Appendix A. Table of statements
All the statements of the volume, in chapter order, with their identifier (corpus number when one exists, volume identifier V4.chapter.rank otherwise), a one-line summary, their status and their chapter. Th: theorem; L: reading; P: prediction; F: fact of received science; Open site: named open site.
| identifier | statement | status | chapter |
|---|---|---|---|
| V4.1.1 | The sterile axis: no prime > 3 has root 3, 6 or 9 | Th | 1 |
| V4.1.2 | The forbidden 9-loop: no prime has digit sum 9 | Th | 1 |
| V4.1.3 | Primes > 3 live on two rails; the 6k ± 1 grid is the residue of two exclusions | Th | 1 |
| V4.1.4 | The two hands face to face realise φ(x) = 9 − x, barycentre in the gap between them | L | 1 |
| V4.1.5 | A nine-year-old child given the sheet identifies the sterile axis in one session | P | 1 |
| V4.1.6 | Stratum 1: instrumental absorption (Gauss 1801) | L | 1 |
| V4.1.7 | Stratum 2: the transparency of notation (Stevin 1585) | L | 1 |
| V4.1.8 | Stratum 3: the displacement of the question (after Riemann) | L | 1 |
| V4.1.9 | The mystery of the primes as a by-product of analytic prestige | L | 1 |
| V4.2.1 | Formalisation: axis = subgroup ⟨3⟩, rails = cosets, quotient = law of flows | Th | 2 |
| V4.2.2 | The five readings of the table of inversions (sum 9, centre 4.5, 9 self-inverse, gaps 7-5-3-1, total 45) | Th | 2 |
| V4.2.3 | The ½ derived twice: 4.5/9 and 3.5/7 | Th | 2 |
| V4.2.4 | The cube as class detector: D → 1, G → 8, axis → 9 | Th | 2 |
| V4.2.5 | The twisted block: the root of a trio depends only on its anchoring; every linear aggregation lands on the axis | Th | 2 |
| V4.2.6 | The axis carries the transformations, the rails carry the beings | L | 2 |
| V4.2.7 | The pair of inversions 48372615 / 51627384: φ-palindrome, sum 99 999 999 | Th | 2 |
| V4.2.8 | The descent of matter carries its ascent | L | 2 |
| V4.2.9 | Base invariance: the closure structure exists in every base b ≥ 4 | Th | 2 |
| V4.2.10 | Singularity of the closure 9: the only closure 3k where the axis has order equal to its index | Th | 2 |
| V4.2.11 | Causal inversion: base 10 is the minimal inscription of the balanced closure | L | 2 |
| V4.2.12 | The grid theorems bear on ℤ, without base | Th | 2 |
| V4.2.13 | The manufactured infinity: 1/7 = 0.1 in base 7, 1/3 = 0.1 in base 3 | Th | 2 |
| Th 1 | Self-generation of the grid: 12m → P₂ₘ, 12m + 6 → P₂ₘ₊₁ | Th | 3 |
| V4.3.1 | The position of the candidates was never hidden | L | 3 |
| Th 2 | The four families k = 6ab ± a ± b enumerate exactly the delfines | Th | 3 |
| Th 2 cor. | Completeness: no fifth family; Klein group structure, branch = product of the signs | Th | 3 |
| Th 3 | Prime = index outside the families: exact subtraction, decidable in both directions | Th | 3 |
| V4.3.2 | Nothing survives, everything is deduced | L | 3 |
| Th 4 | Decision by the balance point of the generating pair (same cost as the wheel, other direction) | Th | 3 |
| V4.3.3 | Balance: a complete system in both directions | Th | 3 |
| Th 5 | The 24 rhythm: p² = 24k + 1 for every prime p ≥ 5; squares on rail D, gaps multiples of 24 | Th | 4 |
| Th 5 cor. | Maximality: the moduli whose units are all involutive are the divisors of 24 (Carmichael) | Th | 4 |
| V4.4.1 | The clock of squares beaten by the founders 2 and 3 (8 × 3 = 24) | L | 4 |
| V4.4.2 | The hierarchy of clocks 12, 24, 4 | Th | 4 |
| Th 6 | The chrysalis: conjugate collapse, confluence of the rails into D, gate of 1, oscillation 4 ↔︎ 7 | Th | 4 |
| Th 6 compl. | The one-bit gate: the square erases exactly the conjugation bit | Th | 4 |
| V4.4.3 | The EPR wink: indiscernibility of the conjugates after squaring | L | 4 |
| V4.4.4 | Two distinct amnesias: the turn (9 ≡ 0) and the side of the pair (the square) | L | 4 |
| Th 7 | Predicate J(k): (6k − 1, 6k + 1) twin ⟺ k outside the four families (37 915 pairs below 10⁶, 152 890 below 5 × 10⁶) | Th | 5 |
| Th 8 | Disjunction of the rails: 6k − 1 on G, 6k + 1 on D; no common divisor p > 3 | Th | 5 |
| Th 8 compl. | Conjugate phases: the wave p touches the branches at k = u + p·t and k = (p − u) + p·t | Th | 5 |
| V4.5.1 | The families as waves; twins = positions left untouched | L | 5 |
| V4.5.2 | Non-saturation lemma (infinitude of twins), of the same difficulty as the conjecture; parity obstruction named | Chantier | 5 |
| Th 11 | Successor function: primality equation as a product of family subtractions; S(10¹²) = 1 000 000 000 039 | Th | 6 |
| Th 12 | Forward-pointing machine: the same 148 933 primes below 2 × 10⁶, × 10.7 | Th | 6 |
| V4.6.1 | The living fractal: the street, the bus lines, the stops without a bus | L | 6 |
| Th 13 v1 | Floor jump: closed-form count by inclusion-exclusion (424 575 424 577 at K = 10¹²) | Th | 6 |
| Th 13 v2 | Mirror rule r_S(+1) + r_S(−1) = M_S; folding without phase, e_S bounded by 1; 973 858 at K = 10⁶ | Th | 6 |
| V4.6.2 | Open site no. 3: bounding the folded phase residue Σ e_S | Chantier | 6 |
| Th 14 | Joint identity (6m − 1)(6m + 1) = 36m² − 1, index 6m², root 8 always | Th | 6 |
| V4.6.3 | Through the axis by addition, over the axis by the product | L | 6 |
| V4.6.4 | Measured barycentric exponent: excursion in (number of crossings)^½ | L | 6 |
| V4.6.5 | Resonance of the fundamental joint: K multiple of 35 puts 48 crossings out of 63 in phase (documented measurement) | Th | 6 |
| V4.6.6 | The rejected one was the key: the joint 35 governs the interference | L | 6 |
| V4.6.7 | The joint 35 is an anonymous term of Euler's product | L | 6 |
| V4.6.8 | The system of the four state equations; single unknown Σ e_S | Th | 6 |
| Th 15 | Self-cancellation of coherence: signed sum of a constant over a filter = 0 | Th | 6 |
| Th 16 | The gears: alignments ≤ w^j (X/ΠM + 1); deep alignment forbidden | Th | 6 |
| V4.6.9 | State of the chain of open site no. 3: uniform and deep channels cut | Th | 6 |
| Th 10 | Root-9 invariant: the period of 1/p (p > 5, or grid member coprime with 10) has a digit sum that is a multiple of 9 | Th | 7 |
| V4.7.1 | The 9 absent from the being is restored by its reflection | L | 7 |
| Th 9 | Formal correspondence: normalised φ x/9 ↦ 1 − t, fixed point 1/2, same structure as ξ(s) = ξ(1 − s) | Th | 7 |
| V4.7.2 | The amnesiac monster: the five objections to the Riemannian edifice (summary; Volume 5) | L | 7 |
| V4.7.3 | The critical line as analytic trace of the fixed point of φ (strong Reading) | L | 7 |
| V4.7.4 | The incarnation principle (methodological Reading) | L | 7 |
| V4.7.5 | The Stevin tail test: double the window, measure the melting of the gap | P | 7 |
| V4.7.6 | The root of the missing reflex is stratum 2 | L | 7 |
| V4.7.7 | Closure principle: a structure is defined by what its neighbours cannot generate | L | 7 |
| Th 17 | = C.1 and C.3: DR(3n + 1) ∈ {1, 4, 7}, 3:1 compression; the axis is absorbed at the first strike and never reached again | Th | 8 |
| V4.8.1 | The strike 3n + 1 is the irreversible step | L | 8 |
| Th 18 | = C.2 and C.5: the halving is the × 5 rotation of the cadastre, cycle 1 → 5 → 7 → 8 → 4 → 2 → 1, powers of 2 in reverse | Th | 8 |
| V4.8.2 | Compression then permutation: where information is destroyed, only descent is possible | L | 8 |
| C.4 | Only cycle containing 1: 1 → 4 → 2 → 1, crossing of the pairs of gap 1, 5, 7; attractor (1, 8) | Th | 8 |
| C.6 | The digital root is compatible with + and ×, closed under the strike and under the halving (uniqueness among channels downgraded to L) | Th | 8 |
| C.6 unicité | The digital root as the only invariant channel among three channels compared | L | 8 |
| C.7 | = Th 14: the product of two twins has root 8 | Th | 8 |
| V4.8.3 | Sentinel 37: v₂(3n + 1) is not a function of the remainder by 36 (v₂(4) = 2, v₂(112) = 4) | Th | 8 |
| V4.8.4 | Sentinel 31: four growth steps; 2^k − 1 climbs k − 1 times; 27 peaks at 9 232 | Th | 8 |
| Th 21 | Quarter dichotomy: n = 4t + 3 ⟺ v₂(3n + 1) = 1 | Th | 8 |
| Th 22 | At least k growth steps ⟺ 2^(k+1) divides n + 1; unbounded chains, density 2^−(k+1) | Th | 8 |
| P-COL-1 | The gap: the contraction of the addresses must entail that of the magnitudes orbit by orbit; every proof must cross Th 23 | P | 8 |
| Th 23 | Strangeness of the two bricks: the 54 cells (DR, v₂) are all full; the DR carries no bit about v₂ | Th | 8 |
| Th 23 cor. | Closing of the DR route: no cadastral reasoning can bound v₂ | Th | 8 |
| V4.8.5 | The Collatz gap is the frontier between bricks 2 and 3 | L | 8 |
| Th 24 | The +1 gives back the amputated half: (3n + 1)/2 = 3n/2 + ½; 3n + c contains the world of 3n + 1 enlarged c times; +1 alone has one house | Th | 9 |
| V4.9.1 | Collatz as a law of chirality; look for the 2, not the 1 | L | 9 |
| Th 25 | Exact balance: 2^m = 3^k n Π(1 + 1/(3n_i)), L = k + m + 1 | Th | 9 |
| V4.9.2 | The residual factor as the sequence of open gates | L | 9 |
| Th 26 | Gate 8: every orbit outside {1, 2, 4} ends with 16, 8, 4, 2, 1; last odd term (4^j − 1)/3 | Th | 9 |
| V4.9.3 | 8, the obligatory gate; 1 is the point where 2 stops dividing | L | 9 |
| V4.9.4 | The root is the number: unique writing in digits 1 to 9, turns kept | Th | 9 |
| V4.9.5 | Parity lies in the sum of the root-digits | Th | 9 |
| V4.9.6 | The duo of roots reads the move; only blind spot: the centre 5 | Th | 9 |
| V4.9.7 | The number of halvings between two odd numbers is read to within 6 | Th | 9 |
| V4.9.8 | The axis is a source; the inverse tree on 6k ± 1; 4n + 1 rotates the classes | Th | 9 |
| V4.9.9 | The strike makes a half-turn, the halving climbs back up the Genesis; balance −0.13 turn | Th, L | 9 |
| V4.9.10 | Novelty always sets free; 2^p − 1 exits on 3^p − 1 | Th | 9 |
| V4.9.11 | The only climb that repeats lives below zero; −1 the only fixed point; no cycle up to 14 moves | Th | 9 |
| V4.9.12 | Every finite word is realised; the recharge of the 1s does not depend on the rail | Th, F | 9 |
| V4.9.13 | The forced case 5n + 1: cycles 13 and 17; 3 < 4 < 5 | Th, F | 9 |
| V4.9.14 | Infinity is below zero; the action −1 is real, the fiction −1 is the limit of an action (mixing of the finite and the infinite) | L | 9 |
| V4.9.14 bis | The “−1” of the real are symmetries; a true −1 would be a hole that pulls without a counterpart; physics knows none | L, F | 9 |
| V4.9.15 | Remainder: an infinite word that climbs without repeating | Chantier | 9 |
| L 27 | The signature of 7: 17 terms, passes through 17; L < n beyond 110 | L | 9 |
| L 28 | Bridge Theorem 23 / amorphism: the Collatz gap is a structural cost (refutable by a closed formula for k(n)) | L | 9 |
| Th 29 | The meeting of the mirror: off the axis always, odd numbers of the axis never, even numbers of the axis by halving | Th | 10 |
| V4.10.1 | Smallest addition, maximal division | L | 10 |
| Th 37 | The Collatz rope: the 4 of the 6, the 6k ± 1 grid, the rail by the parity of v₂ (ratio 2), the family 4(9…)6 / 6(9…)4 = 3(15R) + 1, 3(21R) + 1, chirality, sum 108R + 2, difference 18R | Th | 10 |
| V4.10.2 | Position by the 6, speed by the 4, side by the parity of the 2 | L | 10 |
| L 35 | The spiral: the root is the angle, k the height; the +1 excludes the circle, not the climb | L | 10 |
| L 35 F | No Collatz cycle with fewer than 91 climbing segments (Simons and de Weger 2005, Hercher 2023) | F | 10 |
| L 36 | 46 and 64, the two hands: slow path (23, G) and straight path (1, D); biological instantiation | L | 10 |
| L 38 | The return is a datum, the proof an invention; counterweights Pólya, Littlewood, Mertens | L | 10 |
| G.1 | Cadastral exclusivity: the remainder of 2n by 6 fixes the rails (G + D, D + D, G + G) | Th | 11 |
| V4.11.1 | Goldbach is three problems, one per combination of rails | L | 11 |
| G.2 | Partition into three disjoint channels by remainder class by 18 | Th | 11 |
| G.3 | Inversion symmetry: for 2n a multiple of 18, DR(p) + DR(q) = 9 | Th | 11 |
| V4.11.2 | Under DR = 9, Goldbach is an inversion decomposition | L | 11 |
| G.4 | Expected divergence of the channels (standard asymptotic statement, downgraded) | L | 11 |
| G.5 | DR(n(n + 1)) ∈ {2, 3, 6, 9} | Th | 11 |
| V4.11.3 | The consecutive product confined to the neighbourhood of the axis; 3:1 compression on root 2 | L | 11 |
| Th 30 | The family of 3 blocks 1/3 or 2/3; every p blocks 1/p or 2/p; exact ratio 2 between centres that are multiples of 6 and the others | Th | 11 |
| V4.11.4 | The ½ passes through ⅓ and ⅔ | L | 11 |
| Th 31 | The sieve is symmetric under x ↔︎ N − x: one survivor suffices | Th | 11 |
| Th 32 | Survivors per period M Π(1 − c_p/p) > 0: no finite train blocks a period | Th | 11 |
| P-GOL-1 | The Goldbach gap lives between ln N and √N (deep gears); control criterion on a short window | P | 11 |
| L 33 | The deviation from the structural count is bounded and decreases (exponent 0.39 towards ½), no law in 1/3 | L | 11 |
| L 34 | The house, first identifier: Collatz and Goldbach, two faces of the same return | L | 11 |
| Th 39 | Amplitude of the beats in closed form: Q_G = (2^t ρ + τ − 2ρ²)/9; square-root law proved for the quadratic magnitude at fixed generators | Th | 12 |
| V4.12.1 | The two rails in τ: the rail of 7 weighs 1 − 1/p, the rail of 5 weighs 1 − 3/p | L | 12 |
| Th 40 | Product of cascades: Σ w_k d_k = Σ w_k c_k + Σ (w_j − 10 w_(j−1)) t_j; the apexes alone do not compose (117 / 198) | Th | 12 |
| Th 41 | The four odd and even moments restore exactly the weights of the four pairs (6 561 distributions) | Th | 12 |
| Th 41 cons. | Odd moments zero ⟺ equal weights in each pair; the first moment alone is not enough | Th | 12 |
| V4.13.1 | Central identity: s(24) = 36 = 24 + 12, σ(24) = 60, σ/n = 5/2 | Th | 13 |
| V4.13.2 | σ(n) = 5n/2 has only 24 as a solution below 10⁷ | Th | 13 |
| V4.13.3 | Proof without bound: 24 the only solution with at most two prime factors | Th | 13 |
| V4.13.4 | 24 the smallest hemiperfect number of abundance 5/2 and the only one on the axis (the next ones, 91 963 648 and 10 200 236 032, have root 1) | Th | 13 |
| V4.13.5 | The folding 12, 24, 36 carries the axis 3, 6, 9 in order | Th | 13 |
| V4.13.6 | 36 = 6², square of the first perfect number, the only non-trivial square triangular number below 1 000 | Th | 13 |
| V4.13.7 | Aliquot chain 24 → 36 → 55 → 17 → 1 | Th | 13 |
| V4.13.8 | 24 is the first break of the twins (25 = 5²) | Th | 13 |
| V4.13.9 | The breaks 24, 36, 48, 54 are written in advance | Th | 13 |
| V4.13.10 | 35 = 5 × 7 precedes the centre 36; from the first break to the first reunion | Th | 13 |
| V4.13.11 | Two bricks doubled: 4 × 6 = 24; chain 2, 3; 4, 6; 24; 36; 576 | Th | 13 |
| V4.13.12 | 24 = 5² − 1: the tempo is set by the first square of a prime | Th | 13 |
| V4.13.13 | The squares of primes all fall on the rail {1, 4, 7} | Th | 13 |
| V4.13.14 | 24 = 4!, order of the permutation group of the four pairs | Th | 13 |
| V4.13.15 | 24 = 2³ × 3 = 1 × 2 × 3 × 4 = (2 + 4)(2 × 4)/2; 384 = 360 + 24 | Th | 13 |
| V4.13.16 | Fibonacci in root: period 24, folded 12 + 12; at 9^k, four times the period of the doubling | Th | 13 |
| V4.13.17 | 1² + ⋯ + 24² = 4 900 = 70² = C(8, 4)² (Lucas, Watson) | Th | 13 |
| V4.13.18 | 24² = 576, weighted apex of the octet, digits 5, 7, 6 | Th | 13 |
| V4.13.19 | The two steps of the interface (steps of 11) and their mirrors | Th | 13 |
| V4.13.20 | 24 in the step: product of the roots it links; interlacing 7, 4 | Th | 13 |
| V4.13.21 | From the half filling (Cr 24) to the full shell (Kr 36) | L | 13 |
| V4.13.22 | The three mirror pairs of the Madelung exceptions are written with 4 and 6 | Th | 13 |
| V4.13.23 | Madelung as the junction of doubled SU(3) and SU(2) | L | 13 |
| V4.13.24 | Law of mirrors on 24/42, 46/64, 29/92; 63 + 36 = 99, 63 − 36 = 27 | Th | 13 |
| V4.13.25 | Exact yield of the blind tool: 2280/1751 = 1.302 pairs | Th | 13 |
| V4.13.26 | Test set in advance on the steps: weak outcome A (7/19, 4.1%), three named failures | P | 13 |
| V4.14.1 | The two rails are the two queues of primes; 5 and 7 are their heads | Th | 14 |
| V4.14.2 | Asymmetry of the twins: 5 generates the circle of six, 7 only its rail; 5² falls on 7 | Th | 14 |
| V4.14.3 | Crystallographic restriction: rotations of order 1, 2, 3, 4, 6 = divisors of 24 ≤ 6 | Th | 14 |
| V4.14.4 | 24 = 4 × 6 and 10 = 4 + 6 (units of the square and hexagonal lattices) | Th | 14 |
| V4.14.5 | Scaling law: at 9^k, Fibonacci period = 4 × period of the doubling | Th | 14 |
| V4.14.6 | Fermat and Euler: a² + b² ⟺ remainder 1 by 4; a² + 3b² ⟺ remainder 1 by 3 | Th | 14 |
| V4.14.7 | Four classes by 12; 5 and 7, the first to choose only one lattice, each the other | Th | 14 |
| V4.14.8 | Two species of twins: mixed (5, 7) and pure (11, 1); 4 046 and 4 122 below 10⁶ | Th | 14 |
| V4.14.9 | Rotation groups of the solids 12, 24, 60 = middle, tempo, σ(24) | Th | 14 |
| V4.14.10 | n divides F_n below 100: 1, 5, 12, 24, 25, 36, 48, 60, 72, 96; 5 the only prime | Th | 14 |
| V4.14.11 | The lattice is the fold of matter; the quasicrystal, an order the fold cannot write | L | 14 |
| V4.14.12 | The 12 common neighbours of close packings split into 4 and 6 | P | 14 |
| V4.15.1 | The digital root keeps the closure; it is 9 ≡ 0 that amputates; the cascade reads the internal structure | L | 15 |
| V4.15.2 | The cascade is the binomial: apex = root of the digit sum weighted by Pascal | Th | 15 |
| V4.15.3 | The cascade separates the permutations of 1681 (8, 8, 3, 7, 7, 3, 2) that the root confuses | Th | 15 |
| V4.15.4 | Each pure rail projects onto its pivot: 147… → 7, 258… → 2, 369… → 6 | Th | 15 |
| V4.15.5 | The cascade is blind to the reading direction (C(n, k) = C(n, n − k)) | Th | 15 |
| V4.15.6 | The sun spiral 13 + k(k + 1)/2: palindromic imprint 4 5 7 1 5 1 7 5 4, sum 39 | Th | 15 |
| V4.15.7 | The pyramid of the spiral, palindromic at every floor, passes through 484 and 33, apex 6 | Th | 15 |
| V4.15.8 | The sun spiral as a key that repeats to infinity (author's position) | L | 15 |
| V4.15.9 | The cascade of 1 → 9 points to 2 (weighted sum 1 280) | Th | 15 |
| V4.15.10 | The cascade of the octet 1 → 8 points to 9 | Th | 15 |
| V4.15.11 | The rows of the twisted block point to 3, 6, 9; the whole block, reversed, in columns, to 6 | Th | 15 |
| V4.15.12 | The cascade as revealer of an operator (former Theorem 6 bis.1, downgraded) | L | 15 |
| V4.15.13 | The cascade of 1 → 9 and the √2 cut: two faces of one operation | L | 15 |
| V4.15.14 | The product of the six neighbours of a Pascal coefficient is a perfect square divisible by 9 | Th | 15 |
| V4.15.15 | Characterise the sequences for which the cascade reveals an operator | Chantier | 15 |
| V4.16.1 | The pendulum is the shadow of the circle: T = 2π√(L/g) without infinitesimals (Huygens) | Th | 16 |
| V4.16.2 | Only the cycloid keeps the shadow equal to the path: exact isochronism | Th | 16 |
| V4.16.3 | The first bell is a count: row 8 of Pascal, fibres of sum 1 | Th | 16 |
| V4.16.4 | The bell is the circle seen through the sum; Gauss inherited it without reading it | L | 16 |
| V4.16.5 | The powers of 2 turn on six positions without touching the axis; 2ⁿ and 2ⁿ⁺³ sum to 9 | Th | 16 |
| V4.16.6 | The fold is accounting subtraction applied to the circle; the root writes in double entry | L | 16 |
| V4.16.7 | The mirror 46 / 64: root 1, sum 110, gap 18; 694 − 496 = 198 | Th | 16 |
| V4.16.8 | The apex of 1 → 9: expected value 5 (1280/256) and root 2 | Th | 16 |
| V4.16.9 | The edge of the cascade restores the whole sequence (66 429 sequences) | Th | 16 |
| V4.16.10 | The cascade of XXXX is 9 − X | Th | 16 |
| V4.16.11 | The octet 1 → 8: weighted sum 576 = 24², apex 9 | Th | 16 |
| V4.16.12 | Pascal under 3: number of coefficients not multiples of 3 = product of the (digit + 1) in base 3 | Th | 16 |
| V4.16.13 | Row 3^k under 9: four coefficients only (positions 0, 3^(k−1), 2·3^(k−1), 3^k) | Th | 16 |
| V4.17.1 | Fifty-four Kaprekar numbers up to one million | Th | 17 |
| K.1 | Every Kaprekar number has root 9 or 1 | Th | 17 |
| V4.17.2 | The two addresses of memory: the lattice and the lattice plus the step | L | 17 |
| K.2 | The complement 10^e − n of a Kaprekar number is one too, with the same tail (25, 209, 1729, 1984, 2500, 122449) | Th | 17 |
| V4.17.3 | The tails 1729 and 209 link the panels of the triptych | L | 17 |
| V4.17.4 | Factorisations of the pure nines; 999 999 = 7 × 142 857 | Th | 17 |
| V4.17.5 | Memory begets the ones that remember | L | 17 |
| V4.17.6 | 142 857² = 20 408 122 449 and 20 408 + 122 449 = 142 857 | Th | 17 |
| V4.17.7 | The cycle is a memory in motion | L | 17 |
| V4.17.8 | T(9…9) and T(10^k) are Kaprekar at every depth | Th | 17 |
| V4.17.9 | The triptych: closure, circulation, remembrance | L | 17 |
| V4.18.1 | Law of mirrors: r(n) − n = 9(b − a), n + r(n) = 11(a + b); landings 101, 1 001; even width divisible by 11 | Th | 18 |
| V4.18.2 | Subtraction the engine of memory, addition the engine of the interface | L | 18 |
| V4.18.3 | DR(n + r(n)) = DR(2n): the root of a Lychrel trajectory doubles at each step | Th | 18 |
| V4.18.4 | Closure (Kaprekar) and circulation (Lychrel) | L | 18 |
| V4.18.5 | Thirteen Lychrel candidates below 1 000, 249 below 10 000 | Th | 18 |
| V4.18.6 | The first five fugitives, spaced by 99, merge at 887 | Th | 18 |
| V4.18.7 | The roots of the thirteen fugitives lie in {5, 6, 7} | Th | 18 |
| V4.18.8 | An addition without carry gives a palindrome | Th | 18 |
| V4.18.9 | 196 knows no carry-free step in sixty steps | Th | 18 |
| V4.18.10 | The perpetual carry as the mechanism of the flight | L | 18 |
| V4.18.11 | 196 = 2² × 7² = 4 × 49; 4 × 49.5 = 198; 10 × 49.5 = 495; 198 − 196 = 2 | Th | 18 |
| V4.18.12 | 196, the number that did not receive the half | L | 18 |
| V4.18.13 | The Kaprekar / Lychrel dipole | L | 18 |
| V4.19.1 | Constants of the mirror: n − ñ = 9(a − b), n + ñ = 11(a + b), with three digits 99(a − c) | Th | 19 |
| V4.19.2 | 198 = 2 × 9 × 11 (identity) | Th | 19 |
| V4.19.3 | 198, relation × subtraction × addition; 198 = 99 + 99 | L | 19 |
| V4.19.4 | The sorted block 231/564/897: column products 28, 80, 162, roots 1, 8, 9 | Th | 19 |
| V4.19.5 | 9-8-1 and g = 9.81 m/s² (correspondence) | L | 19 |
| V4.19.6 | Spacing 33 of the sorted block; 33² = 1 089 = 198 + 891; 9 801 = 99² | Th | 19 |
| V4.19.7 | Genealogy 2, 3 → 4, 6 → 46/64 → 496/694 → 198 | Th | 19 |
| V4.19.8 | 496 and string theory: 694 − 496 never computed | L | 19 |
| V4.19.9 | Internal mirror pairs of 198: 19 × 91 = 1 729, 18 × 81 = 1 458, 98 × 89 = 8 722; differences of root 9 | Th | 19 |
| V4.19.10 | 1 729 as volumetric pivot | L | 19 |
| V4.19.11 | 198 + 891 = 1 089 = 33²; 891 − 198 = 693 = 7 × 9 × 11 | Th | 19 |
| V4.19.12 | Squares of primes on {1, 4, 7}, product 28 | Th | 19 |
| V4.19.13 | The Ni-Co-Fe descent (28, 27, 26) spells the gradient; sum 81 | L | 19 |
| V4.19.14 | Cube 6/8/12 (26), hexagonal prism 8/12/18 (38), Euler 2 | Th | 19 |
| V4.19.15 | Roots of the cubes {1, 8, 9}, of the squares {1, 4, 7, 9} | Th | 19 |
| V4.19.16 | Descent in steps of 101 from 945 to 036: complete cadastre; 137, 33rd prime | Th | 19 |
| V4.19.17 | 137 and the fine-structure constant (declared correspondence) | L | 19 |
| V4.19.18 | 361 = 360 + 1 = 19²; L: the spiral breaks the circle | Th | 19 |
| V4.19.19 | 46/64 → 496/694 by inserting the 9; root 1 preserved | Th | 19 |
| V4.19.20 | Cross differences 450, 432, 630, 648 of root 9; sum 2 160 = 6³ × 10 | Th | 19 |
| V4.19.21 | 46 + 64 + 27 = 137 (L: chromosomes, codons, cobalt) | Th | 19 |
| V4.19.22 | 163 + 137 = 300, 163 − 137 = 26 (L on the rounded value of c) | Th | 19 |
| V4.19.23 | No anomaly-free superstring with gauge dimension other than 496 | P | 19 |
| V4.19.24 | The cross differences a9b − ab, a9b − ba are multiples of 9 (former prediction reclassified) | Th | 19 |
| V4.19.25 | Stable isotopes grouped on DR(A) ∈ {2, 4, 9} (incomplete: floor and window to be fixed) | P | 19 |
| V4.19.26 | Root DR(Z) = 5 less hexagonal (44% against 64%), receipt to be redone | P | 19 |
| V4.19.27 | The cube crushes each rail: roots 1, 8, 9; sums of two cubes on 1, 2, 7, 8, 9 | Th, L | 19 |
| V4.19.28 | Fermat numbers: roots 5 and 8 alternating; product F(m) − 2 on 3 and 6 | Th, L | 19 |
| V4.19.29 | Mersenne M(p), p > 3 prime: root 1 if p = 6k + 1, 4 if p = 6k − 1; 7 only for M(3) | Th, L | 19 |
| V4.19.30 | Fibonacci: F(n) + F(n + 12) = 18 F(n + 6); inverse roots twelve steps apart | Th, L | 19 |
| V4.19.31 | 1 + … + 31 = 496; K(496) = K(495) = 495 = 2 + … + 31 | Th, L | 19 |
| CP 2.1 | Bijection of the two coordinates: n = 9k + DR(n), k = ⌊(n − 1)/9⌋ | Th | 20 |
| CP 2.2 | The convention 9 ≡ 0 throws away the circumscribed and mutilates the inscribed | Th | 20 |
| CP 2.3 | DR ∘ DR = DR | Th | 20 |
| CP 2.4 | Decompression in two steps: 10³ − 1, 10¹⁰⁰ − 1, 10¹⁰⁰⁰ − 1 | Th | 20 |
| CP 2.5 | Internal back-projection, the root's regime outside speed | L | 20 |
| CP 3.1 | The bottom (the inner gaze ends) and the ceilingless (k unbounded) | Th | 20 |
| CP 3.2 | The two faces of saudade as exact postures | L | 20 |
| CP 4.1 | An even closure has an occupied centre; the closure 9 has an empty centre | Th | 20 |
| CP 4.2 | The eight digits sum to 36, 36/8 = 4.5, none equals it | Th | 20 |
| CP 4.3 | An unreachable centre is a living attractor | L | 20 |
| CP 5.1 | The inversion of the circle is an involution that exchanges centre and infinity | Th | 20 |
| CP 5.2 | The fifth pair: (4.5; ∞) | L | 20 |
| CP 5.3 | Information bound: the infinite outside in bijection with the finite inside | Th | 20 |
| CP 6.1 | Annexation at full rate: R → R√2 doubles the area | Th | 20 |
| CP 6.2 | The corner of the square lies on the diagonal, which passes through the centre | Th | 20 |
| CP 6.3 | Expansion as ingestion | L | 20 |
| CP 7.1 | Two levels of fixed points: none among the digits, the × 9 table among the tables | Th | 20 |
| CP 7.2 | The tables of the rails sum to 99 999 999; the axis overflows (100 900 899) | Th | 20 |
| CP 7.3 | 99 999 999 = 3² × 11 × 73 × 101 × 137 | Th | 20 |
| CP 7.4 | DR(9k) = 9; 9 + 1 = 10 → 1 | Th | 20 |
| CP 7.5 | The × 9 table, membrane of the fifth pair | L | 20 |
| CP 8 | The table of inversions and the three alignments (9, 10, 11) | Th | 20 |
| CP 9.1 | Every position is a perpendicular bisector | L | 20 |
| CP 9.2 | The fold at every floor | L | 20 |
| CP 9.3 | The cosmological engine: expansion pursues the centre | L | 20 |
| V4.20.1 | Space ingests, time archives: junction with The Law of Sedimentation | L | 20 |
| V4.21.1 | The primordial amputation at six years old: 1 + 1 = 2 presupposes separation | L | 21 |
| V4.21.2 | Subtraction reveals, it does not annihilate (conservation laws) | L | 21 |
| V4.21.3 | The demonised inverse | L | 21 |
| V4.21.4 | The nine-position clock; zero, memory of a completed cycle | L | 21 |
| V4.21.5 | Why nine: writing base and closed structure; bases 12, 16, 60 | L | 21 |
| V4.21.6 | The four pairs, sums 9, differences 7, 5, 3, 1 (functions: L) | Th | 21 |
| V4.21.7 | The barycentre, mirror of itself | L | 21 |
| V4.21.8 | Axiom A0: the root as position in the cycle of nine, three proofs of invariance | Th | 21 |
| V4.21.9 | Rail D is the subgroup of order 3 of the invertible positions | Th | 21 |
| V4.21.10 | G = 2D; G × G = D, D × D = D, D × G = G | Th | 21 |
| V4.21.11 | The axis is the ideal generated by 3 | Th | 21 |
| V4.21.12 | The squares of primes on D, six cases | Th | 21 |
| V4.21.13 | The table of nine writes the four pairs then turns them over | Th | 21 |
| V4.21.14 | The multiplication table in roots: rows a and 9 − a inverse and reversed | Th | 21 |
| V4.21.15 | The paths of ½: six exact, the fifth withdrawn | Th | 21 |
| V4.21.16 | The ½ of spin, of the critical line, of the equilateral triangle | L | 21 |
| V4.21.17 | 4.5 × doubled gaps: 9, 27, 45, 63 (common difference 18) | Th | 21 |
| V4.21.18 | The rails as dynamic cycles around their centres 4, 5, 6 (biological readings: L) | Th | 21 |
| V4.21.19 | 234 + 432 = 666; 567 + 765 = 1 332; 891 + 198 = 1 089 = 33² | Th | 21 |
| V4.21.20 | The extremes of the rows of the block reveal rail D; 6k + 1 on 1, 7, 4 | Th | 21 |
| V4.21.21 | The triplet 578 is missing from the multiples of 123; 5 × 7 × 8 = 280, 5 + 7 + 8 = 20 | Th | 21 |
| V4.21.22 | Four triplets, four dimensions: space, memory, regulation, motion-continuity | L | 21 |
| V4.21.23 | The cascade of the fifteen ruptures (summary; Volume 6) | L | 21 |
Count. 285 numbered statements: 181 Th, 91 L, 9 P, 3 Open site, 1 identified F (the facts of received science cited in the text under the mark F are not numbered, except this one, which carries a conclusion of the Collatz dossier). Theorems 19 and 20 of Tome II are reserved without content (Appendix B).
Chapter 22. The congruence of pairs: the mod that keeps the inverses (piece entered word for word). The statuses are those the piece itself carries, section by section; no statement has been renumbered.