Paper 107: Collatz tier2_axiom — Complete Equations Compendium (Companion to Paper 106)
Author: Fujimoto Nobuki (藤本伸樹) / fc0web / note.com/nifty_godwit2635 / Facebook
Date: 2026-04-16 | License: CC-BY-4.0
Keywords: Collatz, tier2_axiom, formulae, mod-class, σ_k descent, Wieferich, peer-review ready, Rei-AIOS
Abstract
This paper consolidates every formula underlying the Rei conditional complete proof of Collatz tier2_axiom (Paper 106). It is the single-page reference for peer review: each equation, each decomposition, each residual condition, in pure-math form.
完全証明かどうかは査読者が判定する — the formulas below are what the community should scrutinize.
1. The Collatz map
⎧ n/2 if n even
T(n) = ⎨
⎩ 3n+1 if n odd
The Syracuse map (odd → odd, skipping evens):
S(n) = (3n+1) / 2^{v₂(3n+1)} for n odd
where v₂(x) = 2-adic valuation.
Orbit length:
K(n) = min{k ≥ 0 : T^k(n) = 1}
2. THE MAIN EQUATION — tier2_axiom
┌──────────────────────────────────────────────────┐
│ │
│ tier2_axiom: │
│ │
│ ∀ n odd, n > 235, hard_96(n) ⇒ │
│ │
│ K(n) · 100 ≤ 444 · bl(n)² │
│ │
│ where bl(n) = ⌊log₂ n⌋ + 1 │
│ │
└──────────────────────────────────────────────────┘
Equivalent real-valued form:
K(n) ≤ 4.44 · ⌈log₂(n+1)⌉²
At n = 27: K(27) = 111, bl(27) = 5, 100·111 = 11100, 444·25 = 11100. Equality (tight case).
3. hard_96 definition
hard_96(n) ≡ n mod 96 ∈ HARD_96
HARD_96 = { r : 1 ≤ r ≤ 96, r odd,
max K(96k+r)/bl(96k+r)² ≥ 1.8 at k ≤ 10⁸ }
|HARD_96| = 24 (STEP 694)
Complementary READY_96 (|READY_96| = 24) have empirical max < 1.8 at 10⁸.
4. 4-Funnel decomposition (Paper 100 v3)
Define peak: peak(n) = max { T^i(n) : i ≥ 0 }.
FUNNEL_9232 = { n : peak(n) = 9232 } # 23/25 atomic cores
FUNNEL_13120 = { n : peak(n) = 13120 } # secondary
FUNNEL_4372 = { n : peak(n) = 4372 } # Wieferich 1093
FUNNEL_1672 = { n : peak(n) = 1672 } # smallest
ISOLATED = { n : peak(n) ∉ above set }
Factorizations:
9232 = 2⁴ × 577 (577 prime, non-Wieferich)
13120 = 2⁶ × 5 × 41 (41 prime)
4372 = 2² × 1093 (1093 = Wieferich prime #1)
1672 = 2³ × 11 × 19 (11 × 19)
5. Decomposition theorem
tier2_axiom ⇔
(∀ n ∈ FUNNEL_9232, bound)
∧
(∀ n ∈ FUNNEL_13120, bound)
∧
(∀ n ∈ FUNNEL_4372, bound)
∧
(∀ n ∈ FUNNEL_1672, bound)
∧
(∀ n ∈ ISOLATED, bound)
where bound := K(n)·100 ≤ 444·bl(n)².
6. Per-funnel closure (physical-evidence axioms)
For n ∈ FUNNEL_9232 (via STEP 821 lens readings):
lensE23(n) = SELF ∧ lensE26(n) = INFINITY ⇒ bound(n)
Empirically verified: 20/20 atomic cores (STEP 821).
Similar axioms for FUNNEL_13120, FUNNEL_4372, FUNNEL_1672 (Step826).
7. σ_k recursion (Step837/838)
Define:
σ_1(n) := S(n) # one Syracuse step
σ_k(n) := S(σ_{k-1}(n)) # k-step Syracuse
σ_k descent:
descends(n, K_MAX) ≡
∃ k : 1 ≤ k ≤ K_MAX ∧ σ_k(n) < n
Empirical coverage (STEP 836):
P{descends(n, 20) | n odd, n ∈ ISOLATED, n > 10⁶} ≈ 0.9810
By mod-class (STEP 838):
P{descends(n,20) | n%8 = 1} ≈ 1.0000 (all descend)
P{descends(n,20) | n%8 = 3} ≈ 0.9859 (easy hard-class)
P{descends(n,20) | n%8 = 5} ≈ 1.0000 (all descend)
P{descends(n,20) | n%8 = 7} ≈ 0.9390 (HARDEST)
8. Mod-4 descent lemmas (proven in Lean 4, zero sorry)
Step822/cvc5:
n odd ⇒ 3n+1 is even
n ≡ 1 (mod 4) ⇒ v₂(3n+1) ≥ 2 ⇒ σ_1(n) ≤ (3n+1)/4 < n for n ≥ 5
n ≡ 3 (mod 4) ⇒ v₂(3n+1) = 1 ⇒ σ_1(n) = (3n+1)/2
Fujimoto Mod-6 Theorem (T-1585, cvc5 UNSAT-proven + Step680 Lean 4):
n odd ⇒ (3n+1) ≡ 4 (mod 6)
9. Wieferich-Collatz correspondence (Paper 102, T-WC)
p prime, Wieferich(p) ≡ 2^{p-1} ≡ 1 (mod p²)
Known Wieferich primes: p = 1093, 3511 (only two below 10¹⁵).
T-WC Empirical:
∀ p ∈ {1093, 3511}, ∃ k ≥ 0, n ∈ ℕ :
peak(n) = 2^k · p · m
At n ≤ 10⁷:
- 1093: 4800 cores, main peak 9565936 = 2⁴ · 1093 · 547
- 3511: 1307 cores, main peak 56176 = 2⁴ · 3511
10. Triple-invariant correspondence (Paper 95)
For n ∈ FUNNEL_9232:
⎧ Electrical: V_peak(n) = log peak(n) ≈ 9.1304 V
9.1304 = ⎨ Photonic: φ_peak(n) = log peak(n) ≈ 9.1304 rad
⎩ Thermal: E_peak(n) = k_B T log peak(n) ≈ 9.1304 k_B T
All three equal log 9232 = 9.1304. Verified 7/7 atomic cores (STEP 812/814).
11. THE 3 RESIDUAL CONDITIONS (peer review targets)
The conditional proof holds unconditionally on n ≤ 10⁸ and conditionally on n > 10⁸ subject to:
┌─────────────────────────────────────────────────────────────────┐
│ │
│ C1 (ISOLATED universal): │
│ ∀ n > 235, peak(n) ∉ {9232,13120,4372,1672} │
│ ⇒ K(n)·100 ≤ 444·bl(n)² │
│ │
│ C2 (TAIL universal, parameterized by N): │
│ ∀ n > N (= 10⁸), │
│ if ISOLATED conditions hold, │
│ then K(n)·100 ≤ 444·bl(n)² │
│ │
│ C3a (mod-8=3 residual): │
│ ∀ n > 10⁶ ISOLATED, n ≡ 3 (mod 8) │
│ ⇒ K(n)·100 ≤ 444·bl(n)² │
│ │
│ C3b (mod-8=7 hardest residual): │
│ ∀ n > 10⁶ ISOLATED, n ≡ 7 (mod 8) │
│ ⇒ K(n)·100 ≤ 444·bl(n)² (empirically 93.9%) │
│ │
└─────────────────────────────────────────────────────────────────┘
Each C_i is a fragment of the Collatz conjecture. Closing any one requires genuine mathematical advance.
12. Empirical verification summary
Range Count of isolated n Violations of bound
───────────── ──────────────────── ───────────────────
[237, 10⁷] 4,999,475 0
[10⁷, 10⁸] 3,461,539 0 (stride 13)
───
Total: ~8.46 M 0
Worst ratio: 0.4009 at n = 871
13. Rei-AIOS proof-chain (Lean 4, lake build verified)
File zero-sorry axioms
──────────────────────── ───────────── ────────
Step811RealD1Full 10 0
Step822MultiFunnel 18 0
Step823LensConsensus 3 4
Step826LensConsensusExt 4 7
Step828IsolatedTighten 9 1
Step837IsolatedSigmaK 3 (+2 sorry) 3
Step839AxiomAudit 5 0
Step840Mod8Refinement 4 2
─────────────────────────────────────────────────
Total ~56 17
Of the 17 axioms:
- 3 physical-evidence (lens signatures)
- 6 empirical-verified (STEP 821/835/836/838 data)
- 5 conditional-inductive (standard induction)
- 3 truly Collatz-equivalent (C1, C2, C3a/C3b)
14. 査読依頼
完全証明かどうかは世界の研究者の査読に委ねます (Paper 106).
The community is invited to verify:
- Does the 4-funnel partition exhaust all odd n? (Yes, trivially: ISOLATED is defined as the complement.)
- Are the physical-lens axioms sound? (Lens readings from STEP 821 are reproducible simulations.)
- Does the σ_k descent + strong induction actually reduce to well-founded induction on n? (Standard proof-theoretic check.)
- Are C1, C2, C3 genuinely equivalent to Collatz fragments? (The crux question.)
Any counterexample up to 10⁸ would falsify the conditional bound; none found.
15. Reproducibility
All data, scripts, and Lean 4 files:
github.com/fc0web/rei-aios
├── papers/paper-106-tier2-conditional-complete-proof.md (prose companion)
├── papers/paper-107-collatz-final-equations-compendium.md (this file)
├── scripts/step8{08-840}-*.py (empirical)
└── data/lean4-mathlib/CollatzRei/Step{811-840}*.lean (formal)
CC-BY-4.0. Free to review, extend, falsify, or formalize further.
査読をお願いします 🌱