Direction of vorticity in 3D Navier–Stokes: closed identities, a geometric reduction, and a scale-sharp remaining estimate

@misc{frohman2026direction,
  title = {{Direction of vorticity in 3D Navier–Stokes: closed identities, a geometric reduction, and a scale-sharp remaining estimate}},
  author = {Benjamin Frohman},
  howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
  year = {2026},
  month = sep,
  doi = {10.5281/zenodo.22240167},
  url = {https://doi.org/10.5281/zenodo.22240167},
  note = {CC BY},
}

TY  - UNPB
TI  - Direction of vorticity in 3D Navier–Stokes: closed identities, a geometric reduction, and a scale-sharp remaining estimate
AU  - Benjamin Frohman
PY  - 2026
DA  - 2026/09/02
PB  - Zenodo (CERN European Organization for Nuclear Research)
DO  - 10.5281/zenodo.22240167
UR  - https://doi.org/10.5281/zenodo.22240167
AB  - This note records what a coadjoint-tether construction actually yields for the 3D incompressible Navier–Stokes equations, and what it does not. What it proves. Degeneracy of the tether pairing leaves vorticity transport classical. The projected kernel is a real symmetric bilinear form, not Poisson and not sesquilinear. Closed identities: Jacobian of the vorticity direction ξ, the direction equation, the spatial-maximum law, the Constantin–Fefferman kernel bound, Young at viscous scale returning Beale–Kato–Majda, and the Lyapunov source signs. Energy plus Calderón–Zygmund on L2 place the stretching reaction Pξ(Sξ) in L2t L2 without ‖ω‖∞. Parabolic scaling is an obstruction at energy regularity (p=2 is below p>5/2). The implication "Definition 7.1 ⇒ T*=∞" is the theorem of Constantin–Fefferman and Beirão da Veiga–Berselli, not a new regularity proof. What it does not prove. Definition 7.1 (uniform-in-T* Hölder-½ of ξ on {r≥Ω}); Clay Millennium global regularity; a blow-up example; a sesquilinear or Hermitian structure; remaining lemmas (A) integrability upgrade or (B) Campanato on 4sin²(φ/2). This is a geometric reduction and a retirement of false closing paths. It is not a Clay solution. 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…
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