Navier–Stokes Regularity Program (Vol. IX): Interface Law in Taylor–Green Turbulence (Main Paper + Supplementary Diagnostics)

This record contains: (1) Main paper (Vol. IX)Navier–Stokes Regularity Program (Vol. IX): Geometric Sparseness and the Interface Law: DNS-Certified Resolved-Scale Dominance in 3D Navier–Stokes (2) Supplementary notesSupplementary Material for Volume IX: Cross-Quantile Robustness and Extended Diagnostics for the Interface Law in Taylor–Green Turbulence Summary (main result):Using direct numerical simulation (DNS) snapshots of the 3D Taylor–Green vortex at viscosity ν = 0.001 (resolution N = 256), the analysis certifies two empirical laws extracted from a fixed diagnostic pipeline: Law 1 (Interface Law / geometric brake):The fitted interface exponent satisfies θ(t) > 0, equivalently γ̂(t) = 1 + θ(t) > 1, with particular robustness in the late turbulent phase (t ≥ 7). The supplement confirms this across q ∈ {0.95, 0.97, 0.99}. Law 2 (resolved-scale viscous dominance):The scale-dominance ratio Rρ(t0, T) exceeds 1 on all numerically resolved scales ρ ≥ Δx. Any extrapolated crossing Rρ = 1 occurs below the grid scale (subgrid), at the present resolution. Supplement contents:• Cross-quantile robustness scans q ∈ {0.95, 0.97, 0.99}• Snapshot-level tables for θ(t), d_eff(t), and thresholds Λ(t)• Representative log–log fit diagnostics (early and late times)• Scale-dominance tables and heatmaps log10 Rρ(t, ρ)• Cross-quantile overlays and late-time Interface Law statement