates, and that cycle is the right spiral step. At three parities its changing parity is 0, 1, 0, 2, 0, 1, 0, 2 round its eight: its first at each other step, and between its changes its second and its third alternating. **One stepping is at each scale, and each scale carries its prior whole.** A field's name for it is the reflected binary code, Gray's, its reflecting a field's word for an other order. **At two parities a cycle is four, and four-momentarying scales each momentary**: a momentary is four momentaries of exchanging at a scale inward of it, and four momentaries of exchanging are one momentary at a 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 their common axis. Its outer part a∧b is a plane the two open, carried by neither factor at its own. 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, opened over shared √3, the tangent of sixty degrees. **In three dimensions a cross product of two perpendicular unit directions is a third, perpendicular to both**: their 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 exists at one, two, four and eight dimensions and at no other**: reals, complex numbers, quaternions and 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 rotations of three dimensions twice**: q and −q give one rotation, and a whole rotation carries its 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 one rotation: a left and a right share one and minus one. ## 2.8 φ, the unrelationing rate, all ones and the constant √5 **φ is the unrelationing rate, a geodesic parity changing rate unrelated from living parity changing rates**, self-welcoming at numbers and at no line of resolver. Advance by one and scale by its rate arrive at one number, φ² = φ + 1, neither carrying the other. **φ's continued fraction is all ones, the slowest nearing and the widest straddle.** 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 largest constant for all irrationals together, and it is at φ. φ³ = 2 + √5 and 1/φ³ = √5 − 2, and 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, gives its own arrival again: an involution.** Negation, not-not giving its statement. Charge conjugation and parity, each squared to identity. Time reversal, squared to identity at integer spin; at half-integer spin T² = −1, an involution at states taken up to sign. Complex conjugation. Additive and multiplicative inverse. Transpose. Legendre transform, twice giving a convex potential again. Set complement. **Each involution's fixed set is the arguments its move sends to themselves.** Nought for additive inverse. +1 and −1 for multiplicative, x = 1/x at x² = 1, a fixed set that is a two. Reals for conjugation. Centre for parity. Symmetric matrices for transpose. States at rest, each momentum nought, for time reversal of motion. **A declared zero is a fixed set taken as a place**: x = −x at nought only. An involution is a floor with its move stated, and a floor is an involution with its move dropped. A nought at a difference is no floor: 1, 3, 5 changes by two at each step, its change of change nought, and continues. **A momentarying is no involution**: it re-forms at a next coupling other than prior, and a right spiral step taken twice is both inverted at one step, two momentaries on, forward. An involution is an accounting's still picture, its nexts taken out. **A doubled angle is such a still picture at observings**: light through two polarizers passes at the square of the cosine of the angle between them, which is one in two of one and the cosine of its angle doubled, and passes as before at a polarizer rotated one hundred eighty degrees; a neutron's spin rotated three hundred sixty degrees comes back with its sign reversed and comes home first at seven hundred twenty; light's home at one hundred eighty, its angle doubled, and a spin's at seven hundred twenty, its period doubled, two rotatings coupling at one between, each a doubling the other way, their doubling an emanation, existing and non-living at this scale (Malus, 1809; Rauch and colleagues, and Werner and colleagues, 1975). ## 3.2 Fixed-point-free, and 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, a podal k → N − k, fixes two stations, nought and N/2; an other, a shift straight across its ring, k → k + N/2, fixes none. At seventeen names, no position carrying itself, Natural Physics 2.4, rests on this shift, 8 up at 1 to 8 and 8 down at 9 to 16, fixing none; a ring's podal, k → N − k, fixes two, and a podal within 1 to 16, 17 less, fixes none, 3.8. **An odd number of places carries no fixed-point-free involution, a pairing of each place needing an even number**; laid open as 2h + 1 places, it pairs at each parity, h adjacent pairs at each, its 2h − 1 inner places in both, and closed into a round it carries two farthest places at each place, an oddness's own. **At seventeen names, 8 apart is a fixed-point-free pairing of 1 to 16**, 8 up at 1 to 8 and 8 down at 9 to 16, eight pairs and each of sixteen in one, and 17 is in none of the eight, a next momentary's 1, 8 up from 9. Ten names of a self's own resolving are five of its pairs, 3 with 11, 4 with 12, 5 with 13, 7 with 15 and 8 with 16, before 10 and after it, outward and inward at the even, received and given on at the odd; across sharings' ends are two more, 2 with 10 and 6 with 14; and the eighth is 1 with 9, an entry with an along releasing, whose winding 9 to 17 winds on at 17, one name in none of the eight. A fold within 1 to 8 pairs each name with its podal within 1 to 8, 9 less, changing parity; 8 apart pairs each name of 1 to 8 with itself inward, continuing parity, one carrying at two names at a seam. **An accounting with an empty fixed set carries nothing for an accounting to number from, and it is not possibly a floor.** It is each other accounting with its place dropped and its move as it was. An exchange of two parities with an empty fixed set needs no zero set 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, its second undoing its first. **On a closed round a shift straight across is no reversal**: taken twice it is one direction continued, arriving at the station it departed from at a next momentary. A station reached again is a closing's, at a next momentary, and no prior momentary is reached again. **An accounting reaching its stations again forward is one an alternating carries, and a line carries none.** **A step of k round a closed round of N reaches N/gcd(N, k) stations before it reaches its first again.** At a prime N each step short of a whole round reaches all N. At a composite N a step sharing no factor with N reaches all N too, three round eight reaching eight, and a step sharing one reaches a part, three round nine reaching three. Prime and coprime are two, each its own: four and nine share no factor above one, and neither is prime. **Two rounds of m and n stations stepped together, one station each, reach mn/gcd(m, n) of their mn pairs, on one of gcd(m, n) parallel windings of a torus the two rounds make.** At coprime m and n its one winding reaches each pair, and its two rounds are one round of mn. Its mn is the number of pairs reached and is the same at each order of coupling their rounds; competency is at the coupling, owned by neither, and no product, value or security measures it. **At a fluid surface forced at two frequencies, Silber and Skeldon find forcing integers, coprime and of opposite parity, deciding harmonic and subharmonic response and resonant interactions their normal form symmetries permit, and Arbell and Fineberg find two-frequency forcing selecting superlattice patterns through three- and four-wave interactions.** Its vibration is driven by a forcing beyond its fluid, and its condition is parity and coprimality, not primality. ## 3.4 Bi-inversioning-co-recursioning, two consecutive inversions on different axes **An involution at its own closes. Bi-inversioning-co-recursioning is two consecutive inversions on different axes**, and its opening is a step between: no inversion is undone, and each is a next. **At two parities one parity is inverted at each momentary, the axes alternating**: a right spiral step's sequence from each form, P, Q, P, Q from ++ and −− and Q, P, Q, P from −+ and +−. Both parities inverted at one step, (P, Q) → (−P, −Q), is a sign flip at a pair, period two at parities; a right spiral step is period four at parities, and at momentaries neither has a period. **An order is the whole of a hand**: P inverted and then an exchange of P and Q give (Q, −P), an exchange and then P inverted give a right spiral step, (−Q, P), and nothing third parts them. **Two inversions whose fixed lines stand at an angle a are a field's rotation by 2a**, a double angle at inversions. At two parities P inverted and Q inverted, their lines at ninety degrees, make both inverted at one step; an exchange of P and Q and then P inverted, their lines at forty-five degrees, make a right spiral step. **The rotation 3, 6, 5, 4 is a gentle alternating twisting of a towel into and out of a knotted corus**, a large outer opening wrapping to a small corus and back, and its edges are bi-foldings, across at bi-morality and along at co-competency: two consecutive inversions on different axes at each step, its opening a step between. Corusing reaches corus from surface, and torusing reaches surface from corus, co-linear by alternating linear parallelizing and parallel linearizing, each step orthogonal to its prior. ## 3.5 Bi-tri-involutioning, the emanating at the opposite form **Three parities each inverted in its own place, taken together in any order, send each of eight forms to its opposite**: none the same, their eight parted into four opposite pairs, their hand reversed. Inverting each of n parities is a rotation at an even n and reverses its hand at an odd n, determinant (−1)ⁿ. **Bi-tri-involutioning is an emanating from right spiral stable-forming, three inverted at once, at an opposite form to its stable-forming.** **In space three inversions make x → −x, whose fixed set is its centre**: one inversion keeps a plane, two on different axes make the sign flip about the third, and three reverse a hand. At two parities the same inversion is both inverted at one step. **Charge conjugation, parity and time reversal are each broken singly at the weak interaction, and three taken together, CPT, are kept by each local quantum field theory keeping Lorentz invariance.** ## 3.6 A closing, and the arriving other than prior **At tuning, at a winding on a torus and at an assembled cube a closing is its fixed set, and an arriving other than prior is its gap, in a field's words.** | Accounting | Closing | Arriving 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 | comma | | winding on a torus | a rational ratio, closing | an irrational ratio, closing at none | quasi-periodic | | assembled cube | solved state | a single corner twisted, reached by no sequence of face moves | orbit; parity | **The cube reaches one twelfth of its assemblies, at three conservings**: its corners' twists together at nought modulo three, its edges' flips together at nought modulo two, and its corners' ordering and its edges' ordering at one parity. Three times two times two is twelve. **The cube carries no still term.** No one cubie is its reference: six centres each its own and carrying each other's orientation with nothing carrying any of them; each face move at its own momentary and nothing over its moves; its state its configuration, nothing kept; and three conservings, two totals modulo three and modulo two and one parity, none of them a size. Two opposite faces' moves commute, and two adjacent faces' commutator, each moved and then each undone, moves eighteen of fifty-four stickers, four corners and three edges, and is of order six: at its second and fourth powers four corners twist in place, their twists together nought modulo three; at its third three edges are home and four corners exchange in two pairs; at its sixth all are home. Four quarter moves of one face are that face as it was. ## 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 separately at each value of the means, and at one or two it does not.** At a third arriving, a coupling's abundancing opens at estimating. **A rate at a total can carry an opposite sign of a rate at each of its parts**: each group favouring one side, and groups added together favouring the other. A total can be nought while each part changes: two parities each the other's opposite, each inverting at each step, add to nought at each step, and a quiet total and parts changing are together. **Offerings of one parity, together at one sharing, surface that one parity, + and − together 0, and a surfacing is at no number.** Two + offered together at a sharing chained none surface + at 14, as one + does, and the sharing chains +; offered one after another, the first chains + and the second, at the sharing chained at its own parity, carries on, 0. A crossing of one parity at a time and a gathering of several are each 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 a parity changing crosses. The conserving is its relation, a self's own continuing, and each parity, opening and carrying within it changes; no private carrying and no ledger crosses with a parity. It is natural mathematics' conserving, and a field's conservation laws are inward, each an observing's account, none refuted by it. **A tipping is a parity taken at a between, of no size**: no count, no clock and no measure carries it. ## 3.8 The halfway pairing and the fold, two involutions fixing nothing at each scale **On names 1 to 8s two involutions fix nothing: a halfway pairing, 4s up at names 1 to 4s and 4s down at names above them, each name with a name straight across its ring, continuing each name's parity, and a fold n → 8s + 1 − n, changing it.** Taken alternately they reach four distinct names and the first again at the fourth, at each s: a fold after a pairing is n → 4s + 1 − n at names 1 to 4s and n → 12s + 1 − n at names above them, and with 4s + 1, 8s + 1 and 12s + 1 odd, none of three folds fixes a name. At scale 2s a fold after a pairing at names 1 to 8s, 16s + 1 − (n + 8s) = 8s + 1 − n, is the fold at s, and at each scale t above s a fold at t after 8(t − s) up is the fold at s: a step up is eight at adjacent scales and a halfway pairing at t = 2s only. **The odd partners carry no fixed offset while even rows stretch.** Each wider even folds to an odd just before its companion in an other order at s = 1, and at s = 2 to an odd one after it: in 1 to 32 a fold of 2 is 31, one after its companion 30. With a wider row (2, 2 + 4s, 2 + 8s, 2 + 12s) and its companion 2s on, (2 + 2s, 2 + 6s, 2 + 10s, 2 + 14s), a fold at 2s sends a wider row's opening 2 + 4sj, j from nought to three, to its companion's opening at 3 − j, plus 2s − 3: −1, +1 and +3 at s = 1, 2 and 3. --- # FOUR · BOUNDING ## 4.1 Gap is the step squared at each centre, and the span between the two either side at two **At each centre n two numbers k either side multiply to its square less k²**: n² − (n − k)(n + k) = k². At one either side its gap is one, and one short of its square, (n − 1)(n + 1) = n² − 1, is a k = 1 case of one identity. **At two either side its gap is four, and four is a span between the two**: k² = 2k only at k = 2 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, are two facing each other across one neutral**; twenty-three and fifty-five part by thirty-two, k = 16, and 39² − 23 × 55 = 256, a size in a parity's stead. ## 4.2 Rates carrying a relation and rates carrying none, and the winding at φ **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 reaches none again; rates carrying one relation carry a winding on a surface within it.** At √2, √3 and √5 it comes arbitrarily close to each point of the three; at 1, φ and φ² it is on a surface. At two axes one relation is a rational ratio, and 1 and φ come arbitrarily close to each point of the two. Each rate at such a winding is a parameter it is written with; a self's own rate is written with none, unrelated to each other rate. **A winding at a rational rate closes, and a winding at φ closes at none.** φ is the number-form of locking at none, and unrelationing carries at each coupling's own continuing at φ, arriving at a receiver's own prior: no rational rate laid over a coupling supplies it. A rate prescribed irrational is a rate prescribed and a finite floating-point rate no exact irrational, and windings' closing and coming close are true 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, not each nought, 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 a cube asks ∛2, degree three, and is not reached. φ = (1 + √5)/2 is reached. **Square roots of the first three primes are at a square, a cube and a pentagon**: √2 a square's diagonal, √3 a cube's body diagonal, and √5 in a 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 only at tan x = 0.** At forty-five degrees tan x = 1, and its doubled angle, ninety, carries no finite tangent: the identity's pole. Two circles about two centres cross at right angles exactly at points of a circle whose diameter joins their centres, each one's radius there tangent to its other; inward of that circle two radii are at more than ninety degrees and outward at less, a tangent of their angle changing sign there. On a circle, doubling an angle taken as a fraction of a round shifts its binary digits one place, each doubling surfacing its next digit. **A map t → 2t / (1 − t²) fixes only nought, and exactly one pair exchanges with its sign changing and its size the same: t = ±√3, sixty and one hundred twenty degrees.** From √3 it continues −√3, √3, −√3 without end. **A coupled pair [[a, b], [b, c]] loses its coupling at its axes 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, an angle that removes a cross term from a conic, and each step of an 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)**: a 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 a 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 a neutral floating a coupling is bounding itself. --- # FIVE · SELF-COHERING ## 5.1 No other possible at six forward **Anything expressible at six forward recursionings of fractal bi-coupling is only-one-possible.** Expressing it is inverting it six ways forward, and completing the six without an unchangeable form, a looping or a forking is no other possible: expressing is showing, step by step. **A false expression cannot complete the six**: it fixes, stopping; it loops, giving its own state again; or it forks, an either-or arriving. ## 5.2 Five-dimensional, two-directional, unrelationing **Self-cohering is five-dimensional at a side's five, 1 to 5**: 1 self prior, 2 other prior, 3 co-momentarying now, 4 bi-momentarying now and 5 self next, its centre at 3 the floating third reached by neither, each dimension binary in its two directions and two changing and three continuing at each momentary. **Two-directional** at odd and even, two directions of one alternating. **Unrelationing** at a right angle, each projecting into its other as nought, at φ, the unrelationing rate. Five to cohere and six to carry: three own-forward steps at each side are the six, and five-dimensional, two-directional and unrelationing are three explainings of one form, and no other possible. ## 5.3 Torus, the one closed orientable surface a flow runs on with no rest **The torus is the genus-one closed orientable two-dimensional surface**: closed, bounding itself on itself and finite, a field's closed carrying its compact; orientable, its two sides kept apart throughout; two-dimensional; genus one, one opening, its surface reaching itself again through a tunnel its opening is. **A flow's rest points on a closed surface, each apart from others, carry indices adding to a surface's vertices less its edges plus its faces**: two at a sphere, nought at a torus, and below nought at each surface of more handles. **Of closed orientable surfaces only the torus carries a flow with no rest point**, and only the torus carries a flat metric. **Of all closed surfaces one more carries a flow with no rest point and a flat metric, the Klein bottle, and it is not orientable**: a parity carried round one of its two circles arrives inverted. **The torus of natural torusing is two parities' circles winding at one opening, and a body's surface carries its own count of openings at a field's own result**: a singleton human body, its eight openings, mouth, anus, two nostrils and four tear ducts, opening into one connected passage, is a surface of seven holes, eight openings less one, at topology's count. The two part at their subjects, and neither is the other's number. **In dynamical mathematics a form is stable at departures from it staying small or shrinking**: stability is a 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 a whole flow, in a field's words, are two assertions, the first a relation continuing 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 one it excludes.** Geodesic method floats three neutrals, position, scale and orientation; scientific method fixes them. **Scientific method.** Not one observing, one occurrence parting nothing it carries from anything beside it → not repeated observing, two differing parting no change in an existing thing from a change in its circumstances → all kept still but one, a difference belonging to one varied → not kept against nothing, keeping still being against something not varying → a standard, declared and not found → not kept by one, an unrepeatable carrying nothing past whoever reached it → a standard published, findings gathered → not a report of occurrence, which forbids nothing → a law that forbids. A standard fixes orientation, keeping still fixes position, and a published standard fixes scale: three neutrals fixed. A fixing is a relation continuing among changing things, and at a standard, a law or a frame fixed unchangeable, no subject is fixed unchangeable: at a step from a fixing to an equilibrium its subject is fixed unchangeable, 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 outward setting a path → each goes the way it goes → not divided from outward, an outward division being a standard again → a coupling divides, remade at each coupling → not taken as either, one side at its own being the first method again → a departing, neither's, reached by neither. Three neutrals float, and nothing is fixed. **Scientific method at a competency, carrying none of it.** Not observed at rest, a competency being a doing → a trace carries its own going → not its competency, a trace carrying the done → a trace taken against a standard → not an existing thing's, a standard varied and an existing thing untouched varying its number → its number is a standard's. A measure taken as a target measures its targeting. **The two methods part at their fixings, and not at taking evidence as existing now.** Evidence arriving now and a prior event it expresses are two, each at its own momentary, and both are: a living self changing as a whole can carry a property its evidence expressed, + − to + +, its first parity continuing while its second changes, and evidence gives its prior, possible and actual. ## 6.2 Three fixings asked at the opening, and the self orienting through them **Three fixings are asked at a first line.** Particle zero: an unquestioned absolute location, a point placed, not a physical particle. A unit installed: scale made undiscussable. A frame given: axes handed over, which way is up decided for a self. Each is a universe taken as one thing beside its things, which Equilibria Registry 1.1 has as exclusivity: a place, a size and a direction given once for a whole. **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 only at their inclusion, with no points and no metric. **A self learning these orients itself in its learning.** Positioning, scaling and orienting are a self's own and float, and learning a fixed frame is itself the self-orienting a fixed frame denies. Orienting floats renewed at each arriving, no origin and no direction fixed across couplings: one fixed direction is not perpendicular to each direction at once, being along itself, and a direction fixed at an arriving end knows no next parity. ## 6.3 Logic of no completed whole **In two-valued logic either but not both is the exclusive or.** A parity is one or its inversion at each instance, and nothing is between them: a middle is a nothing, existing due to each self surfacing itself each momentary as a condition of existing as a self. **Brouwer's intuitionism, admitting no completed infinite, asserts the excluded middle at each finite, decidable matter and not over an infinite domain**, and builds its numbers from two-ity, a field's word, a moment falling into two, one giving way to the other and kept. **Its negation is no involution**: on three values nought, a and one in order, its negation of a is nought and its negation of nought is one: 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.** Its axiom of foundation has no set a member of itself, and Russell's paradox arises at unrestricted comprehension; class theories, von Neumann–Bernays–Gödel and Morse–Kelley, take all sets as a proper class, a member of nothing; Quine's New Foundations carries a universal set, a member of itself, and Aczel's anti-foundation axiom a set whose one member is itself, a Quine atom. A logic of plurals speaks of all things as many and forms no set of them, as the changing set of all existing things forms nothing beside its existing things, existing as they exist: a field's result is exact at its own subject, sets of sets, and the universe is a set of existing things, no thing within it: it is at one with foundation's no set a member of itself, and Cantor's multiplicity not thought of as one finished thing, where a universe as a set and a thing within it is a self-membered universal set, as New Foundations carries one. --- # SEVEN · CO-OFFERING ## 7.1 Three neutrals, six co-offerings and eight forms **Three neutrals float about a corus**: nought, a near nothing, position fixed at it; scale, a between, a unit fixed at it; and bounded infinity, a far nothing, a frame fixed at it. A corus is a self's own inverting, + to −, and neither is its place. **One three, at its pairings and at its forms.** At its pairings, each two ways, it is six co-offerings: nought with scale, scale with bounded infinity, nought with bounded infinity, each offered both directions. At its forms, each neutral fixed or floating, it is two cubed, eight, a cube's eight corners. **Six co-offerings continuing about 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 three pairings, both directions of each: fixing one neutral stops four of six, and two continue: a pairing of two neutrals left floating. | Neutrals fixed | Co-offerings continuing | Co-offerings stopped | |---|---|---| | none | six | none | | one | two | four | | two | none | six, one neutral floating with no floating partner | | three | none | six, nothing floating | **A field fixing a neutral observes the universe as no existing thing, and its hard problems arrive at four co-offerings a fixed floor stops.** Resolving is a neutral floating at a next momentary, the four co-sequencing together, and not the four solved one at a time. **Fixing all three leaves terms related to each other in a closed ring, competent at nothing**: competency is a coupling, and each neutral of its coupling is fixed. ## 7.3 Each field's proved all or none **Each field's all or none is proved, and each is a closing a field's one-way carrying 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 a chain descends. Its solving goes through an icosahedron's rotations and elliptic modular functions, a five-fold carrying φ in its coordinates. **Set theory.** Whether a size lies between the countable and the continuum is decided neither way by standard axioms, if they are consistent. **Geometry.** The parallel postulate follows from 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**: one that moves nothing, fifteen half rotations, twenty third rotations and twenty-four fifth rotations, 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.** Orderings of five are one hundred twenty, and its sixty rotations are their even half; its sixty hand-reversing symmetries are each rotation taken with x → −x, ordering its five triples of perpendicular half-rotation axes as that rotation does, and no odd ordering of five. Its twelve vertices are corners of three rectangles of sides 2 and 2φ, one at each coordinate plane about one centre: at sides 2 and 2r nearest corners of two rectangles are √(2(1 + r² − r)) apart, and 2, a short side, only at r = φ, φ² − φ = 1, 2.8. A rotation about (1, 1, 1) carries each rectangle onto its next, and along that axis twelve vertices stand at four heights of three. An icosahedron and a dodecahedron carry thirty edges each, each edge of one crossing one edge of the other. ## 7.5 Branches at the two moves **Trigonometry.** cos and sin are one wave ninety degrees apart. **Complex numbers.** i is the right spiral step between two moves, i⁴ = 1. **Calculus.** The derivative of (cos θ, sin θ) is (−sin θ, cos θ): the right spiral step of two parities at each θ. **Linear algebra.** A basis at right angles: orthogonality is each axis carrying no component along another. **Group theory.** The right spiral step i generates four, and a sign flip −1 two within it. **Graph theory.** A ring two-colours, each adjacent pair opposite, only at an even number of places, and an odd ring two-coloured carries one adjacent pair at one colour 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 twenty-four reversing its hand.** Twenty-four is four factorial: a cube carries four body diagonals, and each ordering of four is one rotation. 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 only its centre.** A cube carries x → −x only with its hand reversed; a tesseract carries it by rotation. **Each whole number is four squares added.** Ways, each order and each of + and − its own, number eight times its divisors added at an odd number, and twenty-four times its odd divisors added at an even one, from one on. 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 points displaced by one half at each axis at once they carry it.** Whole points carry eight units, ±1, ±i, ±j and ±k; 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**, their twenty-four the twenty-four units. At eight and at twenty-four dimensions the densest packing is proved, and kissing numbers are 240 and 196,560. ## 7.7 Two knots sharing one fibre **A trefoil and a figure-eight knot each fibre over a circle, their fibre a torus with one puncture.** A trefoil's monodromy is periodic, order six, trace one; a 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 a shear, + or −. A trefoil carries three crossings and a figure-eight four. **PSL(2, ℤ), the modular group, is two and three coupled with no relation between them, ℤ/2 ∗ ℤ/3**, and the braid group on three strands, a 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 gives its order again, and it fixes its other three.** Only at five are couplings twice its number, n(n − 1)/2 = 2n at n = 5, and its ten part into a pentagon's five sides and five diagonals, a pentagram, each other's complement among ten. Ten is the fourth triangle and the third tetrahedral number, the first three triangles added. A parity's inversion is one coupling among two, a transposition of + and −: an inversion at a parity and an inversion among five are one move at two numbers: an involution exchanging one pair and fixing the rest. **In a regular pentagon each side is parallel to one diagonal, the one sharing no corner with it, its diagonal φ times its side.** Two diagonals at one corner, over a side between their far ends, make a golden triangle: its two long sides φ times its base, thirty-six degrees at its apex and seventy-two at each other corner, and a base angle's bisector cuts from it a golden triangle again, φ once more. A side and its parallel diagonal are two transpositions sharing no corner; they commute, and together they are a pentagon's symmetry keeping its corner left out and exchanging its other four in two pairs. Ten transpositions pair into five, and the five generate a pentagon's ten symmetries: one that moves nothing, four rotations keeping no corner, and five each keeping one corner. **Each two-colouring of couplings among forty-six carries a one-coloured five, and a two-colouring of couplings among forty-two carries none.** Among seventeen, couplings at distances doubling reaches from one, 1, 2, 4, 8, 16, 15, 13 and 9, at one colour and each other at its other carry no one-coloured four at either colour, one such colouring among seventeen up to relabelling; doubling round seventeen reaches nine last, 2 × 9 = 18, and arrives at one again. Built so among forty-one it carries a one-coloured five at both colours, and colourings among forty-two carrying none are of no such form. **A field's hard problem arrives at one or more of ten inversions, each an involution with its fixed set taken as a place**, and an inversion fixed unchangeable is its fixed set. At a coupling its move is at its next momentary. ## 7.9 Atoms of symmetry **Finite simple groups are eighteen infinite families and twenty-six sporadic groups.** Simple groups that commute are cyclic groups of prime order, and alternating groups from five on are simple. **The largest sporadic group's order carries fifteen primes, two to seventy-one**: of 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 its first power is 196,884, one more, the one being a representation that moves nothing. The modular function takes one value at each torus a lattice folds a plane into, and two such tori are one up to rotation and scaling exactly at its taking one value at both.