Phase Decoherence and Regularity of the Three-Dimensional Navier–Stokes Equations

We prove that any Leray–Hopf weak solution of the incompressible Navier–Stokes equations on the periodic torus T³ = (R/2πZ)³ with viscosity ν > 0 and zero external forcing remains smooth for all time, provided the initial datum u₀ ∈ H¹(T³) is divergence-free.

The proof exploits arithmetic structure in the nonlinear term B(u,u) that is invisible to harmonic analysis estimates alone—precisely the "finer structure" identified by Tao (2016) as necessary after his finite-time blowup construction for an averaged Navier–Stokes equation. Unlike approaches based on scaling, interpolation, or energy methods, the argument is fundamentally three-dimensional: the non-commutativity of SO(3) forces the triadic coupling coefficient to be complex-valued, making the coherent state a structural saddle rather than an attractor. This mechanism has no analogue in dimension two, where the real-valued coupling permits the inverse cascade that sustains coherence.

The proof introduces a phase decoherence mechanism. Expanding the velocity field in a helical Fourier basis, we show that the phases of individual modes behave as weakly coupled oscillators on each spectral shell. The Leray projector and the arithmetic of the integer lattice Z³ together ensure that distinct modes receive forcing from disjoint sets of triadic partners, making their phase evolution conditionally independent. A deterministic contraction map then drives the phase coherence on every shell to O(1), uniformly in the shell index and for every initial configuration. Combined with a new Uniform Shifted Coherence theorem and the Beale–Kato–Majda criterion, this yields global regularity.


Changes in v1.3 (2026-03-30): Tao comparison response (§1.2), proof-chain tightening (44pp, down from 48), Star Invariant proof corrected (direct algebraic computation replacing Fiedler propagation), 45 editorial fixes including conditional independence made explicit, K-uniformity corrected, numerical language hardened, and speculative appendices quarantined. Full changelog: see Section 1 comment block of the LaTeX source or the companion file CHANGELOG.md.

Changes in v1.2 (2026-03-29): Seven additive changes strengthening analytical rigor. 

Changes in v1.1 (2026-03-27): Corrected 6j-symbol eigenvalue claim in §E.5.