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\ T³= R³/ (2 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-I-II-III/IIIa-IV-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 ₓ ₓ\|A^1/4uN (t) \|₋ℂ²+₀T\|A^3/4uN (t) \|₋ℂ²\, dt C (T, , u₀), \ (C (T, , u₀) \) 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 \ (T³\) 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.