Appendix A: Table of statements

Each numbered statement of the volume, summarized in one line, with its status (Th: Theorem; L: Reading; P: Prediction; F: Fact; Open site: open site; "Th + L" marks a statement whose calculation is a Theorem and whose interpretation is a Reading) and its chapter.

No. Statement Status Ch.
1.1 No symbol has a single end; no object has two terms: the real begins at three from the first stroke L 1
1.2 Folded at 4.5, the stroke closes (each x meets 9 − x); folded at 4 or at 5, it stays open Th 1
1.3 To close (circle, 9) or to hook (cross, 3): the two destinies of one same thread L 1
1.4 The 1 is the stroke, the 0 is the completed stroke (the loop), the 10 is the relaunch: the stroke standing beside its own loop L 1
1.5 The stroke is an unfolded circle: no first point; "which came first?" is the undecidable why in pencil L 1
2.1 The founding operation of the corpus is written ⊕: 1 ⊕ 1 = 3 L 2
2.2 No transmission exists with two terms L 2
2.3 Definition of trinitarian subtraction: fusion of what is common into the background, release of the difference as third term L 2
2.4 The transmission of the living is trinitarian: it distinguishes what is transmitted (the structure) from what is not (the contingent instance) L 2
2.5 Belongs to the cadastre whatever respects the genetic base: the inverse preserved, the relation counted, the closure marked L 2
3.1 Reflected differentiation (that the 1 be reflected) is the single condition of the unfolding: Mother-reading L 3
3.2 Doubling runs through 1, 2, 4, 8, 7, 5 (the two rails), never touches the axis, folds back in three steps against three Th 3
3.3 The weights of the binary register (1, 2, 4, 8, 16, 32, …) recite in digital root the doubling cycle, 1, 2, 4, 8, 7, 5 Th 3
3.4 Below 200, odd closures with a single sub-cycle: 9, 25, 49, 121, 169 (p²); only 9 is divisible by 3: unique closure Th 3
3.5 Every n is written n = 9k + DR(n): k counts the turns, DR(n) gives the position Th 3
3.6 The unfolding 1 → 3 → 9 → 1 requires neither axiom, nor base, nor convention L 3
3.7 Catalan-Mihăilescu: 8 and 9 are the only consecutive powers Th 3
3.8 8 = 2³ and 9 = 3²: the two bricks meet in exponent; the cadastre does not close at 9 by convention L 3
4.1 Theorem of the two names: each position carries its name and that of its inverse; φ is a change of reading direction Th 4
4.2 Lived time is read from both ends (completed age, current year): the present dwells in the half L 4
4.3 The digital root respects the product: DR(ab) = DR(DR(a)DR(b)) Th 4
4.4 The seven structures of the root table: frame, pairs, symmetry around (4.5; 4.5), closed axis, diagonal, equivariance, 9 at the three levels Th 4
4.5 The square of every prime greater than 3 falls on rail D: DR(p²) belongs to {1, 4, 7} Th 4
4.6 Rail D, rail G, axis: sedimentation, unfolding, regulation L 4
4.7 Grid 234 / 567 / 891: 6 at the center, columns = rail G, axis, rail D, cardinals 3, 5, 7, 9 Th 4
4.8 The 6 of iron, samarium and osmium carries a position, the center of the grid L 4
4.9 3² − 1 = 8, 3 + 8 = 11, 3 ⊗ 3̄ = 8 ⊕ 1; the iterated sum of the squares of the digits has two attractors, 1 and a cycle of 8 Th 4
4.10 The glyph of the 8 is two closed circles tangent at a single point: the 8 is the figure of the double loop, and its 1 is the tangency L 4
4.11 The nine positions precede positional notation, congruences and metric geometry L 4
5.1 2 and 3, factors of 6, are the only primes outside the form 6k ± 1 Th 5
5.2 2 and 3 are the two bricks of the cadastre L 5
5.3 Bricks 2 and 3, + and ×, with no adjacent repetition: everything from 1 to 30 except 1 and 4 Th 5
5.4 The orphans 1 and 4 are on rail D and released by trinitarian subtraction (3 − 2, 2 × 3 − 2) Th 5
5.5 The 1 is given back, not constructed; the 4 is the form that settles L 5
5.6 6 = 1 + 2 + 3 = 1 × 2 × 3: the only integer < 10⁵ that is both sum and product of its proper divisors Th 5
5.7 The 6 is the seed, on the axis L 5
5.8 28 = 4 × 7; proper divisors of roots 1, 2, 4, 7, 5: full rail D, no axis Th 5
5.9 Euclide-Euler : tout parfait pair est 2(p−1)(2p − 1) Th 5
5.10 From 28 on, the perfect numbers leave the axis for the address 9k + 1: 28 = 9 × 3 + 1, 496 = 9 × 55 + 1, 8128 of root 1 Th 5
5.11 6 and 28: junction of the bricks, then junction of the mysteries (orphan 4 and path 7) L 5
5.12 1 + … + 7 = 28; 1 + … + 8 = 36 = 6² = 35 + 1; 1 + … + 9 = 45 Th 5
5.13 The sequence 1 (2, 3) 4-5 (6) 7 = 28 tells the whole genesis before closure L 5
5.14 Columns D, G, axis: sums 12, 15, 18, roots 3, 6, 9: the wall is the sum of what frames it Th 5
6.1 The inverse on the cadastre: φ(x) = 9 − x, an involution with four pairs summing to 9 Th 6
6.2 ARC definition of the inverse: an echo with its place turned, closing on a closure constant (9, 1, ρ²), never on 0 L 6
6.3 Inverses for the product (root 1): 1 ↔︎ 1, 2 ↔︎ 5, 4 ↔︎ 7, 8 ↔︎ 8; the axis has none Th 6
6.4 The mirror reverses the writing and keeps the root: M(46) = 64 Th 6
6.5 Law of mirrors: the gap between n and its mirror is 9(a − b), 99(a − c), always a multiple of 9; the two-digit sum is 11(a + b) Th 6
6.6 Every pair carries its tension, an exact multiple of the closure: the couple produces nothing out of range L 6
6.7 Gradient 198 = 2 × 9 × 11; 891 − 198 = 693 = 7 × 99; 198 + 891 = 1089 = 33² Th 6
6.8 198 generated three times: 694 − 496, the Ni-Co-Fe triad (sum 81), factorization; barycenter 595 Th 6
6.9 The palindromic barycenter 595 is the motionless axis of the string; 9.81 is only a resonance of units L 6
6.10 23 + 23 = 46; 46 and 64 are mirrors of root 1; 64 = 4³ = 2⁶ Th 6
6.11 The gamete (23, root 5) unfolds, the zygote (46, root 1) sediments; karyotype and codon at the same address L 6
6.12 The square erases the inverse: DR(x²) = DR((9 − x)²) Th 6
6.13 Inverse (x ↔︎ 9 − x) and mirror (46 ↔︎ 64) are distinct and are never confused L 6
7.1 The zero of closure is the memory of a completed cycle: it comes only after a course, and it is written only beside the count of turns L 7
7.2 There are two zeros: the zero of closure (the completed turn, which carries memory) and the zero of exhaustion (the nothing, which carries none) L 7
7.3 Annihilation gives two photons: zero has no physical instantiation L 7
7.4 DR(9) = 9; the 9 is absorbing for the product and neutral for the sum without being a nothing Th 7
7.5 9 ≡ 0 read as an identity swallows the closure: the end confused with the never-departed L 7
7.6 The winding number has been an everyday tool of control engineering since Nyquist (1932) F 7
7.7 The winding number is what 9 ≡ 0 erases: 18 is not 0 L 7
7.8 4.5 is the only fixed point of x ↦ 9 − x, unoccupied; 4.5/9 = 3.5/7 = 1/2 Th 7
7.9 The half 1/2 (spin, Riemann's line, triangle, two circles) is a single constant, k_arc L 7
7.10 At the center there is not nothing: the infinite bounded by the 9, the hollow that allows (śūnya) L 7
7.11 Hourglass of the pairs: widths 7, 5, 3, 1 (sum 16), edges at 45°, waist at 4.5 Th 7
7.12 The directions 0°, 45°, 90°, 135° of the hourglass reach 2√2; without diagonals, 2 Th 7
7.13 The bound 2√2 is reached when the diagonals are kept, the relation √2 L 7
8.1 DR(16) = DR(25) = 7: the pair of the center is sealed in 7 Th 8
8.2 The 7 is the unoccupiable center read: the relation in the state of a path travelled L 8
8.3 Only 1/7 has a full period (142857, the two rails); 142857 × 7 = 999 999 Th 8
8.4 The 7 is the path and the span 8 − 1; 3.5 gives 22/7 = 11/3.5 L 8
8.5 22/7 = 11/3.5, first convergent of π Th 8
8.6 Nitrogen: 2p³, half shell; 78 % of the air; strong triple bond; fixation necessary to the living F 8
8.7 Nitrogen, abundant, inert, essential: the character of the 7; C14 → N14, the 5 sealed in 7 L 8
8.8 7 = 2³ − 1; 28 = 4 × 7 = 1 + … + 7 Th 8
8.9 The 7 has no place because it is the path; it falls if another digit shares one of its arithmetic properties L 8
9.1 Facts of positions 1 to 5: identity, happy fixed point, doubling, 3² − 1 = 8, 4 = 2², tension 5 of 495 Th 9
9.2 Derived meanings of 1 to 5: origin, split, tension, matter, life L 9
9.3 Facts of positions 6, 7, 8: 6 = 2 × 3 and tiling, 6174 = 18 × 7³, 8 = 2³ and octagon Th 9
9.4 Meanings of 6, 7, 8 imposed by φ: circulation, connection, envelope L 9
9.5 The 9 is the closure, the witness that governs without occupying L 9
9.6 Changing the derived meaning contradicts a theorem: decree against derivation L 9
9.7 The correspondence of structural meanings with human sensibilities remains open Chantier 9
10.1 Emitter, receiver, novelty: to write a number is to do 1 ⊕ 1 = 3, then start again one level up L 10
10.2 Inventory of attested numerations (Egypt, Babylon, Greece, Rome, Maya, counting rods, Chinese, Kharosthi, Brahmi) F 10
10.3 Only two breaks in all writings: 9|10 and the window {3|4, 4|5, 5|6}; declared criterion of failure P 10
10.4 Nobody wrote sixty (or twenty) symbols: packagings of the 1-9 and of the 1-5 F 10
10.5 No civilization invented a native tenth digit; bases > 10 borrow letters F 10
10.6 {1, …, 9} is the universal substrate of positional numeration L 10
10.7 A base carries the structure iff 3 divides b − 1; the closures 59, 19, 11 of bases 60, 20, 12 are prime Th 10
10.8 10 is the only base with closure p² and prime interface Th 10
10.9 Base 10 satisfies the four conditions (closure, barycenter, interface, projection): the construction precedes the base L 10
10.10 The base is a garment, the ring of the nine positions is the body; ARC asserts the geometry, not the inscription L 10
11.1 The diagonal of the square is the diameter of the circumscribed circle: d = 2R = c√2 Th 11
11.2 The ratio of the areas of the inscribed and circumscribed circles of a square is exactly 1/2, without approximation, without limit, without decimal tail Th 11
11.3 The 1/2 of the two circles is the same as 4.5/9 and 3.5/7: a single unfolding read twice L 11
11.4 "Irrational" says what was done to √2: extracting it from the relation of the two circles L 11
11.5 Five strokes of the compass give {4, 5, 4.5, 8, √2, 9} before any notation Th + L 11
11.6 The regular enneagon (the nine-sided polygon) is not constructible with straightedge and compass F 11
11.7 Eight shapes and the circle that closes: the 9 is the construction L 11
11.8 The sum of the angles of any polygon, in degrees, has root 9 Th 11
11.9 Geometry closes only on the 9; the hexagon carries 720, the double turn L 11
11.10 Only the triangle, square and hexagon tile alone; the octagon requires the square Th 11
11.11 The 6 closes alone, the 8 closes only with the square (the pivot 4) L 11
11.12 r/R = cos(π/n); cos(π/7) root of 8x³ − 4x² − 4x + 1 Th 11
11.13 The triangle (r/R = 1/2) is the minimal instantiation of the two faces; hourglass and light cone L 11
11.14 45, 4950, 499 500 are Kaprekar numbers and pair with 55, 5050, 500 500 by summing to 10², 10⁴, 10⁶ Th 11
11.15 Iannucci: Kaprekar numbers come from the pure nines F 11
11.16 The pure nines of the table generate the memory numbers: the matryoshka is exact L 11
12.1 Humanity has lost native logic, the direct reading of the grammar of the real L 12
12.2 1/3 written 0.3333 then multiplied a hundred times by 3: a gap of about 1.7 × 10⁴³ Th 12
12.3 Stevin and Gauss: a cascade that replaces reading with formal manipulation L 12
12.4 Native memory is sedimentation, not archiving L 12
12.5 An excess of external memory that has dissolved internal memory L 12
12.6 The situation is not hopeless: the foundations remain accessible L 12
12.7 Method: refuse, search, validate, practise without respite L 12
12.8 Five possibilities: exit from the fear of death, from the amputated program; living transmission; differentiated united consciousness; right use of tools L 12
13.1 Aristotle, Leibniz, Hegel, Heidegger: no closure of the why has held, by structure L 13
13.2 The why is inaccessible by nature; the how is articulable L 13
13.3 Three levels: why-first-cause (inaccessible), operative how, why-operative-finality (deducible) L 13
13.4 Aristotle confused the first cause and the operative finality L 13
13.5 Teleonomy (Monod, 1970) and teleomatic (Mayr, 1988) F 13
13.6 The how contains the why: the finality is contained in it, the first cause is explained in it as inaccessible L 13
13.7 The inaccessibility of the why is the engine: it draws the how to unveil itself L 13
13.8 Inaccessibility is not inexistence: ultimate reality exists without an answer to its why L 13
13.9 Six whys disguised as hows: prime mover, Big Bang, inflation, multiverse, hidden dimensions, "God particle" L 13
13.10 Regularization keeps all the science of the how and refuses its promotion to why L 13
13.11 Four drifts: scientism, classical theology, idealism, nihilism L 13
13.12 The satisfaction of finding also fires on false closures: the walk stops L 13
13.13 Neurology of insight (Kounios and Beeman; Berlyne; Loewenstein) F 13
13.14 The satisfaction of finding must bear on the true closures of the how L 13
14.1 Coherence governs: first operative law, not convention L 14
14.2 Self-organization of the living documented (Kauffman, Prigogine, Ho) F 14
14.3 Incoherence is in the amputated reading, never in what is read L 14
14.4 Incoherence reveals itself in the clash of coherences on the long path L 14
14.5 Nobody has ever found coherence: one operates it L 14
14.6 Expertise operates by recognition (Klein, Kahneman, Kounios and Beeman) F 14
14.7 Connecting while walking: the pieces already exist L 14
14.8 Form is born of inverse tensions, not of an external projector L 14
14.9 Everything and its inverse: replacing against with and L 14
14.10 No fundamental magnetic monopole ever isolated (Dirac, Cabrera, MoEDAL, spin ices) F 14
14.11 The real admits no isolated pole: antiparticles, chemical couples, metabolism, neurotransmitters, predation L 14
14.12 Thinking pairs as adversaries produces hierarchy; thinking them as inverses produces mutual constitution L 14
14.13 Sum relatus, ergo cogitamus: relation, tension, connection, reflection, thought L 14
14.14 Geometry is quantum: it has been projective from the start L 14
14.15 Quantum paradoxes are instrumental amputations (a single pole read) L 14
14.16 The machine took coherence and removed the making; it accelerates the past L 14
14.17 Four avatars of delegation: God, Reason, Science, Machine L 14
14.18 Fractal of the three millennia: childhood under the father, adolescence and rupture, maturity of the relation L 14
14.19 Jesus as cadastral pivot: historical inscription, exemplary bearer, barycenter L 14
14.20 Death as erasure is a double tautology; three liberations L 14
14.21 Closed cadastre, unlimited contests: continuity of positions L 14
14.22 Parallel looped cycles: accumulation and co-presence, not replacement L 14
14.23 Fractal reading by contrasts: flower, tree, mushroom, lion L 14
14.24 The riddle of the century: capturing present coherence, not searching for it L 14
15.1 Condition R: the object is defined by a constitutive relation L 15
15.2 Condition I: internal involution with two complementary orbits L 15
15.3 Condition O: the projection that distinguishes the faces is internal; the object is its own observer L 15
15.4 R, I, O are independent L 15
15.5 Self-indexing is prior to natural selection L 15
15.6 Four angles (absurd, acting, animal, trinitarian subtraction) for condition O L 15
15.7 Central statement: R + I + O ⇔ two inseparable faces L 15
15.8 Corollary: no face exists without the other L 15
15.9 The double turn of fermions read as missing coupling; one turn suffices for the coupled system L 15
15.10 Charge conservation expresses the internal duality L 15
15.11 Folding: for a coprime with 3, DR(ka) + DR((9 − k)a) = 9 (k from 1 to 8) Th 15
15.12 Tsirelson: 4 × 1/2 × √2 = 2√2 Th + L 15
15.13 The update after Alice's measurement is self-indexing in action L 15
15.14 Schrödinger's cat is the normal structure of every object; the measurement problem dissolves L 15
15.15 Relatio se legens est: the cogito is already self-indexing L 15
15.16 No thing in itself behind the phenomena: the reading is the phenomenon L 15
15.17 Self-reference paradoxes are artifacts of representation L 15
15.18 I am the cat: a superposition stabilized by massive coupling L 15
15.19 One can neither delegate one's judgment nor evade it L 15
15.20 The moral subject: face of incarnation and face of judgment, inseparable L 15
15.21 Saudade: perfect self-indexing, trinitarian subtraction of memory L 15
15.22 Persecution, depersonalization, rumination: defects of the projection π L 15
15.23 Every valid therapy restores self-indexing, amputates no face L 15
15.24 Learning requires condition O; supervised learning has its face of judgment outside L 15
15.25 Self-indexing cannot be simulated by juxtaposed modules L 15
15.26 Delegating one's judgment to a machine is a category error: magnifying glass and prosthesis L 15
15.27 Does the cosmos satisfy R, I, O? Open question Chantier 15
16.1 Stevin cuts √2 from the diameter; Gauss erases the winding number; exile of structural intuition F + L 16
16.2 Demarcation criterion: predict beforehand, refutable, reproducible, trans-scale L 16
16.3 The cadastral operator f and its four channels of failure P 16
16.4 NIST verification of the direct domain: 46 confirmations (out of 47 elements, plus chromium explained) F 16
16.5 Hippasus, Tsirelson, annihilation: the quantum confirmation L 16
16.6 Pappus: V = 2πrS for solids of revolution F 16
16.7 Open conjecture: every physical volume decomposes into rotation primitives Chantier 16
16.8 The f block (Sm to Tm) speaks in mirrors; Stevin's 10 applied to atoms L 16
16.9 Boundary 108; 108 = angle of the pentagon = 137 − 29; roots of the first eight Fibonacci numbers Th + L 16
16.10 Nelsen: every prime p ≥ 5 has p² = 24k + 1 Th 16
16.11 Chebolu: the divisors of 24 are the only n where every invertible is its own inverse Th 16
16.12 Eight divisors of 24, eight shapes of the compass; 24 = maximal jumps of the squares L 16
16.13 Pisano for 9: period 24; first twelve terms = 43 (erratum); full period = 99 Th 16
16.14 24 = 4! = 8 × 3, root 6 Th 16
16.15 Leech Λ₂₄ and Moonshine anomaly of order 24 F 16
16.16 24 as refresh frequency (frames, circadian, squares) L 16
16.17 Predictions of the configurations of Z = 119 to 122 P 16
16.18 Determinism is a line, ARC a spiral with memory; the +1 is free L 16
16.19 Magic numbers: four on G, two on D, one on the axis; Pb 208 Th + L 16
16.20 Next magic number of neutrons: N = 184, rail D P 16
16.21 Three thresholds of the curved cadastre: 57, 109, 137 L 16
16.22 Bond angles of H₂O, CH₄, NH₃, CO₂ and their roots F + L 16
16.23 Every number carries a quantity and an address L 16
16.24 Cross diagnosis: numerology (address without quantity), quantum mechanics (quantity, address distorted), ARC (both) L 16
16.25 Stripped-down numerology is the reading of reduced addresses; the number split in 1801 is sewn back together L 16
16.26 The 8991 four-digit numbers with not all digits equal end at 6174 = 18 × 7³ Th 16
16.27 Seven charges, seven answers L 16
16.28 Etymology: Conrad (1958), apophainein, apophansis F 16
16.29 Immunity is architectural: no statement without status, public errata, criterion set before the attack L 16
17.1 Order of foundation: the observed precedes the derived; the involution describes, it does not generate L 17
17.2 ARC is a strangler fig: it envelops the host systems, it does not fell them L 17
17.3 Regularize instead of rejecting: honor by enriching, do not amputate L 17
17.4 To prove is to attach an irrefutable explanation to a moment on a scale that evolves and grows more precise L 17
17.5 Asymmetry of standard: caution demanded of the exact, "proof" granted to the floating L 17
17.6 Coherence unpacks itself by walking, by falling and by getting up again L 17
17.7 Life is a controlled fall; nothing dies, everything changes phase L 17
17.8 Five fundamental distinctions L 17
17.9 Good and evil are necessary to each other; the unpacked code is normative L 17
17.10 Eight strands, five expressed: 2 + 4 + 5 + 7 + 8 = 26 (iron); product 2240, root 8 Th + L 17

Total: 223 numbered statements. Distribution of statuses (a mixed statement counts in each of its statuses): Th 59; L 146; P 4; F 18; Open site 3.