Navier–Stokes Volume VII: Dynamical Coupling of H2 and H3 via Scale-Dominance Closure

This volume couples the geometric sparseness hypothesis H2 (subcritical tube-volume scaling of high-vorticity interfaces) with the palinstrophy non-concentration hypothesis H3 (strip-law near the interface) by showing that H3 can be used to verify the scale-dominance condition required by the viscous barrier mechanism (Volume VI). Single-focus contribution:H3-driven verification of scale dominance ⇒ propagation of H2 on a nontrivial scale window (under mild threshold drift). Key idea:The barrier inequality involves the stretching amplitude A(t)=||∇u||_{L∞}. Since the mechanism is supported in a thin neighborhood of the interface, we introduce a localized amplitude A_η(t)=||∇u||_{L∞(U_η)} and outline a support-reduction step. An H3-to-amplitude “bridge” (template) is formulated via localized harmonic-analysis estimates, enabling scale-dominance verification from strip palinstrophy control. Numerical pilot (Taylor–Green vortex):A verification-first TG study (512^3, ν=0.01) reports (i) temporal tracking between the fitted H3 exponent θ(t) and interface subcriticality 3−d_eff(t), (ii) strong θ–(3−d_eff) correlation (r≈0.94), and (iii) a resolved-scale certificate for the scale-dominance diagnostic R_ρ(t0,T)>1 over the tested grid of (t0,ρ), with the R_ρ=1 crossing extrapolating below the grid scale. Status:Manuscript / preprint (Navier–Stokes Regularity Program, Volume VII). Templates for the main analytic bridge and strip-to-tube comparison are included as appendices.