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. The argument is genuinely three-dimensional: SO (3) non-commutativity makes the triadic coupling coefficient complex-valued, so the coherent state is not a stable attractor. The 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. 5 (2026-04-29): Single-path restructure. Bootstrap proof of χK = O (1) is the single load-bearing route; independent routes (GRH-conditional, Bessel/DPI, tower property) moved to a companion. §6 reformulated as a Stability Dichotomy via the Pythagorean identity. Adler-to-Giacomin reduction made explicit (Proposition). Bessel-contraction proof corrected (Jacobi-Anger identity). Changes in v1. 4 (2026-04-24): Star Invariant construction (Def 10. 4) rewritten as an honest algebraic cluster-graph model, with alphabet-cage alignment remark deriving 52 = 4×13. Second unconditional proof route added via the Inverse Cascade Trap. Trace-zero proposition closes the Tr (J) =0 gap. CLT distributional sketch removed (replaced with an open-problem remark). Notation and citation hygiene tightened paper-wide. Changelog updated. 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 added: changelog. pdf 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.