Number Theory

Contents

1. Foreword
2. Part I. The sheet and the cadastre
3. Chapter 1. The sheet 1 to 30: the genesis document
   • 1.1 The protocol
   • 1.2 The two founding observations
   • 1.3 The anatomical witness: the two hands face to face
   • 1.4 The pedagogical prediction
   • 1.5 Genealogy of the blindness: three datable strata
   • 1.6 What is known, what is claimed
   • 1.7 The plate
4. Chapter 2. The cadastre: nine positions, one involution, two rails and one axis
   • 2.1 Definition of the cadastre
   • 2.2 The table of inversions: the original demonstration of ½
   • 2.3 The master table of the cadastre
   • 2.4 The twisted block: the axis collects all the sums
   • 2.5 The pair of inversions: 4 and 5
   • 2.6 The priority of the cadastre: Axiom A0 and base invariance
5. Part II. The primes never hidden
6. Chapter 3. The self-generating grid and the four families
   • 3.1 The grid, the pairs, the delfines
   • 3.2 Theorem 1: the self-generation of the grid
   • 3.3 Theorem 2: the four families exhaust the delfines
   • 3.4 Theorem 3: the prime as birth by subtraction of families
   • 3.5 Theorem 4: the decision at the balance point
7. Chapter 4. The squares of primes: the 24 rhythm and the chrysalis
   • 4.1 Theorem 5: the 24 rhythm
   • 4.2 Theorem 6: the chrysalis
8. Chapter 5. The twins: the predicate J(k) and the dynamics of waves
   • 5.1 Theorem 7: the exact characterisation of the twins
   • 5.2 Theorem 8: the disjunction of the rails
   • 5.3 The wave dynamics: the exact state of the infinitude open site
9. Chapter 6. The register: the successor function, forward pointing, the φ folding, the joint and the gears
   • 6.1 Theorem 11: the successor function
   • 6.2 Theorem 12: the forward-pointing machine
   • 6.3 Theorem 13: the floor jump and the φ folding of the register
   • 6.4 Theorem 14: the joint of the pairs and the keystone of the register
   • 6.5 The system of state equations of the register
   • 6.6 Theorems 15 and 16: the bridge and the gears
10. Chapter 7. The Root-9 invariant, Riemann in summary, and the synthesis of Tome I
    • 7.1 Theorem 10: the Root-9 invariant of the inverses
    • 7.2 Theorem 9: the correspondence between the involution and the functional equation
    • 7.3 The amnesiac monster: the critique of the Riemannian edifice, in summary
    • 7.4 The incarnation principle
    • 7.5 The Stevin tail test
    • 7.6 Synthesis of Tome I
11. Part III. Collatz: numbers in motion
12. Chapter 8. Collatz in the cadastre: compression, rotation, and the strangeness of the two bricks
    • 8.1 Compression and absorption: Theorem 17
    • 8.2 The rotation of the halves: Theorem 18
    • 8.3 The attractor (1, 8) and the digital root channel
    • 8.4 The erratum of 8 July 2026: what is withdrawn, and why
    • 8.5 What survives: the quarter dichotomy and the growth chains
    • 8.6 Theorem 23: the strangeness of the two bricks
    • 8.7 The falsification criteria of the dossier
13. Chapter 9. The +1 that gives back the half, the exact balance of the two bricks, gate 8
    • 9.1 Theorem 24: the +1 gives back the amputated half
    • 9.2 Theorem 25: the exact balance of the two bricks
    • 9.3 Theorem 26: gate 8 at the integer level
    • 9.4 Reading 27: the signature of 7
    • 9.5 The corrections to version 2.1 of the Collatz dossier
    • 9.6 Reading 28: the bridge between Theorem 23 and the amorphism theorem
    • 9.7 The root is the number, and novelty sets free
14. Chapter 10. The meeting of the mirror, the Collatz rope and the spiral
    • 10.1 Theorem 29: the meeting of the mirror
    • 10.2 Theorem 37: the Collatz rope
    • 10.3 Reading 35: the spiral, the circle is the projection, the +1 is the height
    • 10.4 Reading 36: 46 and 64, the two hands
    • 10.5 Reading 38: the return is a datum, the proof an invention
    • 10.6 What remains open
15. Part IV. Goldbach
16. Chapter 11. Goldbach as a cadastral partition
    • 11.1 The three problems and the three channels (version 1.0)
    • 11.2 Theorem 30: the third and the two thirds around the empty centre
    • 11.3 Theorem 31: the sieve is a mirror
    • 11.4 Theorem 32: the gears, and where the gap lives
    • 11.5 Reading 33: the deviation is bounded and decreases
    • 11.6 Reading 34: the house, first identifier
17. Part V. The additions of version 0.5
18. Chapter 12. The amplitude of the beats, the product of cascades, the four moments
    • 12.1 Theorem 39: the amplitude of the beats, in closed form
    • 12.2 Theorem 40: the product of cascades, with its carries
    • 12.3 Theorem 41: reading the four pairs separately
    • 12.4 The lexical errata of version 0.4
19. Part VI. 24 and its neighbours
20. Chapter 13. The more-than-perfect number: 24, its middle, its mirror and its shell
    • 13.1 The central identity, and its uniqueness
    • 13.2 The folding carries the whole axis
    • 13.3 The arrival is the square of the first perfect number
    • 13.4 The twins and determinism
    • 13.5 What 24 already is, elsewhere in the corpus
    • 13.6 The two steps of the interface
    • 13.7 Matter: the half filling and the full shell
    • 13.8 The three mirror pairs, and the test that fails
    • 13.9 The open site read (7 October 2026)
    • 13.10 What this chapter does not say
21. Chapter 14. The twins and the two lattices: why 5 and 7 come back
    • 14.1 The twins are the heads of the two rails
    • 14.2 The plane turns in only two ways: 4 and 6
    • 14.3 Each prime chooses its lattice, and the twins choose the other
    • 14.4 The five solids and the identity of 24
    • 14.5 The numbers that carry their Fibonacci
    • 14.6 Where nature counts: crystals, and the forbidden twin
    • 14.7 What this chapter does not say, and the receipt it asks for
22. Chapter 15. The cadastral cascade and the sun spiral
    • 15.1 The cascade: a descent that keeps memory
    • 15.2 Three properties
    • 15.3 The sun spiral
    • 15.4 The cascade reveals the operator
    • 15.5 Anchoring: what the cascade joins
    • 15.6 Delimitation: the known, the new, the open
23. Chapter 16. The cascade and the bell: Huygens, the circle of six, and the bell kept as a count
    • 16.1 Recto: Huygens
    • 16.2 Verso: the circle of six and the cascade
    • 16.3 The edge restores everything
    • 16.4 XXXX gives 9 − X, in one line
    • 16.5 The octet and the axis
    • 16.6 The sharing, in one sentence
24. Part VII. The numbers that remember
25. Chapter 17. The numbers that remember: Kaprekar and the arithmetic of memory
    • 17.1 The two-floor theorem
    • 17.2 The pair theorem: the shared tail
    • 17.3 Generation by the pure nines
    • 17.4 142857, the cycle that remembers itself
    • 17.5 The sums of the cadastre at every depth
    • 17.6 The closed triptych
26. Chapter 18. The 196 enigma: why a number never closes
    • 18.1 The two scaffoldings: 9 at the difference, 11 at the sum
    • 18.2 The doubling of the root: the 142857 cycle
    • 18.3 The three-digit fugitives: one thread, five gates
    • 18.4 The mechanism of the flight: the perpetual carry
    • 18.5 The missing half: 49, 49.5, 50
    • 18.6 The complete dipole
    • 18.7 The epistemic status
27. Chapter 19. 198, 496, 1729 and the fractal bridge: the illusion of recreational mathematics
    • 19.1 The algebraic law of mirrors
    • 19.2 The two blocks
    • 19.3 The genealogy from the founders
    • 19.4 496 and string theory
    • 19.5 1729, inner product of 198
    • 19.6 The rhythm of the squares, the ferromagnetic descent, the polyhedra
    • 19.7 137, 361 and the dimensions of strings
    • 19.8 The fractal bridge: from biology to physics by inserting the 9
    • 19.9 The predictions of the fractal bridge
    • 19.10 What these two articles say, and what they do not say
    • 19.11 Rereadings of the cadastre: the cube, Fermat, Mersenne, Fibonacci and 495
28. Part VIII. The fifth pair
29. Chapter 20. The fifth pair: the core and infinity as partners in involution
    • 20.1 The number with two gazes: inscribed and circumscribed
    • 20.2 The asymmetry of the two gazes
    • 20.3 The empty centre as engine
    • 20.4 The fifth pair: the core and infinity
    • 20.5 The annexation theorem: every expansion pulls the outside inward
    • 20.6 The × 9 membrane: the fixed circle realised in arithmetic
    • 20.7 The table of inversions
    • 20.8 Unified readings
    • 20.9 The falsification criteria
    • 20.10 The sister piece: time that archives
30. Part IX. Book II of the Quantum Bible
31. Chapter 21. What Book II of the Quantum Bible brings
    • 21.1 The primordial amputation, at six years old
    • 21.2 The nine-position clock, and zero
    • 21.3 The four pairs and their functions
    • 21.4 Axiom A0 and its three proofs
    • 21.5 The multiplicative rails and the regulating axis
    • 21.6 The cadastral multiplication table
    • 21.7 The paths of ½
    • 21.8 The three rails as dynamic cycles
    • 21.9 The mirror triplets
    • 21.10 The self-reflection of the 3 × 3 square
    • 21.11 The secret triplet 578
    • 21.12 The cascade of the fifteen ruptures (summary)
    • Coda
32. Chapter 22. The congruence of pairs: the mod that keeps the inverses
    • 0. Statuses and conventions
    • 1. What Gauss's mod throws away
    • 2. The congruence that keeps the pairs
    • 3. Theorem 1: a single internal operation
    • 4. Theorem 2: the congruence is realised by the square
    • 5. Theorem 3: cyclic of order three
    • 6. What the true mod is
    • Appendices
33. Appendix A. Table of statements
34. Appendix B. Withdrawals and corrections
    • B.1 Withdrawals dated by the sources
    • B.2 Downgrades made for this volume
    • B.3 Corrections of fact and of wording
    • B.4 Numbering and form
    • B.5 Corrected wordings from the rereadings of 15 October 2026
35. Appendix C. Merged sources
36. Appendix D. Verifications redone
37. Colophon