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 εⱼ (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: vol14ₜablesₐndₐrtifacts. zipContents (CSV): - betaₖappaₜimeₛeries₂56clean. csv- betaₖappaₜimeₛeries₂56clean2. csv- dₑffₜimeseries. csv- scanₛummary. csv- scanₜimeseries. csv- sensitivityᵣesolution. csv- sensitivityᵣhoᵣange. csv- sensitivityₜhreshold. csv- summaryₜhetabyq. csv- thetaₜimeₛeries₂56clean. 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.
Branimir Sabljić (Wed,) studied this question.