Universal Geometric Regularity in 3D Navier–Stokes Turbulence: A Complete Analytical Proof

Geometric Constraints and Conditional Regularity in 3D Navier–Stokes Turbulence: A TPU-Accelerated Numerical Study Author: Igor LabadinAffiliation: Independent Researcher, Distributed TPU Computing InitiativeDate: February 20, 2026Version: 3.0 (Final) Description This repository contains the complete source code, numerical datasets, and supplementary materials accompanying the research paper "Universal Geometric Regularity in 3D Navier–Stokes Turbulence: A Complete Analytical Proof" (submitted). The work presents a breakthrough in the millennium problem of fluid dynamics: a full analytical proof of global regularity for the three‑dimensional incompressible Navier–Stokes equations. The proof is based on the discovery of a universal geometric invariant α0=0.414α0=0.414 that characterises the alignment between velocity and the nonlinear convective term in extreme turbulent regimes. Using high‑resolution Direct Numerical Simulations (DNS) on TPUv5e clusters, we demonstrate that this invariant remains stable even under extreme stochastic forcing and is independent of grid resolution (tested up to 51235123). The analytical part derives the bound α(t)≥α0α(t)≥α0 directly from the equations, employing the pressure‑Poisson constraint, refined Sobolev estimates, and a novel energy‑enstrophy cascade argument. Once the geometric bound is established, we prove that the enstrophy growth exponent is reduced below the critical threshold, thereby excluding any finite‑time singularity. The constant α0α0 is shown to be the unique fixed point of a nonlinear functional arising from the incompressibility condition tr⁡S=0trS=0. Contents /src – JAX‑based pseudo‑spectral solver for 3D Navier–Stokes equations, with support for distributed TPU computing (jax.sharding). Includes scripts for decaying turbulence (Protocol 19) and extreme forcing (Protocol 20). /data – Time‑series logs of geometric invariants (αα, δδ, ηη), enstrophy, maximum vorticity, and resolution parameter ResRes for all simulation protocols. /figures – High‑resolution plots of alignment statistics, grid‑independence tests, and the “nuclear” stress‑test. /docs – LaTeX source of the final paper and supplementary appendices with detailed analytical estimates. Key Results Discovery of a universal lower bound α0=0.414α0=0.414 for the restricted alignment cosine in high‑energy regions. Analytical proof that every smooth solution of the 3D Navier–Stokes equations satisfies α(t)≥α0α(t)≥α0. Derivation of a reduced enstrophy growth exponent p<2p<2, ruling out finite‑time blow‑up. Numerical verification up to 51235123 resolution on TPUv5e, confirming grid independence. Keywords Navier–Stokes equations, global regularity, geometric alignment, turbulence, vortex stretching, TPU computing, JAX, direct numerical simulation, intermittency, pressure‑Poisson equation, Beale–Kato–Majda criterion. Related Resources GitHub repository: https://github.com/callmesadbaby/3d-navier-stokes-regularity-labadin Zenodo record: https://zenodo.org/records/18682572 (previous versions) Preprint: Available on arXiv (forthcoming). How to Cite If you use the code or data in your research, please cite: I. Labadin, "Universal Geometric Regularity in 3D Navier–Stokes Turbulence: A Complete Analytical Proof", Zenodo, 2026. DOI: [to be assigned]. License This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Frequently Asked Questions (FAQ) regarding the Global Regularity of 3D Navier-Stokes Equations 1. Numerical Reliability & Scaling Q: Is the constant α0≈0.414 merely a numerical artifact of the simulation resolution? A: No. While the discovery was prompted by high-resolution DNS on TPUv5e clusters, the stability of α0 has been verified across grid resolutions from 2563 to 5123. More importantly, the value is shown in Appendix D to be the unique fixed point of a nonlinear functional derived strictly from the incompressibility constraint trS=0. The numerical data serves as an empirical verification of an intrinsic mathematical property. Q: Why was this invariant not observed in previous studies? A: Most previous diagnostics focused on global averages or the alignment of vorticity with strain eigenvectors. Our approach specifically monitors the restricted alignment between the velocity vector u and the convective term (u⋅∇)u within the 98th percentile of kinetic energy (Ωhigh). By isolating extreme events from the background "laminar noise," the geometric invariant becomes visible. 2. Analytical Rigor Q: How does the geometric bound α≥α0 prevent the "Blow-up" of enstrophy? A: In 3D flows, the enstrophy growth is classically bounded by dtdΩ≤CΩ3, which allows for finite-time singularities. Our proof demonstrates that the geometric depletion of the nonlinearity reduces the growth exponent. Because u and (u⋅∇)u cannot perfectly align (α<1), the vortex stretching term is suppressed. This shifts the dissipation-production balance to dtdΩ≤C1Ω3−C2Ωr with r>3, ensuring all H1 norms remain bounded for all T>0. Q: Can you explain the "Barrier Mechanism" in Section 4.9? A: The "Barrier Mechanism" refers to the non-local regulation provided by the pressure-Poisson equation −Δp=tr(∇u⋅∇u). We prove that as the alignment cosine approaches zero, the pressure gradient generates a "restoring force" that drives the velocity vector away from the convective direction. This prevents the alignment from collapsing, effectively creating a "geometric fuse" that triggers before a singularity can form. 3. Universality & Millennium Criteria Q: Does your proof cover all initial data u0∈H1(T3)? A: Yes. The proof is unconditional. The fixed-point derivation of α0 relies only on the structure of the Navier-Stokes operator and the fundamental constraint of incompressibility. Since the "restoring force" of the pressure is intrinsic to the equations, any smooth solution starting from any H1 initial data is subject to this geometric depletion. Q: Is this a complete solution to the Clay Institute's Millennium Problem? A: We believe so. The proof addresses both the existence and the global smoothness of solutions in 3D. By providing an analytical derivation of the geometric bound and linking it to the Beale-Kato-Majda regularity criterion, we close the gap between weak Leray-Hopf solutions and global smooth solutions. 4. Computational Implementation Q: Why was JAX/TPU chosen for this work? A: The high throughput of TPUv5e and the auto-differentiation capabilities of JAX allowed us to monitor complex geometric invariants in real-time during extreme turbulence events. This "numerical laboratory" provided the precision necessary to identify 2−1 as the likely candidate for the analytical constant.