TEBAC Navier--Stokes Program II: Vorticity-to-Shell Decomposition and Resonance Classification

This preprint is the second module of a modular TEBAC-based program toward the three-dimensional incompressible Navier--Stokes existence and smoothness problem. It continues the foundational Stokes--Galerkin--spectral-cascade framework developed in the preceding NS-I module and focuses on the next structural layer: the conversion of the large-data cascade residue into a precise vorticity-to-shell and resonance-classification ledger. The manuscript works on the periodic domain \(\mathbb T^3\). It records the vorticity equation, the enstrophy identity, the vortex-stretching production term, dyadic vorticity-shell balances, transport commutator residues, low--high / high--low / high--high interaction classes, Fourier triadic enstrophy-production formulae, geometric depletion mechanisms, and the first TEBAC resonance-classification framework. The main role of NS-II is not to claim a complete solution of the Navier--Stokes Millennium problem. Rather, it closes the second module as a decomposition and classification module. It translates the large-data obstruction isolated in NS-I into the critical resonance-absorption target exported to NS-III. The decisive exported target is a TEBAC critical resonance absorption estimate of the form \[\Phi_Q^{\mathrm{crit}}(u_N;t)\leq\theta\nu\sum_{q>Q}\|\nabla\omega_N^{(q)}(t)\|_{L^2}^{2}+\mathcal R_{\mathrm{low}}(u_N;t),\qquad 0