GLOBAL REGULARITY FOR THE THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER–STOKES EQUATIONS VIA THE SPECTRAL GAP OF THE RECIPROCAL COST FUNCTIONAL

We establish global-in-time regularity for the three-dimensional incompressible Navier–Stokes equations for initial data compatible with a discrete lattice structure governed by the reciprocal cost functional J(x) = ½(x + x⁻¹) − 1. The proof is organized in three stages. First, we prove unconditional regularity on any finite lattice: the discrete maximum principle is self-improving under the sub-Kolmogorov condition, yielding gradient bounds that are uniform in time and independent of the lattice spacing. Second, we show that spectral Galerkin truncations at resolution N instantiate this discrete framework with effective spacing h = 1/N, producing a family of approximate solutions with a universal vorticity cap. Third, we construct the continuum solution as the limit of this family and prove that the Beale–Kato–Majda integral remains finite for all time, excluding finite-time singularities. The spectral gap Δ = J(φ) = (√5 − 2)/2 > 0, where φ = (1 + √5)/2 is the golden ratio, plays a central role: it ensures that the cascade cutoff constant lies strictly below the minimum excitation cost, making the viscous quenching mechanism structurally stable.