Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Exact Madelung and Bohm Quantum Potential Identities in Core Lean 4 (Zero Mathlib Dependencies)

Benjamin Frohman

Published Oct 25, 2026• 9 citations
Publisher

Abstract

Zero-sorry formalization in core Lean 4 (BohmIdentities.lean, zero external dependencies) proving the cleared algebraic identities underlying the Bohmian guidance law Im(dψ · conj ψ) = dS · |ψ|^2 (Bohm.guidance_cleared) and the quantum potential prefactor identity (Bohm.quantumPotential_cleared). 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.

DOI
10.5281/zenodo.23629496

Related papers

Exact Madelung and Bohm Quantum Potential Identities in Core Lean 4 (Zero Mathlib Dependencies)

Benjamin Frohman

Zero-sorry formalization in core Lean 4 (BohmIdentities.lean, zero external dependencies) proving the cleared algebraic identities underlying the Bohmian guidance law Im(dψ · conj ψ) = dS · |ψ|^2 (Bohm.guidance_cleared) and the quantum potential prefactor identity (Bohm.quantumPotential_cleared).

•Oct 6, 2026•0 citations

Frohman Symplectic Stabilizer Quantum Error-Correcting Codes, Metriplectic Syndrome Decoding, and Fault-Tolerant Logical Gate Synthesis (frohman-symplectic-qec-codes)

Benjamin Frohman

PROPRIETARY COMMERCIAL SOFTWARE & HARDWARE/ALGORITHMIC IP — ALL RIGHTS RESERVED This proprietary quantum architecture specification—authored by Benjamin Stanley Frohman (ORCID: 0009-0006-7068-3718) and anchored to symplectohydronamics commit 8f7db79b5786f3a59a0696a48a675f8a1fe410ca (August 22, 2026, 12:49:03 PM CDT)—formulates the Frohman Symplectic Stabilizer Quantum Error-Correcting (QEC) Code family and Metriplectic Syndrome Decoder. By mapping n-qubit Pauli operators onto the 2n-dimensional binary/qudit symplectic vector space (F_p^2n, omega_Sp), stabilizer codes correspond to isotropic subspaces S subset S-perp, while fault-tolerant logical Clifford gates correspond to symplectic automorphisms U in Sp(2n, F_p) preserving S. We introduce a metriplectic soft-syndrome relaxation flow that suppresses trapping sets in high-rate quantum LDPC (qLDPC) codes and provides formal distance-preservingtransversal gate synthesis. IP Owner & Author: Benjamin Stanley Frohman (ORCID: 0009-0006-7068-3718) Earliest Verified Origin Provenance (America/Chicago, CDT / UTC-5): 2026-08-22 12:49:03 PM CDT (UTC-5) (symplectohydronamics commit 8f7db79b5786f3a59a0696a48a675f8a1fe410ca) Private GitHub Repository: https://github.com/BenFrohman/frohman-symplectic-qec-codes (Release v1.0.0, Commit dead18dcbd6fd2925f666fa090535a1178aa5f6d) SHA-256 Merkle Root: 7778b182a0f10243db4fd6fa8c7c7b84f036d400412ac06296c28f7d498161f1 Commercial Licensing & Federal Acquisition: Available exclusively via Metered Usage License, Annual Enterprise Site License, or Strategic Partnership Agreement countersigned by Benjamin Stanley Frohman (or his designated Texas LLC commercial licensing vehicle). Commercial Computer Software developed 100% at private expense under FAR 12.212 and DFARS 227.7202.

•Aug 22, 2026•0 citations

The quantum potential of the bohmian mechanics as a bridge between classical and quantum mechanics.

Iago Israel

Neste trabalho, buscamos uma interpretação física para o Efeito Aharonov-Bohm, um dos primeiros e mais famosos casos de fase geométrica na mecânica quântica. Para isso, fazemos uso do formalismo bohmiano da mecânica quântica, para podermos formular uma condição precisa (a saber, que a escala de variação espacial do potencial clássico seja muito maior do que a dimensão característica do pacote de onda) para o limite clássico e argumentarmos que o Efeito Aharonov-Bohm depende da violação dessa condição, o que é possível devido à topologia não simplesmente conexa do espaço base acessível ao elétron. Ademais, utilizamos a mecânica bohmiana para recuperar a mecânica quântica a partir da mecânica clássica via abordagem quasi-newtoniana, demonstrando que o potencial quântico, que é a origem da não-classicidade da mecânica quântica, está latente na estrutura da mecânica clássica. Apresentamos uma interpretação termodinâmica para o potencial quântico bohmiano e qual seria a origem machiana do potencial quântico na validade do Princípio de Mach, para compreendermos mais a fundo as bases da mecânica quântica.

•Aug 12, 2022•0 citations

Teilchenzahlformalismus

Udo Scherz

Die in der Quantenmechanik zu fordernde UnUnterscheidbarkeit identischer Teilchen hat, wie in Abschn. 4.5 ausgeführt, das Pauli-Prinzip zur Folge. Danach müssen Mehrteilchenzustände von Fermionen oder Bosonen bezüglich der Vertauschung identischer Teilchen antisymmetrisch bzw. symmetrisch sein. Führt man numerierte Koordinaten von Ort und Spin für die einzelnen Teilchen ein, so müssen die Mehrteilchenzustände durch aufwendige Summen antisymmetrisiert bzw. sym-metrisiert werden, vergl. Gl. (4.31) bzw. (4.32). In diesem Kapitel soll eine andere Methode besprochen werden um das Pauli-Prinzip zu gewährleisten ohne diese Summen wiederholt ausführen zu müssen. Im Rahmen des Teilchenzahlformalismus wird dabei eine Numerierung von vornherein vermieden, indem nur die Anzahl der Teilchen n ν angegeben wird, die sich in einem bestimmten Einteilchenzustand ψ ν befinden. Bei Fermionen können nach dem in Abschn. 4.5 Gesagten diese soge-nannten Besetzungszahlen n ν wegen des Pauli-Prinzips nur die Werte Null oder Eins annehmen, während diese Beschränkung für Bosonen nicht gilt und die n ν entweder Null oder eine natürliche Zahl sind.

Teubner-Studienbücher. Physik•Jan 1, 1999•0 citations

The Emergence of Geometry: π as the Imaginary Phase of Modular Information

José Ignacio Peinador Sala

Abstract We identify the constant $\pi$ as the optimal relative phase for stabilizing chiral quantum channels under $\mathbb{Z}/6\mathbb{Z}$ superselection. Within a modular substrate governed by the quotient ring $\mathbb{Z}/6\mathbb{Z}$, we prove that the phase shift $\phi_2 = \pi$ maximizes the fidelity of the $\mathcal{C}_5$ chiral channel (residue class $5 \pmod 6$) relative to the $\mathcal{C}_1$ channel. This result follows from the exact $\mathbb{Z}_2$ symmetry of the unit group $(\mathbb{Z}/6\mathbb{Z})^\times \cong \mathbb{Z}_2$ and the elementary trigonometric identity $\sin(\theta + \pi) = -\sin(\theta)$. We further demonstrate, via high-precision numerical simulations, that chiral interference under this phase suppresses parity-symmetric Gaussian Unitary Ensemble (GUE) noise by $45.2\%$, yielding a net signal-to-noise ratio (SNR) gain of $+6.07~\text{dB}$. Continuous open-system master equation dynamics (Lindblad bath with 85% collective amplitude damping and 15% local dephasing) confirm that the topologically shielded chiral singlet $\vert{}S\rangle$ preserves long-term state fidelity ($\mathcal{F} = 0.8113$), cutting open-system decoherence by $63.4\%$ ($+4.48~\text{dB}$ net gain). The algebraic core---including the optimal phase theorem and the parity transformation of symmetric noise operators---is formally certified in the Lean 4 proof assistant with zero omitted axioms (sorry-free). 📂 Repository Contents & File Guide 1. Manuscript Files π_as_the_Optimal_Phase_in_MQS.pdf: Final compiled PDF preprint. π_as_the_Optimal_Phase_in_MQS.tex: Complete LaTeX source code. 2. Interactive Notebooks & Executables LEAN_π_as_the_Optimal_Phase_in_MQS.ipynb: Interactive Google Colab notebook for the Lean 4 formal verification suite. Installs elan, pins toolchain v4.11.0, fetches pre-compiled Mathlib4 caches, and compiles all proofs. LEAN_π_as_the_Optimal_Phase_in_MQS.pdf: Static PDF printout of the Lean 4 notebook showing full build execution logs (0 sorrys). PYTHON_π_as_the_Optimal_Phase_in_MQS.ipynb: Interactive Google Colab notebook for numerical simulations (DSP $M=6$ polyphase filters, Qiskit gate-level DFS, Monte Carlo trajectories, and QuTiP open-system master equation). PYTHON_π_as_the_Optimal_Phase_in_MQS.pdf: Static PDF printout of the Python simulation notebook with embedded benchmark reports and plots. 3. Verification Source Code & Publication Figures mst_f1_verification_clean.zip: Clean standalone Lean 4 Lake project source code for local compilation and VS Code integration (lake build). DSP_Polyphase_Filter.pdf: Publication-ready vector graphic (Figure 1 in manuscript) demonstrating GUE noise collapse. lindblad_dynamics_paper_f1.pdf: Publication-ready vector graphic (Figure 2 in manuscript) showing continuous Lindblad fidelity and coherence protection. README.md: Complete instruction manual and build guide. 🛠️ Replication Instructions Cloud Execution (Google Colab - Recommended) Open Google Colab. Upload either LEAN_π_as_the_Optimal_Phase_in_MQS.ipynb or PYTHON_π_as_the_Optimal_Phase_in_MQS.ipynb. Select Runtime > Run all (Ctrl + F9). All dependencies install automatically and results reproduce in under 3 minutes. Local Execution (Lean 4 Command Line) Ensure elan is installed on your local machine. Unzip mst_f1_verification_clean.zip. Run the build commands in terminal: Bash cd mst_f1_verification lake exe cache get lake build 📜 License Code & Proofs: MIT License. Manuscript & Data: Creative Commons Attribution 4.0 International (CC-BY 4.0). 📌 Publication Status: Submitted to *Journal of Physics A: Mathematical and Theoretical* in July 2026 - Manuscript ID: JPhysA-125419.

•Feb 19, 2026•0 citations

162 Lean 4 Theorems for Post-Quantum Infrastructure: Three-Tier Formal Verification of QuantumTimeSandwich, QSSH, and QuantumHarmony

Sylvain Cormier

•Feb 16, 2026•0 citations
Zero-sorry formalization in core Lean 4 (BohmIdentities.lean, zero external dependencies) proving the cleared algebraic identities underlying the Bohmian guidance law Im(dψ · conj ψ) = dS · |ψ|^2 (Bohm.guidance_cleared) and the quantum potential prefactor identity (Bohm.quantumPotential_cleared). 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.
PROPRIETARY COMMERCIAL SOFTWARE & HARDWARE/ALGORITHMIC IP — ALL RIGHTS RESERVED This proprietary quantum architecture specification—authored by Benjamin Stanley Frohman (ORCID: 0009-0006-7068-3718) and anchored to symplectohydronamics commit 8f7db79b5786f3a59a0696a48a675f8a1fe410ca (August 22, 2026, 12:49:03 PM CDT)—formulates the Frohman Symplectic Stabilizer Quantum Error-Correcting (QEC) Code family and Metriplectic Syndrome Decoder. By mapping n-qubit Pauli operators onto the 2n-dimensional binary/qudit symplectic vector space (F_p^2n, omega_Sp), stabilizer codes correspond to isotropic subspaces S subset S-perp, while fault-tolerant logical Clifford gates correspond to symplectic automorphisms U in Sp(2n, F_p) preserving S. We introduce a metriplectic soft-syndrome relaxation flow that suppresses trapping sets in high-rate quantum LDPC (qLDPC) codes and provides formal distance-preservingtransversal gate synthesis. IP Owner & Author: Benjamin Stanley Frohman (ORCID: 0009-0006-7068-3718) Earliest Verified Origin Provenance (America/Chicago, CDT / UTC-5): 2026-08-22 12:49:03 PM CDT (UTC-5) (symplectohydronamics commit 8f7db79b5786f3a59a0696a48a675f8a1fe410ca) Private GitHub Repository: https://github.com/BenFrohman/frohman-symplectic-qec-codes (Release v1.0.0, Commit dead18dcbd6fd2925f666fa090535a1178aa5f6d) SHA-256 Merkle Root: 7778b182a0f10243db4fd6fa8c7c7b84f036d400412ac06296c28f7d498161f1 Commercial Licensing & Federal Acquisition: Available exclusively via Metered Usage License, Annual Enterprise Site License, or Strategic Partnership Agreement countersigned by Benjamin Stanley Frohman (or his designated Texas LLC commercial licensing vehicle). Commercial Computer Software developed 100% at private expense under FAR 12.212 and DFARS 227.7202.
Neste trabalho, buscamos uma interpretação física para o Efeito Aharonov-Bohm, um dos primeiros e mais famosos casos de fase geométrica na mecânica quântica. Para isso, fazemos uso do formalismo bohmiano da mecânica quântica, para podermos formular uma condição precisa (a saber, que a escala de variação espacial do potencial clássico seja muito maior do que a dimensão característica do pacote de onda) para o limite clássico e argumentarmos que o Efeito Aharonov-Bohm depende da violação dessa condição, o que é possível devido à topologia não simplesmente conexa do espaço base acessível ao elétron. Ademais, utilizamos a mecânica bohmiana para recuperar a mecânica quântica a partir da mecânica clássica via abordagem quasi-newtoniana, demonstrando que o potencial quântico, que é a origem da não-classicidade da mecânica quântica, está latente na estrutura da mecânica clássica. Apresentamos uma interpretação termodinâmica para o potencial quântico bohmiano e qual seria a origem machiana do potencial quântico na validade do Princípio de Mach, para compreendermos mais a fundo as bases da mecânica quântica.
Die in der Quantenmechanik zu fordernde UnUnterscheidbarkeit identischer Teilchen hat, wie in Abschn. 4.5 ausgeführt, das Pauli-Prinzip zur Folge. Danach müssen Mehrteilchenzustände von Fermionen oder Bosonen bezüglich der Vertauschung identischer Teilchen antisymmetrisch bzw. symmetrisch sein. Führt man numerierte Koordinaten von Ort und Spin für die einzelnen Teilchen ein, so müssen die Mehrteilchenzustände durch aufwendige Summen antisymmetrisiert bzw. sym-metrisiert werden, vergl. Gl. (4.31) bzw. (4.32). In diesem Kapitel soll eine andere Methode besprochen werden um das Pauli-Prinzip zu gewährleisten ohne diese Summen wiederholt ausführen zu müssen. Im Rahmen des Teilchenzahlformalismus wird dabei eine Numerierung von vornherein vermieden, indem nur die Anzahl der Teilchen n ν angegeben wird, die sich in einem bestimmten Einteilchenzustand ψ ν befinden. Bei Fermionen können nach dem in Abschn. 4.5 Gesagten diese soge-nannten Besetzungszahlen n ν wegen des Pauli-Prinzips nur die Werte Null oder Eins annehmen, während diese Beschränkung für Bosonen nicht gilt und die n ν entweder Null oder eine natürliche Zahl sind.
Abstract We identify the constant π\piπ as the optimal relative phase for stabilizing chiral quantum channels under Z/6Z\mathbb{Z}/6\mathbb{Z}Z/6Z superselection. Within a modular substrate governed by the quotient ring Z/6Z\mathbb{Z}/6\mathbb{Z}Z/6Z, we prove that the phase shift ϕ2=π\phi_2 = \piϕ2​=π maximizes the fidelity of the C5\mathcal{C}_5C5​ chiral channel (residue class 5(mod6)5 \pmod 65(mod6)) relative to the C1\mathcal{C}_1C1​ channel. This result follows from the exact Z2\mathbb{Z}_2Z2​ symmetry of the unit group (Z/6Z)×≅Z2(\mathbb{Z}/6\mathbb{Z})^\times \cong \mathbb{Z}_2(Z/6Z)×≅Z2​ and the elementary trigonometric identity sin⁡(θ+π)=−sin⁡(θ)\sin(\theta + \pi) = -\sin(\theta)sin(θ+π)=−sin(θ). We further demonstrate, via high-precision numerical simulations, that chiral interference under this phase suppresses parity-symmetric Gaussian Unitary Ensemble (GUE) noise by 45.2%45.2\%45.2%, yielding a net signal-to-noise ratio (SNR) gain of +6.07 dB+6.07~\text{dB}+6.07 dB. Continuous open-system master equation dynamics (Lindblad bath with 85% collective amplitude damping and 15% local dephasing) confirm that the topologically shielded chiral singlet ∣S⟩\vert{}S\rangle∣S⟩ preserves long-term state fidelity (F=0.8113\mathcal{F} = 0.8113F=0.8113), cutting open-system decoherence by 63.4%63.4\%63.4% (+4.48 dB+4.48~\text{dB}+4.48 dB net gain). The algebraic core---including the optimal phase theorem and the parity transformation of symmetric noise operators---is formally certified in the Lean 4 proof assistant with zero omitted axioms (sorry-free). 📂 Repository Contents & File Guide 1. Manuscript Files π_as_the_Optimal_Phase_in_MQS.pdf: Final compiled PDF preprint. π_as_the_Optimal_Phase_in_MQS.tex: Complete LaTeX source code. 2. Interactive Notebooks & Executables LEAN_π_as_the_Optimal_Phase_in_MQS.ipynb: Interactive Google Colab notebook for the Lean 4 formal verification suite. Installs elan, pins toolchain v4.11.0, fetches pre-compiled Mathlib4 caches, and compiles all proofs. LEAN_π_as_the_Optimal_Phase_in_MQS.pdf: Static PDF printout of the Lean 4 notebook showing full build execution logs (0 sorrys). PYTHON_π_as_the_Optimal_Phase_in_MQS.ipynb: Interactive Google Colab notebook for numerical simulations (DSP M=6M=6M=6 polyphase filters, Qiskit gate-level DFS, Monte Carlo trajectories, and QuTiP open-system master equation). PYTHON_π_as_the_Optimal_Phase_in_MQS.pdf: Static PDF printout of the Python simulation notebook with embedded benchmark reports and plots. 3. Verification Source Code & Publication Figures mst_f1_verification_clean.zip: Clean standalone Lean 4 Lake project source code for local compilation and VS Code integration (lake build). DSP_Polyphase_Filter.pdf: Publication-ready vector graphic (Figure 1 in manuscript) demonstrating GUE noise collapse. lindblad_dynamics_paper_f1.pdf: Publication-ready vector graphic (Figure 2 in manuscript) showing continuous Lindblad fidelity and coherence protection. README.md: Complete instruction manual and build guide. 🛠️ Replication Instructions Cloud Execution (Google Colab - Recommended) Open Google Colab. Upload either LEAN_π_as_the_Optimal_Phase_in_MQS.ipynb or PYTHON_π_as_the_Optimal_Phase_in_MQS.ipynb. Select Runtime > Run all (Ctrl + F9). All dependencies install automatically and results reproduce in under 3 minutes. Local Execution (Lean 4 Command Line) Ensure elan is installed on your local machine. Unzip mst_f1_verification_clean.zip. Run the build commands in terminal: Bash cd mst_f1_verification lake exe cache get lake build 📜 License Code & Proofs: MIT License. Manuscript & Data: Creative Commons Attribution 4.0 International (CC-BY 4.0). 📌 Publication Status: Submitted to *Journal of Physics A: Mathematical and Theoretical* in July 2026 - Manuscript ID: JPhysA-125419.