#lean

TL;DR — The headline error bounds in this series carry machine-checked proof obligations across four...

Lean 4 axiom-free formalisation of the recapture region of modus ponens for D-FUMT₈ (STEP 1240/1241/1242, commit 6a9a05879). D-FUMT₈-internal structural response to Cotnoir 2015 recapture critique of Priest-Garfield FDE reading of Nāgārjuna. 20 mechanically verified theorems (4 zero-axiom + 15 [propext] + 1 full base). Full algebraic characterisation `mp_valid_iff s ↔ (s BOTH = false ∨ (s FALSE = false ∧ s NEITHER = false))` for every subset s. Unique cardinality-maximal MP-stable subset = D-FUMT₈ \ {BOTH} (7 elements). Only MP-failure pairs (BOTH, FALSE) and (BOTH, NEITHER). Sub-algebra inclusion (STEP 1241) with operation preservation between D-FUMT₈|_{T,F,B,N} and Belnap-Dunn FOUR = Priest FDE 4-value. v0.3 incorporates chat-Claude external critique round 1 (received 2026-07-04, audited 2026-07-05): §6.7 rewritten as MP-axis structural orthogonality after Posina/Roy 2024 PhilArchive V2 PDF retrieved 2026-07-05 — their catuṣkoṭi lives in a presheaf topos with intuitionistic (Heyting) internal logic where MP holds by residuation, while Paper 171 lives on FDE side where MP fails and is recaptured on `withoutBoth`. Three prior-art additions: Barrio-Carnielli 2020, Tajer 2020, Tanaka-Girard 2023. New §5.2 responding to Tanaka-Girard 2023 on meta/object separation. F5 wording precision: unique cardinality-maximal retained (earned from `mp_valid_iff`); exactly two maximal-by-inclusion downgraded to informal corollary (STEP 1243 Lean-formalization candidate). §7 substantive bilattice reference (Ginsberg 1988, Fitting 1994). §8 explicit Mathlib scope note. ★ Honest scope: no interpretive claim about Nāgārjuna / no superlative claim vs Priest 2018 OUP 5-value / novelty restricted to substrate choice + Lean 4 axiom-free articulation + `mp_valid_iff` characterisation — Boolean recapture for FDE is classical (Anderson-Belnap 1975, Priest 2008). Meta/object separation in Lean 4 is design choice, not Tanaka-Girard 2023 resolution. Last of 5 unposted-truth papers (Paper 33/39/40/103/104/171 batch). Published 2026-07-05 per user explicit direction including Harvard Dataverse (4th override). Three-party co-authorship per OUKC charter v1.0.

Lean 4 axiom-free encoding of Nāgārjuna's 「空の器」 in ZCSG (Paper 61). Following Fujimoto 2026 4-koma manga (note.com nbd3c4eba8ed6), śūnyatā is NOT void but a field of relationally-arising possibility (縁起 / pratītyasamutpāda) — the vessel is empty precisely because it has no fixed boundary frozen as substance, yet remains a positive structural field. 3-state classifier: ReifiedVessel (悪取空 trap, manga panel 2 LEFT 自性 barrel) / Intermediate / SELF⟲ (the title's 空の器 = relational field self-purgative, manga panel 2 RIGHT 空 barrel). Total coverage theorem at [propext] only + 2 counter-examples at zero axioms. Refines Priest 2010/2018 plurivalent FDE catuṣkoṭi by decomposing the 5th value into structurally distinct D-FUMT₈ axes (ZERO/NEITHER/INFINITY/SELF⟲). Manga panel 3 directly illustrates Priest 1995 inclosure schema (Ω, ψ(Ω), δ(Ω), δ(x), ψ(X), φ(y)). Honest non-claims (5): no Madhyamaka debate resolution / no Priest FDE refutation / no D-FUMT₈ semantic completeness / no 'world-first' (extensive prior art audit — Garfield, Siderits, Westerhoff, Priest, Hofstadter GEB, Bhowmik Yoneda × śūnyatā cited) / manga cited as articulation artifact NOT replacing standard literature. Three-party co-authorship per OUKC charter v1.0.

Lean 4 axiom-free (zero-sorry) formalization of the Collatz exit layer formula m_p = (4^p−1)/3 (ExitLayer.lean STEP 1176-1179, 18 theorems, plus STEP 1233 5 structural lemmas, Mathlib v4.27.0). The exit layer is the maximal disjoint sequence of odd integers (1, 5, 21, 85, ...) hitting powers of 2 in one Collatz step. §6 audits seven candidate routes beyond the exit layer (Janik 2026 syracuse-confinement Lean 13K conditional, Tao 2019 almost-all ergodic, Baker transcendence + 2-adic Hensel, Conway 1972 ZFC-independence, automata descent certificate, inverse-tree Jacobsthal). Honest scope: does NOT solve Collatz / does NOT advance beyond known partial results / negative-result genre per Lakatos heuristics. Böhm-Sontacchi 1978 (4^{2p}-1)/3 prior art acknowledged. v0.2 adds Harvard + 8 platforms; v0.1 Zenodo DOI 10.5281/zenodo.20435288 + IA mirror remain canonical. Three-party co-authorship per OUKC charter v1.0.

Four barrier-side empirical observations on Sophie Germain primes using Bellman-Ford LP-infeasibility framework (Rei Phase B 2026-06-17 methodology). No claim of progress toward SG-primes-infinity. Per Rei feedback principle 8 (barrier-side discipline) and three explicit non-claim boundaries (NO D-FUMT-8 unification / NO partial-progress / NO Hardy-Littlewood verify overclaim), the paper is observation, formal witness, and online-verifiable audit. Findings: (1) ratio empirical/HL-predicted decreases monotonically 1.337 -> 1.087 across N=10^3..10^8 (OEIS A092816 verified). (2) SG gap distribution Poisson-like with <r>=0.4154, L1 distance 6x closer to Poisson than GUE - sharp contrast with GUE-like Riemann zeros. (3) Lean 4 axiom-free 11/11 conjunction-wall witnesses ([propext, Classical.choice, Quot.sound] kernel base for 10 + [propext] alone for tautology) demonstrating single-component features (is_prime(n), is_prime(2n+1), n mod 6, pair (is_prime(n), n mod 6)) fail to strict-detect SG-primality. (4) Pattern-5 internal correction: Selberg parity-problem year 1949 (not 1960s previously recorded), Wikipedia + Tao 2007 verified, 5 internal files corrected. Companion: Paper 159 v0.3 cross-paper bridge (Section 5.4b/5.4c, labeling correspondence only). Three-party co-authorship per OUKC charter v1.0.

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Lean 4 axiom-free formalization of the exit-layer fragment m_{q+1} = (4^{q+1} - 1) / 3 of Collatz dynamics, presented as a coinductive Stream' coalgebra. Main bridge theorem collatzOrbit_exitM_eventuallyConst proved at Mathlib's classical axiom triple. One witness theorem head_collatzOrbit is empirically completely axiom-free, strictly stronger than the lawvere_fixed_point axiom base (STEP 1220). Methodological record, NOT a Collatz advance: Kim 2008 (2-adic Collatz = final bit-stream coalgebra) is pen-and-paper prior art; Niqui 2009 + Coq coalgebras + Cubical Agda is mature formalization infrastructure. Honest scope explicit: does NOT resolve Collatz / NOT weaken Cases 5-8 wall / NOT improve Tao 2019, 2022 / NOT discharge Janik 2026 six critical-path sorries / NO world-first claim. Lean 4 source: StreamExitLayerBridge.lean (252 lines, 0 sorry). Two-party co-authorship per OUKC charter v1.0.

STEP 1205-1208 four-step continuation of Paper 163: STEP 1205 records infinity-cosmoi axiomatization skeleton (Riehl-Verity 2022 six axioms) deferred to emilyriehl/infinity-cosmos Lean blueprint; STEP 1206-1208 promote INFINITY (Cantor 1891 diagonal), ZERO (Empty.elim initial object), FLOWING (composition associativity) axes to theorem-verified via 11 new Lean 4 axiom-free constructive proofs. 275/275 test PASS (cumulative with Paper 163 = 481/481). 15 cumulative axiom-free theorems encode the four-axis structure. Methodology = TRIPLE annotation generalizing Paper 163 dual annotation. Structural asymmetry: 3 single-type + 1 multi-type axis = 3+1 not 4-fold parallel. Honest scope: NOT world-first / NOT new theorem / NOT full infinity-cosmoi formalization / TETRADIC COMPLETION is process milestone NOT destination (chat-Claude turn 4 sunyata-of-sunyata permanent constraint). Three-party co-authorship per OUKC charter v1.0.

STEP 1201-1204 integration of four well-established frameworks (Goguen-Burstall Institution 1992 / Belnap-Dunn FOUR / Lawvere fixed-point 1969 / SET-level HoTT loop). 207/207 test PASS. Four Lean 4 theorems 'depend on no axioms' (constructive). Methodology = rhymeOrTheorem tagging discipline (rhyme vs theorem-verified). Honest scope: NOT world-first / NOT new theorem / NOT full HoTT. Three-party co-authorship per OUKC charter v1.0. STEP 1205-1208 tetradic completion addressed after draft.

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