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

We identify the mechanism preventing finite-time blow-up of the three-dimensional incompressible Navier–Stokes equations on the periodic torus T³: geometric frustration of the trilinear form on the integer lattice ℤ³. The Lattice–Leray Lemma (proved) shows that the Leray projection Pk = I − k⊗k/|k|² acting on ℤ³ forces a non-vanishing angular variance (Var(cos²α) ≥ 0.073) in the triad interactions at every wavenumber shell—an intrinsic property of the operator, independent of the solution. The Global Frustration Lemma (proved) shows that each Fourier mode is shared by up to 1,855 triads with mutually incompatible alignment demands: no choice of Fourier coefficients can simultaneously maximise stretching across the network. The phase coherence required for blow-up is structurally forbidden by ℤ³. A six-step proof chain connects these geometric facts to bounded enstrophy via Constantin–Doering (1994) variational bounds, Constantin–Fefferman (1993) and Grujić (2009) geometric depletion, Duke (1988) lattice equidistribution, and the Prodi–Serrin criterion. The resulting phase cancellation is 97–99.9% at every tested shell. This explains why Tao’s (2016) averaged system blows up—it removes the frustration—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. The α(φ) curve—ranging from 0.85 (concentrated data) to 3.1 (distributed data), with α > 0.5 at every spectral class—quantifies the margin above the critical Sobolev exponent sc = 1/2. 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 0 at every shell K ≥ 1 on ℤ³. For K ≥ 2: two explicit triads with cos²α = K²/(K²+1) and cos²α = 1. For K = 1: triads with cos²α = 1/2 and 0. Infimum cLL ≥ 0.073 from computation; Duke (1988) equidistribution gives convergence to 4/45 ≈ 0.089 as K → ∞. Global Frustration Lemma (Lemma 8.8, proved). Each mode faces O(N²) conflicting alignment demands from triads with non-proportional optimal directions. 100% frustration at every shell. Even adversarial polarisation gives Raligned 0 → regularity. v15 analytical chain corrected. The v15 proof chain contained a citation error (Bradshaw–Grujić 2017 ARMA does not address vorticity direction) and an incomplete bootstrap closure. v16 replaces this with the Lattice–Leray / Global Frustration framework, which proves the geometric mechanism directly and identifies the quantitative bridge as a computational verification rather than an unproved assumption. Paper tightened to 51 pages (from 67 in v15). Literature sections rewritten for concision—connections to existing work, not summaries of it. Changes in v15 Minimal triad system, time-reversal, helicity conservation, Galerkin monotonicity. 67 pages. Changes in v14 Defence and validation release. Cross-language verification, v2→v3 narrative, FFT cross-validation. 63 pages. Changes in v13 Arithmetic correction s > 1/2 → s > 1. Direct lattice evaluation, bootstrap dynamics, semi-transparent shells. 60 pages. Changes in v12 Step 5 closed: δ > 0 in 72/72 configs. Two-scale reinforcement mechanism. 57 pages. v8–v11 −i bug discovered (v8). Alias-free architecture (v9). CF criterion tested and rejected (v10). Six-step chain (v11). v3–v7 Initial development: solver, Cauchy–Schwarz bounds, universality sweep, Sobolev threshold s > 7/2.