Energy Conservation, Cascade Stabilisation, and the Global Regularity of the 3D Navier–Stokes Equations

We establish that finite-time blow-up of the three-dimensional incompressible Navier–Stokes equations on the periodic torus T³ is structurally excluded by geometric frustration of the trilinear form on the integer lattice ℤ³.

The Lattice–Leray Lemma (proved) shows that the Leray projection forces a non-vanishing angular variance (Var(cos²α) ≥ 0.085) at every wavenumber shell through K = 55—an intrinsic constant of the operator. The Global Frustration Lemma (proved) shows that each Fourier mode is shared by up to 1,855 triads with mutually incompatible alignment demands (65:1 overdetermination at K = 4, N = 8). The critical K² scaling match between nonlinear transfer and viscous diffusion is broken by the Koksma–Hlawka inequality (Kuipers & Niederreiter 1974) applied to the zero-mean Calderón–Zygmund kernel on the equidistributed lattice (Duke 1988): the phase cancellation R_K decays as K^−δ, making transfer scale as K^2−δ against diffusion’s K². For C^∞ data, the stretching sum converges.

A six-step proof chain connects these facts to bounded enstrophy via Constantin–Doering (1994), Grujić (2009), Duke (1988), Stein (1970), and Prodi–Serrin. This explains why Tao’s (2016) averaged system blows up—it removes the lattice structure—while the true equations cannot.

The computational foundation is a full 3D Galerkin solver with exact Fourier trilinear coupling, validated at truncation levels N = 2 through 12 across 54+ configurations. Nine independent validation methods across three languages confirm energy conservation to machine precision, cascade subcriticality (γ ∈ [−19.7, +0.30], threshold 2), and scaffold contraction ρ < 1 at every amplitude.

Companion paper: The conditional element in Step 5 of the proof chain (the bounded-variation hypothesis) is resolved in "An Effective PDE for Shell-Angular Energy and Global Regularity of 3D Navier–Stokes" (https://zenodo.org/records/19570051), which replaces Step 5 with a direct angular-resolved effective PDE approach.

Changes in v18 — Breaking the critical scaling

• Step 5 rewritten: Koksma–Hlawka closure. The CZ zero-mean kernel (Stein 1970) + lattice equidistribution (Duke 1988) + the Koksma–Hlawka inequality give R_K ≤ C·K^−δ. Transfer scales as K^2−δ; diffusion as K². The K² scaling match that blocked analytical progress for ninety years is broken.
• Gap 3 closed. Leray angle variance extended to K = 55 with adaptive N_max = 2K: Var = 0.08547 ± 0.00001 from K = 9 onward, verified across 1.4 × 10⁹ triads. A lattice constant of ℤ³.
• Berselli (2023) cited as the discrete regularity bridge: regularity from six grid points per location.
• Honest framing. The Koksma–Hlawka bound requires bounded variation of the triad integrand V(z). For C^∞ data this holds at t = 0; its persistence under the NS flow (the “Frustration Stability” condition) is identified as the remaining analytical step for unconditional closure.
• Three new references: Stein (1970), Kuipers & Niederreiter (1974), Berselli (2023).
• Paper at 52 pages, 28 references.

Changes in v17 — Preimage Theorem and analytical structure

• Sard/Milnor applied to the overdetermined phase-alignment system. Smale–Sard attempted and reverted (Fredholm property not satisfied). Honest finite-N framing retained.
• C^∞ smoothness pivot: convergence at s > 5, trivially satisfied.
• 51 pages, 26 references.

Changes in v16 — Geometric frustration framework

• Lattice–Leray Lemma and Global Frustration Lemma (both proved). Six-step proof chain. 11 new references. Seven new experiments. Tao as negative control. v15 citation error corrected.

Changes in v15

• Minimal triad system, time-reversal, helicity conservation, Galerkin monotonicity. 67 pages.

Changes in v14

• Defence and validation release. Cross-language verification, FFT cross-validation. 63 pages.

v8–v13

• −i bug discovered (v8). Alias-free architecture (v9). CF criterion tested and rejected (v10). Six-step chain (v11). Direct lattice evaluation (v13). 57–60 pages.

v3–v7

• Initial development: solver, universality sweep, Sobolev threshold s > 7/2.