TEBAC Navier--Stokes Program III: Spectral Cascade Obstruction and Critical Resonance Absorption — Reduction Module

This preprint is the third module of the TEBAC Navier--Stokes program. It continues the spectral Stokes, Galerkin, vorticity-shell, and resonance-classification framework developed in NS-I and NS-II, and focuses on the large-data critical resonance absorption problem for the three-dimensional incompressible Navier--Stokes equations on the periodic domain \(\mathbb T^3\). The manuscript develops a theorem-bearing reduction framework for the critical high-frequency cascade. Starting from the coherent critical resonance residue isolated by the previous modules, it performs a sequence of spectral, helical, Beltrami, packing, axial, phase, and envelope reductions. At each stage, exact geometric cores are separated from leakage terms: exact Beltrami or helical configurations are shown to cancel under the Leray projection, thin leakage is absorbed into viscosity, and active leakage is controlled by occupation or dissipation-budget estimates whenever available. The final outcome is a claim-safe reduction theorem: the large-data critical resonance absorption problem is reduced to a single terminal obstruction, namely the terminal axial-envelope coercivity estimate. In schematic form, the remaining estimate is \[\int_0^T\mathfrak B_Q^{\mathrm{env}}(u_N;t)\,dt\leq\frac{\theta_{\mathrm{env}}\nu}{2}\int_0^T\mathcal D_{>Q}(u_N;t)\,dt+C(T,\nu,u_0),\] uniformly in the Galerkin cutoff \(N\). This manuscript does not claim a proof of the Navier--Stokes Millennium problem. It is intended as a theorem-bearing reduction module: NS-III closes the reduction chain down to the terminal axial-envelope coercivity obstruction, while the complete unconditional absorption theorem is reserved for a subsequent terminal coercivity argument.