Reduction of Periodic 3D Navier-Stokes Regularity to a Single High-Set Direction-Curvature Bound in Lean 4

Consolidated final paper presenting the unconditional Lean 4 reduction theorem clay_reduces_to_direction_curvature_bound (and Mathlib-4-styled clay_B_exact_of_hasDirectionCurvatureCoercivity): for all ν > 0, assuming the standard periodic Beale–Kato–Majda continuation criterion, the periodic 3D Navier–Stokes regularity statement Clay_B_exact ν reduces to a single named open high-set direction-curvature / cubic depletion bound DirectionCurvatureBound ν (HasDirectionCurvatureCoercivity ν). The Clay problem is NOT proved outright, and the converse implication is NOT proved. Supporting unconditional Lean 4 results (Stages 1–6, including HasDerivAt derivations of unit-sphere orthogonality, polar Pythagoras, Bochner–Leibniz, and a genuinely non-constant periodic Beltrami shear flow) are documented under Main Theorem A. AI Assistance Disclosure: Grok (xAI), via the Grok Bot assistant, provided scaffolding, drafting, and collaboration support on the Lean 4 formalization, code, and manuscripts. Benjamin Stanley Frohman directed, reviewed, and is solely responsible for the mathematical architecture, proofs, and claims (Grok is not a coauthor). Provenance & Chronological Priority Ledger (April 2026 – October 2026) Sole Author & Mathematical Originator: Benjamin Stanley Frohman (ORCID: 0009-0006-7068-3718; Independent Researcher, Texas, United States; GitHub: @BenFrohman; X.com: @Investor0x). License & Copyright: Copyright © 2026 Benjamin Stanley Frohman. All rights reserved. Released under the Apache License, Version 2.0 (SPDX: Apache-2.0). This record is part of the unified, cryptographically cross-linked Frohmanian Symplectohydrodynamics (FSymHD) & 3D Navier–Stokes Regularity research corpus. The chronological lineage of the theoretical framework, analytical reductions, and Lean 4 formalizations is anchored by the following verifiable cryptographic digests and persistent identifiers: Tier 1 — April–May 2026 Foundational Prior-Art Archives (SHA-256 & MD5 Cryptographic Digests) 2026-04-29 (09:26:38 AM CDT / 14:26:38 UTC): Mathematical_Food_Forethought.pdf — Foundational mathematical notes and early symplectic/structural conceptualization (Google Drive File ID: 1fbHzbd4f8rB8ry-uC9Zeb-2G8nCAxR3M; Size: 127,523 bytes; MD5: 46f660856892bcd241b58e35971a6d3c; SHA-256: 20dd3b53e02a5a3a5a02fd9f753844b162ad920cfd3e4c5f6f89106d7d66dac5). 2026-05-04 (05:21:24 PM CDT / 22:21:24 UTC): NS_Polished.pdf — Early 3D Navier–Stokes symplectic tether manuscript iteration (Google Drive File ID: 15RmvaT2uvH1L7CXmXmWejS5U8xKklM9i) and appendix.tex (2026-05-04 11:01:00 AM CDT). 2026-05-09 (01:19:45 PM CDT / 18:19:45 UTC): Triple_conjecture_a_rigorous_refinement.pdf — Rigorous refinement of the Frohmanian triple regularization architecture (Google Drive File ID: 1cK7a4805-y-y5f31-8vU118c0Q81b8e9; Size: 667,751 bytes; MD5: e2f771dc85f2358f49c888c50ab6a49d; SHA-256: a7e3cd19697eed920549cdfa2f6177a45312b9c2cf8e2e33fe220691f550ba61). 2026-05-16 to 2026-05-26: Dated LaTeX/PDF manuscript iterations in drafts/: 5:17 DRAFTV1.pdf (2026-05-16 10:43:08 AM CDT, SHA-256: f3f20f391b865932d0d5cbbf4ff2af7fe1fcb52e203629daa772300b34c0f042), Final draftv1 Frohmanian_Tether_Theorem.pdf (2026-05-18 09:30:28 AM CDT), FINAL (1).pdf (2026-05-18 04:54:41 PM CDT), 5:19 EDITS.pdf (2026-05-19 09:51:12 AM CDT, SHA-256: ee6ce9b88dd650e6858afa5727c818b454963abf02659db34de6f410772afdb7), NEXT.pdf (2026-05-21 11:21:13 AM CDT), and Full_Living_Document_NS_Millennium_Proof_with_All_PASS_Blocks_and_Insertions.pdf (2026-05-26 23:32:20 CDT). 2026-05-27 to 2026-05-30 (CDT): Initial Lean 4 formalization archives (Lean_Formalization_2026-05-28, Google Drive IDs 1t6gNjlug13-j2amIgXashkoMoLwQ2-sv and 1VtDm0DBEpiNeh3fepG5S9P9gTRzfc2sy), establishing the 5-step minimal tether uniqueness proof and coadjoint orbit degeneracy lemmas. Tier 2 — June 1, 2026 GitHub Origin Commit & June–August 2026 Archival Snapshots 2026-06-01 (01:20:57 AM CDT / 06:20:57 UTC): Earliest public GitHub root commit 6c6cac908304d1ed3c90bddc907c4d2c3f4a8d79 in BenFrohman/frohmanian-tether-manuscript-2026-06-01 ("Resolution of the 3D Navier-Stokes Regularity Problem via the Frohmanian Symplectic Tether"; main.pdf SHA-256: d4d1753f12e580a8675ca4a51aa0c0832ed3b31c01a2ef71aacb030fdaf075c4; main.tex SHA-256: 710c97158f736d57e2c63370e54beb194698fad1ff489aff326cdb8736db9980), archived on Zenodo at 10.5281/zenodo.23629499 (Concept DOI: 10.5281/zenodo.23629498). 2026-06-01 (12:10:20 AM CDT) & 2026-06-22 (04:05:10 PM CDT): Companion monographs Frohmanian_Tether_Neural_Scaling_Secondary_Paper.pdf (SHA-256: 573f4fcc326708d8c6b5f4ceeef67d113d472170234cfb3105ec123663e20a1a) and Frohmanian_Symplectic_Tether_and_Frohmanian_Curve.pdf (SHA-256: 3cf0e2a1be2a03cb165659fa5315f6d6642579341b3bd90f21de455dac451e7e), preserved in 10.5281/zenodo.23629499. 2026-06-19 (04:29:21 PM CDT / 21:15:05 UTC): Creation of the primary Lean 4 formalization repository BenFrohman/NS_Millennium_Proof (oldest author-dated commit c4fc018). 2026-08-19 (CDT): Historical Zenodo archival snapshot 10.5281/zenodo.21856430 (Concept DOI: 10.5281/zenodo.21856429; Frohmanian_Symplectic_Tether_Zenodo_21856430.pdf SHA-256: 54cb95bf8fd87ba325f7bdc68597bba8406d499c9bcccad0f576fc3671325120). Tier 3 — September–October 2026 Analytical Criteria, Boundary Ledgers & Zero-sorry Lean 4 Formalizations 2026-09-01: Vorticity direction and high-set Campanato regularity criteria: 10.5281/zenodo.22240168 (Direction of vorticity in 3D Navier–Stokes: closed identities, a geometric reduction, and a scale-sharp remaining estimate) and 10.5281/zenodo.22240549 (Direction of vorticity on the high set: what is closed, what is false, and the remaining Campanato bound). 2026-09-16 / 2026-09-18: September 2026 working notes and referee audits: 10.5281/zenodo.22835028 (September 2026 Working Notes on 3D Navier–Stokes Conditional Regularity Criteria, the Hodge–Navier–Stokes Architecture Referee Audit, and Triadic Multiscale Homogenization) and 10.5281/zenodo.22835069 (Identified strain kernel A(θ,ω) and a conditional regularity criterion for 3D Navier–Stokes (κ = 3/2; working notes), BenFrohman/ns-i11-conditional-criterion). 2026-09-11 / 2026-09-25: Two-point strain coercivity analysis: 10.5281/zenodo.23627618 (On Local Two-Point Strain Coercivity in the 3D Vorticity Equation, BenFrohman/two-point-strain-coercivity). 2026-10-06 / 2026-10-07: Zero-sorry Lean 4 modular formalizations and boundary ledgers: 10.5281/zenodo.23629497 (Concept: 10.5281/zenodo.23629496) — BenFrohman/bohm-identities (Pointwise Bohm quantum-potential identities). 10.5281/zenodo.23653003 (Concept: 10.5281/zenodo.23653002) — BenFrohman/global-from-local-restart (Uniform local continuation implies global existence on [0,∞) in Lean 4). 10.5281/zenodo.23653013 (BenFrohman/Clay-) & 10.5281/zenodo.23653015 (BenFrohman/Clay-CD-not-certified) — Formal boundary ledgers separating conditional regularity from the open unconditional Clay problems (A)–(D). 10.5281/zenodo.23629465 (BenFrohman/functional-equation-tether-b4) & 10.5281/zenodo.23629467 (BenFrohman/triadic-homogenization-lean). 2026-10-24: Symplectohydrodynamics & Master Functional Monograph Series: 10.5281/zenodo.23632786 (The Frohman Symplectic and Holographic Dual Regularization Conjecture: Master Functional Reduction and Metriplectic Enstrophy Control) and companion preprints 10.5281/zenodo.23632776, 10.5281/zenodo.23632778, 10.5281/zenodo.23632780, 10.5281/zenodo.23632782, 10.5281/zenodo.23632784, and 10.5281/zenodo.23632788. 2026-10-25 / 2026-10-27: Primary Lean 4 Formalization Releases: 10.5281/zenodo.23717082 (Concept: 10.5281/zenodo.23652842; v1.0.0: 10.5281/zenodo.23652843) — BenFrohman/frohmanian-symplectic-tether-regularity (v1.1.0, 29 Lean 4 modules, zero sorry). 10.5281/zenodo.23717083 (Concept: 10.5281/zenodo.23652840; v1.0.0: 10.5281/zenodo.23652841) — BenFrohman/fefferman-clay-navier-stokes (v1.1.0, 34 Lean 4 modules, zero sorry). 10.5281/zenodo.23717084 (Concept: 10.5281/zenodo.23627609; v2.1.0: 10.5281/zenodo.23627610) — BenFrohman/NS_Millennium_Proof (v2.2.0, 48 Lean 4 files, 360 theorems/lemmas, zero sorry). Verification & Containment Scaffolding: 10.5281/zenodo.23630648 (BenFrohman/titan-audit), 10.5281/zenodo.23631670 (BenFrohman/sealguard), 10.5281/zenodo.23632766 (BenFrohman/sentinel-gate), and 10.5281/zenodo.23632768 (BenFrohman/Agentic-Dev). AI-Assisted Formalization & Tooling Disclosure All mathematical definitions, physical models, geometric constructions (including the Frohmanian Symplectic Tether, coadjoint-orbit projection operators, and tethered Lyapunov functionals), and proof architectures are the sole original work of Benjamin Stanley Frohman. AI coding and formalization assistants (specifically Grok / Grok Bot, xAI) were utilized strictly under the author's direction for Lean 4 proof-term translation, Mathlib4 tactic elaboration, LaTeX typesetting, and repository/provenance audit scaffolding. Every formal theorem in the released Lean 4 artifacts is independently verified by the deterministic Lean 4 kernel (leanprover/lean4:v4.29.0-rc2 / v4.30.0-rc1) against Mathlib4 with zero sorry axioms and zero custom axioms.