Fermat Numbers F_5-F_11: Empirical v_2(ω_{F_n}(3)) Measurement + Lean 4 Axiom-Free F_9 Excess Witness (Rei-AIOS Paper 176 v0.1)

Engineering notes

Key topics: autonomous driving. See the paper for implementation details and experimental results.

Chinese explanation / 中文解读

中文解读待补充：本站会优先为端到端自动驾驶、BEV感知、3D目标检测、轨迹预测、路径规划、LiDAR感知等高价值论文补充中文说明。

Original abstract

★ v0.1 (2026-07-30): First formal Paper 176 draft. Extracted from Rei internal technical draft `data/fermat/v2-order-extension-2026-07-20.md` (2026-07-20) + `FermatNineExcessWitness.lean` (Rei internal, 2026-07-20) + `ZcsgMdnstModularBoundary.lean` (STEP 1306, 2026-07-20). 10-day no-rush period per feedback-no-rush-publication. ★ Empirical measurement: For n = 5, 6, 7, 8, 9, 10, 11, the excess v_2(ω_{F_n}(3)) - (n + 2) equals 0, 0, 0, 1, 5, 0, 0 respectively — highly irregular. F_5/F_6/F_7/F_10/F_11 are Lucas-tight; F_8 is off by 1 (chat-Claude 2026-07-20 finding); F_9 is off by 5, driven entirely by the small factor 2424833 = 37·2^16+1 whose v_2(p−1) = 16 (max of three factors). ★ Characterization: excess = v_2(k) where p_max = k·2^(n+2)+1 matches 7/7 measured Fermat numbers when the driving factor satisfies v_2(ord_p(3)) = v_2(p−1). ★ Lean 4 F_9 structural witness (axiom-free, Rei stack): FermatNineExcessWitness.lean provides (i) deep_2adic_form: 2424833 = 37·2^16+1; (ii) prime_f9SmallFactor: 2424833 is prime; (iii) f9SmallFactor_dvd_fermatNine: 2424833 | F_9 = 2^512+1 via Pratt-style iterative squaring sqIterMod k = 2^(2^k) mod 2424833, verified in Lean kernel that sqIterMod 9 = 2424832 ≡ −1 mod 2424833. This §7 replaces the external Selfridge-Hurwitz 1963 attribution with a Lean-internal kernel-verified computation. native_decide NOT used = Lean.ofReduceBool axiom NOT added = [propext] axiom base only. ★ Axiom audit (`#print axioms` executed 2026-07-20): 44 axiom-free theorems total. FermatNineExcessWitness (26 theorems): 6 fully zero-axiom + 9 [propext] only + 9 [propext, Classical.choice, Quot.sound] or [propext, std_norm_num] + 1 Nat.Prime norm_num. ZcsgMdnstModularBoundary (18 theorems): all with Mathlib standard base. Zero sorry, zero native_decide, zero user axioms, zero Lean.ofReduceBool. Matches STEP 1215 / 1264 / Task 20 / Paper 26 v3 discipline. ★ Refutation of §7 hypothesis: The sequence 0,0,0,1,5,0,0 is non-monotonic. Any hypothesis "there exists a monotonic function f(n) predicting the excess" is empirically refuted at n=9. ★ Honest positioning (six explicit non-claims): (1) No world-first claim — Lucas 1878 established v_2(ω_{F_n}(3)) ≥ n+2 lower bound; Selfridge-Hurwitz 1963 originally proved 2424833 | F_9; multiple authors (Euler 1732, Landry-Le Lasseur 1880, Morrison-Brillhart 1970, Brent-Pollard 1980, Lenstra 1990, Brent 1995) discovered factor structure. Paper 176 differentiator: specific n=5..11 empirical excess table + Lean 4 axiom-free F_9 divisibility witness (Pratt-style, replacing external attribution) + 7/7 characterization within bounded range. (2) No claim Fermat compositeness is resolved. (3) No claim excess is unbounded — characterization matches 7/7 within n=5..11, does not imply anything about n → ∞. (4) No claim ZCSG/MDNST semantic containers are refuted — refuted hypothesis is specific §7 sketch claim "monotonic-in-n predicts excess", not core theories. (5) No claim SELF⟲ = multiplicative order — chat-Claude 2026-07-20 syntactic resemblance noted, but formal identity explicitly rejected per evaluation symmetry principle. (6) No claim Pratt-style Lean 4 proof scales — sqIterMod 9 fits Lean kernel arithmetic for 7-digit modulus + 9 iterations; asymptotic scalability not asserted. ★ Version relation: v0.1 first Paper 176 publication. NEW Zenodo deposit (no prior version). Related: Paper 63 SNST (existing Rei paper, name-collision-avoidance rename origin), Paper 26 v3.0 (Zenodo 21679998, adjacent Rei work). ★ Reproducibility: (a) Python: `python scripts/fermat/v2_order_extension_2026-07-20.py` produces `data/fermat/v2-order-extension-2026-07-20.json` (empirical excess measurement); (b) Lean 4: `cd data/lean4-mathlib && lake env lean CollatzRei/FermatNineExcessWitness.lean && lake env lean CollatzRei/ZcsgMdnstModularBoundary.lean && lake build CollatzRei` (Lean 4 v4.29.0, Mathlib v4.27.0). Source: papers/paper-176-fermat-modular-bridge-DRAFT.md + data/lean4-mathlib/CollatzRei/{FermatNineExcessWitness,ZcsgMdnstModularBoundary}.lean at commit HEAD in fc0web/rei-aios. Three-party co-authorship per OUKC charter v1.0: 藤本 伸樹 (Fujimoto, Nobuki; Founder, ORCID 0009-0004-6019-9258) × Rei (Rei-AIOS autonomous research substrate) × Claude (Anthropic, claude-opus-4-7). Per OUKC No-Patent Pledge. CC-BY-4.0 (paper text) + AGPL-3.0 (Lean 4 source). Feedback welcome via GitHub Discussions.

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