This preprint is the fifth module of a modular TEBAC approach to the three-dimensional incompressible Navier--Stokes regularity problem on the periodic torus\ T³= R³/ (2 Z) ³. \ The preceding modules establish the periodic Leray--Stokes--Galerkin foundation, the vorticity-to-shell decomposition, the critical resonance absorption mechanism, the terminal axial-envelope packing closure, and the cutoff-uniform critical \ (H^1/2\) -Stokes estimate for Galerkin approximants. The purpose of the present module is to perform the final Clay-compatible assembly step from the NS-IV critical norm closure theorem to global smooth periodic solutions. The central implication assembled in NS-V is\₀ ₓ ₓ\|A^1/4uN (t) \|₋ℂ²+₀T\|A^3/4uN (t) \|₋ℂ²\, dt C (T, , u₀) in the Galerkin cutoff \ (N\), followed by the Galerkin compactness passage, synchronization with the local classical branch, weak--strong uniqueness, continuation beyond any finite maximal time, and smoothness bootstrap. The module proves that, relative to the upstream NS-I--NS-IV exports, the cutoff-uniform critical estimate excludes finite-time blow-up and yields a unique global smooth solution for smooth divergence-free mean-zero initial data on \ (T³\). The pressure is reconstructed by the periodic Poisson equation\- p=ᵢⱼ (uᵢ uⱼ), ₓ℃p (x, t) \, dx=0. \ Status statement: NS-V is a final assembly module in the TEBAC Navier--Stokes candidate proof chain. It closes the Clay-compatible global regularity assembly relative to the upstream module exports. It should not be read as an independently refereed or officially certified solution of the Clay Millennium problem; the full external status depends on referee-level verification of the entire NS-I--NS-V chain, especially the critical NS-IIIa/NS-IV interface.
Tosho Lazarov Karadzhov (Thu,) studied this question.