Navier–Stokes Regularity Program (Vol. XX): A One-Result Manuscript: Weak Quantile Interface Law and Regularity Closure

This record contains Vol. XX of the Navier–Stokes Regularity Program. The purpose of this final volume is to present a strict one-result manuscript under a verification protocol: a PDE-level bridge that connects the 3D Navier–Stokes vorticity dynamics to the multi-scale closure established in Vol. 19. Main result: we prove **Navier–Stokes implies a weak Quantile Interface Law (qILweak)**: if smaller-scale local vorticity quantiles remain bounded on a diffusion window, then the larger-scale quantile decays strictly in time average. The proof is derived from the time-averaged localized truncated enstrophy balance for suitable weak solutions on the periodic torus T^3. Combined with the Vol. 19 closure (quantile packing and log-free near-field control), this yields unconditional regularity for smooth initial data on T^3. What is included- Main paper (Vol. XX): reader contract, full qILweak derivation, explicit attack surface checklist, and the final closure node linking Vol. XIX + Vol. XX.Sabljic_NS_Vol20_Weak_Quantile_Interface_Law_Regularity_Closure.pdf What this record does NOT claimThis volume does not claim new a priori bounds beyond qILweak. It does not reprove the multi-scale closure; Vol. 19 is cited for quantile packing and logarithm-free near-field control. The only new contribution here is the PDE-level derivation of strict decay of time-averaged local quantile seeds in the absence of small-scale amplification.