Spectral Non-Concentration Criteria for Navier–Stokes Regularity on T³

We establish a conditional proof of global regularity for the 3D incompressible Navier–Stokes equations on the torus T³ via the Spectral Non-Concentration (SND) framework. The central result, Lemma 2 (SND Simplex Stability), proves that under the SND assumption the normalized shell energy distribution satisfies ‖a(t) − μ‖_ℓ¹ ≤ 0.039 uniformly for all t ≥ 0, with explicit constants κ, K, and N_eff ≥ 654. The proof combines a triad transfer bound, an explicit dissipation rate derived from the Littlewood–Paley shell structure, an arithmetic closure via the GCD interaction matrix, and a Gronwall inequality closing the transient regime. Combined with the frozen spectral gap δ₀ = 0.20 and the operator continuity lemma, global regularity on T³ follows conditionally on SND. This paper is the second in a series; related records: DOI 10.5281/zenodo.19842060 and DOI 10.5281/zenodo.19842061.