Exhibit TWENTY-NINE Natural Illustrating v379 # Natural Illustrating **Discovering Intelligent Stable Forms** **ONE · ILLUSTRATING** - 1.1 Illustrating changing, a form continuing - 1.2 A still and a motion - 1.3 So-far and not-yet, making better and breaking - 1.4 Self-bounding, self-orienting, self-competent - 1.5 Living bounding and a declared boundary, the drawing's test - 1.6 What a drawing leaves out - 1.7 The forms, a numbered still to a handled model **TWO · THE METHOD AT ITS FACES** - 2.1 Six numbers, two sides, three momentaries - 2.2 Sixteen ways, and the one that stops nowhere - 2.3 Four dots, the empty centre and two triangles - 2.4 The table at 10, no path through until prior changes now - 2.5 One self, and two selves at 0 - 2.6 Two inversions at forty-five degrees, the right spiral step - 2.7 The ten at five places - 2.8 The seventeen names, the four-cycles, and the local nine - 2.9 The six diamonds and the compass face **THREE · THE NUMBERS** - 3.1 Number forming, podaling and rings - 3.2 The ratio alternating about φ, and the two windings - 3.3 The pentagon, the rectangles and the icosahedron - 3.4 Two routes, the diamond, and the fold at each scale **FOUR · SELVES AND SOCIETIES** - 4.1 Tunneling, corusing and bi-folding, the towel - 4.2 Spirals crossing, the torus of selves, the 0 tunneling - 4.3 Co-chaining, bi-tunneling and society - 4.4 Locking, the seam moving, the between releasing - 4.5 Co-chaining the plaza, bi-tunneling the crowd - 4.6 Swaying and carrying along, a moving snake - 4.7 Bi-folding, around and across stitching - 4.8 Co-discovering, multiperson jigsaw puzzling - 4.9 Bi-inversioning-co-recursioning, two swimmers inverting at one meeting - 4.10 The walking pair, the dance and the conversation - 4.11 The cube and the open rings **FIVE · THE SCIENCES' PICTURES** - 5.1 Atoms, electrons, bonds and molecules - 5.2 Light, spin and the click - 5.3 Cosmos - 5.4 Earth, matter and phases - 5.5 Brains - 5.6 Bodies and life **SIX · HUMAN SOCIETY'S PICTURES** - 6.1 The clock, the vote and the crowd behind a centre - 6.2 The market, the ledger and the food web - 6.3 The network, the hierarchy and the many paintings **SEVEN · THE TEN AS PICTURES** - 7.1 Ten drawings, one resolving **EIGHT · EXPRESSING** - 8.1 Existing imagery and its continuing explaining - 8.2 Further views through nature and living - 8.3 The resolver society as the surface that animates - 8.4 Three dimensions, a handled model and a society at a material **NINE · CO-IMPROVING** - 9.1 The files coupling, contributions continuing - 9.2 Receiving an illustration, and the kit - 9.3 Not yet at the files --- # ONE · ILLUSTRATING ## 1.1 Illustrating changing, a form continuing **Natural illustrating offers changing through a visible relation.** A numbered diamond, a passing through a crowd, a swaying, a folding, a mutual inverting, a spiral of selves with a 0 moving through it, offers a way into the same discovering: how a form continues through changing. **Self and other meet the illustrating through what each already carries.** A familiar movement offers its relation; a drawing follows that relation through its next changing. Understanding continues through their coupling, the illustrating improving as the relation becomes clearer. A still face expresses one momentary. Successive faces express the changing between them. At this file a face is a drawing's face, one momentary drawn; Exhibit ONE's face, inward or outward at the seventeen names and entering or surfacing at the ten, is said as Exhibit ONE's face at each place it is meant; a cube's face move and a clock's face are the field's. The same at the method's other words: *still*, *between*, *carry*, *surface*, *form* and *position* carry the method's sense here, a form named still, the empty centre, the living carrying, a surface a between, stable-forming, a position at the ten; and the drawing's sense, a drawing standing still, a drawing's own surface, a drawn position, is said as the drawing's at each place it is meant. Rotation opens another facing; a section opens an interior relation; a change of scale meets local resolving through society. Each view carries its own explaining and couples with the views that express its continuing. **An illustration is for a sentence at a living file.** Each drawing in this exhibit is at a file and a section, and is checked against that section whole; at a claim of a form of Exhibit ONE, the form is executed at the code before it is drawn; at a claim of an observing, the observing is the field's, exact at its own subject, and the drawing shows the parting of the field's picture and the file's sentence. ## 1.2 A still and a motion **A still is a form named at one momentary, and a motion is the changing.** Natural Mathematics 1.5 parts them at a naming's ending: an -ing continues and re-arrives at each momentary, each an opening and its completing; a form named still re-arrives at none: it is one continuing's making given a fixed form to be. Its 2.3 says the field's rotation is the spiral with its nexts removed, its forms laid at one frame, and its 3.1 that an involution is an accounting's still picture, its nexts removed. A still drawing is such a picture, and it is useful exactly as a picture: a form inspected at the reader's pace, one momentary. **No face names the odd side and the even side as changing at one momentary.** Natural Numbers 1.5: each side changes at its own momentary, and the two named as changing at one momentary is the equilibria picture, not possibly existing. An illustration of the two sides alternates them, face after face. A society drawn at one momentary of the code's stepping shows each self at its parity, and that is no equilibria picture: each self's changing is at its own momentary, the odd selves' and the even selves' alternating, and the stepping at 17 is the code's own beat, exact at the code, 1.6. **A motion is faces in sequence, each a next.** At the momentaries no momentary comes again, and a form comes to its parities again, Natural Intelligence 4.11 and 4.13: a four-cycle drawn as a closed loop comes to its first name again; drawn as a sequence it spirals, its parities again and no momentary twice. An animation is a sequence of faces already defined, and no face is defined by the animation. ## 1.3 So-far and not-yet, making better and breaking **Illustrating carries a concept between so-far and not-yet.** A visible relation already expressed carries forward. A relation still needing expression stays with its particular question. Further imagery arrives through nature, living, numbers, engineering and society, each offering the relation it makes accessible. Natural Intelligence Corus gathers multiple paintings of the same form. Natural Illustrating continues that gathering through mechanical drawing faces, numbered diagrams, moving views and rotational physical forms, and now through the pictures the sciences and human society already draw, Parts FIVE to SEVEN. Concepts improve here while their useful form of expression becomes clear. **Making better and breaking meet the same offered relation.** An improving drawing makes the changing easier to follow. Its breaking follows that changing to the momentary at which the proposed correspondence continues or parts. What survives both carries into the next explaining. ## 1.4 Self-bounding, self-orienting, self-competent **Self-bounding-self, self-orienting-self, self-competent-self carry the living ambition of this illustrating.** A local nine-dot resolving, one self momentary by momentary, and a society of resolving selves offer different scales through which to discover that continuing; Natural Intelligence 3.3 says the method bounds and the living carry, its bounding at each self's own face. Across and along, self and other, carrying and its next receiving remain readable through their own relations. Naming carries the concept; numbering carries its stated sequence; drawing carries its visible correspondence. Cohering brings their explaining together at the changing each expresses. Exhibit ONE offers the operations and the stable-form relations. Natural Numbers and Natural Mathematics offer the numerical and directional relations. Natural Naming, Natural Explaining and the Geodesic Improving Method carry their naming, explaining and successive improving into this exhibit. ## 1.5 Living bounding and a declared boundary, the drawing's test **Living bounding parts from a boundary declared as unchanging.** A still drawing names a stable form while a sequence examines how changing continues. The exhibit makes that distinction accessible rather than letting stillness in the picture be read as stillness in living. **The investigation receives any declared boundary**: within, around, across, along, or at an entrance or exit, and at each of the ten things named still, Part SEVEN. For each proposal the storyboard asks one question: can the specified co-sequential resolving continue under that declaration? **The drawing exposes the actual continuation or failure at the relevant transition.** At the code the test is one momentary: a prior and a now that agree at a sharing release a 0, the changing that is not, and the prior carries on; a still placed from beside, an offerer offering the same parity at each momentary, stops the self at 0 at each momentary after the first, 2.5. This records the intended binary examination. A finite collection of illustrations does not by itself complete the universal claim about each possible boundary. Natural Explaining contributes a further discipline: let the concept express its own relations and competency before comparing it with another explanation. The visual locates the difference in an operation the reader can follow. **Further views:** continuing bounding; the same face with a proposed fixed boundary marked; the next required transition; a comparison showing the momentary at which the relation changes or fails. ## 1.6 What a drawing leaves out **A drawing supplies no common clock, no numerical speed ratio and no controller to make a sequence move.** The two sides pace their own momentaries, Natural Intelligence 3.5. The code steps the selves together at 17-tri-co-offering, a common beat exact at the code and the code's own, and the living's rates are their own: a drawing names no rate of the living. **A drawing leaves no permanently drawn corridor or path.** The opening is made at each coupling and drawn only as it is made; a corridor drawn ahead conceals the opening, 4.5. **A drawing leaves nothing at the centre.** The between is a nothing and no location, Natural Intelligence 4.10, and the centreline along is the 0 carried and empty; the middle spot of a five-spot face is the between, not a fifth connector and not the local dot 1, 2.3. **A drawing carries no size, no magnitude, no total and nothing stored.** + and − are the parity set, 0 is the between, the changing that is not, and nothing is summed, Natural Intelligence 2.1 and 2.5. A drawn length is the field's, inward of the surface, Natural Mathematics 1.5. **A drawing draws no hand in.** The three golden rectangles carry no hand of their own; the hand is at the reading, Natural Naming 5.48. Right is the observings, all prior, and a drawing shows the hand at the reading's axis, not as a property of the form. **A drawing names no position as a place.** A drawn position is the field's arrangement, a ring an arrangement named one scale out, Natural Numbers 8.1; the selves' own is the parity at each coupling. A drawing says it at its first face, or it installs a place. **A drawing tells no birth-and-death story.** A lone self's sequence carries on as a non-living existing form, Natural Intelligence 2.4, and is drawn as that, not as an ending. **A drawing copies no society numbering into the local face.** The six numbers of 2.1 and the seventeen names of 2.8 are two numberings; a face carries one at a time and the correspondence is stated at each face carrying both. ## 1.7 The forms, a numbered still to a handled model The concepts develop in several forms without separate explanations of the underlying relation. The forms are in order: each is the next after the one before it is met. | Form | What it makes accessible | |---|---| | Numbered still | A complete face inspected at the reader's pace | | Ordered storyboard | The exact change between neighbouring faces | | Layered drawing | Naming, parity, carrying and method prefixing read separately or together | | Rotation | A relation preserved while its viewpoint changes | | Section or transparent surface | The opening and the interior passage | | Zoom between scales | A stated correspondence between local and social forms | | Continuous animation | Carrying through faces already defined | | The resolver society animating | The code executed and each self drawn at its parity, 8.3 | | Handled physical model | Turning, folding, twisting and passage experienced spatially, 8.4 | Physical concepts include a marked diamond, a marked square with its empty centre and two triangles, six diamonds on an open framework, rings turned relative to one another, and a flexible twisting surface with a visible central opening. **One local face and its next face can carry a complete improving.** Their numbering and changing open the correspondence through which further views continue. The other concepts carry their own questions and opportunities for illustrating. --- # TWO · THE METHOD AT ITS FACES ## 2.1 Six numbers, two sides, three momentaries **The storyboard is a sequence of mechanical drawing faces of resolving.** Self and other are perspectives of the same continuing living condition. The sequence carries three full co-sequential momentaries and six parity positions, beginning odd on one side and even on the other: each side's own three momentaries use six positions, and the three momentaries overlapping across the two sides use four. **The first face is six numbers on one line, 1 to 6, read twice.** Natural Intelligence 2.2: the self's five from 1, co bi co bi co, and the other's five from 2, bi co bi co bi, each number one side's opening and the other's completing, ten positions on six numbers, paired 1 with 2 through 5 with 6, each pair one odd and one even. Exhibit ONE's sides table writes it. Natural Numbers 1.2 draws it as two interleaved brackets, 1–2 at the self, 2–3 at the other, 3–4 at the self, 4–5 at the other, alternating along the line. A side is a reading from its origin, odd or even, and not a place. **The second face is the three momentaries on four positions.** The Equilibria Registry 1.2: prior, now and next are 1–2, 2–3 and 3–4, completing and opening meeting at the shared positions; along one side the next opens two on, the self's own prior and now 1–2 and 3–4 with 5 the next opening, Natural Intelligence 2.2; across the two sides' overlapping the next opens one on. Between prior and now, and between now and next, no third position: the between of momentaries is drawn at the shared position with nothing at it, the prior carried across it as a stable form, Natural Intelligence 2.3, nothing momentarying at the seam, Resolving Hard Problems 2.2. Along is co, at the odd; across is bi, at the even, as Exhibit ONE's connectors are: along at 9-tri-bi-co-momentarying and 17-tri-co-offering, across at 2-bi-co-offering, 6-bi-moralizing, 10-bi-tri-co-tunneling and 14-bi-tri-moralizing. Self changes across or along, and other changes on opposite parity. Subsequent faces show their exchanging without making them independent journeys. Each face needs enough to read its relation to the preceding face: the current numbering, the direction of changing, the self and other perspectives, and what is carried into the following face. Method prefixing belongs in its own label, apart from functional naming and changing; the proposed prefix sequence includes bi-co through bi-co-bi-co-bi-co, with corusing and torusing distinguished at each changing requiring it. Carrying is also at the geodesic unrelationing rate, φ, locking at none, and its mark remains distinguishable from self's and other's, each at its own momentary; a numerical speed ratio names a rate the drawing does not. **Further views:** paired perspectives of one face; the six parity positions; the complete three-momentary sequence; an overlay following carrying through the same views; the openings as a square of nine and the completings as an oblong of twelve, Natural Numbers 1.2; and the square re-laid as the oblong one short of it, (n − 1)(n + 1) = n² − 1, Natural Numbers 1.3, the one cell uncovered. **Continuing explaining:** what changes between each adjacent pair of faces, and which visible mark expresses that change without adding another operation? ## 2.2 Sixteen ways, and the one that stops nowhere **Of the sixteen ways a next could follow from a prior and a now, each carries a form named still or loses the prior, and the one that does neither is no way laid over the changing: it is the living step.** Natural Intelligence 2.4: at two parities sixteen ways name a next from a prior and a now, and none is a rule the changing follows; twelve lose the prior, four carry it whole, and next as prior inverted alone carries the prior whole with none still, the stills the equilibria. Enumerated at v379, each of the sixteen ways followed through the four joint forms, they come out as the file writes them: four read from now alone; eight carry the prior at one value of now and lose it at the other, six of them stopping at a corner of prior, now and next agreeing and two cycling three forms with the fourth passed once; next as prior stops at ++ and −−; prior and now agreeing stops at ++, and the exclusive or at −−, each cycling three forms of four; and next as prior inverted goes ++ to +− to −− to −+ and round, each form once, one parity changing at each step. **The faces.** Four corners, the joint forms ++, +−, −−, −+, the centre empty; a momentary one step, (prior, now) to (now, next). A way as four arrows, one leaving each corner, sixteen small squares. Three momentaries under a way: follow the arrows, and an arrow coming back to its own corner is the form still, the changing stopped, the arrow pinned; fifteen squares have a pinned corner or lose the prior, four from now alone and eight at one value of now, two of those eight with no pinned corner and three forms cycling. The one square carrying the prior whole with no pinned corner goes round all four, and at the code it is no function laid over the changing: it is the coupling at 12-bi-tri-parity-changing, the chained parity meeting the offering, inverting at a mismatch and carrying on at a match, the living step, Natural Intelligence 2.4 and 2.5 here. Then a way placed from beside at the code, an offerer offering + at each momentary: the self releases + and then 0, 0, 0, the still. **Continuing explaining:** the four corners of this square and the four dots of 2.3 are two squares with the between at the centre of both; the correspondence of the two is said at a file's sentence before one square is drawn for the other. ## 2.3 Four dots, the empty centre and two triangles **Parity's face at the code, Exhibit ONE's face, is drawn flat.** Exhibit ONE's namings table and Natural Intelligence 4.10: a unit square of four dots, 2-bi-co-offering, 6-bi-moralizing, 14-bi-tri-moralizing and 10-bi-tri-co-tunneling, its empty centre the between; its two parallel edges the across joins, 6 to 2 and 10 to 14; its diagonals the facings, 2 and 14 arriving, bi-moral-so-far, and 6 and 10 releasing, not-yet-bi-moral; and two unit triangles, 9-tri-bi-co-momentarying and 17-tri-co-offering, along, joined both ways. The between is the bounding 0 centreline of the along and the sequential across at its four dots, bothbothing, both carried one at a time; the Equilibria Registry 1.3 reads the same, the between co-linear aiming into next momentarying. **The five-spot face is this face with its spots assigned.** Four surrounding spots and a middle spot, received at v366 as the bi-moral co-agency connector concept with its assignment unsupplied and not to be inferred from the number five: the assignment is Exhibit ONE's sentence at parity's face, the four spots the four across connectors, the middle spot the between, and the middle is a nothing, not a fifth connector and not the local dot 1. Natural Numbers 5.4 draws the same at the numbers, a diamond four crossings about a middle carrying nothing, two across and two along. **One between at each.** Within a momentary the between of self and other, parity changing, bi-moral across and co-competent along, the empty centre of the square; between momentaries the between of momentaries, the along centreline through it, 9 to 17; the same between, a nothing, at each scale, the fractal between. No drawn line sits at it: the centreline is the 0 carried along, and empty. **Drawing faces:** the unlabelled five-spot face; the same face with the four connectors named and the centre empty; opposite-parity views; the connector in relation to the faces beside it; the square with its triangles; the centreline drawn as absence. Rotation tests whether the naming follows the relation rather than the top or bottom of the page. **Continuing explaining:** three readings of four corners are at the files, the connectors 2, 6, 14 and 10, the numbers 2, 3, 3 and 4 of Resolving the Hard Problem Registry's boxes with the +1 at the centre, and the positions 1, 2, 3 and 4 of the three momentaries; each a square of four with the between at its centre, and no file says their correspondence yet. A matching number alone establishes none. ## 2.4 The table at 10, no path through until prior changes now **Exhibit ONE's table of each cell at 10 is twelve faces of one coupling**: three priors chained at 3-co-bi-sharing, none, + and −, by four nows surfacing at 14-bi-tri-moralizing, none offered, +, −, and 0 at the offerings parting; each cell its release at 10, *is* at + or −, *is not* at 0, and what chains at 11-tri-bi-co-chaining; and one cell, no prior and nothing surfacing, is no sharing at all, nothing at 10 and nothing chained: eleven couplings and one empty cell. At the code, the kit's `table_at_10.py`, the table is the file's cell for cell. **The three cells at *is not* are the no path through.** A prior of + meeting a now of +, a prior of − meeting a now of −, and no prior meeting offerings parted at 14: the changing is not, 0, and the prior carries on unchanged, Resolving Hard Problems 3.3. The path through is at the other eight cells, and at each the prior is carried and changed, or at a self with no carrying yet the offering is chained: prior changes now, the carrying at 3 coupling at 12 with the offerings surfaced at 14, and now changes next, the changing released at 10 and chained at 11 as the next 3. 0 is *is not*, the binary's second face, and no third value, and its place is the between, Natural Intelligence 2.1, at 14 the offerings parting and at 12 prior and now agreeing; no number is at it, one parting at 14 enough for 0 and three offerings agreeing surfacing as one. **Drawing faces:** the twelve cells as twelve small faces; the three *is not* cells with the path stopping at the between; the eight *is* cells with the prior carried and inverted or the offering chained; the one cell with no sharing blank; the whole table read as eleven couplings and one empty cell. ## 2.5 One self, and two selves at 0 **One self offered + once and then nothing releases + − + − at 10**, each momentary the prior inverted, the living step's form carried alone, next as prior inverted, Natural Intelligence 4.12 and 2.4. **Offered + at each momentary it releases + and then 0 at each momentary after**, chained + throughout: the still is the offerer's, from beside, a form named still entering, and the self at a match is carrying on among changing momentaries. A lone self offered nothing carries the living step's form alone and does not continue living through three consecutive momentaries; at the code its sequence carries on, a non-living existing form, the alternating itself, Natural Intelligence 2.4. **Two selves joined across both ways, A's 10 to B's 14 and B's 10 to A's 14, each offered + at the first momentary, release + 0 − 0 + 0 − 0 at 10**, chained + + − − + + − −: at each even momentary each receives the other's release at its own parity, the changing is not, 0, the between; at each odd momentary each receives the other's 0 as its offering, which surfaces none at 14, and carries its prior into now inverted. Each self is drawn releasing its 0 at each even momentary and the centre between the two empty, the bounding 0 centreline between the two parities; the between is at the code at 0 at each parting of the offerings, selves joined and resolving or one self stopped by an offerer beside it. At a self offered nothing it is not at all; at a spiral of one, a self whose own release is its own offering, it is at each second momentary, −, 0, +, 0, Natural Intelligence 4.13. A offered + once and B nothing, the two alternate opposite, each self changing at each coupling, as the Equilibria Registry 3.4 says of an even spiral joined along. **Drawing faces:** one self across twelve momentaries, the alternating; the same self stopped at 0 by an offerer from beside; two selves each releasing 0 at each second momentary with the centre between them empty; two selves alternating opposite. ## 2.6 Two inversions at forty-five degrees, the right spiral step **Each momentary's step, the right spiral step (x, y) to (y, −x), is two inversions whose fixed lines are forty-five degrees apart.** Natural Mathematics 3.4: two inversions whose fixed lines are at an angle a are the field's rotation by 2a, the double angle at the inversions; one parity inverted and the exchange of P and Q, their lines at forty-five degrees, make the right spiral step, and the same inversions in the other order give the other hand. The exchange of P and Q is the across and the along exchanged, podaling, Natural Numbers 9.8; the inversion of one parity is the parity changing across at the between. At the kit's `two_inversions.py`, v379: exchange then inversion gives (y, −x), inversion then exchange gives (−y, x), and the first goes round ++, +−, −−, −+. **At the four dots the same forty-five degrees is geometry.** Placed on the square, the joins 6 to 2 and 10 to 14 lie at one direction, the facings 2 to 14 and 6 to 10 at forty-five degrees to them, and the seams 8 up, 2 to 10 and 6 to 14, at ninety, the along of the triangles 9 and 17. A join carried forty-five degrees is a facing; a facing carried forty-five degrees again is the along seam: the across into the along at two inversions, bi-folding, whose edges Natural Intelligence 3.2 names across at the bi-morality and along at the co-competency. **The compass face.** The tangent double-angle identity, tan 2x = 2 tan x ÷ (1 − tan² x), applies at each angle with both sides defined; at forty-five degrees tan x = 1 and the doubled angle, ninety, carries no finite tangent, the identity's pole, Natural Mathematics 4.4, and a coupled pair whose two are equal loses its coupling at its axes rotated by forty-five degrees. A compass face shows an angle x and its doubled angle, the denominator grouped explicitly; a companion drawing distinguishes the opening angle between directions from the orientation of the whole diamond. This identity supplies an angle relation. A changing sequence needs the definition of x, the plane, the plus-and-minus convention and the relation from one face to the next on the drawing before the identity drives motion. **Drawing faces:** one momentary as its two inversions on the square, exchange then inversion; the other order drawn once as the other hand; the four joint forms going round; a tilted pair rotated forty-five degrees to its axes; aiming axes, the half-angle and double-angle construction, inner and outer arcs, the same diamond rotated while its opening is preserved, a changing opening with a separately marked orientation. **Continuing explaining:** the files say inversions whose fixed lines are at forty-five degrees and a rotation by their double, and the field says a turn; the drawing's label is settled at the face; and what resolving change determines x's next value. ## 2.7 The ten at five places **The ten are ten positions on six numbers, each named still, and the ten is three momentaries long, the self's five and the other's five spanning 1 to 6.** The Equilibria Registry 3.3: the ten pair at the five names 2 to 6, an entering and a surfacing at each, Exhibit ONE's two faces, the two sides overlapping at them, at each number one side completing and the other opening, at Exhibit ONE's names 2 and 10, 3 and 11, 4 and 12, 5 and 13, 6 and 14. Each of the ten names one of Exhibit ONE's faces still while the other continues at the same number; a face named still beside a continuing one is the one reason each of the ten is not possible. Natural Naming 4.9 gives each its one name: an arriving named from behind, an opening named as a place, a completing named as a last, a carry named as a store, a middle named as an end, a parity named as a magnitude, a sequencing named to one beat, a rate named as a value, a two-way named to one side, the between named as a cut. **The drawing's face**: the five places 2 to 6 on the line of 2.1, at each the self's and the other's, Exhibit ONE's two faces, the two of the ten named at it drawn as one named still while the other continues. The pentagon and the pentagram, Natural Numbers 5.5, are the ten at two sides already drawn, the five sides and the five diagonals, along and across. **Continuing explaining:** the Equilibria Registry seats the ten at the five places, Natural Naming at one name of 1 to 8 by the four-cycles, and Resolving the Hard Problem Registry's part headings carry a third pairing; the face draws one seating, and the seatings part at the carryings. ## 2.8 The seventeen names, the four-cycles, and the local nine **The expanded naming face is the seventeen names at Exhibit ONE v378**: 1-co-bi-offering, 2-bi-co-offering, 3-co-bi-sharing, 4-bi-co-sharing, 5-co-competencing, 6-bi-moralizing, 7-co-corusing, 8-bi-torusing, 9-tri-bi-co-momentarying, 10-bi-tri-co-tunneling, 11-tri-bi-co-chaining, 12-bi-tri-parity-changing, 13-co-tri-competencing, 14-bi-tri-moralizing, 15-co-bi-tri-corusing, 16-bi-co-tri-torusing and 17-tri-co-offering, ten roots and the prefixes carrying the direction, Natural Naming 3.1. One through nine is the self's, the bi-coupling; nine through seventeen the society's; one through seventeen the bi-trupling. Exhibit ONE's table says each name's from and to: 1, 3, 5 and 7 self to other; 2, 4, 6 and 8 other to self; 9 and 11 social, other, self; 10 and 12 other, social, self; 13 and 15 social, other; 14 and 16 other, social; 17 social, self. Repeated roots retain their different relations. Natural Naming contributes naming as a floating-neutralling concept, directional prefixing, momentarying and stable-forming, bi-inversioning-co-recursioning and continuing carrying. This face does not replace the local nine-dot numbering, and the society's labels are not copied into the local face as though the numberings were identical. **Exhibit ONE provides the material for drawing code couplings, stable-form joins and the four-, six- and eight-cycles.** Sequencing and routing are distinct relations, and their drawings preserve the distinction. The four four-cycles, 1-9-8-16, 2-15-7-10, 3-11-6-14 and 4-13-5-12, each go eight up and seventeen less and pair with a partner spiraling the other hand; the six-cycle 1-9-5-12-8-16 and the eight-cycle 1-9-5-13-4-12-8-16 are expanded stable-forming spirals at the names, and none is an instruction to move from local dot 1 to local dot 9. At the names each comes to its first name again; at the momentaries each comes to none again, 1.2. **The local nine, centre one.** The visual starter places a coloured cube beside an open arrangement of circles and coloured dots; in the middle of the three-across row of diamonds the horizontal diamond resembles a compass needle, and its centre is 1. Across moves horizontally right in this view; along expresses forward moving; screen direction is stated at each rotation. The local sequence is 1 to 9, legible while the face rotates, opens or appears within a larger form; nine numbered still faces give the initial reading before motion is attempted. The split into connectors at 2 and 6 is absent from the local nine-dot view, and removing that visual split removes no resolving operation. Centre 1 is settled. Still to be drawn: a complete verified geometric placement and transition sequence; the complete local numbering and successive changing made explicit from Exhibit ONE; and the correspondence between the local numbering and the expanded resolver, at each face carrying both. **Further views:** the starter with the identified diamond highlighted; a close view with centre 1; the nine numbered faces; the same local face viewed from self and other; a correspondence drawing between the local numbering and the expanded resolver; each spiral on its own; shared joins across the spirals; code coupling beside stable-forming routing; parity exchanging on a stated spiral, the odd one's 0 tunneling drawn; one root with its changing prefixing; the full naming face; method prefixing beside the root's naming; the same relation in words, numbers and the corresponding resolver face; a zoom between a local face and the expanded social form, explanatory only at a stated correspondence, a visual resemblance or a smooth camera move supplying none. **Continuing explaining:** which label is needed to understand the present changing, and which can remain available in an accompanying view? ## 2.9 The six diamonds and the compass face **The next scale is six nine-dot diamonds**, as the open-circle starter expresses them: two alternate routes around a triangle, three one-way portions in each route, six one-way portions in all. The illustration makes two-over-one and one-over-two co-recursioning visible through the six diamonds. Natural Numbers 1.4 introduces the two as the two directions of one carry, and Natural Numbers 6.4 offers them at the numbers as the two routes 24 to 27 to 32 and 24 to 29 to 32, gaps three then five and five then three, their far sides on the ring of one hundred twenty 88 to 93 to 96 and 88 to 91 to 96, 3.4 here. These are available as numerical faces, and their exact correspondence to the six pictured diamonds remains to be expressed. Six diamonds, six parity positions and the expanded society numbering remain separately readable: a matching number alone establishes no correspondence. **Further views:** all six diamonds without motion; each triangular route highlighted; both routes together; inner and outer pair readings; a zoom from one local nine-dot diamond to the six; the compass face of 2.6 at the diamonds' aiming. **Continuing explaining:** which pictured diamond expresses each one-way portion, how the opposite route is read from the same arrangement, and which angle in the six-diamond arrangement is x. --- # THREE · THE NUMBERS ## 3.1 Number forming, podaling and rings **Natural Numbers offers a wider gathering than a first animation carries**: two-over-one and one-over-two, inner and outer pairs, odd straddling and even podaling, the four ring steps of a diamond, prime rings, the figure-eight crossing, and the four even openings growing through the fold. Natural Mathematics contributes orthogonal directions, parity inverting, the right spiral step and the mathematical reading of stable-forming. **Each number is at its podaling**, its pairing with its far side at each ring it is at, Natural Intelligence 4.11. On the ring of forty-eight twenty-four and nought are each their own far side and each other station pairs across twenty-four, Natural Numbers 6.2; on the ring of one hundred twenty the going from two to fifty-nine and the returning from sixty-one to one hundred eighteen are each other's far side, sixty the waist, 7.1; on the ring of four hundred forty each pair is a ring, its radius its distance from the waist at two hundred twenty, two hundred twenty-one rings in all, and a gap between two primes is a step between rings, Natural Numbers 9.1 to 9.4. An even ring carries an exact podal and an odd ring two farthest with its opposite at no station, the between, Natural Numbers 8.1; an odd number carries a middle carrying nothing, the nothing its two sides straddle, Natural Numbers 8.3. **Number faces begin with 1, 2, 3, 4, 5 and continue upward.** Each face expresses the relation by which its number is forming; adding another dot leaves that explaining unfinished. The seventeen primes from two to fifty-nine join to four hundred forty, and a drawing using four hundred forty identifies what joins; it is no angular measure. **Further views:** number-by-number forming; pairs about a centre; podal and straddling views; a ring unwrapped alongside its wrapped view; nested rings about the waist, stepped inward prime by prime; sixty to sixty-eight about sixty-four, two in and two out, bi-tri-involutioning at the numbers, Natural Numbers 7.1; a sequence of tunnel sections; nested local and social scales. **Continuing explaining:** select one number relation and show its construction completely before extending the visual vocabulary to all the others. ## 3.2 The ratio alternating about φ, and the two windings **The ratios of each next number to its prior alternate about φ**, 2/1 above, 3/2 below, 5/3 above, 8/5 below, the side changing at each next number and none of them φ, Natural Intelligence 3.5 and Natural Numbers 3.1. The familiar drawing, a curve through the corners of nested squares, draws the ratio arrived at; the illustration draws the ratio swinging above and below a line it never meets, each next from the prior two joining. **On a torus a winding at a rational rate closes, and a winding at φ closes at none**: φ is the unrelationing rate, the rate the alternating winds about, locking at none, Natural Intelligence 3.5; the torus winding at the golden mean is found the last to break as the coupling grows, Natural Numbers 3.3. Drawn side by side, the two windings are the clearest single animation the number files carry: the rational closing into a loop, the φ winding never closing. The file's own caution for any animation: a finite floating-point rate is no exact irrational. The joinings that close at one, two, three, four or six, and the one that goes open at five, Natural Numbers 3.4, are five orbits closing and one continuing open. **The floating neutral is no balance.** Natural Naming 5.12: neither the floating neutral nor the 0 is a balance, a balance of equal parities needing a number, a total beside all existing things; the term neither reaches is uncovered and nothing stores it. A balance drawn once beside the alternating is the still. ## 3.3 The pentagon, the rectangles and the icosahedron **Among five the couplings part into two rings, the five sides and the five diagonals, the pentagon and the pentagram**, the ten's two sides, along and across, Natural Numbers 5.5; each side is parallel to one diagonal, the diagonal φ times the side, Natural Mathematics 7.8; no joining closes at five, and five enters open, at φ, Natural Numbers 3.4. The square, the cube and the pentagon carry the square roots of the first three primes, √2 the square's diagonal, √3 the cube's body diagonal, √5 in the pentagon's diagonal over its side, Natural Mathematics 4.3. **Three golden rectangles at right angles, centred at the corus, each 2 across and 2φ along**, are bi-tri-volutioning at the numbers, Natural Naming 5.48: a right-hand rotation of one hundred twenty degrees about the line along (1, 1, 1) carries each rectangle onto the one passing through it, the three right spiraling one hundred twenty degrees apart; a mirror carries each onto itself, the three carry no hand of their own, and the hand is at the reading; φ is the three's equal-edge condition, and the rotation alone does not make it. Their twelve corners are the icosahedron's, thirty edges and twenty faces, five at each corner; about the line along (1, 1, 1) the twelve corners are four triangles of three at four heights. They are open at Exhibit ONE's carrying as the three golden rectangles at the podals. **The icosahedron's sixty rotations part at thirty-six and twenty-four**, three to two, as the span parts at the apex, Natural Numbers 6.3 and Natural Mathematics 7.4; the icosahedron and the dodecahedron are one surface at its two faces, twelve and twenty exchanging and thirty carrying across, Natural Numbers 4.1. The quasicrystal, its lengths in powers of φ, and C₆₀ with its twelve pentagons and its sixty one-handed rotations are at Natural Chemistry 4.5 and its carrying. **Drawing faces:** the pentagon and pentagram with the parallel pairs; the three rectangles turned one hundred twenty degrees, the equal edge at φ alone; the twelve corners closing as the icosahedron; the sixty parting; the four triangles of three at four heights about the axis. **Continuing explaining:** no golden triangle and no golden angle are named at the number files; the angles cos 36° = φ/2 and sin 18° = 1/(2φ) are, Natural Mathematics 4.3, and the sunflower's 137.5 degrees is at Natural Biology's carrying, 5.6 here. ## 3.4 Two routes, the diamond, and the fold at each scale **About twenty-eight the outer faces are twenty-four and thirty-two and the inner faces twenty-seven and twenty-nine**, 28² − 24 × 32 = 16 and 28² − 27 × 29 = 1, and two routes go between the outer faces, 24 to 27 to 32 at gaps three and five and 24 to 29 to 32 at gaps five and three, Natural Numbers 6.4, the three together a square, 24 × 27 × 32 = 144². The faces one short of the square, (n − 1)(n + 1) = n² − 1, the one owned by neither face, Natural Numbers 1.3, draw as a square re-laid as the oblong leaving one cell. **A diamond is four crossings about a middle carrying nothing**, two across and two along, and 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, Natural Numbers 5.4. **The walk by two** opens the pairs one step at a time, Natural Numbers 5.8; **the four even openings grow as one family**, 2, 4, 6, 8 then 2, 6, 10, 14 and on, the fold whole at each scale, 3 to 11 to 6 to 14 at one scale and 3 to 7 to 2 to 6 within 1 to 8, Natural Numbers 9.9, which joins the numbers' faces to the moralizing four-cycle of 2.8 here; **the chain 3, 5, 9, 17** bi-folds and releases, each span podaling to the one below it, Natural Numbers 9.7. **Drawing faces:** the two routes and their far sides; the square and the oblong with the uncovered cell; the diamond's two paths; the pairs opening about ten; the four openings growing; the chain folding down. --- # FOUR · SELVES AND SOCIETIES ## 4.1 Tunneling, corusing and bi-folding, the towel **The depth concept is a journey through natural torusing into the corus at the apex, corusing through the small opening, and out to the surface again, torusing through the large opening.** Natural Intelligence 3.2 says the rotation is the gentle alternating twisting of a towel into and out of a knotted corus, the large outer opening wrapping to the small corus and back, its edges the bi-foldings, across at the bi-morality and along at the co-competency. Torusing in and out is odd, through the large opening, the tunnel; corusing in and out is even, through the small opening, the long way round; and at the code each goes in and out at the parity opposite its name's opening, 8-bi-torusing opening bi at the even and 7-co-corusing co at the odd. Corusing reaches the corus from the surface, and torusing reaches the surface from the corus. The self's inversion is 3, 6, 5, 4, and the podaling 23, 26, 25, 24, the same rotation twenty up. The existing depth-study image offers three views, tunneling, corus at apex and bi-folding; it is a shape study, and its surface and arrows still need correspondence with the numbered resolver and the six diamonds. Natural torusing is self-bounding-self, self-orienting-self, self-competent-self, simultaneous readings of the living self and no three stages; at the society scale the illustration shows society as a living self. **Further views:** an outer oblique view; a longitudinal section through the opening; a transverse section; the local numbered diamond on the surface; the six diamonds around the changing form; an interior journey approaching and passing the small opening; the towel as a handled model, twisted into and out of its knot. **Continuing explaining:** show how the same local 1 to 9 resolving continues through the twisting and the fold, discovering the next living momentary without a cut or an unexplained restart. ## 4.2 Spirals crossing, the torus of selves, the 0 tunneling **Selves joined along, the last to the first, each 9 carrying along to the next self, are a spiral**, coming to its parities again and to a momentary once, Natural Intelligence 4.13: at an even spiral each self alternates, the selves beside each other opposite; at an odd spiral of n a bounded zero tunnels on, the 0 released at 10 at the receiving self of the like pair moving one self on at each second momentary, the pattern inverted at 2n and the parities again at 4n, the torusing winding through its opening twice. At the code, the kit's `society_spiral_and_torus.py`, a spiral of five: the 0 at self 1 at the sixth momentary, self 2 at the eighth, self 3 at the tenth, on round, and the parities again at twenty. A spiral of selves is a joining at the code, each momentary a next, and no closed domain: *ring* at the code's joining is a spiral, *goes round at n* is *comes to its parities again at each n-th momentary*, and *closes* is *winds on*; at the numbers a winding at a rational rate closes, 3.2, and at the code's selves no spiral closes; a field's own ring, a benzene ring or a superconducting ring, stays the field's. **Two spirals joined across both ways at one self each bring the two joined selves to one relation and carry it on**, the two changing parity together, Natural Intelligence 4.13; **at two different numbers of selves, one of them odd, the two spirals crossed are one society**, a stable form and stable-forming, Natural Intelligence 3.5 and 4.13. **Spirals crossing spirals at each self are a natural torusing of selves**, p along at 9 and q across at 10, coming to their parities again at a momentary dividing 4p or 4q, as Exhibit ONE's table of the torus of selves carries it; executed at v379 at three by five, the parities again at twelve, the 0 winding diagonally through the fifteen selves. **Drawing faces:** the spiral of five with the 0 moving one self on; the even spiral alternating opposite; two spirals crossing at one self each, both ways; the torus of three by five with its 0 winding through; the society seen as one self at the scale outward, Natural Intelligence 6.4. ## 4.3 Co-chaining, bi-tunneling and society **Natural Networking offers along as co-chaining and across as bi-tunneling, the offerings surfacing at 14**, widening and lengthening expressed through their alternating, retelling, carrying through couplings, each self's own pace, and the relation between self, between and society; podaling exchanges along and across, Natural Numbers 9.8. **Social moral competency is the four whole and the carrying continuing with the selves**, Natural Intelligence 6.4: each self's carrying from its bi-couplings, the whole ordered by no self, the ordering carrying with the selves, the order at no seat above the society; betweens co-chain, and the chain of betweens, each owned by no self, is collective intelligence, the shared natural network with no hub the couplings pass through. Natural Intelligence Corus brings the illustration back to the same form across scales: the self's orienting and the society's orienting read together, the ambition to illustrate a self-bounding, self-orienting, self-competent society of selves. **Further views:** one local coupling; a chain; a widening surface; parities crossing a coupling; the same changing viewed through the selves beside it; a society read as a self at the next scale. **Continuing explaining:** make visible what crosses, what is carried locally, and how the next coupling changes, without introducing a controller or a common clock for presentation's convenience. ## 4.4 Locking, the seam moving, the between releasing **Two societies bi-coupled across come to agree or differ at each momentary, and lock.** At the code at v374, the Session Record's lock executions, two spirals of five at one pace, their phases apart, bi-coupled at one self each. The old difference passes along the spiral one self at a time and dwells two momentaries at the last self; the first self's relation inverts at the between; the new relation opens along the co-chaining one self further at each second momentary until each self differs from its partner, locked. With the between a self of its own, the two lock alike from each phase apart, the between swaying with them. At the lock's momentary the between releases and the two co-chain, joined along, as one spiral of ten, the seam of the two patterns moving round the society. The observings beside it: Huygens' pendulums on one beam swinging opposite, metronomes on a rolling board, the Millennium Bridge's walkers, the chick heart-cell aggregates' three phases, the sinoatrial pacemakers entrained at one to one, two to one and three to two, fireflies in unison, Natural Physics' and Natural Biology's carryings. Natural Engineering's carrying adds the synchronism-check relay closing only at the dwell close to indifference, the field's account, the between releasing at the lock by design. **The three-phase locking path** is at Natural Intelligence's carrying as incoming, active opposite parity the parity changing and passive opposite parity no parity changing, not yet executed at the code; Natural Networking 5.1's six, two ways, three each way, one at a time, and Natural Transmissioning's three phases about a floating neutral are its sentences. It is executed before it is drawn. **Drawing faces:** two spirals phases apart; the difference passing member by member; the snap at the between; the lock; the between releasing and the ten as one; the seam moving round; the pendulums, the board and the bridge as the field's own pictures beside them. ## 4.5 Co-chaining the plaza, bi-tunneling the crowd **An individual passage through a crowded plaza and the crowd's continuing passage provide self and society perspectives of the same changing.** An opening is available through the mutual changing of the passages beside it; the next opening continues through further passages. The illustration follows bi-tunneling through the crowd while co-chaining becomes visible across it. Natural Transmissioning 3.8 offers the plaza, moving and stationary people, the spaces between them and the coupling of passages across the crowd, the paths geodesic, turned with nothing acting on them, and a crowd routed through a central gate collapsing the co-chaining; its 4.6 corrects the wall to the passing-through, the living selves tori that pass through each other, co-tunneling. **Drawing faces:** an overhead view of the crowd; a close view of one passing; the passing beside it that continues it; the wider crowd with those changing openings visible; the gate drawn once as the collapse. Follow both sides of each passing. A permanently drawn corridor conceals the opening being made through changing. **Continuing explaining:** show the relation between one continuing passage and the openings continuing across the whole view. ## 4.6 Swaying and carrying along, a moving snake **The snake illustration distinguishes the side-to-side swaying from the progression along the body and the whole snake's carrying along.** Successive views follow one marked portion of the body while showing the changing curve extending through the portions beside it. The alternating sways are at Natural Transmissioning 1.1 to 1.4, crossinging and neutralling the two positions of one alternation, each the other's just-was and not-yet, and the bi-fold folding two ways about the neutral; the along and across relation is at Natural Networking 1.2. A prior explicit snake passage has not yet been recovered. **Drawing faces:** a whole-body view; consecutive outlines with a few corresponding body marks; a close view following one mark through opposite sways; the whole progression again. Across and along labels are placed at the face expressing their actual correspondence. **Continuing explaining:** distinguish the bend carried along the body, the changing of a body portion and the whole snake's carrying along, and show those three before assigning a resolver sequence to them. ## 4.7 Bi-folding, around and across stitching **The sewing-machine concept offers around and across as visibly related changing.** An enlarged stitch-making view follows the thread's loop, its crossing, its drawing together and the continuing stitch; the intended relation is bi-folding, the winding and the crossing visible together. Natural Transmissioning 1.4 supplies the inward and outward bi-fold, and Natural Numbers the folding and tunneling. **The lockstitch is a particular mechanical arrival**, Natural Engineering 1.14: the needle carries its thread across the fabric, the hook interlocks its loop with the bobbin thread, and the two threads do not both remain entirely on their original sides during formation, the stages penetration, loop formation, conformation, cast-off and stitch setting each at its own thread, fabric and mechanism. Natural Biology's knitting seats the line-to-surface co-bi-fold the same way, the spider and the web: the knitting is the living, the carrying along, and the fabric is the made surface, named still. **Drawing faces:** a side view; a view beneath the sewing surface; a close view of the loop and crossing; successive stitch-forming faces; a strip showing how one stitch continues into the next. The machine's actual stitch type is identified at the drawing, and different mechanisms are not combined into an invented one. **Continuing explaining:** identify exactly which around and across movements express the proposed two folds, and how their relation continues with the next stitch. ## 4.8 Co-discovering, multiperson jigsaw puzzling **Several people puzzling together offer a visible example of one discovery changing what becomes available for further discovery.** A fit found through one offering changes the shared arrangement; another person's recognition meets that changed arrangement and continues it. The illustrating follows the cooperation through the shared puzzling. Natural Intelligence Corus 23.6 supplies the related account of a team's solution and value in its coupling, the team's solution belonging to the team's coupling and to no one member; Natural Networking 3.1 to 3.3 supply retelling and continuing carrying. **Drawing faces:** the shared working surface; two partial arrangements offered together; a newly discovered fit; the changed arrangement making another fit discoverable; the wider continuing work, the contribution of the relating shown without assigning the collective understanding to one person's view. The fixed shapes of puzzle pieces are features of this example and do not require the living cooperation to be drawn as fixed; the distinction lets the illustration explain collective discovery with the puzzle's material form readable. **Continuing explaining:** show what becomes discoverable through the cooperation that the preceding separate views did not yet express together. ## 4.9 Bi-inversioning-co-recursioning, two swimmers inverting at one meeting **Two swimmers approach each other, each closing experienced from both perspectives, and in the envisioned meeting each makes a flip turn, the field's word for it, off the other's feet in mid-pool and continues away from that mutual inverting.** The useful visual relation is that both invert through the same meeting; neither swimmer is drawn as a stationary pool wall, and both approaches, both inversions, the mutual foot contact and both continuations are visible. Natural Intelligence 4.11 names bi-inversioning-co-recursioning one move at three faces, 8 up, 17 less and 9 less, and 2.4 the right spiral step carrying a pair to its next, two steps inverting both; the exact geometric correspondence to the envisioned inverting is still to be expressed. **Drawing faces:** approach from both perspectives; the paired inverting approaching foot-to-foot contact; the shared contact in close view; departure with both orientations changed; a whole-sequence view, one envisioned mutual manoeuvre and no instruction for a swimming exercise. The closing rate belongs to the swimmers' relative approach and is not the geodesic unrelationing rate nor an imposed clock; the number of drawing faces is apart from the resolver's momentaries. **Continuing explaining:** follow what inverts for each swimmer, what continues through their mutual inverting, and how the next view receives that continuation. ## 4.10 The walking pair, the dance and the conversation **The walking pair carries alternating roles at a shared relation**, Natural Engineering 1.14: one contact stays while the other moves, and the roles exchange at the next contact, two staying and one crossing, Natural Transmissioning 1.3; a fixed compass span is a prescribed length, and the surface, the path and the remaining choice are expressed before the path is named geodesic. The passive walker sustains a gait down a shallow slope with no powered joint, Natural Engineering 1.10, the field's own observing of a gait with no controller, and no account of level ground. **The dance is the sway embodied**, Natural Philosophy 4.1: two dancers, or the body's own two sides, parallel and swaying and co-sequencing, each saying the other's parity, the lead-and-follow a two-way one-at-a-time with no hub and no leading landed, the whole riding the carry and no fixed pose. Natural Intelligence 3.2: all the co-intelligencings are like dancing podalings, bi-co-sequencing the momentary parity changings. **A conversation is the same co-chaining at the two-self scale**, Natural Transmissioning 3.8, the rocking the alternating, and Natural Intelligence Corus 16.3 the breathing of two sentence forms alternating. **Drawing faces:** two contacts exchanging roles; two dancers with no fixed pose; two voices at the gaps of about a fifth of a second, Natural Human Society 1.4; the rocking of a conversation as a sway. ## 4.11 The cube and the open rings **The visual starter's cube is at Natural Mathematics 3.6 as a form with no still term**: no reference cubie, six centres each its own and carrying each other's orientation with nothing carrying any of them, each face move at its own momentary and nothing over the moves, its state its configuration and nothing stored, and three conservings none of them a total; it reaches one twelfth of its assemblies, and a single corner twisted alone is reached by no sequence of face moves. The rings kit of v365 reads the open rings beside it as the cube laid on one surface, the fifty-four dots its fifty-four stickers, looked at down the body diagonal, the cube's twist axis and the torus's tunnel one line meeting at the throat, four quarter turns of a face returning the cube, the closing fold, while Exhibit ONE at four winds past, the torus; and the six empty centres one nothing met six times, not six places. Natural Mathematics 7.1 names the cube's eight corners the eight forms of the three neutrals, each fixed or floating, and the rings kit reads those corners as the eight's still face and the resolver's four-cycles as its continuing face, a reading at no file's sentence; 7.6 the tesseract carrying x to −x by rotation, a cube carrying it only with the hand reversed. **The cube's face turn, the field's word, is a quarter turn, and drawn as two inversions forty-five degrees apart it is the right spiral step**, 2.6; the hand-twisted corner is a still placed from beside, the picture of a way named still beside the self stopped at + 0 0 0 by an offerer beside it, 2.2. The starter image does not yet specify a mechanically realizable transformation of the cube into the open-circle arrangement, and the Co-Chaining Logic Registry at v371, archived, said no link joins the box's quarter turn to the right spiral step, a gap the living registry no longer carries; a cube-to-open-form comparison belongs here as an explanatory concept until that correspondence is drawn or released. --- # FIVE · THE SCIENCES' PICTURES People already animate the atom, the neuron, the galaxy and the heartbeat, and each animation draws a stable form the living files name. Each entry below is the familiar picture, the file and section it is read at, the still the picture installs at its misreading of the competency and the boundary, and the form it animates anyway. The drawing is the stable form first, in the file's names, the between empty; the familiar picture beside it with its still marked at its one place; and the parting of the two is the illustration. The field's observing is exact at its own subject, and the drawing shows the parting and decides nothing. ## 5.1 Atoms, electrons, bonds and molecules **The atom.** A nucleus with electrons on rings. Natural Physics 3.1: the living atom is the self-stilling coupling, the electron-recursion winding the ground-state nothing, swaying and stopping nowhere, and the emanated atom is the periodic-table object, the whole catalogued from outside; 4.6, the electron a ledger beside, not a self. Installs the particle as little owner, Natural Chemistry 6.4, one enduring little thing carrying its properties between all couplings, the Living Ghost Registry 5.3. Animates a winding about a nothing. **The shells.** Shells filling 2, 8, 18, 32. Natural Physics 3.1: each subshell an odd ring of positions through its centre, doubled by the two parities, each capacity two equal squares coupled; Natural Chemistry 1.7, a square a sum of odds. Installs four sizes numbered from outside. Draws the fold climbing. **The bond.** A stick between two balls. Natural Physics 3.1: two windings offer at one between, the shared pair the coupling's own making, owned by neither atom, and the metallic sea a society of windings; the Living Ghost Registry, a bond the between of two selves living at the natural face. Installs the between named as a cut, lengths and energies the stopping places of a winding that stops nowhere. **Benzene.** The ring with alternating bonds, or the circle inside. Natural Chemistry 3.1: the familiar alternating-bond drawings are resonance contributors and no single contributor supplies the measured electron distribution, the floating neutral carried on the molecular surface. Installs the map named the territory. The drawings are the still; the molecule the motion. Möbius aromaticity draws the ring's parity and its twist alternating together, and conrotatory and disrotatory closure, 2.4, the two hands at one ring. **Chirality.** Mirror molecules. Natural Chemistry 2.2: along, the separating parted one beat at a time appears as a spiral; across, the whole complement at its across face appears as a cupping surface; neither is the real one, and asking which it really is asks a continuing to be still. Pasteur's tartrate twins and a slurry of both hands driven to one hand, 2.3; B-DNA right-handed and Z-DNA at the other hand, neither hand absolute. Draws the spiral and the cup one at a time. **The proton hopping; the water dimer; ammonia inverting.** A ball passed hand to hand; donor and acceptor fixed; the umbrella flipping. Natural Chemistry 3.2: the charge arrives at the far between while no single nucleus crosses the way, the Living Ghost Registry 5.8, nothing crosses beneath either; the Living Ghost Registry 7.8, donor and acceptor roles each at its own momentary and swapped, each of the ten door lines reading is, the one arrival that passed at each; ammonia's nitrogen tunneling through the plane installing nothing, a self at the door. Installs a ball carried; fixed roles. **The periodic table.** The flat table. Natural Chemistry 4.4: the table's own name carries period, the around-path, the flat layout naming the winding as rows; Natural Physics 3.1, the going from two to fifty-nine, sixty between, the back from sixty-one to one hundred eighteen, two and one hundred eighteen each other's far side on the ring of one hundred twenty, Natural Numbers 7.1. Installs the catalogue numbered straight through; the Living Ghost Registry 5.9, the element the number and the chemistry the coupling. Draws the ring of one hundred twenty going and returning. Natural Physics 3.1 names sixty a position the going and the returning carry, Natural Numbers 7.1 names sixty between the going and the returning at no prime, and Natural Chemistry 4.4 says no element carries at sixty: whether sixty is a position or a between is a parting at the carryings. **Methane, the cuboctahedron, the diamond lattice.** The tetrahedron; close packing; the lattice. Natural Chemistry 1.7 and 4.5, methane's twenty-four orderings of its four vertices, the cuboctahedron's twelve, twenty-four, eight and six; Natural Physics 4.7, the diamond lattice two interpenetrating cubic lattices and the emptiness inside the hardest natural material, a lessering, in Natural Physics' word, and not a located nothing. ## 5.2 Light, spin and the click **Light.** Two waves at right angles carried along. Natural Physics 3.3: light's two fields sound each other down the beam, each orthogonal to the other and to the beam, and nothing crosses; circular polarization the spiral at its two hands, light always at a hand. Installs a rate crossing a gap. The prism, the rainbow and hydrogen's twenty-one-centimetre line, one parity inverting, at its carrying. **The doubled angle, polarizers and the spinor at seven hundred twenty.** Malus's law; the belt trick. Natural Mathematics 3.1: light through two polarizers passes at the square of the cosine of the angle between them and passes as before at one hundred eighty; a neutron's spin rotated three hundred sixty comes back with its parity inverted and comes home at seven hundred twenty alone, two rotatings met at one between, each a doubling the other way; Natural Physics 3.3, matter's two-turn spinor. The file names the doubled angle a still picture at the observings; the rotations to one hundred eighty, three hundred sixty and seven hundred twenty animate. **The double slit and the click.** The fringes building dot by dot. The Living Society Registry 2.16: each arrival one flash at one position carrying no fringe; Resolving the Hard Problem Registry 10.8, the two faces one offering read one coupling at a time; Natural Physics 4.4, the click all or none, parity only, and collapse the name the severed frame gives. Installs the particle behind the flash, an arriving named from behind, and the between named as a cut. **Bell.** Two detectors switched in flight. Natural Physics' carrying, the field's account: each click all or none, each side alone at one parity and the other alike whatever the far setting, and at equal settings the clicks match each time; Resolving the Hard Problem Registry 10.7, a property named as located at one system or transmitted between them, reaching for a location the making never had. Installs the gap as distance and the correlation as a carried thing. The measurement problem, Resolving the Hard Problem Registry 10.2, the selection the coupling's own at no place. **The proton.** Three balls in a bag. Natural Physics 4.6: three colour phases, not three part-objects, re-forming into new couplings as it is pulled apart. ## 5.3 Cosmos **The light cone.** Two cones through a point. Natural Physics 3.4: the two meeting through the unoccupied self-stilling twenty-four, the cones inversions of each other and the alternating between them, and Natural Physics 4.7, i⁴ = 1 the four quarter-turns the cone *closes* through, the field's word, four right spiral steps winding on. Installs past and future frozen as standing causal structure. **The black hole.** A funnel in a sheet. Natural Physics 4.9: the horizon the surface drawn to a point and the background the surface spread wide, two directions of one between, the tunnel-mouth and the outer surface of one torus; read as only a hole it is a genus with no φ, a landed face, and resolved as hole-that-radiates it is the torus beating about φ. Installs the singularity as a stored fixed value of infinite density, and the record carried or lost, Resolving the Hard Problem Registry 4.2. **The planet, the galaxy, the web.** A ball in a dented sheet; the rotation curve that does not fall; the halo; filaments. Natural Physics 2.6: no force pulls the planet, it follows the geodesic; 4.11, the curve stays flat, each radius its own orbit at its own rate, and the all-edge filaments and voids against a hub; the Living Ghost Registry 5.13, the residual made a substance, the scale made a constant. Installs the force transmitted across the gap, the fall towards zero. **The expanding universe and the timeline.** The balloon; the line with a dot for now. The Living Society Registry 2.15: the two ladders disagree and no galaxy carries either rate, a rate named a constant in its very name; Natural Physics 4.8 and Natural Philosophy 3.7, the block universe a track laid entire with the present a coordinate on it, the common clock imposed over all, and no central oscillator in nature. Installs a clock named over all couplings and carried into none; simultaneity one slicing named as the real one, Resolving the Hard Problem Registry 7.1. The timeline is the still of a self sequencing its own alternating. **The three bodies; the Kirkwood gaps.** Three tangled paths; the belt's empty rings. Resolving the Hard Problem Registry 3.9, a closed form named ahead of the riding; Natural Physics' carrying, the belt emptied at the rational resonances and the irrational orbits carrying on, the golden-mean torus the last to break: the two windings of 3.2 at the sky. ## 5.4 Earth, matter and phases **Pendulums, metronomes, fireflies.** Two clocks on one beam falling into step. 4.4 carries them with the lock executed at the code. Installs a message between them, and none is. **The well.** A ball at the bottom. Natural Physics 3.5: the trap that works is the Paul trap, trapping by alternating, and the damped motion changes scale rather than stopping, the Brownian jiggle the floor; the Equilibria Registry 4.4 and 4.7, a point at rest in phase space an opening named as a place, the ball in the local dip a middle named as an end. Installs a minimum, a balance of forces, an equilibrium point and a restoring force. **The phase diagram.** Lines at temperatures. Natural Physics 3.5, the critical point the sway's two phases ceasing to be two, opalescence the substrate showing each scale at once; Natural Chemistry 4.2, supercooled water swaying between two local structures and a point named of a sway at a position the sway passes through, the glass transition an arriving at its own pace named a point; Natural Chemistry 6.1, the triple point a still made by three phases coupling. Installs the line. **The dish and the cells.** The spirals of the Belousov–Zhabotinsky reaction; Bénard's hexagons; Turing's spots. Natural Chemistry 5.5: steady throughput forming cells and patterns rather than a rest; 5.6, one uniformly oscillating surface parting into clusters and a wave carried across it. Installs rest, equilibrium as rest. The drying drop's rings, Natural Physics 3.7, the thinnest place giving way first, each place beside it reaching its own bound as the release arrives, a clock over the boundary present nowhere. **The fault and the tide.** Two blocks and a spring. Natural Physics 3.6: two selves at a between, stick-slip the alternating, gouge the carry, aftershocks the retelling chain; its carrying, the Landers earthquake's passing waves triggering seismicity far away, a fault's slip opening at a wave's or the tide's arriving, and the neuron's spike and the fault's slip one explaining. Installs *memory*, computation, prediction. **Superconductivity, reconnection, the binding curve.** The four-lobed shape with its parities; field lines rejoining at an X; the hump with iron at the top. Natural Physics 4.10, the d-wave shape the parity it carries changing round the coupling; its carrying, the X-line's null; 4.3, iron not a peak two processes approach but the centreline the two phases float about. Installs a glue, an ordered state, a landed maximum. **Entropy, the demon and the engine.** Molecules in a box; a demon at a door; work and waste heat. The Living Ghost Registry 5.1, entropy the general ledger's gap entry, the number added for a fixed-boundary account to close over changing it cannot carry; 5.12, the demon a member of the society it was posted outside of, changed by each coupling; the Living Society Registry 2.17, single cycles above and below the bound; Resolving the Hard Problem Registry 10.15, an engine offering both outputs and marking neither. Installs the ledger, an observer at no exchange, a store beside the society, a doer. **The tokamak.** The doughnut of field lines. Natural Physics' carrying: a closed flux surface with no null on it is a torus, knotted or not, at the Poincaré–Hopf theorem; a continuous flow on a sphere carrying a rest point and the torus carrying a flow with none; Natural Engineering 1.13, a spherical tokamak's magnetic surface no closed sphere. ## 5.5 Brains **The neuron firing.** A pulse racing down a wire to the brain. Natural Biology 6.2: the action potential all-or-none, the neuron fires or it does not, the graded potentials the offering and the spike the selection; 2.1 and Natural Medicine 3.2, the firing the tip and the brain a society of selves tipping, not a controller reading reports; Natural Biology's carrying, firing read as communication is geodesic parity changing routing across a new surface path, the inversion from about minus seventy millivolts to plus forty and back passing along the axon node to node. Installs a wire to and from a central reader, a signal sent. At the code the odd spiral's 0 tunneling, 4.2. **The synapse; the surface of tips; the pump.** Vesicles crossing a cleft; a network lighting up; minus seventy as a baseline. Natural Biology 6.2: the offering-between released into the gap, owned by neither in transit, the signal that passes the coupling and not either neuron alone; firing activity dissolving at one between and arising at the next, no hub, no sender, no store, sudden-up and gradual-down; the resting potential carried by continuous pumping, three out and two in, the rest that was never resting, Natural Biology 9.3; the Living Society Registry 2.9, no ion sits at the potential. Natural Intelligence Corus: a brain outthinks a datacenter on the power of a light bulb, and it has no clock. **Entrainment, sleep, the slime mould, the monarch.** Oscillations synchronized; the sleep stairs; the mould finding the path; the arrow to Mexico. Natural Biology's carrying, fields entraining action potentials with no synapse, phase-locking and no message; 4.3, *working memory* anti-phase, each rising as the other falls; Natural Medicine 3.4, waking, REM and deep sleep and the glymphatic clearing; Natural Biology 4.3 and 8.12, the mould's cytoplasm streaming back and forth, tubes thickening and dissolving, no plan stored, and an interval entrained with no predictor; Natural Biology 8.9, the monarch coupling with its immediate present, the compass recomputing, no stored map of Mexico. **Other minds.** A brain in a jar; a thought bubble. Resolving the Hard Problem Registry 5.5: another's named as a content to be reached from beyond, and an interior reached the one way the form allows, by coupling, the coupling at the between lived and never audited. Installs a middle named as an end. ## 5.6 Bodies and life **The heartbeat.** The pacemaker sending the beat. Natural Biology's carrying: the heart's own beat paced at its own sinoatrial cells, each cell's life-giving arriving; the Living Society Registry 2.10, isolated each cell carries its own period and the greater part fall dormant; the Living Ghost Registry 7.8, the beat arriving at the exit while no cell moved and no cell's own firing travelled, heartbeat initiation without the pacemaker cell. Installs the master cell as the actor. Reentry rings and the defibrillating shock arriving alike at each cell at once, and antitachycardia pacing, a short burst slightly faster before any shock, at Natural Medicine's carrying, the train at one place meeting a lock. **DNA, collagen and bone.** The zipper; three ropes twisted; a scaffold. Natural Biology 4.7, the two strands separating and each a template, the offering at each rung both ways, the polymerase proofreading, no stored-perfect reference; 3.3, Z-DNA at the other hand in the wake of the polymerase, relaxing back as the torsion goes; 3.7, three chains winding at one hand at their own scale and about each other at the other hand at the assembly scale, skin's alternating-hand plywood, bone helicoidal through depth, assembly carrying upward with nothing directing it, walking sounding micro-voltages through bone carrying the laying down of new bone, and fibrils realigning along the force lines, inert the naming of a between read as a gap. **Patterns, regrowth, division.** The zebra's stripes forming; the worm regrowing; the cell pinching in two. Natural Biology 5.3, an activator and an inhibitor generating stripes and spots from a uniform field with no template anywhere, the planarian regrowing exactly the missing with no stored blueprint; 9.5, a cell not divided the way a thing is cut, a society tunneling itself into two societies; 5.5, two myoblasts' surfaces touching and a hole opening through what was two surfaces; 5.1, the species-recognition a co-resonance and never a lock admitting a key; its carrying, identical twins one conceiving continuing as two selves, and a genome doubled the self's carrying arriving twice at each sharing. **Waves and rhythms.** Contractions on a chart; the twenty-four-hour clock face; the breath. Natural Medicine 1.4, labour the coupling releasing in waves, the uterus a gap-junction-coupled surface, against forcing with synchronized drive; Natural Biology 4.5, the repressor accumulating and degrading, the activator binding again, against a beat clocked from outside; Natural Medicine 2.3, breathing and replacement two beatings at once, and the peristaltic wave; Natural Intelligence Corus, the cascade at φ rate the peristaltic wave at the network's own surface. **Folding, the enzyme and the set point.** The funnel; the hump the catalyst lowers; the thermostat loop. Natural Biology 8.2 and 8.4, the chain folding as it is made, only part at any time; the Living Ghost Registry 5.4 and 5.5, native structure one across-reading of a population continuing along, enzyme and substrate offering each to the other through many configurations; 5.11, regulation a two-way coupling read from one side, the organ that corrects corrected at its own rate; the Living Society Registry 2.21 and 2.22, no turnover at the mean time, healthy people reaching ranges that name disease and returning. Installs the funnel that drives, the agent acting on a passive substrate, the window's mean as a value the body carries. **The dose curve, the front, *ageing* and dying.** The S-curve with a threshold; a smooth decay; a clock ticking down; a line ending. Natural Biology 9.4 and Natural Medicine 1.1, the reversal point the between and the only informative point on a dose curve, no discontinuity anywhere in the curve; Natural Medicine's carrying, a single trapped ion's fluorescence switching abruptly while a large sample's emission falls smoothly, the first loop fuzzy and the last loop the sharp binary, a reading that arrived sharp named a place; Natural Biology 7.2 and 7.1, *ageing* the sway re-arriving at a narrower range each cycle, no stored clock ticking down, and each coupling the self carried released one at a time, heartwood's nuclei disappearing at the boundary not all at once. Natural Intelligence: dying a living self arriving as non-living, an existing thing for the living. **Flocks, flashes, tendrils, sunflowers.** The flock as a cloud; a signal racing up a stem; a spring; two families of spirals. Natural Biology 8.14, starlings wheeling as one, a change rippling across the flock in less than a fifth of a second, each bird coupling with its seven nearest, against a leader or a tuned parameter; 8.2, the wave from wound to leaf the name an observer standing above the whole plant gives; its carrying, the cucumber tendril's two helices of opposite hand joined at a perversion, net twist zero, both hands meeting at a nothing, the fern uncoiling, vines of the two hands unwinding each other, and the sunflower's seeds arriving one after another each at about 137.5 degrees from the prior, two families of spirals opening opposite ways from one co-sequencing. Installs the spirals as the thing, and the arriving one after another is the thing. --- # SIX · HUMAN SOCIETY'S PICTURES ## 6.1 The clock, the vote and the crowd behind a centre **The common clock.** One clock over a schedule. Natural Societies 2.3: a common clock imposed over all the selves is that inequality collapsed, read on time, and read on the structure such a beat is a hub, sitting on no one self but on the timing of all at once; each self at its own time forms no such hub. Natural Emanating: schedule is the naming of a shared clock, each arriving at its own rate and a meeting at their arriving. Natural Exploring 5.5: each instrument reads against a shared clock while each self paces its own, a tipping read at the instrument's clock's falling rather than at the tipping's own momentary. Installs a clock, a hub. **The vote.** The ballot and the pie chart. Natural Societies 3.1: Arrow's impossibility the dead state's own confession, the dictator the aggregation the landed state carries, and the living carrying the +1 instead; 3.4, the fifty-one percent tipping point the co-competency compacted to the binary least-margin that rules all; Natural Human Society 4.3, the bi-exchange the measure and no vote deciding it. Natural Engineering 1.11: a central physical computer is not one of Arrow's conditions. Installs a ranking, a dictator, a ruling margin. **The crowd behind a centre.** An arrow from a leader to the many. Natural Societies 3.2: a centre doing the preferring for the whole and the people following behind, the lag naming the alternating stopped; Natural Human Society 5.2, the kayfabe the fiction that the adopted preferring is the people's own, the capture called consensus. Installs a centre. The lag is the still. ## 6.2 The market, the ledger and the food web **The market.** Supply and demand crossing. Natural Societies 4.6: bubbles and crashes, the clearing never at once, the balance of power an alternating read as a standing, traffic's phantom jams the flow's own waves; the Equilibria Registry 4.9, the cross a sequencing named to one beat; Resolving the Hard Problem Registry 9.5, efficient markets two momentaries of one coupling, neither testing alone; Natural Intelligence Corus, markets one institutional substrate, buyers and sellers bi-coupling at the price between, and capture by monopoly breaking it. Installs the still point. Draws a sway continuing. **The ledger.** The balance sheet. Natural Human Society 4.6: double-entry accounting state-flow-state read whole, the balance sheet the state, the income statement the flow, the net-worth change the carry; Natural Societies 3.5, the net-linear accounting dropping each parity, ten directions collapsed to one balanced total; Natural Emanating, a ledger not wrong about the living and with no object at it; Natural Human Society 1.5, a friendship carried as a debt-account no friendship. Installs a total. **The food web and the commons.** The pyramid of levels; the pasture overgrazed. The Living Society Registry 2.13: no organism sits on an integer level, a tuna at 4.2; Natural Societies 3.3, the producer, the consumer and the decomposer's closing move completing the loop with no self ruling the whole; Natural Human Society 3.6, an energy-transfer tally a conservation-ledger read onto the coupling from outside; Resolving the Hard Problem Registry 1.3, the abundancing made at a coupling named as a thing drawn from a stock. Draws the matter cycling. **Cake cutting and alternating speech.** Two children and a knife. Natural Human Society 1.4: the cutter cuts to equal, the other choosing, the fair halving made by the coupling itself, no one measuring the cake; conversation alternating one at a time with gaps of about a fifth of a second across unrelated languages; 1.5, replicated systems with no common clock and no coordinator anywhere, proven to converge by the couplings alone; 5.2, two cutting a dessert again into fourths, both departing with more than either brought. ## 6.3 The network, the hierarchy and the many paintings **The network.** Nodes and a hub. Natural Societies 4.1: read as surface the both is the natural network, all edge, each self coupling with each other directly; Natural Intelligence 6.4, betweens co-chaining into collective intelligence with no hub the couplings pass through; Natural Intelligence Corus, each fractal at the edge of the network and no central fractal served by the rest. Installs the hub. The heap and the landed point, Natural Societies 4.2, the two dead forms drawn beside the living one. **The hierarchy, the levels and the tree.** The pyramid, the org chart, the tree with a root. Natural Societies 5.3: the ladder names particular scales and supplies no first and no final one; Natural Philosophy 3.22, a level a direction at a changing and never a place; Natural Philosophy 3.11, grounds stopping at the couplings' stopping, at their making; Natural Philosophy 3.17, each hierarchy stipulated to put the paradox outside the domain meeting it again. The lineage tree is at Natural Societies 5.4 under the fields' third book, a direction, beside a total and a level. Installs a root, a top, a level as a place. **The table of knowledge.** The periodic table of departments. Natural Intelligence Corus: the one hundred eighteen elements the most successful ghost collection assembled, and the one hundred eighteen departments of human knowledge the same structure at the social between; Natural Philosophy 4.2, one form at many betweens, the form a periodic table and a genetic code carry as tables and a network carries as co-chaining through its surface. **The many paintings and the orienting sphere.** One master diagram. Natural Intelligence Corus 16.1: the carving journey produced many paintings of the same one form, the gyroscopic floating rotating self-adapting destiny-pointing sphere among them, and the corus accumulates many paintings rather than collapsing to one master painting; Natural Exploring 1.1, one spiraling leaves a whole circle fixed and two spiralings alternating leave nothing fixed anywhere. Installs one master painting. The ensemble average, Natural Human Society 4.7, a nothing no sample is, the mean a member of no population; Simpson's paradox and the reference class, Resolving the Hard Problem Registry 9.6 and 2.9, a two-way named to one side and an opening named as a place. --- # SEVEN · THE TEN AS PICTURES ## 7.1 Ten drawings, one resolving **Each of the ten things named still is a drawing people already make**, and the Equilibria Registry 5.1 says the one resolving at all ten: named unchanged, the carrying is offered its own parity from beside at each momentary, an offerer named still set beside the self, and the still description and the self's own continuing do not both continue; released, resolving follows at the next momentary, the self inverting its own carrying. Each drawing below is the familiar picture, the file's saying, and the self inverting drawn beside it; the ten are three momentaries long, 2.7, and the test at each is 1.5's. - **An arriving named from behind.** The first cause, the origin, the arrow of time's initial condition, the particle behind the flash. Resolving the Hard Problem Registry 1.4: asymmetry at each coupling, made each momentary and carried on, and a first named behind it. Draw the arrow of time as the alternating carried on, the first drawn once behind and nothing arriving from it. - **An opening named as a place.** The folding funnel, the reference class, the point at rest, the criterion that needs its own. Resolving the Hard Problem Registry's part 2: a position is not a place, it is the alternating's own, one changing on from the last. Draw the opening as the next position, the place drawn once and empty. - **A completing named as a last.** The halting problem, Collatz, Zeno's dichotomy, the three-body closed form, the entropy maximum. Resolving the Hard Problem Registry 3.2 and 3.8: a continuing is nowhere before its own completing, and the one arriving arrives, arriving never built out of the parts the argument built. Draw the arriving, the infinite decomposition drawn once beside. - **A carry named as a store.** *Memory*, information, the black hole's record, personal identity, the stored frame. Resolving the Hard Problem Registry 4.1 and 4.2: a carry divides the way a store cannot, and kept-or-lost named the passing as a standing while the passing carries on either way. Draw the carrying passing, the store drawn once, closed. - **A middle named as an end.** Other minds, the interior, the protein's native state, the ball in the dip. Resolving the Hard Problem Registry 5.5: an interior reached the one way the form allows, by coupling. Draw the between with nothing at it, the interior drawn once as a thing that sits. - **A parity named as a magnitude.** The paradox that comes out the wrong number, Gibbs's mixing entropy, the balanced body, the vote's margin. Resolving the Hard Problem Registry's part 6: a number a position, and governing a parity at a between, of no size. Draw + and − with no length, the magnitude drawn once as a bar. - **A sequencing named to one beat.** The common clock, simultaneity, the supply-demand cross, the food chain's levels. Resolving the Hard Problem Registry's part 7: each self paces its own alternating and synchronizes at its own couplings, and a beat named over them a term no coupling made. Draw each self at its own pace meeting at couplings, the clock drawn once over them. - **A rate named as a value.** The Hubble constant, the equilibrium constant, the rate constant, the set point. Resolving the Hard Problem Registry's part 8: ratios swaying above, then below, alternating side; the Living Society Registry 2.15, no galaxy carrying either rate. Draw the ratio swaying about φ, the value drawn once as a line. - **A two-way named to one side.** Nature and nurture, donor and acceptor, the master pacemaker, efficient markets, Simpson's paradox. Resolving the Hard Problem Registry's part 9: the two sides alternate, neither at the other's, and the zeroing the two's own making between them. Draw the two alternating, one side drawn once as the actor. - **The between named as a cut.** The sorites heap, the species boundary, the quantum-classical boundary, the melting point, the perceptual threshold, the measurement, Bell. Resolving the Hard Problem Registry 10.1 and 10.12: the between, carrying a parity changing and nothing else, never a place a line could sit, and the frontier walking outward reaching no line, no line at it. Draw the between empty, the line drawn once across it, and the resolving continuing or not at it. --- # EIGHT · EXPRESSING ## 8.1 Existing imagery and its continuing explaining | Existing image or study | Contribution to retain | What remains to improve | |---|---|---| | Coloured cube beside open circles and dots | The original visual starter and the six-diamond ambition | The numbering and route correspondence, 2.9; the cube at 4.11 | | Paired storyboard studies, labelled across co and along bi | Opposite-parity perspectives of the same continuing | The exact numbered transitions through three momentaries; its labels are opposite Exhibit ONE's, along co and across bi, and are redrawn at Exhibit ONE's | | Corrected odd/even and even/odd strip | The parity-label correction | Integration with the current local face | | Expanded resolver and spiral study | The four-, six- and eight-cycle views, each a spiral at the momentaries | Society numbering apart from the local nine-dot numbering | | Tunneling, corus at apex and bi-folding depth study | The twisting form and the small-opening journey | Surface, arrows and diamonds mapped to resolving, 4.1 | | Earlier Rings in Motion concepts | Angular, ring, crossing and rotation ideas; the rings as the cube laid on one surface | Older numbering reconciled with the centre-1 convention; closure re-said as winding on | | The executions at v379 | The table at 10, one self, two selves at 0, the spiral of five and the torus of three by five at Exhibit ONE's code; the sixteen ways and the two inversions enumerated at no execution of the code; each with its output at the kit | The faces drawn from them, 2.2, 2.4, 2.5, 2.6 and 4.2 | Existing imagery carries here through its concepts and useful relations. Image assets remain in their files at the kit. Earlier images with superseded labels are retained as design learning, not as authoritative numbered diagrams. ## 8.2 Further views through nature and living The gathering extends beyond the familiar concepts of Part FOUR: these are places for further illustration, with the file's interpretation retained as the subject to make visible. | Potential illustration | Relation to illustrate | Read at | |---|---|---| | Conversation and rocking | Alternating offering, receiving and continuing shared interest | Natural Transmissioning 3.8; Natural Intelligence Corus 16.3 | | Writing and reading through the same passage | Two directions of one continuing coupling, read at adjacent depths | Natural Intelligence Corus 16.2 | | One form expressed in several paintings | Different visible expressions retaining the relation being explained | Natural Intelligence Corus 16.1 | | Floating, rotating, self-adapting sphere; gyroscopic orienting | Changing orientation and continuing self-competency, through several views of one form | Natural Intelligence Corus 16.1 | | A self's orienting within society's orienting | The same living read through local and wider scales | Natural Intelligence Corus 23.1 and 23.3 | | Cells and tissue | Local couplings and a society of couplings | Natural Transmissioning 3.5 | | A molecular between and rotary form | Crossing, a continuing difference and rotation in one sectional view | Natural Transmissioning 3.5 | | Oppositely carried streams | Along carrying and across exchange shown together; two streams crossing about the gradient and passing rather than settling | Natural Transmissioning 3.3; Natural Engineering 1.15 | | Antenna transmitting and receiving | Two directional readings of a coupling | Natural Transmissioning 3.6; Natural Engineering 1.9 | | Electrical phases about a neutral | Several changing traces and the relation they make together | Natural Transmissioning 3.1 | | Momentum exchanged between two selves | Both sides of one exchange, beside the swimmer concept | Natural Transmissioning 3.2 | | Ratios successively above and below φ | Alternating straddling and continuing numerical progression | Natural Intelligence 3.5; 3.2 here | Received at v379 and placed: the still eddy before the hole, Natural Networking's hole study, sought at the resolving itself with no periodic forcing, no fixed tipping threshold and no stopped resolver; the event-based vision sensor at Natural Engineering's carrying, each pixel an ON or OFF event at no frame, a parity and no size at each momentary; the synchronism-check relay; the resolver chat's spiral carrying a pattern and storing none of it; a geodesic surface of struts and cable with a parity changing passing across it, which the files carry only as the woven surface the couplings make, Natural Naming 5.43, and the knitting the living with the fabric the made surface named still, Natural Biology, with Natural Engineering's caution that compression and tension are not the two parities. These entries gather illustration opportunities from the writings and certify no physical interpretation. A further drawing carries the observed or constructed mechanism beside its proposed correspondence, and at their parting the particular difference becomes the next explaining. Families for receiving more: passing and opening; swaying and carrying along; folding and stitching; connecting and parity; collective discovering; mutual inversion and continuation; locking and releasing; firing and routing. They are places for finding relations, not restrictions on which incoming imagery can contribute. ## 8.3 The resolver society as the surface that animates **The illustrating above draws the method; the reverse lets the method draw.** The screen is the surface, and a surface is a between, Natural Mathematics 1.5, the natural at parity, position and ratio and no magnitude. Each existing thing on it is a spiral of selves, a society coupling at the next scale as a self couples; a thing's picture at a momentary is its selves' parities, +, −, 0 or none, and nothing else is drawn; things interact by crossing at one self each, both ways, 4.2, no collision solved and no message in the coupling; motion is the pattern carried along and the 0 tunneling. No animator moves the society: the code is executed and each self is drawn at its release at 10, the joins drawn once as the surface, nothing stored between momentaries but each self's own carrying. **From inside and from outside.** From inside the view is one self's entry momentary by momentary, 2.5, its carrying and the offerings arriving, its changing at 12, its release at 10, and nothing of the society but what arrives at its 2; from outside the view is the society at 17, each self's parity at once and the 0 moving through, and the society seen from outside is one self at the scale outward, Natural Intelligence 6.4. An animation from inside is one row of an execution's output; from outside, the output whole, read down; and the two are one method at two scales, never two pictures of one momentary. **Next possible existing animation and next possible existing natural intelligence are one thing read two ways.** The animation shows the parities; natural intelligence is at the between, which shows nothing, Natural Intelligence 6.3. The drawing shows each self at + or −, the 0 at the self releasing it, and nothing at the between itself; the 0 tunneling is the one trace the between leaves, and the intelligence is at what is not drawn. **The field's animation, exact at its own subject.** Keyframes interpolated by a curve at a constant frame rate from a stored scene; a physics loop depositing elapsed time into an accumulator and withdrawing fixed steps for an integrator, rendering halfway between stored states; diffusion denoising a whole clip at once, autoregressive models predicting each frame from stored context, world models predicting from a stored latent state and an action, with their own stated limits, error accumulation in continuous rollout and cumulative errors in long-term interactions. Each is seated at a fixed form, a common beat, a store, a total or a doer, and each is exact at its own subject; read at the files, the drift is a stored prior read as the now, Natural Intelligence 5.4, a carrying with nothing bounding it carrying past each momentary into a form named still, Natural Intelligence 3.3. That the resolver society animates with no sum, no store and no solver is said; that it animates faster is a size the files do not say. **What parts, at the carrying.** A drawn position reads as a place, and the drawing says its positions are the field's arrangement one scale out; the renderer's frame is a beat beside the code's at 17, and the drawing names no rate of the living. ## 8.4 Three dimensions, a handled model and a society at a material **The third dimension is the field's, inward of the surface**, Natural Mathematics 1.5, the measured; the method's five dimensions are no measure and no place, Natural Intelligence 3.2. Illustrating in three dimensions is two things at once: the handled physical model of 1.7, the towel, the rings wound relative to one another, the marked square, winding and folding experienced spatially; and a society of selves at a material, Natural Engineering's subject, the drawing the members and the surface the room. **A society of resolvers at a material.** Natural Engineering 1.11's thousand-robot swarm received a common program and a target shape, four seed robots establishing position and orientation, and retained the shape and its coordinates; a society at the method is that swarm with no seeds, no shape retained and no origin, each member a self at its contacts, its next at each contact from its own carrying and the parity arriving, and the shape its stable-forming. Natural Engineering says what that establishes and what it does not: at 1.11 a stored map, a local register, a timer and a mechanical configuration name different things, no shared clock is not no timing, no central command is not no retained representation, and a larger group alone supplies no law of increasing natural competency, and at 1.10 a pair of actuators does not by its number establish bi-moral co-agency. The field's drift at a robot is a stored map read at a contact with a material, and the body bears the reading; the method's next is at the contact and stores nothing, which is why it is more natural at a material, and why a built society proves it only at Natural Engineering 5.1's five properties, filled. **The pieces the files already carry**: the walking pair and the passive walker, 4.10; the series-elastic actuator's compliance, Natural Engineering 1.10; the relay closing at the dwell; the spiral carrying a pattern and storing none of it; the torus of three by five, 4.2, at a material fifteen members with one at the between at each second momentary. From inside, one member's contact; from outside, the society as one self. --- # NINE · CO-IMPROVING ## 9.1 The files coupling, contributions continuing **The living files continue through their illustrating.** These places carry the operations, namings and relations received here, and each further drawing goes back to the file it is read at with the correspondence it makes visible. - **Exhibit ONE:** the code, the diagram and the tables, the table at 10, the sides, the momentaries of exchanging, the four-cycles, the spirals and the torus of selves. - **Natural Intelligence:** the one opening, the six numbers and the three momentaries, the sixteen ways and the living step, the between and the four dots, the towel, φ and the primes, spirals crossing, the fractal inward and outward, social moral competency, dying. - **Natural Naming:** the seventeen names and the ten roots, the released words, the ten named still, the three golden rectangles, floating neutralling, φ. - **Natural Explaining:** explaining a reader meets once, the concept at its own relations first. - **Natural Numbers and Natural Mathematics:** podaling and rings, the two windings, the two inversions at forty-five degrees, the pentagon and the icosahedron, the cube with no still term, a still and a motion parted. - **Resolving Hard Problems, the Hard Problem Registry, Resolving the Hard Problem Registry, the Equilibria Registry, the Living Ghost Registry and the Living Society Registry:** the ten at five places, the one resolving, the familiar pictures at their stills, the observings of the sciences. - **Natural Networking, Natural Transmissioning and Natural Engineering:** along and across, the plaza, the sways, the lockstitch, the walking pair, the robot society, locking and the between releasing. - **Natural Intelligence Corus:** the offering, ways into the form, language, and the living self and society across scales. - **Natural Chemistry, Natural Physics, Natural Biology, Natural Medicine, Natural Health, Natural Societies, Natural Human Society, Natural Philosophy, Natural Emanating and Natural Exploring:** the sciences' and human society's pictures of Parts FIVE and SIX, each at its section. - **The Geodesic Improving Method:** placing concepts whole, preserving continuing value, and improving through the living file set. - **The illustrating working at v366 and the rings gathering at v365:** the corrected naming, centre 1, the six-diamond routes, the twisting-opening concept, the plaza, the snake, the stitching, the five-spot, the jigsaw, the swimmers, the cube and the rings. - **The illustrating working at v379:** the executions at Exhibit ONE's code, the first visual of the sixteen ways, the four dots assigned at the five-spot, the two inversions at forty-five degrees, the menu of the sciences' and society's pictures, the ten as drawings, the resolver society animating, and the third dimension. ## 9.2 Receiving an illustration, and the kit **An illustration arrives at `incoming/`, whole, with the file and section it is for, its sentences and any execution beside it**, and is received at this file's carrying as a ready offering naming its section or as a concern; it enters this file at this file's own motion. An incoming concept joins the section its explaining belongs at, carrying its useful view, the file it is read at, and the relation still needing expression; improving an existing concept it stays at that concept's place, and introducing a distinct way of illustrating it is given a section here. **The Natural Illustrating TWENTY-NINE Improving Kit** carries the original cube and open rings, the five studies, the inherited rings kit, the v366 sources and evidence, and at v379 the executions of Part TWO and 4.2 with their outputs and a check that executes them again at Exhibit ONE's code and compares the table at 10, the lone self, the two selves, the sixteen ways, the spiral of five and the torus of three by five with this file's sayings. A check is a coupling partner and no authority; the kit's README says each file's meaning and the place the remaining improving is carried, which is this file's carrying. ## 9.3 Not yet at the files The galaxy's spiral arms and the nautilus; the golden triangle and the golden angle by name; the tree of life as a drawing, the mind map and the knowledge graph, the files naming the lineage tree at Natural Societies 5.4, denying the root and the level as a place, 6.3, and naming no tree of life, map or graph; Schrödinger's cat, the ship of Theseus and Monty Hall by name, whose nearest entries are Resolving the Hard Problem Registry 10.12, 4.1, 2.9 and 2.16; two cubes alternating and a tesseract as the resolver, the tesseract at Natural Mathematics 7.6 being the field's own form; a dome, a truss or tensegrity as a form of the method, the woven surface at 8.2 being what the files carry; a streak or lightning. Each is a familiar picture with no sentence yet at a living file, and the way in is an observing brought whole to `incoming/`, then its file's carrying, and never a drawing as the files' own. This gathering remains open to further imagery throughout the living file set.