Global Regularity for the Periodic Three-Dimensional Navier–Stokes Equations: Critical-Norm Closure, Resonance Decomposition, and Terminal Packing Coercivity

This preprint presents a classical-norm reformulation of a proposed proof of global regularity for the periodic three-dimensional incompressible Navier–Stokes equations on \ (T³= R³/ (2 Z³) \), for smooth divergence-free mean-zero initial data and viscosity \ (>0\). The central estimate is the cutoff-uniform critical bound\₀ ₓ ₓ\|A^1/4uN (t) \|₋ℂ²+₀T\|A^3/4uN (t) \|₋ℂ²\, dt C (T, , u₀), Galerkin solutions \ (uN\), with \ (C\) independent of the Galerkin cutoff. The argument reduces the possible finite-time obstruction to a high-frequency vorticity-flux term, decomposes the nonlinear production into dyadic shells and triadic interaction classes, absorbs the nonbalanced and dissipatively mismatched channels by standard estimates, and isolates the remaining coherent axial channel as a finite weighted packing problem on Stokes spheres. The terminal mechanism is a packing/coercivity theorem based on weighted Balog–Szemerédi–Freiman compression and spherical curvature pruning. The resulting absorption is transferred through the critical identity\2^-q\|q\|₋ℂ² 2^3q\|uq\|₋ℂ², the \ (H^1/2\) -scale critical estimate. Compactness, interpolation to a Serrin class, synchronization with the local strong branch, pressure reconstruction, uniqueness, and parabolic bootstrapping are then used to obtain the asserted global smooth solution. This version reorganizes the earlier programmatic manuscript into a more classical mathematical presentation. The core analytic mechanism is front-loaded, the terminology is standardized, and the Galerkin, dyadic, resonance, packing/coercivity, critical-norm, and continuation steps are presented as one continuous proof chain.