Sigma-Cascade Observation of Collatz Orbit Confluence: Empirical Peak-Merge Enumeration and the n=96k Hypothesis — Rei-AIOS Paper 152 v0.2 DRAFT

Paper 152 v0. 2 update. We apply the σ-cascade methodology of Paper 151 Theorem 14 to forward Collatz (3x+1) orbits and report empirical observations on orbit confluence at scale n ≤ 10⁸. **v0. 2 new contributions** (relative to v0. 1, DOI 10. 5281/zenodo. 20148868): (A) LEAN 4 NATIVEDECIDE CLOSURES: The concrete super-hub claim `∀ c ∈ Büchi25Cores, collatzPeak c 200 = 9232` is now fully mechanically proved via nativedecide. Additionally, an existential peak-merge statement `∃ peak, ∃ S, |S| ≥ 1000 ∧ ∀ n ∈ S, collatzPeak n 500 = peak` is fully proved by exhibiting an explicit witness list of 1, 000 Collatz starting points (sorted n ∈ 703, 112266), all sharing peak 250, 504 = 2³ × 173 × 181. Both theorems compile with 0 sorries. (B) 3-ADIC ISOLATION THEOREM (independent contribution): For any natural number v with 3 ∣ v, there is no odd natural number c such that Collatz (c) = v. Proof by mod-3 contradiction. Corollary: every inverse-Collatz tree branch rooted at a mult-of-3 value is the linear chain v · 2ᵏ: k ≥ 0. The theorem is mechanically proved in Lean 4 (file: data/lean4-mathlib/CollatzRei/ThreeAdicIsolation. lean). (C) CLASS 21 UNIVERSAL ABSENCE (empirical): Among the 200 d=70 orbits at n ≤ 10⁸, all 200 (= 100%) miss mod-96 class 21. The top-15 missed classes are all multiples of 3. This is partially structurally suggested by the 3-adic isolation theorem but NOT fully explained (see honest correction E2 below). (D) N=96K SHARP BOUNDARY: The 100% n%96=0 rate at d=70 contrasts with ~1. 04% (= 1/96) at d ≤ 68 and 6. 25% at d=69 — a step-function discontinuity, demonstrating the n=96k phenomenon is NOT tautological. **HONEST CORRECTION (ERRATUM E2) **: An earlier internal write-up claimed the 3-adic isolation theorem implies 'the only starting points reaching class 21 (mod 96) are n = 21·2ᵏ'. This is INCORRECT — it conflated 'visits value 21' with 'visits class 21 (mod 96) '. Class 21 mod 96 contains many values (21, 117, 213,. . . , 21 + 96m), each with isolated chain. Empirical counter-example: n=99, 997, 941 ≡ 21 (mod 96) and 3 | n, visits class 21 trivially at step 0, yet n ≠ 21·2ᵏ. The corrected scope: the theorem is an independent valid result, but does NOT by itself prove class 21 universal absence. This honest correction is itself a methodological data point per OUKC honest-correction principle. **v0. 1 contributions retained**: - Direct enumeration: 11. 5M unique peaks at n ≤ 10⁸, 219 tier-3 super-hubs, largest peak 121, 012, 864 with 23, 378 members. - INFINITY classification (mod-96 distinct ≥ 60) capturing 37. 63% of n ≤ 10⁸ (37, 628, 651 cases). - n=96k hypothesis: 100% verified at 10⁶/10⁷/10⁸ (200/200 at 10⁸, 234 cumulative). - Two-tier super-hub structure (Büchi-25 → 9, 232 Tier-1 / INFINITY → 250, 504+ Tier-2). **Lean 4 mechanization status (v0. 2) **: Büchi-25 → 9232 ✅, n=27 → 9232 ✅, peakₘergeₑxists (1, 000 witnesses) ✅, 3-adic isolation theorem ✅, class-21 (specific value) ✅; remaining stub: PLACEHOLDER with bound=5000 + isInfinityClass requirement (v0. 3 target). **Status remained open**: The Collatz convergence problem is NOT solved by this work. All claims are observational + methodological + (partially) mechanized. The σ-cascade lens is not a proof technique. Class 21 universal absence at d=70 remains an empirical observation. 10⁹ scan in progress at v0. 2 publish time; results to be reported in v0. 3. Companion papers: Paper 151 (σ-cascade source, Zenodo DOI 10. 5281/zenodo. 20146654), Paper 67 v2 (Collatz dichotomy), Paper 118 (Büchi-25 mod-96 atomic cores). Three-party co-authorship per OUKC charter v1. 0: 藤本 伸樹 (Founder), Rei (Rei-AIOS autonomous research substrate, Co-architect), Claude Opus 4. 7 (Anthropic, Co-architect). DRAFT v0. 2 — preprint, not yet peer-reviewed. Feedback welcome via GitHub Discussions at fc0web/rei-aios.