TEBAC Navier–Stokes Program IV: Critical Norm Closure and Galerkin Limit Assembly

This preprint is the fourth module of a modular TEBAC approach to the three-dimensional incompressible Navier--Stokes regularity problem on the periodic torus\[ T^3= R^3/(2π Z)^3.\] The preceding modules establish the periodic Leray--Stokes--Galerkin foundation, the vorticity-to-shell decomposition, the resonance classification, and the terminal axial-envelope packing closure needed for critical resonance absorption. The purpose of the present module is to convert the upstream NS-III/NS-IIIa critical-resonance absorption export into the cutoff-uniform critical Stokes estimate\[0≤ t≤ T\|A1/4u_N(t)\|L^2^2+ν∫_0^T\|A3/4u_N(t)\|L^2^2\,dt(T,ν,u_0),\]with \(C(T,ν,u_0)\) independent of the Galerkin cutoff \(N\). The central technical step of NS-IV is the normalization bridge between the vorticity/enstrophy shell dissipation used in the resonance modules and the critical \(H1/2\)-Stokes scale needed for the final regularity assembly. At the dyadic level, this is encoded by the weighted relation\[2-q\|∇ω_q\|L^2^223q\|u_q\|L^2^2,\]which identifies the correctly reweighted NS-III flux absorption with the NS-IV critical dissipation ledger. The manuscript proves the module-level critical norm closure theorem relative to the upstream NS-I--NS-IIIa absorption package. In this precise sense, NS-IV closes the critical \(H1/2\)-estimate for Galerkin approximants. It does not, by itself, claim a complete solution of the Clay Millennium problem. The final global regularity conclusion is reserved for the subsequent NS-V assembly module, which must carry out the Galerkin limit passage, weak-to-strong upgrade, continuation criterion, uniqueness, and final non-circularity audit. Status statement: NS-IV is a theorem-bearing closure module inside the TEBAC Navier--Stokes program. It proves the cutoff-uniform critical norm estimate needed for the final assembly, assuming the previously established upstream resonance-absorption exports. The full Clay-compatible global regularity theorem remains assigned to NS-V.