Navier–Stokes Regularity Program (Vol. VIII): Technical Closure of the H3→ Aη Bridge and Strip–Tube Estimates

This volume provides the deferred technical proofs used as templates in Vol. VII. We (i) formalize the interface-layer reduction that isolates the viscous barrier term to a thin strip Uη(t) around the interface Σ(t), (ii) prove the harmonic-analytic bridge from the H3 (palinstrophy non-concentration) hypothesis to an L∞-type control of Aη(t) = ||∇u(·, t)||_{L∞(Uη(t))}, and (iii) give a rigorous strip–tube comparison needed to interpret interface geometry via tube-volume scans. What this volume closes (relative to Vol. VII)(i) Interface-layer reduction: the barrier contribution depends only on a thin strip Uη(t) where the cutoff gradient is supported.(ii) H3→Aη bridge: control of the localized amplitude Aη(t) from H3-type non-concentration.(iii) Strip–tube comparison: quantitative bounds relating strip covers and tube-volume scans, used to estimate effective interface dimension from data.