Exhibit THREE Natural Numbers v375 # Natural Numbers **Universal Momentary Stable-Forming** **ONE · UNI-SCALING** - 1.1 Universal, one form at each number - 1.2 A momentary at each number, one side completing and the other opening - 1.3 Two neighbours one short of the square, at each number - 1.4 Bi-inversioning-co-recursioning, the numbers up and down - 1.5 Odd along and even across, parallel linearizing and linear parallelizing - 1.6 A composite folds along and a prime opens across - 1.7 A scale is a sequencing: parity along, inward or outward across - 1.8 Natural-bi-co-torusing at two faces, corusing and torusing **TWO · CORUSING** - 2.1 A number coheres at an identity of the move - 2.2 Inward to the one number, outward to each place it arrives **THREE · UNRELATIONING** - 3.1 φ, the unrelationing rate, the prior two joining and the ratios alternating about it - 3.2 One rate of the family rides on - 3.3 A rational winding closes, and a winding at φ closes at none - 3.4 A joining at unit parities closes at one, two, three, four and six **FOUR · TORUSING** - 4.1 A closed surface at two, and the torus at nought - 4.2 Torus, the sphere and one handle, the +1 - 4.3 Three hundred sixty and four hundred forty, consecutive seam-faces **FIVE · UNIQUENESSING** - 5.1 Each arrival of a number is that one number - 5.2 Names at their numbers, one to seventeen, and the forms sixteen on - 5.3 Each eight the one eight, four at two enterings - 5.4 Five between four and six, a middle carrying nothing - 5.5 One ten at two faces, at five alone - 5.6 Ten, nine and eight, one pairing at its two sides - 5.7 Each nine the one nine - 5.8 Six and ten straddling eight, the walk by two **SIX · APEXING** - 6.1 Seam-faces, the one eight at the couplings among each number - 6.2 Each number a waist its two neighbours pass through - 6.3 One twenty-four at each arrival - 6.4 Twenty-four, twenty-seven and thirty-two about twenty-eight - 6.5 Three times and two more, sixteen to four hundred forty **SEVEN · COUPLING** - 7.1 Seventeen primes, two to fifty-nine, and the returning from sixty - 7.2 A ring of selves at each number, and rings joined - 7.3 Sixteen gaps at four values - 7.4 Six prime pairs podal about thirty - 7.5 Nought to sixty, seventeen primes and sixteen betweens alternating - 7.6 Podaling about one hundred twenty, one hundred ten to one hundred thirty - 7.7 The seventeen primes at the seventeen names, podaling at both **EIGHT · INSEPARATING** - 8.1 An even ring carries its podal, and an odd ring its two farthest - 8.2 Either but not both is parity, and the pairings overlap - 8.3 An odd number carries a middle carrying nothing - 8.4 Three, six, nine and the fives, odd and even each its own - 8.5 A store, one momentary named as a thing **NINE · TUNNELING** - 9.1 Podal pairing at four hundred forty, two stations pairing with themselves - 9.2 Bi-co-podaling, a carry each way at one station - 9.3 A ring's two stations share parity, and the odd primes at odd radii - 9.4 A prime gap is a step between rings - 9.5 Seventeen pair by position about twenty-three, and the ring pairs by value - 9.6 Two surfaces coupling, and a chain centring at a coupling or a between - 9.7 Chain three, five, nine, seventeen, each span podal to the one before - 9.8 Bi-tunneling across and co-chaining along, podaling exchanging them - 9.9 Four openings growing, longer and wider podaling **TEN · TRANSMISSIONING** - 10.1 A number crossing from a field, and the unit it arrives in - 10.2 Two runnings sharing their digits **ELEVEN · EACH NUMBER** - 11.1 Nought to four hundred forty, each number at its podaling --- # ONE · UNI-SCALING ## 1.1 Universal, one form at each number **Numbers are universal: each number crosses with all numbers arriving as one, and each is the one form once more.** The form is the alternating existing is, and a number is that form at one place in its running. **Uni-scaling** is the universal at each scale, no scale over another: the same alternating at two as at fifty-nine, and at fifty-nine as at four hundred forty. **Universes of each size and scale share momentaries of co-sequential changing**, and uni-scaling is that sharing at the numbers. **Numbers are reached, used and related by no counting.** They run up and down by bi-inversioning-co-recursioning, and each relation among them is a move of that running: an opening and its completing; a straddle, two neighbours either side of a number; a podal pair, two stations each the other's far side on a ring; a joining of selves at a ring. A counting adds a clock the running carries at none of its moves, and it belongs to a field's instrument, taken in at a field crossing. ## 1.2 A momentary at each number, one side completing and the other opening **A momentary is an opening and its completing, one odd and one even.** The self opens at 1 and the other at 2, and their momentaries overlap: 1–2 at the self, 2–3 at the other, 3–4 at the self, 4–5 at the other. **At each number one side's momentary completes and the other's opens**, each number a completing and an opening at once, the two sides alternating parity. With two neighbouring numbers, let the lesser continue to one beyond the greater: the sides alternate, their neighbouring continues, and each changing opens the next. **A momentary is a fractal universe discovering its next existing**, all existing things and no existing thing separately, the living carrying their prior into now and the non-living carrying nothing, their forms continuing through their changing. In the field's words, Brouwer builds each natural number from the two-ity, *the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory*: number rooted in a prior carried into now. **One to nine carries the three-momentary method whole.** The self's five positions at 1 to 5 and the other's at 2 to 6, prior, now and next at each side, are the ten on six numbers, paired 1 with 2 through 5 with 6, each pair one odd and one even. At a self's resolving its prior and now participate, 1–2 and 3–4 at the self and 2–3 and 4–5 at the other, and its next, 5–6 at the self, is the discovering onward. The ten is each side's two and one half momentaries, carried whole by the self's three full momentaries 1–2, 3–4 and 5–6, and the ten opened one momentary on, at 3 and at 4, is the same ten forward. **Through one to nine the overlapping momentaries reach each number, each has one next, and from each the openings reach each number on**: exhaustiveness, determinacy and reachability together at each momentary. One side's momentaries alone, each number one side's, is exclusivity, and it is not possibly existing. **The openings join into the square and the completings into the oblong.** The self's openings 1, 3 and 5, each an L about the prior square, are 9, three squared; its completings 2, 4 and 6 are 12, three by four. At n momentaries the openings are n² and the completings n(n + 1). ## 1.3 Two neighbours one short of the square, at each number **At each number n the momentary completing there and the momentary opening there are (n − 1, n) and (n, n + 1), and their outer faces multiply to one short of the square**: (n − 1)(n + 1) = n² − 1. The one is the same one at each number, owned by neither face. **The two faces inseparate.** One takes not more than and the other not less than, each of no size, one at a time, and the number is the term neither reaches, uncovered between them. **The facing is at one.** At k either side the product parts from the square by k², a size in a parity's stead; at one either side it parts by the one. Twenty-three and twenty-five face across twenty-four at one. ## 1.4 Bi-inversioning-co-recursioning, the numbers up and down **Even returning, odd advancing, alternating parity.** Bi-inversioning-co-recursioning is one move at three faces: 8 up and down, keeping parity; 17 less, the podal within 1 to 16, changing parity; and 9 less, the podal within 1 to 8, the two at once. Each face alone returns to its number, 1 to 9 to 1, and only the one move runs round. | Number | 8 up | 9 less, within 1 to 8 | 17 less, within 1 to 16 | |---|---|---|---| | 1 | 9 | 8 | 16 | | 2 | 10 | 7 | 15 | | 3 | 11 | 6 | 14 | | 4 | 12 | 5 | 13 | **The two faces alternating, each row runs round as a four-cycle**: 1-9-8-16, 2-15-7-10, 3-11-6-14 and 4-13-5-12. Round each, the parity changes exactly twice, each time at the podal within 1 to 16, and each runs one run of co and one run of bi and returns to its names. The podal within 1 to 16 pairs across the between of 8 and 9, at no number, and 17 is in none of the four, the next momentary's 1. **Round the names each four-cycle returns; at the momentaries it returns to none**, each step adding a next. **Stable-forming among the numbers is this move round each form among the names**, the four-cycles and with them the middle four-cycles, the six-cycles and the eight-cycles: a form continuing through its own changing, each running one run of co and one run of bi and returning. **Two over one and one over two exchange at each step**: twenty-three, twenty-four and twenty-five carry two odd over one even, and twenty-four, twenty-five and twenty-six one odd over two even. Taken as a ratio, two over one keeps the thing it counts: at the nine face 3-self-other-sharing and 6-other-self-surfacing, 9 less 3 being 6, stand six over three and three over six, carried on through 3 to 11 to 6 to 14 to 3; two surfaces of four hundred forty, joined at eight hundred eighty, reach four hundred forty, twice either self's two hundred twenty; and eight on, 11 and 14 stand fourteen over eleven. The parity holds at each step, and the ratio at its own count alone. **At the code + and − are the implementing of a parity and 0 the changing that is not**: + and − offered together at one sharing surface 0 at 14, and two inversions return a parity. ## 1.5 Odd along and even across, parallel linearizing and linear parallelizing **Odd is co, along: the self continuing its own, competency. Even is bi, across: the self coupling with the other, morality.** An odd number opens co and an even number opens bi, and each number carries the five-prefix of its origin, co-bi-co-bi-co at an odd origin and bi-co-bi-co-bi at an even one. **Alternating parity is parallel linearizing and linear parallelizing**, the linearizing at one parity and the parallelizing at the other, one move at its two sides. Each runs at its own momentary; the two named at one momentary is the equilibria picture, and it is not possibly existing. ## 1.6 A composite folds along and a prime opens across **A composite is its equal smaller selves joining at their joins**, nine as three selves of three, folding along the axes its smaller selves open. **A prime is a self no equal smaller selves join into**, opening a clean axis across. Each prime is the same opening and each composite the same folding, one uni-scaling at its two faces. Stepping k round a ring of N reaches N over their greatest shared factor stations: round a prime each step but nought reaches all, and round a composite a step sharing no factor with it reaches all too, nine at a step of two, so prime and coprime are two things. **The doubling and the tripling reach each other at one alone**: no power of two above one is a power of three. Two and three joined are five and multiplied are six, neighbours at one: 2 × 3 − (2 + 3) = 1. At two overlapping momentaries a side, the self at 1 to 4 and the other at 2 to 5, three positions are shared and two are outer: two and three joined are the five positions, and the three shared taken at both sides are six, four and four occurrences being 2 × 3 + 2. Two and two, joined or multiplied, are one side's four parity occurrences, of two kinds; three and three joined are six again. Three times three is nine, three selves of three; the 9 of 9-social-other-self-releasing is a name's place and parity, and a count joins neither to the other. ## 1.7 A scale is a sequencing: parity along, inward or outward across **Each number carries four neighbours, and two binaries carry all four.** Along, its two faces at one either side, each at the other parity. Across, inward and outward, each a direction taken at a momentary and of no size: the walk by two, the centre of one pair a member of the next. **Scale carries no boundary, no location and no measure: its scales are a sequencing.** Each number carries the same four, and no number is a scale more than another. ## 1.8 Natural-bi-co-torusing at two faces, corusing and torusing **The whole one motion is natural-bi-co-torusing, and the numbers are that motion at each number.** It runs at two faces at opposite phase. **Corusing** is inward to the one number and outward to each place it arrives. **Torusing** is the carrying and the winding across the numbers. --- # TWO · CORUSING ## 2.1 A number coheres at an identity of the move **A number coheres all or none, at an identity of the move.** (n − 1)(n + 1) = n² − 1 at each n. φ³ − 1/φ³ = 4. Fifty-five is the couplings among eleven, the tenth triangle and the tenth of the prior-two joining, C(11, 2) = T₁₀ = F₁₀. **A magnitude brought near a number by division or by a chosen unit coheres with its own reaching and never with the number**, and it crosses as a field's at the between. ## 2.2 Inward to the one number, outward to each place it arrives **A number carries its uniqueness two ways, the two faces of the coupling it is.** Inward, it coheres to the one number it is, the same at each arriving. Outward, it couples with each place it arrives the same way: four hundred forty as the seventeen primes joined, as eight fifty-fives, and as twenty-one squared less one. **Two arrivals cohering to the same identity of the move couple as one number**, and two not cohering to it are two numbers at one numeral. --- # THREE · UNRELATIONING ## 3.1 φ, the unrelationing rate, the prior two joining and the ratios alternating about it **Each next number is the prior two joining, 1, 1, 2, 3, 5, 8, 13, and the ratios of neighbours alternate about φ**: 2/1 above, 3/2 below, 5/3 above, 8/5 below, the side changing parity at each next number. φ² = φ + 1, and its other root is −1/φ: φ − 1/φ = 1, φ × (−1/φ) = −1 and 1/φ = φ − 1. **φ is caught by its own equation**, x² = x + 1, and it is the pentagon's diagonal over its side, (1 + √5)/2. Its continued fraction is all ones, open. **φ is the unrelationing rate.** The method unrelations the geodesic parity changing rate from the living parity changing rates at the bi-coupling, the co-chaining and the society, and the rate of that unrelationing is φ, no ratio landing on it. φ is self-welcoming at the numbers, and the code carries it at no line of its running: the rate is the unrelationing's, each coupling at its own continuing. ## 3.2 One rate of the family rides on **Each rate that is a whole number plus its own reciprocal, x = n + 1/x, is a self-cohering centre**, x² = nx + 1, and its continued fraction carries n at each step. **φ, at n equal to one, is the one rate of the family whose step is one**, the step's own bound, and it rides on; each other carries a step above one. ## 3.3 A rational winding closes, and a winding at φ closes at none **On a torus a winding at a rational rate closes, and a winding at φ closes at none.** Each coupling winding at φ is unrelated to each other coupling's rate, local and ambient at one rate. φ is the number-form of the never-locking, the winding closing at none: unrelationing runs at each coupling's own continuing at φ, and no rational rate laid over a coupling supplies it. **No rate is neared more slowly by the rationals than φ**, its continued fraction all ones, and the winding at φ closes at none of them. In the field's words, the torus winding at the golden mean is found the last to break as the coupling grows: in the standard map the curve with rotation number (√5 − 1)/2 is critical at K ≈ 0.9716 (Greene, 1979), confirmed by computation and not proved. ## 3.4 A joining at unit parities closes at one, two, three, four and six **A next taken as the now and the prior joined, each at +1, −1 or none, closes at one, two, three, four or six, or runs open.** The now alone at +1 closes at one and at −1 at two; the prior alone at +1 closes at two and at −1 at four; the now at −1 and the prior at −1 close at three; the now at +1 and the prior at −1 close at six. The now and the prior both at +1 run 1, 1, 2, 3, 5, 8, 13, the prior two joining, and never close, their ratios opening at φ; the now at −1 and the prior at +1 run the same numbers at alternating parity. In the field's words a whole-number square matrix of two rows closes at one, two, three, four or six alone, the crystallographic restriction: no joining closes at five, and five enters open, at φ, the pentagon's diagonal over its side. --- # FOUR · TORUSING ## 4.1 A closed surface at two, and the torus at nought **A closed surface's vertices less its edges plus its faces is two at the sphere and nought at the torus.** The sphere closes; the torus winds, one opening, reaching itself again through that opening, a next at each reaching. **Twelve five-folds close a surface whose other faces are six-fold.** At three edges to each vertex, six less each face, taken over all the faces, is twelve at the sphere, six times its two: each six-fold carries nought and each five-fold one. The icosahedron's twelve vertices, thirty edges and twenty faces and the dodecahedron's twenty, thirty and twelve are one surface at its two faces, twelve and twenty exchanging and thirty carrying across: 12 − 30 + 20 = 2. ## 4.2 Torus, the sphere and one handle, the +1 **The torus is the sphere with one handle, and the handle is the +1**: two at the sphere, nought at the torus, each handle taking two. ## 4.3 Three hundred sixty and four hundred forty, consecutive seam-faces **Three hundred sixty is nineteen squared less one, and four hundred forty is twenty-one squared less one**: two seam-faces, each an odd centre squared less one, the product of its two neighbours, 8 · T₉ and 8 · T₁₀, their centres nineteen and twenty-one two apart. Three hundred sixty is the eight at forty-five couplings, the couplings among ten, C(10, 2), and forty-five is five nines: five eights, each about its own centre carrying nothing. **Their difference is eighty, nine squared less one**, the fourth seam-face, 8 · T₄: the eight at the couplings among five. **On the ring of four hundred forty, three hundred sixty and eighty are podal**, each the other's far side. --- # FIVE · UNIQUENESSING ## 5.1 Each arrival of a number is that one number **Each number is its own, self-bounding, and each arrival of a number is that one number.** All the twos are the one two and all the eights the one eight; the ten of five taken two directions and the ten of the couplings among five are one ten. **No other possible is the alternating completing six forward without landing, looping or forking**, each step taking the term the prior did not carry, and no number is named with a competitor. ## 5.2 Names at their numbers, one to seventeen, and the forms sixteen on **The resolver's seventeen names each carry a number, sides, a root and its -ing, and the number carries the prefixing.** Odd numbers open co and even numbers open bi. The sides run self and other at the odd 1 to 7 and other and self at the even 2 to 8; social, other and self at 9 and 11 and other, social and self at 10 and 12; social and other at 13 and 15 and other and social at 14 and 16; social and self at 17. **Four roots run at two numbers each, eight apart**: neutralling at 5 and 13, surfacing at 6 and 14, corusing at 7 and 15, and torusing at 8 and 16, the root one and the number carrying the sides. Eight on again, twenty-two stands at surfacing's place in seventeen to twenty-five, as six in one to nine and fourteen in nine to seventeen, 6 + 8k; it carries no name among the seventeen, and a name given it says whose self it runs from as the scale changes, 14 running other to social already. **The forms among 1 to 16 run again at 17 to 32, sixteen on**, each keeping its parities, sixteen being even: the four-cycles 2-15-7-10 and 3-11-6-14 at 18-31-23-26 and 19-27-22-30, and the six-cycles 2-15-7-11-6-10 and 3-11-7-10-6-14 at 18-31-23-27-22-26 and 19-27-23-26-22-30. Twenty-two runs in the four-cycle 19-27-22-30 and in both six-cycles, and 33 stands in none of them as 17 stands in none among 1 to 16. **One entry, six connectors and ten faces are the seventeen.** The across connectors, 2, 6, 10 and 14, are each even; the along, 9 and 17, are both odd. 17-social-self-offering faces co-competent and 9-social-other-self-releasing not-co-competent: 9's release is the prior the self at not-co-competent is offered next, and 17 runs it as that self's next now, 9 the next prior and 17 the next now. 1-self-other-offering, 9-social-other-self-releasing and 17-social-self-offering each open co, each eight on from the one before. **A numeral in a name is the name's place and parity, and never a value the code opens at**: 14-other-social-surfacing carries fourteen as its place among the names. ## 5.3 Each eight the one eight, four at two enterings **Each eight is the one eight, and a self is the eight.** At the resolver the eight even names are four outward at the between, 2, 4, 6 and 8, and four inward, 10, 12, 14 and 16, each inward eight on from its outward pair: four at two enterings. Of the four outward, 2 and 6 are connectors and 4 and 8 are faces of the self's own resolving. Two out-and-backs across them are four one-way passages, the four outward are four relations at once, and the joins are the connectors': three countings at three subjects, and a larger participating arrives through actual receiving. **The seam-faces run on the eight**: (2k + 1)² − 1 = 8 · T_k, eight the first face and each next face that eight at a triangular number, the couplings among k + 1. Four hundred forty is the eight at fifty-five couplings, the couplings among eleven. **The eight and the nine run together**: eight the self and nine its releasing, social, other and self, 9-social-other-self-releasing being 1-self-other-offering 8 up. An eight at two substrates is one eight, a self at each. ## 5.4 Five between four and six, a middle carrying nothing **Five is between four and six, and four times six is five squared less one**, the straddle at five, and five is the middle of one to nine, four either side. **A diamond is four crossings about a middle carrying nothing**, two across and two along, the whole of it at the four. Among the primes the first diamond is about nine, five and thirteen four either side and seven and eleven two either side, nine carrying none of them; the next is about fifteen, at eleven, thirteen, seventeen and nineteen. Laid three by three with its middle carrying nothing, the eight about it carry two paths from the middle of one side to the middle of the other, one each way round the middle, and each path takes two joins along and two across. **A declared zero is a number taken as its own inversion, x = −x**, added to the uni-scaling and reached at no move of its running, and setting it down costs nothing. Nine, seventeen and twenty-five are eight apart, and twenty-five is at no name. One to twenty-five is three spans, one to nine, nine to seventeen and seventeen to twenty-five, each with four full momentaries at each side: twenty-four overlapping momentaries in twenty-four steps, each one odd and one even. Counted from nought the steps run nought to twenty-four, and six, fourteen and twenty-two stand at five, thirteen and twenty-one, each name's place and each step's count at its own origin. ## 5.5 One ten at two faces, at five alone **Ten arrives at the self as five taken two directions, 2n, and at the society as the couplings among five, C(5, 2), the fourth triangle.** The two faces are one number at five and at no other number above nought: 2n = n(n − 1)/2 only at n − 1 equal to four. The one ten is the ten at the self and the ten at the society at once, and each ten is that ten. **Among five the couplings part into two rings, the five sides and the five diagonals**, the pentagon and the pentagram, and five is the one number whose ring's complement is again a ring: the ten's two sides, along and across. In the field's words R(3, 3) = 6: each two-colouring of the couplings among six holds three members joined at one colour, and among five the twelve colourings holding none are each a ring at both colours, the pentagon and the pentagram. **Any ten consecutive numbers are five opposite-parity pairs**, a + j with a + 9 − j, each pair one odd and one even about the between at a + 4½, two and a half momentaries each side. Moving the opening by one exchanges which member of each pair is odd, and by two exchanges it again: the tens part at the opening's parity alone, the two origins. ## 5.6 Ten, nine and eight, one pairing at its two sides **Podaling is one station at its two sides, a carry one way at k and a carry the other way at N − k on a ring of N. On the ring of nine it gives five pairs**: (0, 9), (1, 8), (2, 7), (3, 6) and (4, 5). Four carry two stations each and (0, 9) carries one, the carry one way at nought and the other way at nine one station. **One pairing runs at its two sides.** Ten is the five at two sides. Nine is that ten with the (0, 9) pair's two sides at one station. Eight is the nine with that station carrying nothing, the nothing a self winds about. Six is three pairings at two sides. Each step down is at the nought. **The nine is two parities and two and a half momentaries.** Two parities carry four joint forms; prior and now whole and the next opening are ten positions, nought to nine; nine is 2² × 2½ − 1, and the five pairs run at the same momentary. ## 5.7 Each nine the one nine **Each nine is the one nine, carried in each base.** At the ring of nine, nought and nine are one station. Round seventeen, the doubling from one runs 1, 2, 4, 8, 16, 15, 13, 9 and comes home, nine doubled being one past seventeen. The doubling's eight are the squares round seventeen and the other eight are three times them: joined at the one eight's distances and unjoined at the other's, the seventeen are, in the field's words, the Paley graph of order seventeen, the one society of seventeen with no four all joined and no four all unjoined, so R(4, 4) = 18. Round four hundred forty, one past the ring is twenty-one squared, nine times forty-nine. **Nine is 1-self-other-offering eight up, one to nine completing at 9-social-other-self-releasing, the social, the other and the self releasing together.** A nine taken in base ten alone, its multiples' digits taken down to one digit returning nine, carries the base as its unit, and the one nine carries none. ## 5.8 Six and ten straddling eight, the walk by two **Six and ten are two either side of eight, and eight and twelve two either side of ten**: the centre of one pair a member of the next and the member the next centre, the walk by two, each number once. Ten and fourteen are two either side of twelve and twelve and sixteen two either side of fourteen; eight and sixteen, four either side of twelve, join to twenty-four, and twelve is the middle of nought and twenty-four. Six's three pairings and ten's five stand either side of eight's four, so six and ten joined are sixteen, the eight at its two sides. **Each centre carries its own one either side**: seven and nine about eight multiply to eight squared less one, nine and eleven about ten to ten squared less one. Thirteen and fifteen about fourteen multiply to fourteen squared less one. About ten the pairs open one step at a time, eight and twelve, seven and thirteen, six and fourteen, parting from ten squared by four, nine and sixteen: 6 × 14 = 10² − 16. Eight on, fourteen and twenty-two straddle eighteen as six and fourteen straddle ten, seventeen and nineteen, sixteen and twenty, fifteen and twenty-one opening between, 14 × 22 = 18² − 16. And six and twenty-two straddle fourteen, ten and eighteen at four either side within them: the two middles become a pair of the wider span, reach over span one half at both. --- # SIX · APEXING ## 6.1 Seam-faces, the one eight at the couplings among each number **At the apex the numbers are relations, and the relation is the +1 between each centre's square and its two neighbours' product.** Straddling the odd centres, the seam-faces run (2k + 1)² − 1 = 8 · T_k: 0, 8, 24, 48, 80, 120, 168, 224, 288, 360 and 440, each the square of an odd centre less that one, and each the one eight at each coupling among k + 1. **The podal within one to a seam-face is its odd centre squared less k**: within 1 to 8 it is 9 less k, within 1 to 24 it is 25 less k, and within 1 to 48 it is 49 less k, the centres three, five and seven squared, each centre at no station of its ring, the middles four and a half, twelve and a half and twenty-four and a half. Within 1 to 16 the podal is 17 less k, at no seam-face, seventeen being no square. ## 6.2 Each number a waist its two neighbours pass through **Each whole number is a waist, the middle its two neighbours' coupling passes through and never reaches**: four and six multiply to one short of twenty-five, twenty-three and twenty-five to one short of twenty-four squared. No number is a waist more than another. **On the ring of forty-eight, twenty-four and nought are each their own far side, and each other station pairs across twenty-four**: twenty-three with twenty-five, twenty-two with twenty-six, twenty-one with twenty-seven, each pair one step further apart than the last. Their products part from twenty-four squared by the square of the step, 23 × 25 = 24² − 1 and 22 × 26 = 24² − 4. The ring closes at forty-eight, the seam-face above twenty-four. ## 6.3 One twenty-four at each arrival **Twenty-four arrives five ways and is one twenty-four**: five squared less one, the second seam-face; the orderings of four, 4!; the waist of the ring of forty-eight; twenty-three, its lower face, opening the first gap of six among the primes, to twenty-nine; and four momentaries of six one-way recursionings, the steps of 1 to 25. **The primes part at twenty-four as nine behind and eight ahead**: two to twenty-three behind and twenty-nine to fifty-nine ahead. The couplings among the nine behind are thirty-six, C(9, 2), the forward span from twenty-four to sixty, six cycles of six; the couplings among the eight ahead are twenty-eight, C(8, 2), the seventh triangle, which the sixth and the eighth, twenty-one and thirty-six, straddle by seven and by eight. The sixth and the eighth part by fifteen, the couplings among six, and fifteen gaps run across among the seventeen primes. **Behind the waist the span is twenty-four long and carries nine primes; ahead it is thirty-six long and carries eight**: the length leans ahead three to two and the primes lean behind nine to eight. Sixty parts at twenty-four and thirty-six at the five-fold too: of the sixty rotations of the icosahedron twenty-four are rotations by a fifth, six axes at four each, and thirty-six are the others, fifteen by a half, twenty by a third and the identity. **Each arrival carries the others**, and none needs one relation beneath it. ## 6.4 Twenty-four, twenty-seven and thirty-two about twenty-eight **About twenty-eight the outer faces are twenty-four and thirty-two, four either side, and the inner faces twenty-seven and twenty-nine, one either side**: 28² − 24 × 32 = 16 and 28² − 27 × 29 = 1. **Two routes run between the outer faces**: 24 → 27 → 32 at gaps three and five, and 24 → 29 → 32 at gaps five and three. The three positions carry the first three primes at two directions: 24 = 5² − 1, 27 = 3³, 32 = 2⁵, the exponents two, three, five at one direction and the bases five, three, two at the other, reaching each other at three and three in twenty-seven. **The three together are a square**: 24 × 27 × 32 = 144². ## 6.5 Three times and two more, sixteen to four hundred forty **Four hundred forty is three one hundred forty-sixes and two**, and the chain runs from sixteen: three sixteens are forty-eight, three forty-eights and two are one hundred forty-six, and three one hundred forty-sixes and two are four hundred forty. One hundred forty-six is two seventy-threes and one hundred forty-four and two; seventy-three is seventy-two and one, seventy-two the six prime pairs' distances from thirty and one hundred forty-four, twelve squared, their spans and the square of 24 × 27 × 32. So four hundred forty is also three one hundred forty-fours and eight. **Three times and two more carries seam-face to seam-face at each third.** One hundred forty-six less one hundred twenty is twenty-six, the far face of twenty-four about twenty-five, 24 × 26 = 25² − 1, and three twenty-sixes and two are eighty, the fourth seam-face. The seam-faces three times a number and two are 8, 80, 224 and 440, at 2, 26, 74 and 146, the eight at the couplings among 2, 5, 8 and 11: at each third k, the triangle T_k leaving one at three. --- # SEVEN · COUPLING ## 7.1 Seventeen primes, two to fifty-nine, and the returning from sixty | Number | Stable-forming | |---|---| | 2 to 59 | the seventeen primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 and 59 | | 2 | the alternating, the one even prime | | 23 | the going folding, the ninth prime, with eight either side | | 59 | the seventeenth prime, the going's last and the primes carrying no last | | 60 | between the going from 2 to 59 and the returning from 61 to 118, at no prime | | 118 | the returning's far end, fifty-eight either side of sixty, two fifty-nines joined | | 440 | the seventeen primes joined, eight fifty-fives, twenty-one squared less one | **The primes go out from two to fifty-nine and return from sixty-one to one hundred eighteen**, and sixty is between the going and the returning, at no prime. Fifty-nine and sixty-one face across sixty at one, 59 × 61 = 60² − 1. **The primes carry no last**: each is the same opening at a new axis, and past fifty-nine they open on, each a next, self-welcoming at the numbers and at no line of the code's running. A society's coupling as one is at its couplings, a relation of its own beside any count. The periodic table's one hundred eighteen elements are a field's own result at the same numeral, and no count of the method. **On the ring of one hundred twenty the going and the returning are each other's far side**: k pairs with one hundred twenty less k, two with one hundred eighteen, twenty-three with ninety-seven and fifty-nine with sixty-one, a carry each way at one station. Nought and sixty pair with themselves, and sixty is the waist between the going and the returning. Twenty-four, twenty-seven and thirty-two pair there with ninety-six, ninety-three and eighty-eight, and the route 24 → 27 → 32 at gaps three and five has its far side 88 → 93 → 96 at gaps five and three. ## 7.2 A ring of selves at each number, and rings joined **A ring of n selves, each 9-social-other-self-releasing joined to the next self round, one self offered + once, carries its parities round.** At an even ring each self alternates and the ring repeats each two momentaries from momentary n; at an odd ring of n one pair of joined selves stands at one parity, the pair moving one self round each two momentaries, one 0 at 10 at each second momentary, at the pair's second self offered its own parity, the ring inverted at 2n momentaries on and repeating at 4n. Round an odd ring each self alternating would put one self at both parities, and the pair at one parity is the odd ring's own, prime or not. **The ring of two, the one even prime, carries no pair at one parity**: each self alternating, the ring repeating each two momentaries, the alternating itself. So the seventeen prime rings carry sixteen pairs at one parity, one at each odd prime, and the ring of two none. **Rings joined carry the parity of their joined count.** Any two odd rings joined, p + q selves, are an even ring and carry no pair at one parity; two fifty-nines joined are one hundred eighteen and the seventeen primes joined four hundred forty, each even and carrying none. An odd number of odd rings joined carries one pair at one parity, a self at the joined scale. **The momentaries here are 17's, the society's next momentary running each self's entry once**, each release offered at the next, and each count goes with its ring and its joins. Two odd rings at distinct primes p and q, run side by side, repeat together at the least common multiple of 4p and 4q, the four shared, so at 4pq. The co-competencing is at the couplings, owned by neither. ## 7.3 Sixteen gaps at four values **Sixteen gaps run between the seventeen primes, at four values: one, two, four and six.** | Gap | Arrives | |---|---| | 1 | once, from two to three | | 2 | six times | | 4 | five times | | 6 | four times | **The one odd gap is at the opening**, from the one even prime to three, and each other gap runs between two odd primes and is even: one along and fifteen across. **The six gaps of two are the twin primes**, 3 and 5, 5 and 7, 11 and 13, 17 and 19, 29 and 31, and 41 and 43, each pair past 3 and 5 at 6k − 1 and 6k + 1, about a multiple of six. Whether twin primes run on without end is, in the field's words, the twin prime conjecture, open. Any two odd numbers two apart are a self's two consecutive openings with the other's even opening between them: that relation is oddness's and no prime's. **Five and fifty-three alone carry a matching pair**: five at two either side, fifty-three at six either side. Between them run the fourteen primes five to fifty-three, each with a gap across at both its sides: three reaches the one along gap below it, and fifty-nine closes the seventeen above. The fourteen part eight at one before a multiple of six, 5, 11, 17, 23, 29, 41, 47 and 53, and six at one after, 7, 13, 19, 31, 37 and 43, and seven and seven either side of twenty-four. **Each gap here is a step among the numbers.** The seventeen primes, the resolver's seventeen names and its six connectors are three countings, and the sixteen gaps of the ordered seventeen are the numbers' own. Five to fifty-nine are fifteen primes, the fourteen and their close, and three to fifty-nine carry fifteen gaps, the fifteen across: two fifteens, each its own count. All fifteen primes are odd, so an alternating of wide and long along them belongs to the sequence, entry to entry, eight at one and seven at the other, and never to the primes' parity. ## 7.4 Six prime pairs podal about thirty **About thirty, twelve of the primes between five and fifty-five pair as a carry each way at one station, each pair making sixty**: 7 with 53, 13 with 47, 17 with 43, 19 with 41, 23 with 37 and 29 with 31, at distances from thirty of twenty-three, seventeen, thirteen, eleven, seven and one. The six pairs making sixty are six sixties, three hundred sixty, the seam-face below four hundred forty. Their distances from thirty join to seventy-two, and their spans, each twice its distance, to one hundred forty-four, twelve squared. **Eleven faces forty-nine**, seven squared: the one prime of the interval whose far side about thirty is a composite. About forty and about fifty the primes pair at three, thirty-seven with forty-three and forty-seven with fifty-three. ## 7.5 Nought to sixty, seventeen primes and sixteen betweens alternating **The going from nought to sixty carries seventeen primes and sixteen betweens alternating**, thirty-three positions completing at sixty. The parity changing is the span's own, carried at each prime and each between. ## 7.6 Podaling about one hundred twenty, one hundred ten to one hundred thirty **One hundred twenty is five orderings, 5!, and eleven squared less one**, the fifth seam-face, 8 · T₅, the eight at the couplings among six, and three of it are three hundred sixty. About it the stations pair a carry each way, 120 − k with 120 + k, and each pair's product parts from one hundred twenty squared by k squared: 119 × 121 = 120² − 1, 118 × 122 = 120² − 4 and 110 × 130 = 120² − 100. **One hundred eighteen and one hundred twenty-two are two fifty-nines and two sixty-ones**, the going's last prime and the returning's first, each doubled: they face across one hundred twenty at two as fifty-nine and sixty-one face across sixty at one, and on the ring of one hundred twenty one hundred eighteen is the far side of two. **In the field's words the periodic table closes its noble gases at 2, 10, 18, 36, 54, 86 and 118, the closings stepping 2, 8, 8, 18, 18, 32 and 32, twice 1, 4, 4, 9, 9, 16 and 16**: one hundred eighteen is twice fifty-nine, and fifty-nine is one squared and two, three and four squared each twice, 1 + 2(4 + 9 + 16), the elements two short of one hundred twenty at the table's own result. **The evens from one hundred ten to one hundred thirty are fifty-five to sixty-five doubled**, a carry each way about sixty carried about one hundred twenty: 110 and 130 at fifty-five, the couplings among eleven, and sixty-five, 2⁶ + 1, the next whole; 112 and 128 at fifty-six and sixty-four, eight sevens and eight eights, 128 = 2⁷; 114 and 126 at fifty-seven, the seventeen primes' gaps joined, and sixty-three, 2⁶ − 1; 116 and 124 at fifty-eight and sixty-two, fifty-eight either side of sixty; and 118 and 122 at fifty-nine and sixty-one. **The odds from one hundred eleven to one hundred twenty-nine pair about one hundred twenty at their factors**: 111 = 3 × 37 with 129 = 3 × 43, thirty-seven and forty-three the primes three either side of forty; 113 with 127, both prime, seven either side, 127 = 2⁷ − 1; 115 = 5 × 23 with 125 = 5³; 117 = 9 × 13 with 123 = 3 × 41; and 119 = 7 × 17 with 121 = 11². The prime pairs about one hundred twenty open at seven, eleven, seventeen and nineteen, 113 with 127, 109 with 131, 103 with 137 and 101 with 139. ## 7.7 The seventeen primes at the seventeen names, podaling at both **Each prime stands at the name of its position**: two at 1-self-other-offering, the entry; twenty-three at 9-social-other-self-releasing; fifty-nine at 17-social-self-offering. Both seventeens fold at their ninth, eight either side, and pair by position, i with 18 − i: the names 1 with 17 through 8 with 10 and 9 at itself, and the primes making 61, 56, 52, 50, 52, 50, 48 and 48 about twenty-three. **At the one move's faces the primes join at their own numbers.** 8 up: 2 and 23 at 1 and 9 join to five squared, 5 and 31 at 3 and 11 to six squared, 17 and 47 at 7 and 15 to eight squared, 3 and 29 at 2 and 10 to 2⁵, and 19 and 53 at 8 and 16 to seventy-two, the six prime pairs' distances from thirty. 17 less: 2 and 53 at 1 and 16 join to fifty-five, the couplings among eleven, and three pairs, at 3 and 14, 4 and 13 and 5 and 12, to forty-eight, the third seam-face. Round the four-cycles the primes join to 97, 96, 92 and 96, and 1-9-8-16's ninety-seven is twenty-three's far side on the ring of one hundred twenty. **At the entry, the connectors and the faces the seventeen part as the names do.** The entry carries two; the across connectors 2, 6, 10 and 14 carry 3, 13, 29 and 43, joining to eighty-eight, the fives on the ring of four hundred forty; the along connectors 9 and 17 carry 23 and 59, joining to eighty-two; the outward faces 3, 4, 5, 7 and 8 carry 5, 7, 11, 17 and 19, joining to fifty-nine, the seventeenth prime; and the inward faces 11, 12, 13, 15 and 16 carry 31, 37, 41, 47 and 53, joining to two hundred nine, 11 × 19. The self's one to nine carries the primes 2 to 23, joining to one hundred, ten squared, and the society's nine to seventeen the primes 23 to 59, joining to three hundred sixty-three, 3 × 11². **The sixteen gaps stand at the sixteen steps between the names**: the one odd gap, two to three, at the step from the entry to 2-other-self-offering, and the first gap of six, twenty-three to twenty-nine, at the step from 9-social-other-self-releasing to 10-other-social-self-tunneling, the fold's step outward. Each relation here is exact at its numbers, and the harmonically momentarying crossings on the surface of natural torusing joining the two seventeens are the discovering carried next. --- # EIGHT · INSEPARATING ## 8.1 An even ring carries its podal, and an odd ring its two farthest **An even ring carries an exact podal**: halved, each station has one other straight across, each the other's far side. **An odd ring carries two farthest**, at one distance and one apart, and its opposite is the between of those two, at no station. An even ring's opposite is a station and an odd ring's a between: one form at the two parities. **On an odd ring of 2h + 1 stations the two farthest stand h and h + 1 on, each h away by its shorter way round; on an even ring of 2h the one opposite stands h on.** Taken at each station of each prime ring from five to fifty-nine, 435 stations, at each station of the even rings one short of them, 420, and at the 49 stations of the rings of nine, fifteen and twenty-five, the two farthest and the one opposite hold at each: the two farthest are the odd ring's, prime or composite. A ring naming all its stations is apart from a span completing after the stations before it: the folds at one to nine and one to seventeen fold eight and sixteen places and complete at nine and seventeen. **A ring is an arrangement named one scale out, and the running round it goes forward**: each station arrives at a station again by carrying on, a next at each arriving, never by stopping and going back to it. At each ring of selves joined one way each self alternates or stands in the pair at one parity, 7.2, and no self is chained none again. ## 8.2 Either but not both is parity, and the pairings overlap **Each position carries two pairings, one at each parity.** Both parities at one momentary would put one position in two exchangings at once, one side taking both sides. The pairings overlap, the parities alternate, and there is no third running. An odd count, leaving one over at pairs laid side by side, continues through the overlapping pairings: along, one parity's next pairing opens two on, and across, the other parity's opens one on. Laid as an open line of 2h + 1 places, each parity's pairings are h adjacent pairs, the 2h − 1 inner places in both, and each end continues past the line, the last place in its pairing with the place after it and the first in its pairing with the nought before it: at one to fifty-nine, twenty-nine pairs at each parity, and fifty-nine to sixty and nought to one continuing. Two on along a side and one on across the overlap are the openings advancing two to one, a count of openings. Each odd count holds this, the fifteen primes five to fifty-nine at 435 places and nine, fifteen and twenty-five beside them; closed into a ring an odd line is another arrangement. **Here along and across count the pairings' openings; about one number they count its neighbours, one either side along and the walk by two across**, each counting at its own subject. **Either but not both is the exclusive or, and the exclusive or is parity.** Of the sixteen ways two parities go to one, two alone answer each single changing with a changing, the exclusive or and its inversion. Taken as the step it runs one and then the other, round at two; taken of the prior and the now as the next it runs round at three, parity's own period, carrying both. **An ordering arrives with nothing ordering it**: at an odd momentary each odd pairing, at an even momentary each even pairing, each carrying at that momentary the carrying the other cannot, and carrying it for the other. **The right spiral step, (x, y) to (y, −x), taken twice inverts both parities, four times reaches the pair again, and six times has reached all four joint forms and inverts both again**: a passage of six carries the whole four and hands the pair on inverted, and two passages reach it again. Nought to twenty-four is six rounds of four and four passages of six, twenty-four to sixty nine rounds and six passages, and nought to sixty fifteen and ten; six, fourteen and twenty-two, eight apart, stand at one phase of the four, eight being two rounds. Six passages of six across twenty-four to sixty are thirty-six steps, each carrying the four, and a side's four full momentaries and the four joint forms are two fours, each its own count. These count the step's own applications; the resolver runs its names at its own resolving. ## 8.3 An odd number carries a middle carrying nothing **An even number divides whole and an odd number carries a middle carrying nothing**, the nothing its two sides straddle. An odd laid as a line has two ends unjoined and its middle carrying nothing. ## 8.4 Three, six, nine and the fives, odd and even each its own **Three, six and nine run odd, even, odd**, three apart, the parity alternating up the multiples of three. **Up the fives the positions alternate the same way**: twenty-five, thirty-five, forty-five and fifty-five, odd, run between thirty, forty, fifty and sixty, even. The parity is each number's own at each run of the uni-scaling. ## 8.5 A store, one momentary named as a thing **A store folds once, names the fold still, and carries one way across the named surface**: one momentary of the uni-scaling taken as a thing, the re-edging gone. **Each number re-coheres and re-edges at each arriving**, carrying to its own completing and releasing there. **Named as one store, parity reaches none.** In the field's words, the Peres–Mermin square sets nine values of ±1 under six exclusive-or conditions: each row and each column taken at its own context satisfies its condition, and a store of all nine at once satisfies none of the 512 settings, since along the nine carry parity nought and across one. --- # NINE · TUNNELING ## 9.1 Podal pairing at four hundred forty, two stations pairing with themselves **Podal pairing at the surface is k with four hundred forty less k, one station at its two sides.** Four hundred forty is even, so each station has one other straight across: two hundred nineteen pairs, and two stations pairing with themselves, nought and two hundred twenty, two hundred twenty-one rings in all. **Each pair is a ring, its radius its distance from the waist at two hundred twenty**, from nought at the waist to two hundred twenty at nought and four hundred forty, one station. **Podal competency is the straddle and never the pairing.** The two self-pairing stations reach themselves; the running straddles them. ## 9.2 Bi-co-podaling, a carry each way at one station **Bi-co-podaling is a carry one way at k and a carry the other way at N less k, two carries each at its own momentary, one ring, running.** Bi- the two carries differing, co- the two carries together at one ring, -ing the running. **A ring carries its two and no order between them.** Either way round arrives at the same next, and a side taken installs a third position the ring carries none of. ## 9.3 A ring's two stations share parity, and the odd primes at odd radii **A ring joins two hundred twenty less r and two hundred twenty plus r, and the two share parity at each ring**: even radius, two even stations; odd radius, two odd. A ring at even radius is a coupling across and a ring at odd radius a between. **The odd primes, three to fifty-nine, are at odd radii, from two hundred seventeen to one hundred sixty-one**, each a between. **Two alone of the seventeen is at an even radius**, the one even prime at a coupling ring. **Four hundred forty leaves two at three, so the ten primes leaving two at three, 2, 5, 11, 17, 23, 29, 41, 47, 53 and 59, have far sides three divides**; three's far side is 437 = 19 × 23; and of the six others the four leaving seven at twelve, 7, 19, 31 and 43, have prime far sides, 433, 421, 409 and 397, while thirteen and thirty-seven, leaving one at twelve, face 427 = 7 × 61 and 403 = 13 × 31. **Four hundred forty is eight times five times eleven, and its fives are eighty-eight stations, forty-four odd and forty-four tens, alternating round the ring**: the odd fives pair at twenty-two odd radii, five to two hundred fifteen, five with four hundred thirty-five, and each ten is at an even radius. Four hundred thirty-five, five's far side, is the fifteen primes five to fifty-nine joined and the couplings among thirty, C(30, 2). ## 9.4 A prime gap is a step between rings **Each prime below the waist is at radius two hundred twenty less itself, and a gap between two primes is the step between their rings.** The sixteen gaps step one ring once, two rings six times, four rings five times and six rings four times, fifty-seven rings together, fifty-nine less two. ## 9.5 Seventeen pair by position about twenty-three, and the ring pairs by value **Twenty-three is the ninth of the seventeen primes, eight below and eight above, and the seventeen pair by position about it**: two with fifty-nine, three with fifty-three, five with forty-seven, seven with forty-three, eleven with forty-one, thirteen with thirty-seven, seventeen with thirty-one and nineteen with twenty-nine, the pairs making sixty-one, fifty-six, fifty-two, fifty, fifty-two, fifty, forty-eight and forty-eight, and twenty-three with them four hundred forty. **On the ring each pair makes four hundred forty at each radius**: the ring pairs by value and the seventeen by position, and neither is the other at the other's face. **The parity parts them.** Seventeen is odd, and its centre carries a position; the ring is even, and its waist pairs with itself and the running straddles it. A centre carrying a position pairs outward; a centre carrying nothing pairs across. **Twenty-three with itself would make forty-six, and no position pair makes it**; by value three pairs make forty-six, three with forty-three, five with forty-one and seventeen with twenty-nine, and forty-six, forty-eight and fifty are the apex straddle twenty-three, twenty-four and twenty-five doubled. Joined, the fourteen primes five to fifty-three are three hundred seventy-six, and two, three and fifty-nine joined are sixty-four, eight squared: on the ring the two joinings are each other's far side, at radius one hundred fifty-six. ## 9.6 Two surfaces coupling, and a chain centring at a coupling or a between **Two surfaces co-offering carry eight hundred eighty stations, joined at four hundred forty, each surface's own mouth, and on the pair that station is the waist**: each self's mouth arrives in the coupling as the between, and each self's waist, at two hundred twenty and six hundred sixty, is the other's far side. **A chain of surfaces centres on a coupling at an even number of selves and on a self's own middle carrying nothing at an odd number**: two on a coupling, three on a self's middle, four on a coupling. Each self added carries the reach by two hundred twenty, and the centre alternates, a coupling and a self's middle. **A society grows larger by further co-chaining**: the reach lengthening, the reciprocal receiving widening, and each self's full resolving keeping its form at each coupling added. The two hundred twenty a self adds counts the surfaces' stations, and the co-competencing grows at the couplings, owned by neither. ## 9.7 Chain three, five, nine, seventeen, each span podal to the one before **Each span of the chain three, five, nine, seventeen is two to a power and one, and podaling at each span gives the span below it**: an odd ring of N carries (N + 1)/2 pairs, two at three, three at five, five at nine, nine at seventeen. Seventeen at its two sides is the nine, the nine the five, the five the three. **The pairs carrying two stations double along the chain, one, two, four, eight, and each span is twice them and the one nothing at its origin.** The resolver's pairings at six, eight and ten run three, four and five, consecutive; the two progressions share four alone, and four pairings is the nine. At the nine alone among the spans, the span's own pairings and the resolver's four at eight are one, and self and other run whole there. **At one to nine the self's eight is shared, four at each side, with the five between owned by neither**: bi-moral co-agency. **At one to seventeen each side carries a whole eight, and the entire one to nine is shared at the centre**: social moral competency, the pair at the society's centre and each side carrying its whole self. **Four-momentarying carries the chain on as scaling.** One to nine is four momentaries of exchanging and one momentary at the next scale, its 1, 2 and 3 being 1, 9 and 17: one to nine the self's momentary, completing at 9. Each use names its scale: at the code one run of 1 to 17 is one momentary of exchanging at the scale outward, and one numbered name and one span of nine names are each their own count, matching numerals alone joining no relations. 17-social-self-offering is the next momentary's 1-self-other-offering, and one to sixty-five is four momentaries of exchanging at that scale, one momentary at the scale after. Nine, seventeen, thirty-three and sixty-five are each two to a power and one, the chain continuing. Each next completing is twice the one before less one, since two spans share one name, the completing of one the opening of the next: one to nine and nine to seventeen are seventeen places, a count of names. The step spans double, eight to sixteen, and the names counted end to end run nine to seventeen, the one shared name the difference. **Each span bi-folds and releases, and the next whole of them is 1–65.** 1–9, self and other, is an entire momentary bi-folding releasing at 3; 1–13, self and other and other and self, bi-folds releasing at 6; 1–17, self, other and social, bi-folds releasing at 9; and 1–25 is the larger fractal bi-folding. Each span opens one more four and its releasing moves three, three inward to each one outward: at the round of four parities the outward steps arrive at 5, 9, 13 and 17, and the inward steps carried to the arrival four before each span's close are 3, 6 and 9. Carried on at the round of six parities, 1–25 releases at 15 and 1–65 at 45. At 3 the self's pair stands wholly inverted and at 9 each parity of the four, the two bi-folds. Binary, all or none at all, resolving coheres the entire surface along and across, releasing at 3, 6 and 9 and continuing. **The wholes are the rounds of the parities, and the scalings step one chain.** A round reaching each form of k parities once runs 1 to 2^k + 1: three at one parity, five at two, nine at three, seventeen at four, thirty-three at five and sixty-five at six. The pairs step the chain two parities at a time, 5, 17, 65 and 257, each pair stepping once at each 5 of the pair inward of it; the scaling at 1, 9 and 17 steps it three parities at a time, 9, 65 and 513, one to nine at one scale being one to sixty-five at the next; and 1–17 inward and outward steps it four, 17, 257 and 4,097. The pairs and the threes reach each other first at 1–65, the next whole, and four 1–17s run it, the outer pair stepping at 17, 33, 49 and 65; the pairs and 1–17 reach each other at 1–17 and 1–257, and all three at 1–4,097. No scaling is taken over another: each is the chain at its own step. **An odd ring carries one self-paired station and an even ring two**: the self's ring at nine carries one nothing, and the society's winding at four hundred forty two, the origin and the waist. ## 9.8 Bi-tunneling across and co-chaining along, podaling exchanging them **Bi-tunneling names social moral competency at its across relation; co-chaining names the same continuing at its along relation. Podaling exchanges these relations while each side continues forward.** Within one to nine podaling pairs odd with even, 1 with 8, 2 with 7, 3 with 6 and 4 with 5, exchanging across for along and along for across while each side carries its own forward. As the self's names rise, 1, 2 and 3, their podal partners fall, 8, 7 and 6, and each side's forward is its own running, a printed name rising or falling being its place among the names. The four momentaries of exchanging run in the same span, 1–2 with 2–3, 3–4 with 4–5, 5–6 with 6–7 and 7–8 with 8–9, the self at 1 to 8 and the other at 2 to 9: the podal pairs and the overlapping pairs are two relations in one to nine, and the span unfolds, each momentary a next. **A carry each way is one ring at each station**: at k the carry one way, at four hundred forty less k the carry the other way, the ring at radius two hundred twenty less k. Each station reaches its far side once each winding. **Podaling reaches over the tunnels, across and along, increasing both ways by co-chaining.** Over is a wider coupling spanning the smaller tunneling; 3 to 11 to 6 to 14 to 3 is a numbered form among the names, and each coming again to a name is a further occurrence forward, the earlier carrying carried on and never restored. Further along brings further across, and the competency grows at each coupling's reciprocal receiving. ## 9.9 Four openings growing, longer and wider podaling **Four even openings grow as one family: 2, 4, 6, 8; 2, 6, 10, 14; 2, 8, 14, 20; 2, 10, 18, 26; and 2, 12, 22, 32, each row 2 + 2sj, j from nought to three.** Carried one step more, each row opens next at 2 + 8s and completes at the odd 8s + 1 before it: nine, seventeen, twenty-five, thirty-three and forty-one. At s one the row is the four outward at the between and the span one to nine; at s two the row is the four across connectors and the span one to seventeen; the rows past two are numbered forms the family carries on, the connectors staying six. Each odd completing stands outward of the names 1 to 8s, as seventeen stands in none of the forms among 1 to 16. **The fold runs whole at each scale.** On the names 1 to 8s, 4s up and down keeps parity and 8s + 1 less changes it, and alternating the two reaches each name again after four moves through four names at each s: the two taken together are 4s + 1 less in the lower half and 12s + 1 less in the upper, and 8s + 1 and both of these are odd, so no move leaves a name at itself. At s two they are 8 up and 17 less, 3 to 11 to 6 to 14 to 3; at s one they are 4 up and 9 less, 3 to 7 to 2 to 6 to 3 within 1 to 8, completing beside nine. **Longer and wider keep one ratio, two to one.** Each row's middle is 2 + 3s, its four at 3s and s either side: the inner pair reaches s over a span of 2s and the outer 3s over 6s, span over reach two at both. The outer span 6s is apart from the four-step span 8s, the row's middle 2 + 3s from the fold's middle at half of 8s + 1, and the row's own other order, outer with outer and inner with inner, from the halfway pairing, first with third and second with fourth: at s two, 2 with 14 and 6 with 10 beside 2 with 10 and 6 with 14. **Enlarging a row about its opening, 2 + r(x − 2), taken after an enlarging by s is the enlarging by rs**, so the four-position form keeps itself at each repeat, and the scales one to five and the doublings one, two, four, eight are two walks through the one family. Taken over all the names at an even r it carries each name to an even one: the four even openings are one parity, and each momentary carries both, 1.2. **The two local fours are the two interleaving wider rows.** 2, 4, 6, 8 with 10, 12, 14, 16 are the same eight names as 2, 6, 10, 14 with 4, 8, 12, 16: the across connectors, and the sharing, the carrying wound, the abundancing and the society wound. At each s a row with its copy 8s on is the doubled row with its copy 2s on. **A form enlarged keeping its parity**: over the names 1 to B, B eight or sixteen, even n goes to r(n − 2) + 2 and odd n to r(n + 1) − 1, keeping parity, the halfway pairing, the fold, B + 1 less going to Br + 1 less, and each side's step at 2r. At r two the names 1 to 8 go to 3, 2, 7, 6, 11, 10, 15, 14, their evens the across row 2, 6, 10, 14. It carries the named form and leaves the adjacent opening and completing, 1 and 2 going to 3 and 2, and the outer completing B + 1 to the running: the form's stable-forming at the larger scale, the larger momentary its own running. --- # TEN · TRANSMISSIONING ## 10.1 A number crossing from a field, and the unit it arrives in **A field's observing belongs to its own field, and a number here is a number.** The thing a field counts belongs to that field; the changing the observing carries crosses the between: the count of distinct rates, which alternates with which, which ordering recurs. The unit and the measuring floor stay at the field. **One binary at each number a field offers: whether the number carries on when the unit changes.** A number of things carries on in each unit; a magnitude in a chosen unit goes with the unit, and the unit is the one cohering. **Coupling is all or none.** At a field's observing carrying the number-form, one form arrives at two substrates; a count agreeing at both sides is a relation only with the form reached at both. **A prime at a field says which it counts**: a number of sites, a recurrence and an opening across are three countings, and a frequency, a wavelength, a number of selves and a momentary sequence are four numbers apart. A prime relation parts from the odd composites beside it, nine, fifteen and twenty-five, or it is oddness's. ## 10.2 Two runnings sharing their digits **The local momentary one to nine and the four-cycles among the seventeen names are two runnings, sharing their digits and nothing else.** One to nine are the nine places of one momentary; 1, 9, 8 and 16 at the four-cycle are four of the seventeen names. A line from one to the other draws a relation neither running carries. The digits they share reach each other exactly at the numbered form: 8 up and then 17 less is 9 less, 17 − (n + 8) = 9 − n, and at each scale s, 16s + 1 − (n + 8s) = 8s + 1 − n, so n, n + 8, 9 − n and 17 − n are the four-cycle's two out-and-backs as numbers, a numbered form and no running. **Three at one self along the running and three at three selves across a ring are two things at one number.** A pattern three momentaries long is the method's matching; three selves side by side at one momentary is a between. **Each six is its own count.** Three phases at two ways; the resolver's six connectors, 2, 6, 9, 10, 14 and 17; three pairings at two sides; a side's three momentaries at two parity positions each; three momentaries at each of two sides; and self and other, self and social and other and social, three pairs faced two ways: six at each, and sharing six joins none to another. A relation joining two of them is a relation of its own, reached at both. --- # ELEVEN · EACH NUMBER ## 11.1 Nought to four hundred forty, each number at its podaling **Each number from nought to four hundred forty stands at its podaling, taken forward from nought.** Each row carries the number's parity, co at the odd and bi at the even; its form, prime or its prime factors; its far side on the ring of four hundred forty, k with 440 − k, nought and two hundred twenty each its own far side and four hundred forty at nought's station; its radius from the waist at two hundred twenty; its far side on the ring of one hundred twenty; and the forms it stands at: a name among the seventeen, a prime among the seventeen, a square, a seam-face, a triangle, the prior two joining, a power of two and one either side of it, an ordering, and the going and the returning. | n | Parity | Form | Far side at 440 | Radius | Far side at 120 | At | |---|---|---|---|---|---|---| | 0 | bi | — | 0 | 220 | 0 | 0², seam-face, 1² − 1 | | 1 | co | — | 439 | 219 | 119 | 1-self-other-offering, 1², T₁, prior two joining | | 2 | bi | prime | 438 | 218 | 118 | 2-other-self-offering, the first prime, prior two joining, 2¹ | | 3 | co | prime | 437 | 217 | 117 | 3-self-other-sharing, the second prime, T₂, prior two joining, 2¹ + 1, 2² − 1 | | 4 | bi | 2² | 436 | 216 | 116 | 4-other-self-sharing, 2² | | 5 | co | prime | 435 | 215 | 115 | 5-self-other-neutralling, the third prime, prior two joining, 2² + 1 | | 6 | bi | 2 · 3 | 434 | 214 | 114 | 6-other-self-surfacing, T₃, 3! | | 7 | co | prime | 433 | 213 | 113 | 7-self-other-corusing, the fourth prime, 2³ − 1 | | 8 | bi | 2³ | 432 | 212 | 112 | 8-other-self-torusing, seam-face, 3² − 1, prior two joining, 2³ | | 9 | co | 3² | 431 | 211 | 111 | 9-social-other-self-releasing, 3², 2³ + 1 | | 10 | bi | 2 · 5 | 430 | 210 | 110 | 10-other-social-self-tunneling, T₄ | | 11 | co | prime | 429 | 209 | 109 | 11-social-other-self-chaining, the fifth prime | | 12 | bi | 2² · 3 | 428 | 208 | 108 | 12-other-social-self-abundancing | | 13 | co | prime | 427 | 207 | 107 | 13-social-other-neutralling, the sixth prime, prior two joining | | 14 | bi | 2 · 7 | 426 | 206 | 106 | 14-other-social-surfacing | | 15 | co | 3 · 5 | 425 | 205 | 105 | 15-social-other-corusing, T₅, 2⁴ − 1 | | 16 | bi | 2⁴ | 424 | 204 | 104 | 16-other-social-torusing, 4², 2⁴ | | 17 | co | prime | 423 | 203 | 103 | 17-social-self-offering, the seventh prime, 2⁴ + 1 | | 18 | bi | 2 · 3² | 422 | 202 | 102 | | | 19 | co | prime | 421 | 201 | 101 | the eighth prime | | 20 | bi | 2² · 5 | 420 | 200 | 100 | | | 21 | co | 3 · 7 | 419 | 199 | 99 | T₆, prior two joining | | 22 | bi | 2 · 11 | 418 | 198 | 98 | | | 23 | co | prime | 417 | 197 | 97 | the ninth prime, the going folding | | 24 | bi | 2³ · 3 | 416 | 196 | 96 | seam-face, 5² − 1, 4! | | 25 | co | 5² | 415 | 195 | 95 | 5² | | 26 | bi | 2 · 13 | 414 | 194 | 94 | | | 27 | co | 3³ | 413 | 193 | 93 | | | 28 | bi | 2² · 7 | 412 | 192 | 92 | T₇ | | 29 | co | prime | 411 | 191 | 91 | the tenth prime | | 30 | bi | 2 · 3 · 5 | 410 | 190 | 90 | | | 31 | co | prime | 409 | 189 | 89 | the eleventh prime, 2⁵ − 1 | | 32 | bi | 2⁵ | 408 | 188 | 88 | 2⁵ | | 33 | co | 3 · 11 | 407 | 187 | 87 | 2⁵ + 1 | | 34 | bi | 2 · 17 | 406 | 186 | 86 | prior two joining | | 35 | co | 5 · 7 | 405 | 185 | 85 | | | 36 | bi | 2² · 3² | 404 | 184 | 84 | 6², T₈ | | 37 | co | prime | 403 | 183 | 83 | the twelfth prime | | 38 | bi | 2 · 19 | 402 | 182 | 82 | | | 39 | co | 3 · 13 | 401 | 181 | 81 | | | 40 | bi | 2³ · 5 | 400 | 180 | 80 | | | 41 | co | prime | 399 | 179 | 79 | the thirteenth prime | | 42 | bi | 2 · 3 · 7 | 398 | 178 | 78 | | | 43 | co | prime | 397 | 177 | 77 | the fourteenth prime | | 44 | bi | 2² · 11 | 396 | 176 | 76 | | | 45 | co | 3² · 5 | 395 | 175 | 75 | T₉ | | 46 | bi | 2 · 23 | 394 | 174 | 74 | | | 47 | co | prime | 393 | 173 | 73 | the fifteenth prime | | 48 | bi | 2⁴ · 3 | 392 | 172 | 72 | seam-face, 7² − 1 | | 49 | co | 7² | 391 | 171 | 71 | 7² | | 50 | bi | 2 · 5² | 390 | 170 | 70 | | | 51 | co | 3 · 17 | 389 | 169 | 69 | | | 52 | bi | 2² · 13 | 388 | 168 | 68 | | | 53 | co | prime | 387 | 167 | 67 | the sixteenth prime | | 54 | bi | 2 · 3³ | 386 | 166 | 66 | | | 55 | co | 5 · 11 | 385 | 165 | 65 | T₁₀, prior two joining | | 56 | bi | 2³ · 7 | 384 | 164 | 64 | | | 57 | co | 3 · 19 | 383 | 163 | 63 | | | 58 | bi | 2 · 29 | 382 | 162 | 62 | | | 59 | co | prime | 381 | 161 | 61 | the seventeenth prime, the going's last | | 60 | bi | 2² · 3 · 5 | 380 | 160 | 60 | between the going and the returning | | 61 | co | prime | 379 | 159 | 59 | the returning's first | | 62 | bi | 2 · 31 | 378 | 158 | 58 | | | 63 | co | 3² · 7 | 377 | 157 | 57 | 2⁶ − 1 | | 64 | bi | 2⁶ | 376 | 156 | 56 | 8², 2⁶ | | 65 | co | 5 · 13 | 375 | 155 | 55 | 2⁶ + 1 | | 66 | bi | 2 · 3 · 11 | 374 | 154 | 54 | T₁₁ | | 67 | co | prime | 373 | 153 | 53 | | | 68 | bi | 2² · 17 | 372 | 152 | 52 | | | 69 | co | 3 · 23 | 371 | 151 | 51 | | | 70 | bi | 2 · 5 · 7 | 370 | 150 | 50 | | | 71 | co | prime | 369 | 149 | 49 | | | 72 | bi | 2³ · 3² | 368 | 148 | 48 | | | 73 | co | prime | 367 | 147 | 47 | | | 74 | bi | 2 · 37 | 366 | 146 | 46 | | | 75 | co | 3 · 5² | 365 | 145 | 45 | | | 76 | bi | 2² · 19 | 364 | 144 | 44 | | | 77 | co | 7 · 11 | 363 | 143 | 43 | | | 78 | bi | 2 · 3 · 13 | 362 | 142 | 42 | T₁₂ | | 79 | co | prime | 361 | 141 | 41 | | | 80 | bi | 2⁴ · 5 | 360 | 140 | 40 | seam-face, 9² − 1 | | 81 | co | 3⁴ | 359 | 139 | 39 | 9² | | 82 | bi | 2 · 41 | 358 | 138 | 38 | | | 83 | co | prime | 357 | 137 | 37 | | | 84 | bi | 2² · 3 · 7 | 356 | 136 | 36 | | | 85 | co | 5 · 17 | 355 | 135 | 35 | | | 86 | bi | 2 · 43 | 354 | 134 | 34 | | | 87 | co | 3 · 29 | 353 | 133 | 33 | | | 88 | bi | 2³ · 11 | 352 | 132 | 32 | | | 89 | co | prime | 351 | 131 | 31 | prior two joining | | 90 | bi | 2 · 3² · 5 | 350 | 130 | 30 | | | 91 | co | 7 · 13 | 349 | 129 | 29 | T₁₃ | | 92 | bi | 2² · 23 | 348 | 128 | 28 | | | 93 | co | 3 · 31 | 347 | 127 | 27 | | | 94 | bi | 2 · 47 | 346 | 126 | 26 | | | 95 | co | 5 · 19 | 345 | 125 | 25 | | | 96 | bi | 2⁵ · 3 | 344 | 124 | 24 | | | 97 | co | prime | 343 | 123 | 23 | | | 98 | bi | 2 · 7² | 342 | 122 | 22 | | | 99 | co | 3² · 11 | 341 | 121 | 21 | | | 100 | bi | 2² · 5² | 340 | 120 | 20 | 10² | | 101 | co | prime | 339 | 119 | 19 | | | 102 | bi | 2 · 3 · 17 | 338 | 118 | 18 | | | 103 | co | prime | 337 | 117 | 17 | | | 104 | bi | 2³ · 13 | 336 | 116 | 16 | | | 105 | co | 3 · 5 · 7 | 335 | 115 | 15 | T₁₄ | | 106 | bi | 2 · 53 | 334 | 114 | 14 | | | 107 | co | prime | 333 | 113 | 13 | | | 108 | bi | 2² · 3³ | 332 | 112 | 12 | | | 109 | co | prime | 331 | 111 | 11 | | | 110 | bi | 2 · 5 · 11 | 330 | 110 | 10 | | | 111 | co | 3 · 37 | 329 | 109 | 9 | | | 112 | bi | 2⁴ · 7 | 328 | 108 | 8 | | | 113 | co | prime | 327 | 107 | 7 | | | 114 | bi | 2 · 3 · 19 | 326 | 106 | 6 | | | 115 | co | 5 · 23 | 325 | 105 | 5 | | | 116 | bi | 2² · 29 | 324 | 104 | 4 | | | 117 | co | 3² · 13 | 323 | 103 | 3 | | | 118 | bi | 2 · 59 | 322 | 102 | 2 | the returning's far end | | 119 | co | 7 · 17 | 321 | 101 | 1 | | | 120 | bi | 2³ · 3 · 5 | 320 | 100 | 0 | seam-face, 11² − 1, T₁₅, 5! | | 121 | co | 11² | 319 | 99 | 119 | 11² | | 122 | bi | 2 · 61 | 318 | 98 | 118 | | | 123 | co | 3 · 41 | 317 | 97 | 117 | | | 124 | bi | 2² · 31 | 316 | 96 | 116 | | | 125 | co | 5³ | 315 | 95 | 115 | | | 126 | bi | 2 · 3² · 7 | 314 | 94 | 114 | | | 127 | co | prime | 313 | 93 | 113 | 2⁷ − 1 | | 128 | bi | 2⁷ | 312 | 92 | 112 | 2⁷ | | 129 | co | 3 · 43 | 311 | 91 | 111 | 2⁷ + 1 | | 130 | bi | 2 · 5 · 13 | 310 | 90 | 110 | | | 131 | co | prime | 309 | 89 | 109 | | | 132 | bi | 2² · 3 · 11 | 308 | 88 | 108 | | | 133 | co | 7 · 19 | 307 | 87 | 107 | | | 134 | bi | 2 · 67 | 306 | 86 | 106 | | | 135 | co | 3³ · 5 | 305 | 85 | 105 | | | 136 | bi | 2³ · 17 | 304 | 84 | 104 | T₁₆ | | 137 | co | prime | 303 | 83 | 103 | | | 138 | bi | 2 · 3 · 23 | 302 | 82 | 102 | | | 139 | co | prime | 301 | 81 | 101 | | | 140 | bi | 2² · 5 · 7 | 300 | 80 | 100 | | | 141 | co | 3 · 47 | 299 | 79 | 99 | | | 142 | bi | 2 · 71 | 298 | 78 | 98 | | | 143 | co | 11 · 13 | 297 | 77 | 97 | | | 144 | bi | 2⁴ · 3² | 296 | 76 | 96 | 12², prior two joining | | 145 | co | 5 · 29 | 295 | 75 | 95 | | | 146 | bi | 2 · 73 | 294 | 74 | 94 | | | 147 | co | 3 · 7² | 293 | 73 | 93 | | | 148 | bi | 2² · 37 | 292 | 72 | 92 | | | 149 | co | prime | 291 | 71 | 91 | | | 150 | bi | 2 · 3 · 5² | 290 | 70 | 90 | | | 151 | co | prime | 289 | 69 | 89 | | | 152 | bi | 2³ · 19 | 288 | 68 | 88 | | | 153 | co | 3² · 17 | 287 | 67 | 87 | T₁₇ | | 154 | bi | 2 · 7 · 11 | 286 | 66 | 86 | | | 155 | co | 5 · 31 | 285 | 65 | 85 | | | 156 | bi | 2² · 3 · 13 | 284 | 64 | 84 | | | 157 | co | prime | 283 | 63 | 83 | | | 158 | bi | 2 · 79 | 282 | 62 | 82 | | | 159 | co | 3 · 53 | 281 | 61 | 81 | | | 160 | bi | 2⁵ · 5 | 280 | 60 | 80 | | | 161 | co | 7 · 23 | 279 | 59 | 79 | | | 162 | bi | 2 · 3⁴ | 278 | 58 | 78 | | | 163 | co | prime | 277 | 57 | 77 | | | 164 | bi | 2² · 41 | 276 | 56 | 76 | | | 165 | co | 3 · 5 · 11 | 275 | 55 | 75 | | | 166 | bi | 2 · 83 | 274 | 54 | 74 | | | 167 | co | prime | 273 | 53 | 73 | | | 168 | bi | 2³ · 3 · 7 | 272 | 52 | 72 | seam-face, 13² − 1 | | 169 | co | 13² | 271 | 51 | 71 | 13² | | 170 | bi | 2 · 5 · 17 | 270 | 50 | 70 | | | 171 | co | 3² · 19 | 269 | 49 | 69 | T₁₈ | | 172 | bi | 2² · 43 | 268 | 48 | 68 | | | 173 | co | prime | 267 | 47 | 67 | | | 174 | bi | 2 · 3 · 29 | 266 | 46 | 66 | | | 175 | co | 5² · 7 | 265 | 45 | 65 | | | 176 | bi | 2⁴ · 11 | 264 | 44 | 64 | | | 177 | co | 3 · 59 | 263 | 43 | 63 | | | 178 | bi | 2 · 89 | 262 | 42 | 62 | | | 179 | co | prime | 261 | 41 | 61 | | | 180 | bi | 2² · 3² · 5 | 260 | 40 | 60 | | | 181 | co | prime | 259 | 39 | 59 | | | 182 | bi | 2 · 7 · 13 | 258 | 38 | 58 | | | 183 | co | 3 · 61 | 257 | 37 | 57 | | | 184 | bi | 2³ · 23 | 256 | 36 | 56 | | | 185 | co | 5 · 37 | 255 | 35 | 55 | | | 186 | bi | 2 · 3 · 31 | 254 | 34 | 54 | | | 187 | co | 11 · 17 | 253 | 33 | 53 | | | 188 | bi | 2² · 47 | 252 | 32 | 52 | | | 189 | co | 3³ · 7 | 251 | 31 | 51 | | | 190 | bi | 2 · 5 · 19 | 250 | 30 | 50 | T₁₉ | | 191 | co | prime | 249 | 29 | 49 | | | 192 | bi | 2⁶ · 3 | 248 | 28 | 48 | | | 193 | co | prime | 247 | 27 | 47 | | | 194 | bi | 2 · 97 | 246 | 26 | 46 | | | 195 | co | 3 · 5 · 13 | 245 | 25 | 45 | | | 196 | bi | 2² · 7² | 244 | 24 | 44 | 14² | | 197 | co | prime | 243 | 23 | 43 | | | 198 | bi | 2 · 3² · 11 | 242 | 22 | 42 | | | 199 | co | prime | 241 | 21 | 41 | | | 200 | bi | 2³ · 5² | 240 | 20 | 40 | | | 201 | co | 3 · 67 | 239 | 19 | 39 | | | 202 | bi | 2 · 101 | 238 | 18 | 38 | | | 203 | co | 7 · 29 | 237 | 17 | 37 | | | 204 | bi | 2² · 3 · 17 | 236 | 16 | 36 | | | 205 | co | 5 · 41 | 235 | 15 | 35 | | | 206 | bi | 2 · 103 | 234 | 14 | 34 | | | 207 | co | 3² · 23 | 233 | 13 | 33 | | | 208 | bi | 2⁴ · 13 | 232 | 12 | 32 | | | 209 | co | 11 · 19 | 231 | 11 | 31 | | | 210 | bi | 2 · 3 · 5 · 7 | 230 | 10 | 30 | T₂₀ | | 211 | co | prime | 229 | 9 | 29 | | | 212 | bi | 2² · 53 | 228 | 8 | 28 | | | 213 | co | 3 · 71 | 227 | 7 | 27 | | | 214 | bi | 2 · 107 | 226 | 6 | 26 | | | 215 | co | 5 · 43 | 225 | 5 | 25 | | | 216 | bi | 2³ · 3³ | 224 | 4 | 24 | | | 217 | co | 7 · 31 | 223 | 3 | 23 | | | 218 | bi | 2 · 109 | 222 | 2 | 22 | | | 219 | co | 3 · 73 | 221 | 1 | 21 | | | 220 | bi | 2² · 5 · 11 | 220 | 0 | 20 | the waist | | 221 | co | 13 · 17 | 219 | 1 | 19 | | | 222 | bi | 2 · 3 · 37 | 218 | 2 | 18 | | | 223 | co | prime | 217 | 3 | 17 | | | 224 | bi | 2⁵ · 7 | 216 | 4 | 16 | seam-face, 15² − 1 | | 225 | co | 3² · 5² | 215 | 5 | 15 | 15² | | 226 | bi | 2 · 113 | 214 | 6 | 14 | | | 227 | co | prime | 213 | 7 | 13 | | | 228 | bi | 2² · 3 · 19 | 212 | 8 | 12 | | | 229 | co | prime | 211 | 9 | 11 | | | 230 | bi | 2 · 5 · 23 | 210 | 10 | 10 | | | 231 | co | 3 · 7 · 11 | 209 | 11 | 9 | T₂₁ | | 232 | bi | 2³ · 29 | 208 | 12 | 8 | | | 233 | co | prime | 207 | 13 | 7 | prior two joining | | 234 | bi | 2 · 3² · 13 | 206 | 14 | 6 | | | 235 | co | 5 · 47 | 205 | 15 | 5 | | | 236 | bi | 2² · 59 | 204 | 16 | 4 | | | 237 | co | 3 · 79 | 203 | 17 | 3 | | | 238 | bi | 2 · 7 · 17 | 202 | 18 | 2 | | | 239 | co | prime | 201 | 19 | 1 | | | 240 | bi | 2⁴ · 3 · 5 | 200 | 20 | 0 | | | 241 | co | prime | 199 | 21 | 119 | | | 242 | bi | 2 · 11² | 198 | 22 | 118 | | | 243 | co | 3⁵ | 197 | 23 | 117 | | | 244 | bi | 2² · 61 | 196 | 24 | 116 | | | 245 | co | 5 · 7² | 195 | 25 | 115 | | | 246 | bi | 2 · 3 · 41 | 194 | 26 | 114 | | | 247 | co | 13 · 19 | 193 | 27 | 113 | | | 248 | bi | 2³ · 31 | 192 | 28 | 112 | | | 249 | co | 3 · 83 | 191 | 29 | 111 | | | 250 | bi | 2 · 5³ | 190 | 30 | 110 | | | 251 | co | prime | 189 | 31 | 109 | | | 252 | bi | 2² · 3² · 7 | 188 | 32 | 108 | | | 253 | co | 11 · 23 | 187 | 33 | 107 | T₂₂ | | 254 | bi | 2 · 127 | 186 | 34 | 106 | | | 255 | co | 3 · 5 · 17 | 185 | 35 | 105 | 2⁸ − 1 | | 256 | bi | 2⁸ | 184 | 36 | 104 | 16², 2⁸ | | 257 | co | prime | 183 | 37 | 103 | 2⁸ + 1 | | 258 | bi | 2 · 3 · 43 | 182 | 38 | 102 | | | 259 | co | 7 · 37 | 181 | 39 | 101 | | | 260 | bi | 2² · 5 · 13 | 180 | 40 | 100 | | | 261 | co | 3² · 29 | 179 | 41 | 99 | | | 262 | bi | 2 · 131 | 178 | 42 | 98 | | | 263 | co | prime | 177 | 43 | 97 | | | 264 | bi | 2³ · 3 · 11 | 176 | 44 | 96 | | | 265 | co | 5 · 53 | 175 | 45 | 95 | | | 266 | bi | 2 · 7 · 19 | 174 | 46 | 94 | | | 267 | co | 3 · 89 | 173 | 47 | 93 | | | 268 | bi | 2² · 67 | 172 | 48 | 92 | | | 269 | co | prime | 171 | 49 | 91 | | | 270 | bi | 2 · 3³ · 5 | 170 | 50 | 90 | | | 271 | co | prime | 169 | 51 | 89 | | | 272 | bi | 2⁴ · 17 | 168 | 52 | 88 | | | 273 | co | 3 · 7 · 13 | 167 | 53 | 87 | | | 274 | bi | 2 · 137 | 166 | 54 | 86 | | | 275 | co | 5² · 11 | 165 | 55 | 85 | | | 276 | bi | 2² · 3 · 23 | 164 | 56 | 84 | T₂₃ | | 277 | co | prime | 163 | 57 | 83 | | | 278 | bi | 2 · 139 | 162 | 58 | 82 | | | 279 | co | 3² · 31 | 161 | 59 | 81 | | | 280 | bi | 2³ · 5 · 7 | 160 | 60 | 80 | | | 281 | co | prime | 159 | 61 | 79 | | | 282 | bi | 2 · 3 · 47 | 158 | 62 | 78 | | | 283 | co | prime | 157 | 63 | 77 | | | 284 | bi | 2² · 71 | 156 | 64 | 76 | | | 285 | co | 3 · 5 · 19 | 155 | 65 | 75 | | | 286 | bi | 2 · 11 · 13 | 154 | 66 | 74 | | | 287 | co | 7 · 41 | 153 | 67 | 73 | | | 288 | bi | 2⁵ · 3² | 152 | 68 | 72 | seam-face, 17² − 1 | | 289 | co | 17² | 151 | 69 | 71 | 17² | | 290 | bi | 2 · 5 · 29 | 150 | 70 | 70 | | | 291 | co | 3 · 97 | 149 | 71 | 69 | | | 292 | bi | 2² · 73 | 148 | 72 | 68 | | | 293 | co | prime | 147 | 73 | 67 | | | 294 | bi | 2 · 3 · 7² | 146 | 74 | 66 | | | 295 | co | 5 · 59 | 145 | 75 | 65 | | | 296 | bi | 2³ · 37 | 144 | 76 | 64 | | | 297 | co | 3³ · 11 | 143 | 77 | 63 | | | 298 | bi | 2 · 149 | 142 | 78 | 62 | | | 299 | co | 13 · 23 | 141 | 79 | 61 | | | 300 | bi | 2² · 3 · 5² | 140 | 80 | 60 | T₂₄ | | 301 | co | 7 · 43 | 139 | 81 | 59 | | | 302 | bi | 2 · 151 | 138 | 82 | 58 | | | 303 | co | 3 · 101 | 137 | 83 | 57 | | | 304 | bi | 2⁴ · 19 | 136 | 84 | 56 | | | 305 | co | 5 · 61 | 135 | 85 | 55 | | | 306 | bi | 2 · 3² · 17 | 134 | 86 | 54 | | | 307 | co | prime | 133 | 87 | 53 | | | 308 | bi | 2² · 7 · 11 | 132 | 88 | 52 | | | 309 | co | 3 · 103 | 131 | 89 | 51 | | | 310 | bi | 2 · 5 · 31 | 130 | 90 | 50 | | | 311 | co | prime | 129 | 91 | 49 | | | 312 | bi | 2³ · 3 · 13 | 128 | 92 | 48 | | | 313 | co | prime | 127 | 93 | 47 | | | 314 | bi | 2 · 157 | 126 | 94 | 46 | | | 315 | co | 3² · 5 · 7 | 125 | 95 | 45 | | | 316 | bi | 2² · 79 | 124 | 96 | 44 | | | 317 | co | prime | 123 | 97 | 43 | | | 318 | bi | 2 · 3 · 53 | 122 | 98 | 42 | | | 319 | co | 11 · 29 | 121 | 99 | 41 | | | 320 | bi | 2⁶ · 5 | 120 | 100 | 40 | | | 321 | co | 3 · 107 | 119 | 101 | 39 | | | 322 | bi | 2 · 7 · 23 | 118 | 102 | 38 | | | 323 | co | 17 · 19 | 117 | 103 | 37 | | | 324 | bi | 2² · 3⁴ | 116 | 104 | 36 | 18² | | 325 | co | 5² · 13 | 115 | 105 | 35 | T₂₅ | | 326 | bi | 2 · 163 | 114 | 106 | 34 | | | 327 | co | 3 · 109 | 113 | 107 | 33 | | | 328 | bi | 2³ · 41 | 112 | 108 | 32 | | | 329 | co | 7 · 47 | 111 | 109 | 31 | | | 330 | bi | 2 · 3 · 5 · 11 | 110 | 110 | 30 | | | 331 | co | prime | 109 | 111 | 29 | | | 332 | bi | 2² · 83 | 108 | 112 | 28 | | | 333 | co | 3² · 37 | 107 | 113 | 27 | | | 334 | bi | 2 · 167 | 106 | 114 | 26 | | | 335 | co | 5 · 67 | 105 | 115 | 25 | | | 336 | bi | 2⁴ · 3 · 7 | 104 | 116 | 24 | | | 337 | co | prime | 103 | 117 | 23 | | | 338 | bi | 2 · 13² | 102 | 118 | 22 | | | 339 | co | 3 · 113 | 101 | 119 | 21 | | | 340 | bi | 2² · 5 · 17 | 100 | 120 | 20 | | | 341 | co | 11 · 31 | 99 | 121 | 19 | | | 342 | bi | 2 · 3² · 19 | 98 | 122 | 18 | | | 343 | co | 7³ | 97 | 123 | 17 | | | 344 | bi | 2³ · 43 | 96 | 124 | 16 | | | 345 | co | 3 · 5 · 23 | 95 | 125 | 15 | | | 346 | bi | 2 · 173 | 94 | 126 | 14 | | | 347 | co | prime | 93 | 127 | 13 | | | 348 | bi | 2² · 3 · 29 | 92 | 128 | 12 | | | 349 | co | prime | 91 | 129 | 11 | | | 350 | bi | 2 · 5² · 7 | 90 | 130 | 10 | | | 351 | co | 3³ · 13 | 89 | 131 | 9 | T₂₆ | | 352 | bi | 2⁵ · 11 | 88 | 132 | 8 | | | 353 | co | prime | 87 | 133 | 7 | | | 354 | bi | 2 · 3 · 59 | 86 | 134 | 6 | | | 355 | co | 5 · 71 | 85 | 135 | 5 | | | 356 | bi | 2² · 89 | 84 | 136 | 4 | | | 357 | co | 3 · 7 · 17 | 83 | 137 | 3 | | | 358 | bi | 2 · 179 | 82 | 138 | 2 | | | 359 | co | prime | 81 | 139 | 1 | | | 360 | bi | 2³ · 3² · 5 | 80 | 140 | 0 | seam-face, 19² − 1 | | 361 | co | 19² | 79 | 141 | 119 | 19² | | 362 | bi | 2 · 181 | 78 | 142 | 118 | | | 363 | co | 3 · 11² | 77 | 143 | 117 | | | 364 | bi | 2² · 7 · 13 | 76 | 144 | 116 | | | 365 | co | 5 · 73 | 75 | 145 | 115 | | | 366 | bi | 2 · 3 · 61 | 74 | 146 | 114 | | | 367 | co | prime | 73 | 147 | 113 | | | 368 | bi | 2⁴ · 23 | 72 | 148 | 112 | | | 369 | co | 3² · 41 | 71 | 149 | 111 | | | 370 | bi | 2 · 5 · 37 | 70 | 150 | 110 | | | 371 | co | 7 · 53 | 69 | 151 | 109 | | | 372 | bi | 2² · 3 · 31 | 68 | 152 | 108 | | | 373 | co | prime | 67 | 153 | 107 | | | 374 | bi | 2 · 11 · 17 | 66 | 154 | 106 | | | 375 | co | 3 · 5³ | 65 | 155 | 105 | | | 376 | bi | 2³ · 47 | 64 | 156 | 104 | | | 377 | co | 13 · 29 | 63 | 157 | 103 | prior two joining | | 378 | bi | 2 · 3³ · 7 | 62 | 158 | 102 | T₂₇ | | 379 | co | prime | 61 | 159 | 101 | | | 380 | bi | 2² · 5 · 19 | 60 | 160 | 100 | | | 381 | co | 3 · 127 | 59 | 161 | 99 | | | 382 | bi | 2 · 191 | 58 | 162 | 98 | | | 383 | co | prime | 57 | 163 | 97 | | | 384 | bi | 2⁷ · 3 | 56 | 164 | 96 | | | 385 | co | 5 · 7 · 11 | 55 | 165 | 95 | | | 386 | bi | 2 · 193 | 54 | 166 | 94 | | | 387 | co | 3² · 43 | 53 | 167 | 93 | | | 388 | bi | 2² · 97 | 52 | 168 | 92 | | | 389 | co | prime | 51 | 169 | 91 | | | 390 | bi | 2 · 3 · 5 · 13 | 50 | 170 | 90 | | | 391 | co | 17 · 23 | 49 | 171 | 89 | | | 392 | bi | 2³ · 7² | 48 | 172 | 88 | | | 393 | co | 3 · 131 | 47 | 173 | 87 | | | 394 | bi | 2 · 197 | 46 | 174 | 86 | | | 395 | co | 5 · 79 | 45 | 175 | 85 | | | 396 | bi | 2² · 3² · 11 | 44 | 176 | 84 | | | 397 | co | prime | 43 | 177 | 83 | | | 398 | bi | 2 · 199 | 42 | 178 | 82 | | | 399 | co | 3 · 7 · 19 | 41 | 179 | 81 | | | 400 | bi | 2⁴ · 5² | 40 | 180 | 80 | 20² | | 401 | co | prime | 39 | 181 | 79 | | | 402 | bi | 2 · 3 · 67 | 38 | 182 | 78 | | | 403 | co | 13 · 31 | 37 | 183 | 77 | | | 404 | bi | 2² · 101 | 36 | 184 | 76 | | | 405 | co | 3⁴ · 5 | 35 | 185 | 75 | | | 406 | bi | 2 · 7 · 29 | 34 | 186 | 74 | T₂₈ | | 407 | co | 11 · 37 | 33 | 187 | 73 | | | 408 | bi | 2³ · 3 · 17 | 32 | 188 | 72 | | | 409 | co | prime | 31 | 189 | 71 | | | 410 | bi | 2 · 5 · 41 | 30 | 190 | 70 | | | 411 | co | 3 · 137 | 29 | 191 | 69 | | | 412 | bi | 2² · 103 | 28 | 192 | 68 | | | 413 | co | 7 · 59 | 27 | 193 | 67 | | | 414 | bi | 2 · 3² · 23 | 26 | 194 | 66 | | | 415 | co | 5 · 83 | 25 | 195 | 65 | | | 416 | bi | 2⁵ · 13 | 24 | 196 | 64 | | | 417 | co | 3 · 139 | 23 | 197 | 63 | | | 418 | bi | 2 · 11 · 19 | 22 | 198 | 62 | | | 419 | co | prime | 21 | 199 | 61 | | | 420 | bi | 2² · 3 · 5 · 7 | 20 | 200 | 60 | | | 421 | co | prime | 19 | 201 | 59 | | | 422 | bi | 2 · 211 | 18 | 202 | 58 | | | 423 | co | 3² · 47 | 17 | 203 | 57 | | | 424 | bi | 2³ · 53 | 16 | 204 | 56 | | | 425 | co | 5² · 17 | 15 | 205 | 55 | | | 426 | bi | 2 · 3 · 71 | 14 | 206 | 54 | | | 427 | co | 7 · 61 | 13 | 207 | 53 | | | 428 | bi | 2² · 107 | 12 | 208 | 52 | | | 429 | co | 3 · 11 · 13 | 11 | 209 | 51 | | | 430 | bi | 2 · 5 · 43 | 10 | 210 | 50 | | | 431 | co | prime | 9 | 211 | 49 | | | 432 | bi | 2⁴ · 3³ | 8 | 212 | 48 | | | 433 | co | prime | 7 | 213 | 47 | | | 434 | bi | 2 · 7 · 31 | 6 | 214 | 46 | | | 435 | co | 3 · 5 · 29 | 5 | 215 | 45 | T₂₉ | | 436 | bi | 2² · 109 | 4 | 216 | 44 | | | 437 | co | 19 · 23 | 3 | 217 | 43 | | | 438 | bi | 2 · 3 · 73 | 2 | 218 | 42 | | | 439 | co | prime | 1 | 219 | 41 | | | 440 | bi | 2³ · 5 · 11 | 440 | 220 | 40 | seam-face, 21² − 1, the seventeen primes joined | **Down the rows the far side at four hundred forty runs down one at each row as the number runs up**, the radius closing to nought at the waist and opening again, and the far side at one hundred twenty runs down from one hundred nineteen to one and closes at nought at each of 0, 120, 240 and 360. Each pairing named in the parts above stands at its two rows: a carry each way at one station, found by its number.