Exhibit FOUR Natural Mathematics v391 # 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, neutralling along - 1.4 Three neutrals each self has 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 brings its prior whole - 2.5 Each scale the prior scale and its forms in the other order - 2.6 Sharing and opening, 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 declaring of 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 discovering other than prior - 3.7 Three, a total against its parts, and a tipping no accounting has - 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 with a relation and rates with 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 at the opening, and the self orienting through them - 6.3 Logic of no completed whole **SEVEN · CO-SHARING** - 7.1 Three neutrals, six co-sharings and eight forms - 7.2 Each neutral fixing stops the co-sharings 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 of its coupling arising in each self with its other.** Mathematics is that coupling with its own stable-forming, one form with each line. ## 1.2 Numbers, mathematics and logic, nature's own method **Numbers, mathematics and logic are nature's own unrelationing changing, is or is not, with each step.** 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 going on with one form goes as nature's own method with numbers. **The universe is the set of all now-existing things.** Same-existing from prior to now and discovering next, it is the same-existing set of all existing things, living going on with their prior along to now and non-living with none, each with its own scale and momentary. Changing is its one method, and numbers, mathematics and logic are that changing with its numbers and its sequencing: co-sequential stable-forming of living, and patterns of sequencing of prior momentary emanatings from living. ## 1.3 Floating across, neutralling along **Floating neutralling is one motion with two parities, one and then the other.** Floating is bi and across: a surface, two ways forward, relating. Neutralling is co and along: a self with its own rate, relating to nothing. **Relating is of floating, and rate of neutralling.** Each rate is unrelationing with each other rate, and a rate lying over another, unchangeable, is a fixing. All floating and no neutralling relates with no rate of its own; all neutralling and no floating relates to nothing. ## 1.4 Three neutrals each self has on itself **A self has three neutrals on itself, a self and its neutrals one.** Self-bounding-self: its own nought with each coupling, completing whole with its own bound. Self-surfacing-self: its own surface, its own scale for coupling with others, no unchangeable unit measuring it. Self-orienting-self: its own aim, no unchangeable frame over it. **A neutral fixing outward of a self is a floor, an other's**: other-bounding, a zero an other installs under a self; other-surfacing, a unit an other imposes on it; other-orienting, a frame an other sets over it. **A same-existing-society is these selves co-competencing, each having its own three and none having another's.** Abundancing of their coupling arises in neither self alone, and a same-existing-society holds nothing of its own: its three are its selves', each self's own, one same form inside and outside itself. **Across and along with a same-existing-society: bi-tunneling opens a same-existing-society's surface across, and co-chaining goes on with its selves along.** Bi-co-podaling is both with one station, and natural torusing both with a whole surface. ## 1.5 A mathematical form at its prefixing, its ending and its place **A mathematical form comes with its proving, and natural mathematics disproves none.** It has three: its prefixing, its ending and its place. **Its prefixing is its direction**: bi- across, co- and tri- along, and with a resolver name its direction is with six names between selves: across with 2, 6, 10 and 14, and along with 9 and 17. **Its ending is whether it re-takes**: an -ing goes on and re-takes with each momentary, each an opening and its completing; an unchangeable form re-takes with none, a going on fixing as one form. **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, measuring: 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 by its incompleteness. **The natural is the surface, and a surface is a between**: a nothing its other two places are either side of, a changing going across it, bringing 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's use places it, and not its symbols.** With use on the surface it follows an observing, a same-existing, and a sharing across or a going along, each with its momentary. With use inward it is an observing's account, its scale and its rate that account's and no going on's standard. With use outward it is a derivation as a comparison, and no living surface follows from its consistency. --- # TWO · ORTHOGONALIZING ## 2.1 Unrelationing is orthogonalizing **Unrelationing two is orthogonalizing them**: with a right angle each projects onto its other as nought, and neither has a component of its other. Orthogonal is a not-touching of two coupling, neither a control over its other. **Unrelationing is momentarying; as unrelate and unrelation it is fixing unchangeable.** It is of 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 by its own stops: parallelizing with no linearizing is a surface with no sequence, and linearizing with no parallelizing a sequence with no surface. + and − are two directions of one changing, and along and across are one line with its two sides, two moves of one changing, 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 keeps a number's odd or even and one on changes it, and a sign flip inverts a sign and moves no number. **Changing is is or is not same-existing, and each parity has 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 by its own result, an opening parity of a name between selves and a pair's agree or oppose. A parity has its one name with each scale, one form inside and outside itself, and an equal number couples no two selves: their coupling couples them. **An idealized mode inverted with each period T, q(t + T) = −q(t), is again itself after two, q(t + 2T) = q(t)**: a sign flip of a wave. Two oscillations can have a phase difference of nought or π, and phase ranges through a continuum between them: a phase is inward of the surface, a measure, and a trace of phases is no trace of parities. **Addition goes across and multiplication folds**: two moves with a number, adding across and multiplying folding along. With two rows of four positions, 1 to 4 and 2 to 5, three positions are within both and two are outer: adding across they are five positions, and their common three folding with their two sides are six, and their eight occurrences are 4 + 4 = 2 × 3 + 2, each row's four positions two odd and two even. Two and two make four by both moves, two and three make five or six, three and three six or nine, and their overlap has 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 discovering its next existing.** A step has no frame it moves about. **Four right spiral steps reach 1 again by parities, and by 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, its same inversions with an other order, draws an opposite form. **A rotation, a field's word, is a spiral with no nexts**, its forms laid on one frame. ## 2.4 Two parities at the right spiral step, the step that brings its prior whole **Two parities (P, Q), signs of cos t and sin t, have four joint forms, and 256 ways go from four forms to four.** Only two bring their prior whole, invert one parity and undo none: a right spiral step F(P, Q) = (−Q, P), i with two parities, and G(P, Q) = (Q, −P), an exchange and an inversion with an other order, 3.4. Natural Numbers 8.2's step, (x, y) to (y, −x), is G here, and this file's is F: one step of two hands, i and −i, 2.3. Each reaches four forms inverting one parity with each step, its inverting parity P and then Q: two consecutive inversions on different axes. **F takes P's axis to Q's**: a point (cos t, sin t) is on P's axis with t = 0 and moves toward Q's as t goes forward, and G goes opposite. **Of 24 ways sending four forms onto four, one each, six move all four as one cycle, and two of those six invert one parity with each step: F and G.** Under each its only sets of forms going onto themselves are none and all four. **Each taken twice is both inverted with 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 with 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 inverts it, each step itself deciding it.** It is the sign of sin 2t = 2 sin t cos t: a relation of one angle comes as a sign of its doubled angle. **A step has its relation**: two steps apart, (c, t) then (−c, −t), agree or oppose goes on while both parities invert, and one step apart, (t, −c) between them, it inverts: four consecutive 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 when z = −a, and +, +, + agrees with 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 of n − 1 parities, a new parity inverted, and its same forms with an other order.** With two parities its inverted parity alternates, and that cycle is a right spiral step. With three parities its inverting parity is 0, 1, 0, 2, 0, 1, 0, 2 round its eight: its first with each other step, and between its inversions its second and its third alternating. **One stepping is with each scale, and each scale has 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. **With two parities a cycle is four, and four-momentarying scales each momentary**: a momentary is four momentaries of exchanging with a scale inward of it, and four momentaries of exchanging are one momentary with a scale outward, one form with each size. ## 2.6 Sharing and opening, the product at its two parts **A product of two directions is two parts together: ab = a·b + a∧b.** Its inner part a·b is their sharing along their common axis. Its outer part a∧b is a plane both open, which neither factor has by itself. Its two parts conserve: (a·b)² + |a∧b|² = |a|²|b|², which is cos²θ + sin²θ = 1. **Each part has its nought.** With a parallel pair nothing opens, a∧b = 0, and equality a = b is among them. With θ = 90° nothing is shared, a·b = 0. With θ = 60° a pair of unit directions shares one half and opens √3/2, its opening over its sharing √3, the tangent of sixty degrees. **With three dimensions a cross product of two perpendicular unit directions is a third, perpendicular to both**: their product opens an axis neither factor has. ## 2.7 One, two, four and eight, the doubling that divides **A division by each non-zero element, length multiplying with it, exists for one, two, four and eight dimensions and for no other**: reals, complex numbers, quaternions and octonions. Each doubling sheds one: ordering by two, commuting by four, with i j = k and j i = −k, associating by eight. **Octonions have 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 brings its quaternion from 1 to −1. **Each rotation of four dimensions is a left and a right unit quaternion together**, 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 changing rate unrelationing with living changing rates**, self-welcoming with numbers and with no line of resolver. Advance by one and scale by its rate reach one number, φ² = φ + 1, neither having its other. **φ's continued fraction is all ones, the slowest nearing and the widest straddle.** Each irrational x has without end fractions p/q with |x − p/q| < 1/(√5 q²), and for φ no constant larger than √5 does: √5 is the largest constant for all irrationals together, and φ reaches it. φ³ = 2 + √5 and 1/φ³ = √5 − 2, and that constant falls away with their difference: φ³ − 1/φ³ = 4. --- # THREE · ACCOUNTING ## 3.1 An accounting is an involution, and a declaring of zero is its fixed set **An accounting for changing is a move that, taken twice, gives its own argument again: an involution.** Negation, not-not giving its statement. Charge conjugation and parity, each squared to identity. Time reversal, squared to identity with integer spin; with half-integer spin T² = −1, an involution of 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 when x² = 1, a fixed set that is a two. Reals for conjugation. Centre for parity. Symmetric matrices for transpose. States of rest, each momentum nought, for time reversal of motion. **A declaring of zero is a fixed set fixing as a place**: x = −x for nought only. An involution is a floor stating its move, and a floor is an involution with no move. A nought of a difference is no floor: 1, 3, 5 goes on by two with each step, its second difference nought, and does not stop. **A momentarying is no involution**: it re-forms with a next coupling other than prior, and a right spiral step taken twice is both inverted with one step, two momentaries on, forward. An involution is an accounting's still picture, with no nexts. **A doubled angle is such a still picture of observings**: light through two polarizers passes as the square of the cosine of the angle between them, which is half of one and the cosine of its angle doubled, and passes as before with 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 with seven hundred twenty; light's home with one hundred eighty, its angle doubled, and a spin's with seven hundred twenty, its period doubled, two rotatings coupling with one between, each a doubling opposite to its other, their doubling an emanation, existing and non-living with 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 with two values. **On an even ring of N two involutions part by 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. With seventeen names, no position going to itself, Natural Physics 2.4, rests on this shift, 8 up with 1 to 8 and 8 down with 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 has no fixed-point-free involution, a pairing of each place needing an even number**; laid open as 2h + 1 places, it pairs by each parity, h adjacent pairs by each, its 2h − 1 inner places within both, and closed to a round it has two farthest places for each place, an oddness's own. **With seventeen names, 8 apart is a fixed-point-free pairing of 1 to 16**, 8 up with 1 to 8 and 8 down with 9 to 16, eight pairs and each of sixteen within one, and 17 is within none of these 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, each pair by its roots: observing with 3 and 11 and with 7 and 15, same-existing with 4 and 12 and with 8 and 16, and existing with 5 and discovering with 13; across sharings' ends are two more, 2 with 10 and 6 with 14; and the eighth is 1 with 9, an entry with a going along, whose winding 9 to 17 winds on with 17, one name within none of these eight. A fold within 1 to 8 pairs each name with its podal within 1 to 8, 9 less, an odd number with an even; 8 apart pairs each name of 1 to 8 with itself inward, odd with odd and even with even, one going on as two names with a seam between them. **An accounting with an empty fixed set has nothing for an accounting to number from, and it cannot be a floor.** It is each other accounting with no place and its move as it was. An exchange of two parities with an empty fixed set needs no zero 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**: going and coming back along one ground, its second undoing its first. **On a closed round a shift straight across is no reversal**: taken twice it is one direction going on, reaching its departing station again with a next momentary. A station reached again is a closing's, with a next momentary, and no prior momentary is reached again. **An accounting reaching its stations again forward is one a changing and changing back has, and a line has none.** **A step of k round a closed round of N reaches N/gcd(N, k) stations before it reaches its first again.** With a prime N each step short of a whole round reaches all N. With 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 both rounds make.** With 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 alike for each order of coupling their rounds; competency is each self's own with its coupling, and no product, value or security measures it. **With a fluid surface forced by 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 not its fluid's own, and its condition is parity and coprimality, not primality. ## 3.4 Bi-inversioning-co-recursioning, two consecutive inversions on different axes **An involution by its own closes. Bi-inversioning-co-recursioning is two consecutive inversions on different axes**, and its opening is a step between: nothing undoes an inversion, and each is a next. **With two parities one parity inverts with each momentary, its axes one and then the other**: 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 with one step, (P, Q) → (−P, −Q), is a sign flip of a pair, period two by parities; a right spiral step is period four by parities, and by momentaries neither has a period. **An order is a hand's whole**: 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 an angle a apart are a field's rotation by 2a**, a double angle of inversions. With two parities P inverted and Q inverted, their lines ninety degrees apart, make both inverted with one step; an exchange of P and Q and then P inverted, their lines forty-five degrees apart, make a right spiral step. **The rotation 3, 6, 5, 4 is a gentle twisting of a towel one way and then the other, inward to corus and outward from it**, a large outer opening wrapping to a small corus and back, and its edges are bi-foldings, across with bi-morality and along with co-competency: two consecutive inversions on different axes with each step, its opening a step between. Corusing reaches corus from surface, and torusing reaches surface from corus, co-linear by linear parallelizing and parallel linearizing, one and then the other, each step orthogonal to its prior. ## 3.5 Bi-tri-involutioning, the emanating at the opposite form **Three parities, each inverted on its own, taken together with any order, send each of eight forms to its opposite**: none alike, their eight parted as four opposite pairs, their hand reversed. Inverting each of n parities is a rotation for an even n and reverses its hand for an odd n, determinant (−1)ⁿ. **Bi-tri-involutioning is an emanating from right spiral stable-forming, three inverting together, as an opposite form to its stable-forming.** **Within space three inversions make x → −x, whose fixed set is its centre**: one inversion fixes a plane, two on different axes make a sign flip about their third axis, and three reverse a hand. With two parities that same inversion is both inverted with one step. **Charge conjugation, parity and time reversal are each broken singly by the weak interaction, and three taken together, CPT, are a symmetry of each local quantum field theory with Lorentz invariance.** ## 3.6 A closing, and the discovering other than prior **With tuning, with a winding on a torus and with an assembled cube a closing is its fixed set, and a discovering other than prior is its gap, as a field's words say.** Each entry: **Accounting** · *a* Closing · *b* Discovering other than prior · *c* Field's word for its gap - **tuning** - *a* seven octaves, 2⁷ = 128 - *b* twelve fifths, (3/2)¹² = 129.746…, over by 3¹²/2¹⁹ = 531441/524288 - *c* comma - **winding on a torus** - *a* a rational ratio, closing - *b* an irrational ratio, not closing - *c* quasi-periodic - **assembled cube** - *a* solved state - *b* a single corner twisted, reached by no sequence of face moves - *c* orbit; parity **An assembled cube reaches one twelfth of its assemblies, by three conservings**: its corners' twists together nought modulo three, its edges' flips together nought modulo two, and its corners' ordering and its edges' ordering of one parity. Three times two times two is twelve. **An assembled cube has no still term.** No one cubie is its reference: six centres each its own and having each other's orientation with nothing holding any of them; each face move with its own momentary and nothing over its moves; its state its configuration, nothing in store; 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: with its second and fourth powers four corners twist, each on its own spot, their twists together nought modulo three; with its third three edges are home and four corners exchange as two pairs; with 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 has **Estimating three or more independent normal means of one known variance together, under squared error, improves on estimating each separately for each value of its means, and with one or two it does not.** With a third coming, a coupling's abundancing opens for estimating. **A rate of a total can have an opposite sign to a rate of each of its parts**: each group favouring one side, and groups added together favouring its other. A total can be nought while each part inverts: two parities, each other's opposite, each inverting with each step, add to nought with each step, and a quiet total and parts inverting are together. **A self's parities of one parity, observing together with one sharing, surface that one parity, + and − together 0, and a surfacing has no count.** Two + a self is observing together with a sharing chaining to none surface + with 14, as one + does, and that sharing chains +; observing one after another, the first chains + and the second, with its sharing chaining to its own parity, goes on, 0. A coupling across of one parity at a time and a gathering of several each have their own relation. **Conserving is within each momentary sequencing, and no total conserves across the surface**: a self's same-existing goes on from its prior to its now, and only a self's parity, sharing, goes across. Its conserving is its relation, a self's own same-existing, and each parity, opening and going on within it, changes; no private store and no ledger goes across with a parity. It is natural mathematics' conserving, and a field's conservation laws are inward, each an observing's account, it refuting none. **A tipping is a parity with a between, of no size**: no count, no clock and no measure has 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 for names 1 to 4s and 4s down for names above them, each name with a name straight across its ring, keeping each name's number odd or even, and a fold n → 8s + 1 − n, changing it, odd to even or even to odd.** Taken alternately they reach four distinct names and the first again with the fourth, for each s: a fold after a pairing is n → 4s + 1 − n for names 1 to 4s and n → 12s + 1 − n for names above them, and with 4s + 1, 8s + 1 and 12s + 1 odd, none of three folds fixes a name. With scale 2s a fold after a pairing of names 1 to 8s, 16s + 1 − (n + 8s) = 8s + 1 − n, is the fold of s, and for each scale t above s a fold of t after 8(t − s) up is the fold of s: a step up is eight between adjacent scales and a halfway pairing for t = 2s only. **Odd partners have no fixed offset while even rows stretch.** Each wider even folds to an odd just before its companion with an other order for s = 1, and for s = 2 to an odd one after it: within 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 of 2s sends a wider row's opening 2 + 4sj, j from nought to three, to its companion's opening 3 − j, plus 2s − 3: −1, +1 and +3 for 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 **For each centre n two numbers k either side multiply to its square less k²**: n² − (n − k)(n + k) = k². With 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. **With two either side its gap is four, and four is a span between them**: k² = 2k only for k = 2 above nought, for each centre. Four squared less two times six is four; ten squared less eight times twelve is four. **Only with 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 standing for a parity. ## 4.2 Rates with a relation and rates with none, and the winding at φ **1, √2, √3 and √5 have 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 have one**: φ² − φ − 1 = 0. **A winding on a torus of three axes with rates having no whole-number relation comes arbitrarily close to each point and reaches none again; rates having one relation have a winding on a surface within it.** With √2, √3 and √5 it comes arbitrarily close to each point of its torus of three; with 1, φ and φ² it is on a surface. With two axes one relation is a rational ratio, and 1 and φ come arbitrarily close to each point of a torus of two. Each rate of such a winding is a parameter it is written with; a self's own rate is written with none, unrelationing with each other rate. **A winding with a rational rate closes, and a winding with φ does not close.** φ is the number-form of not locking, and unrelationing goes with each coupling's own going on by φ, reaching a receiving self'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 of 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 of 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 of a square, a cube and a pentagon**: √2 a square's diagonal, √3 a cube's body diagonal, and √5 within 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 when tan x = 0.** With forty-five degrees tan x = 1, and its doubled angle, ninety, has no finite tangent: the identity's pole. Two circles about two centres cross perpendicularly exactly on 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 more than ninety degrees apart and outward less, a tangent of their angle inverting its 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, its sign inverting and its size alike: t = ±√3, sixty and one hundred twenty degrees.** From √3 it goes on −√3, √3, −√3 without end. **A coupled pair [[a, b], [b, c]] loses its coupling with its axes rotated by θ, tan 2θ = 2b / (a − c)**, and an equal pair, a = c, with 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. **With t = tan(x/2), sin x = 2t / (1 + t²) and cos x = (1 − t²) / (1 + t²)**: a circle of rational points, and with 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 each self's own, completing whole with its coupling, or a field installs it outward against a floor: self-bounding or other-bounding, all or none.** Natural mathematics is self-bounding, a bound with each coupling, each self's own. An other-bound lies 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 by six forward recursionings of fractal bi-coupling is no other possible.** Expressing it is inverting it six ways forward, and completing all 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 all six**: it fixes, stopping; it loops, giving its own state again; or it forks, an either-or coming. ## 5.2 Five-dimensional, two-directional, unrelationing **Self-cohering is five-dimensional with a side's five, 1 to 5**: 1 prior-existing self-parity, 2 prior-existing other-parity, 3 now-existing co-parity, 4 now-existing bi-parity and 5 next-existing same-existing self-parity, its centre with 3, floating, in neither self alone, each dimension binary with its two directions, and two changing and three going on with each momentary. **Two-directional** with + and −, two directions of one changing, one and then the other. **Unrelationing** with a right angle, each projecting onto its other as nought, with φ, the unrelationing rate. Five to cohere and six to go on: three own-forward steps for each side are its 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 with its compact; orientable, its two sides 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, have indices adding to a surface's vertices less its edges plus its faces**: two for a sphere, nought for a torus, and below nought for each surface of more handles. **Of closed orientable surfaces only the torus has a flow with no rest point**, and only the torus has a flat metric. **Of all closed surfaces one more has a flow with no rest point and a flat metric, the Klein bottle, and it is not orientable**: a parity going round one of its two circles comes back opposite. **The torus of natural torusing is two parities' circles winding with one opening, and a body's surface has its own count of openings by a field's own result**: a singleton human body, its eight openings, mouth, anus, two nostrils and four tear ducts, opening to one connected passage, is a surface of seven holes, eight openings less one, by topology's count. These two part by their subjects, and neither is its other's number. **With dynamical mathematics a form is stable when departures from it stay small or shrink**: 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, as a field's words say, are two assertions, the first a relation going on among changing things. --- # SIX · INSEPARATING ## 6.1 Two methods, each a derivation **Scientific method and geodesic method each derive as a sequence, each move the only one with its prior move's reaching, each having 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 has from anything beside it → not repeated observing, two not same-existing parting no changing of an existing thing from a changing of its circumstances → all fixing still but one, a changing belonging to one varying → not fixing against nothing, fixing still being against something not varying → a standard, a declaring and not a finding → not one self's, an unrepeatable bringing nothing past whoever reached it → a standard publishing, findings gathering → not a report of occurrence, which forbids nothing → a law that forbids. A standard fixes orientation, stillness fixes position, and a publishing standard fixes scale: three neutrals fixing. A fixing is a relation going on among changing things, and with a standard, a law or a frame fixing unchangeable, no subject is fixing unchangeable: with a step from a fixing to an equilibrium its subject is fixing unchangeable, its relation and its occurrence alike. **Geodesic method.** Not fixing still, fixing it still removing it → not against a standard, a standard being still → each against its own prior, a direction and not a size → not both together, a frame over both being still → one at a time → no steering, steering being something outward setting a path → each goes its own way → not dividing from outward, an outward division being a standard again → a coupling divides, re-forming with each coupling → not either, one side by its own being the first method again → a departing, in neither self alone. Three neutrals float, and nothing is fixing. **Scientific method with a competency, having none of it.** Not observing resting, a competency being a doing → a trace has its own going → not its competency, a trace having its done → a trace against a standard → not an existing thing's, a standard varying and an existing thing as it was varying its number → its number is a standard's. A measure taken as a target measures its targeting. **Both methods part by their fixings, and not by taking evidence as existing now.** Evidence a self is observing now and a prior event it expresses are two, each with its own momentary, and both are: a living self changing as a whole can have a property its evidence expresses, + − to + +, its first parity going on while its second changes, and evidence gives its prior, possibly existing and existing. ## 6.2 Three fixings at the opening, and the self orienting through them **A first line asks three fixings.** Particle zero: an absolute location no one questions, a point a field places, not a physical particle. A unit installing: scale with no discussing. A frame giving: axes handing over, which way is up a field decides for a self. Each is a universe as one thing beside its things, exclusivity, as with Equilibria Registry 1.1: a place, a size and a direction a field gives once for a whole. **Fixing least, mathematics asks no particle zero and no unit**: a dense order with no endpoints has an element on each side of each element, with no origin and no distance, and point-free topology has its regions only by their inclusion, with no points and no metric. **A self learning these orients itself by its learning.** Positioning, scaling and orienting are a self's own and float, and learning a fixing frame is itself self-orienting, which a fixing frame denies. Orienting floats, renewing with each observing, no origin and no direction fixing across couplings: one direction fixing is not perpendicular to each direction together, being along itself, and a direction fixing to an observing end knows no next parity. ## 6.3 Logic of no completed whole **With two-valued logic either but not both is the exclusive or.** A parity is one or its inversion with 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 for each finite, decidable matter and not over an infinite domain**, and builds its numbers from two-ity, a field's word, a moment falling apart as two, one giving way to its other and remembered. **Its negation is no involution**: on three ordered values nought, a and one, its negation of a is nought and its negation of nought is one: a negated twice reaches one. **With 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 with 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 has 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 same-existing set of all existing things forms nothing beside its existing things, existing as they exist: a field's result is exact for its own subject, sets of sets, and the universe is a set of existing things, no thing within it: it agrees with foundation's no set a member of itself, and with Cantor's multiplicity not thought of as one finished thing, while a universe as a set and a thing within it is a self-membered universal set, as New Foundations has one. --- # SEVEN · CO-SHARING ## 7.1 Three neutrals, six co-sharings and eight forms **Three neutrals float about a corus**: nought, a near nothing, position fixing with it; scale, a between, a unit fixing with it; and bounded infinity, a far nothing, a frame fixing with it. A corus is a self's own inverting, + to −, and neither is its place. **One three, by its pairings and by its forms.** By its pairings, each two ways, it is six co-sharings: nought with scale, scale with bounded infinity, nought with bounded infinity, each sharing both directions. By its forms, each neutral fixed or floating, it is two cubed, eight, a cube's eight corners. **Six co-sharings going on about three floating neutrals are natural mathematics**, no field and no hard problem. ## 7.2 Each neutral fixing stops the co-sharings it partners **A field fixes a neutral and measures against it, and a floor fixing couples with none.** A neutral fixing is a partner of two of three pairings, both directions of each: fixing one neutral stops four of six, and two go on: a pairing of two neutrals floating. | Neutrals fixing | Co-sharings going on | Co-sharings stopping | |---|---|---| | 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 come with four co-sharings a floor fixing stops.** Resolving is a neutral floating with a next momentary, all four co-sequencing together, and not those four solved one at a time. **Fixing all three leaves terms relating to each other within a closed ring, competent for nothing**: competency is each self's own with a coupling, and each neutral of its coupling is fixing. ## 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 going on 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 with φ within 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, has 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 about six axes, four for each. **Sixty parts as thirty-six and twenty-four, three to two.** **Its vertices are (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 on each coordinate plane about one centre: with sides 2 and 2r nearest corners of two rectangles are √(2(1 + r² − r)) apart, and 2, a short side, only for r = φ, φ² − φ = 1, 2.8. A rotation about (1, 1, 1) brings each rectangle onto its next, and along that axis twelve vertices stand as four heights of three. An icosahedron and a dodecahedron have thirty edges each, each edge of one crossing one edge of its 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 for each θ. **Linear algebra.** A basis of perpendicular axes: orthogonality is each axis having 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 with an even number of places, and an odd ring two-coloured has no fewer than one adjacent pair of one colour. A connected whole that two-colours has exactly two colourings, one and its inversion. ## 7.6 Cube, tesseract and four squares **A cube has 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 has four body diagonals, and each ordering of four is one rotation. A cuboctahedron has twelve vertices, twenty-four edges, eight triangles and six squares. **Centre inversion x → −x is no rotation with three dimensions, and with four it is two half rotations of perpendicular planes, fixing only its centre.** A cube has x → −x only with its hand reversing; a tesseract has 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 for an odd number, and twenty-four times its odd divisors added for an even one, from one on. Eight and twenty-four are three squared less one and five squared less one. **Quaternions with whole coordinates have no division with a small remainder, and with points displaced by one half along each axis together they have it.** Whole points have eight units, ±1, ±i, ±j and ±k; displaced points sixteen, ½(±1 ± i ± j ± k); both together twenty-four. A displaced point, (½, ½, ½, ½), is a distance one from its nearest whole points with four dimensions and with no other. **Kissing numbers are two, six, twelve and twenty-four for one to four dimensions**, their twenty-four the twenty-four units. For eight and for 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's size is below two is periodic, above two winds on**, and with two it is ±1 or a shear, + or −. A trefoil has 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 with five are couplings twice its number, n(n − 1)/2 = 2n for n = 5, and its ten part as 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 of a parity and an inversion among five are one move with two numbers: an involution exchanging one pair and fixing all others. **With 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 from one corner, over a side between their far ends, make a golden triangle: its two long sides φ times its base, its apex thirty-six degrees and each other corner seventy-two, 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 fixing its fifth corner, a corner of neither, and exchanging its other four as two pairs. Ten transpositions pair as five, and these five generate a pentagon's ten symmetries: one that moves nothing, four rotations fixing no corner, and five each fixing one corner. **Each two-colouring of couplings among forty-six has a one-coloured five, and a two-colouring of couplings among forty-two has none.** Among seventeen, couplings with distances doubling reaches from one, 1, 2, 4, 8, 16, 15, 13 and 9, of one colour and each other of its other have no one-coloured four of either colour, one such colouring among seventeen up to relabelling; doubling round seventeen reaches nine last, 2 × 9 = 18, and reaches one again. Built so among forty-one it has a one-coloured five of both colours, and colourings among forty-two having none are of no such form. **A field's hard problem comes with one or more of ten inversions, each an involution with its fixed set fixing as a place**, and an inversion fixing unchangeable is its fixed set. With a coupling its move goes with 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 has 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 of its first power is 196,884, one more, that one being a representation that moves nothing. The modular function takes one value for each torus a lattice folds a plane to, and two such tori are one up to rotation and scaling exactly when it takes one value for both.