TEBAC Navier--Stokes Program III: Spectral Cascade Obstruction and Critical Resonance Absorption — Final Closure Upgrade

This preprint is the final closure-upgrade version of Program III / NS-III in the TEBAC Navier--Stokes program. It upgrades the earlier NS-III reduction module by importing the terminal arithmetic-packing coercivity mechanism supplied by the companion module NS-IIIa. The manuscript works in the periodic three-dimensional setting \( T^3\) and focuses on the large-data critical-resonance absorption problem for the incompressible Navier--Stokes equations. Building on NS-I, NS-II, and the terminal NS-IIIa companion, it assembles the full spectral-cascade reduction chain and replaces the previously isolated terminal axial-envelope obstruction with the terminal packing-coercivity theorem imported from NS-IIIa. The central upgraded export is the NS-III critical-resonance absorption theorem. In schematic form, the critical high-frequency resonance flux is bounded by a viscous absorption term plus an energy-controlled lower-order residue: \[Φ_Qᶜʳⁱᵗ(u_N;t)≤θν D>Q(u_N;t)+ Rlow(u_N;t), 0<θ<1,\] with the lower-order residue controlled uniformly in the Galerkin cutoff \(N\). The file includes an explicit NS-IIIa import contract, a final non-circularity audit, acceptance tests, dependency ledgers, and a module-level closure statement. The result should be read as the closure of NS-III as a critical-resonance absorption module within the TEBAC modular Navier--Stokes program. This manuscript does not claim a standalone proof of the Navier--Stokes Millennium problem. Its role is to export the critical-resonance absorption estimate required by the subsequent NS-IV critical norm closure module.