TEBAC Navier–Stokes Program: A Modular Proof of Global Regularity on the Periodic Three-Torus

This master preprint presents the unified TEBAC Navier--Stokes program for the three-dimensional incompressible Navier--Stokes equations on the periodic torus\[ T^3= R^3/(2π Z)^3.\] The manuscript integrates the full NS-I--NS-V modular chain into a single continuously numbered proof dossier with one common abstract, one final bibliography, unified references, and a final theorem-facing structure. The proof architecture is\[{NS-I}{NS-II}{NS-III/IIIa}{NS-IV}{NS-V}.\] The first module establishes the periodic Leray--Stokes--Galerkin foundation, the Stokes scale, the classical energy ledger, and the critical cascade ledger. The second module converts the vorticity stretching obstruction into a dyadic shell and triadic resonance classification. The third module reduces the high-frequency cascade obstruction to critical resonance absorption, while the companion NS-IIIa module supplies the terminal axial-envelope coercivity and arithmetic packing closure. The fourth module converts the resulting absorption export into the cutoff-uniform critical estimate\[0≤ t≤ T\|A1/4u_N(t)\|L^2^2+ν∫_0^T\|A3/4u_N(t)\|L^2^2\,dt≤ C(T,ν,u_0),\]with \(C(T,ν,u_0)\) independent of the Galerkin cutoff \(N\). The fifth module performs the final Clay-compatible assembly: Galerkin compactness, synchronization with the local classical branch, weak--strong uniqueness, continuation, pressure reconstruction, and smoothness bootstrap. The master theorem states that every smooth divergence-free mean-zero initial datum on \( T^3\) generates a unique global smooth solution of the periodic three-dimensional incompressible Navier--Stokes system. The manuscript is presented as a complete TEBAC modular proof dossier and is intended for independent referee-level verification of the full chain, especially the NS-IIIa/NS-IV critical estimate interface. My manuscript “TEBAC Navier-Stokes Program: Global Regularity on the Periodic Three-Torus” has been submitted to Acta Mathematica and is under editorial consideration.