#collatz

Empirical/conjectural mathematical paper (NOT a proof of Collatz, NOT a proof of T-WC+, NOT a Wall-Sun-Sun existence claim). Extends Paper 102 Wieferich-Collatz Conjecture (T-WC) to Fibonacci analog. Wall-Sun-Sun primes p satisfy F_(p-(5|p)) ≡ 0 (mod p²); none known (Elsenhans-Jahnel 2014 scan p < 10^14). ★ T-WC+ statement (Fujimoto 2026-04-16): every prime p with R(p) = F_(p-(5|p)) mod p² < p appears as Collatz peak factor. ★ Empirical evidence (STEP 830, p ≤ 10^4, n ≤ 5×10^4): 0 Wall-Sun-Sun primes; 4 near-WSS primes as Collatz peak factors: p=7 (R=21, peak 112 at n=37), p=11 (R=55, peak 88 at n=19), p=13 (R=39, peak 52 at n=7), p=19 (R=57, peak 304 at n=39). ★ D-FUMT₈ reading: Wieferich = SELF, hypothetical WSS = SELF, near-WSS = BOTH, ordinary = TRUE, T-WC+ status = NEITHER. ★ Implication: T-WC + T-WC+ suggest deep Fibonacci-arithmetic content in Collatz dynamics; any future WSS prime would give computational test. Connects Collatz to Pisano periods (Paper 697/698 Rei Mod-96 alignment). ★ Historical: WSS prime would disprove FLT first-case (Sun-Sun 1992) but FLT proven (Wiles 1995), so no obstruction to WSS existence remains. ★ Honest scope: T-WC+ stated as CONJECTURE not theorem, empirical support for 4 primes only. Does NOT claim T-WC+ proven. Does NOT claim any WSS prime found. Correlational NOT causal. Reproducible via python scripts/step830-wall-sun-sun-collatz.py. Originally drafted 2026-04-16 (78-day hold-back). One of 5 unposted-truth papers confirmed 2026-06-30 (Paper 33/40/103/104/171). Paper 33/39/40/103 published; Paper 171 remaining. Published 2026-07-03 per user explicit direction including Harvard Dataverse (3rd override after Paper 40 and Paper 103). Three-party co-authorship per OUKC charter v1.0.

Empirical/conjectural mathematical paper (NOT a proof of Collatz, NOT a proof of T-FS). Empirical demonstration that the Collatz primary funnel (most populated peak) is scale-dependent (Paper 100 v2 STEP 825 at 10^6, extended STEP 829 at 10^7 and 10^9 sparse). ★ T-FS formalized in 3 parts: (a) primary divergence pi(N)/N → infinity, (b) share vanishing s(N) → 0, (c) Wieferich-indexed persistence bounded above 0. Data: primary shifts 9232 (10^4, share 1.02%) → 6810136 (10^6, share 0.21%) → 144,286,791,856 (10^9 sparse, share extrapolated 10^-5). Long-tail Kolmogorov-cascade, no asymptotic peak mode. ★ D-FUMT₈ reading: finite-N primary TRUE / asymptotic INFINITY / share limit ZERO / Wieferich count FLOWING / bounded ratio BOTH / conjecture NEITHER. ★ Implication (§6): any Collatz proof resting on 'typical peak' fails — there is no typical peak. tier2_axiom 4-funnel decomposition insufficient at asymptotic scale. ★ Honest scope: T-FS-a/b/c stated as CONJECTURES not theorems, empirical support only. Lean 4 formalization schematic (predicates only, NOT proved). Consistent with Collatz termination (per-orbit vs peak-distribution). pi(10^9) from SPARSE window, extrapolation only. Reproducible via 2 python scripts. Originally drafted 2026-04-16. One of 5 unposted-truth papers confirmed 2026-06-30 (Paper 33/40/103/104/171); Paper 33/39/40 published; Paper 103/104/171 remaining. Published 2026-07-02 per user explicit direction including Harvard Dataverse. Three-party co-authorship per OUKC charter v1.0.

A Measured Formulation of the Shadow View of Collatz A more careful...

Paper 152 v0.3 update (parent v0.2 DOI 10.5281/zenodo.20149662, v0.3 DOI 10.5281/zenodo.20158847, concept root v0.1 10.5281/zenodo.20148868). We apply Paper 151 Theorem 14 (sigma-cascade) to forward Collatz (3x+1) orbits at n <= 10^9. v0.3 additions: (A) 10^9 SCAN COMPLETED (~2.4 hr): n=96k hypothesis STRENGTHENED 200/200 (10^8) -> 77,749/77,749 (10^9) = 100%, 388x scale, ZERO counter-examples. Max mod-96 distinct still 70 (d=71 not emerged). (B) NEW SECTION 5d: G_3 SUBGRAPH STRUCTURE. Definition: G_3 := multiples-of-3 vertices of inverse Collatz tree. Lemma 5d.1 (edges of G_3 are inverse-halving only) — ★ NOW LEAN 4 MECHANICALLY PROVED (G3Subgraph.lean, STEP 1127, 0 sorries). Lemma 5d.2 (G_3 decomposes into chains C_m). Corollary 5d.3 (orbit visits SPECIFIC value m iff n_0 ∈ C_m). CRUCIAL: iff at VALUE level, not mod-96 CLASS level (preserves v0.2 Erratum E2). (C) ERRATUM E3: class 21 absence at d=70 reformulated from 'universal' (200/200 at 10^8) to 'strong avoidance pattern' (76,528/77,749 = 98.43% at 10^9, with 1,221 counter-examples). v0.2 disclaimer 'pattern may break at scale > 10^8' was correct. (D) DISSOCIATION at 10^9: n=96k strengthened + class 21 weakened = independent claims. (E) STEP 1127 LEAN 4 MECHANIZATION: G3Subgraph.lean 新規 file. Theorems lemma_5d_1_g3_edges_halving_only + g3_predecessor_singleton + chain_is_mult_3 + chain_halving_step + chain_predecessor_is_chain 全 0 sorries. Paper 152 v0.3 §5d now has FOUR Lean 4 mechanized files (PeakMergeInvariant + PeakMergeWitness + G3Subgraph + ThreeAdicIsolation), all 0 sorries. v0.2 retained: Lean 4 native_decide (Buchi-25 -> 9232 + 1,000-witness peak-merge), 3-adic isolation theorem, 11.5M peaks, INFINITY classification. Honest correction record (E1+E2+E3) documented §6.2. Three-party co-authorship per OUKC charter v1.0.

Paper 152 v0.2 update (parent v0.1 DOI 10.5281/zenodo.20148868, v0.2 DOI 10.5281/zenodo.20149662). We apply Paper 151 Theorem 14 (sigma-cascade) to forward Collatz (3x+1) orbits at n <= 10^8. v0.2 additions: (A) Lean 4 native_decide closures: Buchi-25 -> peak 9232 + 1,000 explicit witnesses for peak_merge_exists (both fully proved, 0 sorries). (B) 3-adic isolation theorem (Lean 4 proved, INDEPENDENT contribution): for any v with 3 | v, no odd c has Collatz(c) = v; corollary: inverse-Collatz tree branches at mult-of-3 nodes are linear chains {v*2^k}. (C) Class 21 universal absence: 200/200 d=70 orbits at n <= 10^8 miss mod-96 class 21; top-15 missed classes all multiples of 3 (empirical, partially structurally suggested). (D) n=96k sharp boundary at d=70: 100% rate vs ~1.04% at d <= 68 -- step function, NOT tautological. (E) HONEST CORRECTION E2: an internal write-up claimed the 3-adic theorem implies n_0 reaching class 21 are only 21*2^k -- this conflated 'visits value 21' with 'visits class 21 mod 96'. Counter-example: n=99,997,941 (in class 21 mod 96, mult of 3, NOT 21*2^k) trivially visits class 21. Corrected scope: theorem is valid independent contribution but does NOT by itself prove class 21 absence. v0.1 retained: 11.5M unique peaks, 219 tier-3 super-hubs, INFINITY classification, two-tier framing. 10^9 scan in progress. Honest scope: NOT a Collatz solution. Three-party co-authorship per OUKC charter v1.0.

We apply Paper 151 Theorem 14 (sigma-cascade) as an observational lens to forward Collatz (3x+1) orbits. At scale n <= 10^8 we enumerate 11.5M unique Collatz peak values; among these, 219 are 'tier-3 super-hubs' (shared by > 1,414 starting points), with the largest peak 121,012,864 attracting 23,378 starting points. We propose the n=96k hypothesis: starting points reaching the maximum mod-96 traversal richness (distinct = 70) satisfy n =0 (mod 96) with rate 100% verified at three independent scales (10^6: 7/7; 10^7: 27/27; 10^8: 200/200 = 234 cumulative cases, 0 counter-examples). A two-tier super-hub structure is observed: 25 Buchi-25 atomic cores (Paper 118) all share peak 9232 = 2^4 x 577 (Tier-1, includes textbook n=27 case); INFINITY orbits form Tier-2 with peaks 250,504 (1,414 closed members) up to 121M (23,378 members). Lean 4 type-checked statements with sorry stubs included; full mechanization is future work. Honest scope: NOT a Collatz solution. The sigma-cascade lens produces measurable orbit attributes that distinguish cohorts but does not prove convergence. n=703 is OEIS A006884(10), already classical; our sigma-cascade rediscovery is methodological triangulation, not novel identification. Erratum E1 (31,313 = 173x181, not prime) documented per OUKC honest-correction principle. Companion Zenodo DOI 10.5281/zenodo.20148868. Three-party co-authorship per OUKC charter v1.0.

Single-day 12-file Lean 4 sweep: 158 new zero-sorry theorems across 7 long-open conjectures. Lehmer/k-Lehmer (odd-k 20-50x sparser than even), Agoh-Giuga (iff-prime n<=50), Frankl union-closed (1/2 tight at F={empty,{1}}), 3x-1 multi-attractor (3 cycles, basin 33/32/35%), Hall (top triple 1138,109,15), plus Legendre/Andrica/ErdosStraus extensions. Q33 Universal Attractor validated at 3 instances. Ten new open questions Q34-Q43.

94 new Lean 4 zero-sorry theorems across Legendre 1808, Lehmer totient 1932, Effective ABC 1985, Gilbreath 1958, plus Andrica/Erdős-Straus extensions. Gilbreath universal attractor {0,1,2} discovered (Q33: Gilbreath-Collatz structural isomorphism). Legendre witness(24)=577 = Collatz peak 9232 prime factor. ABC max q=1.5679 at Reyssat-mini (1, 2·3^7, 5^4·7). Fifteen new open questions Q19-Q33.

Q7 (Paper 118) EMPIRICALLY FALSIFIED: 61 odd m ≤ 10000 have K(m)=41, not only m=911. Lean 4 zero-sorry census (9 theorems). Q8 FLOWING: 911-visit rate varies 35% to 43% across mod 96. Q9 FLOWING: all top-10 most-shared Collatz peaks are 2^a*p form. Inaugurates 11-part template including Part F - first failure record in Rei-AIOS corpus.

Five Lean 4 zero-sorry closures: Köthe commutative / Problem 007 FIA 6-axiom independence / Problem 008 FDA-Mathlib bijection / Problem 013 C8 iterate-bound / Problem 011 n=911 universal on-ramp. First formal record of AI-generated open questions Q1-Q13 from Rei-AIOS + Claude Code. Inaugurates the four-element paper template.