Navier–Stokes Regularity Program (Vol. XIV): Unified Coherence & Gain Diagnostics for DNS

Problem. The 3D Navier–Stokes regularity problem motivates continuation criteria that can be tested at resolved scales. This volume supplies a DNS-facing measurement layer for the Interface–Law / coherence–Morrey–BMO route by translating abstract hypotheses into concrete diagnostics. Main result (what this volume establishes). A unified empirical certification framework consisting of:(i) a Gain certificate verifying positive interface gain exponent θ > 0 via dyadic Poincaré scaling on vorticity interfaces; and(ii) a Coherence certificate verifying scale-averaged direction-coherence decay ε_j(t) → 0 on enlarged interfaces Σ♯(t).The volume specifies standardized output formats (Tables 1–3, Figures 1–4), robustness tests across threshold quantiles q ∈ {0.95, 0.97, 0.99}, and practical pass/fail criteria (P1)–(P3). Complete implementation details are provided in the Supplement. How to verify / reproduce (minimum). Using DNS vorticity snapshots:1) Choose a statistically steady late-time window I (example: t ≥ 7 in the included tables).2) Compute the gain diagnostic (Section 3) and the coherence diagnostic (Section 4 + Supplement).3) Produce Tables 1–3 and Figures 1–4 and apply criteria (P1)–(P3) with a chosen tolerance Tol.4) Cross-check robustness across q ∈ {0.95, 0.97, 0.99}. Status of this upload- Main paper: Vol. XIV (protocol + certificate definitions + standardized outputs + pass/fail criteria).- Supplementary material: full implementation protocol, parameter ranges, tables, workflow, and validation checks. - ZIP: vol14_tables_and_artifacts.zipContents (CSV):- beta_kappa_time_series_256_clean.csv- beta_kappa_time_series_256_clean2.csv- d_eff_timeseries.csv- scan_summary.csv- scan_timeseries.csv- sensitivity_resolution.csv- sensitivity_rho_range.csv- sensitivity_threshold.csv- summary_theta_by_q.csv- theta_time_series_256_clean.csv Verification pointer:The reported diagnostics in Vol. XIV can be cross-checked directly against these CSV tables (time-series and summary tables). The CSVs are intended as the source-of-truth numerical outputs for the figures/tables presented in the PDFs. What this work does NOT claim (explicit non-claims)- This volume does NOT prove global-in-time regularity of 3D Navier–Stokes.- The certificate provides DNS-testable evidence for the stretching-depletion hypothesis at resolved scales, but the implication “depletion ⇒ regularity” requires rigorous continuation estimates developed elsewhere in the multi-volume program.