What question did this study set out to answer?

The central aim is to develop a coherence model that circumvents the need for pointwise coherence in the vorticity direction within the Navier-Stokes context.

January 14, 2026Open Access

Navier–Stokes Regularity Program (Vol. XII): Average Coherence from Interface Dissipation — SACO Entry without Pointwise Coherence (with Technical Supplement)

Key Points

The central aim is to develop a coherence model that circumvents the need for pointwise coherence in the vorticity direction within the Navier-Stokes context.
Introduced the SACO entry inequality using average directional coherence from an interface dissipation budget.
Utilized localized interface dissipation on stopping-time selected active parabolic boxes.
Presented angle-energy estimates and fractional bookkeeping to support new theoretical claims.
Establish the capability to close the SACO entry inequality without pointwise coherence.
Confirmed revised near-field bounds leading to a reduction in coherence requirements.
Provided a clear pathway for verifying theoretical claims through detailed technical proofs in the supplementary material.

Abstract

Problem: Near-field vortex-stretching control in the Interface–Law regularity program for 3D Navier–Stokes, previously requiring pointwise Hölder coherence of the vorticity direction.Main result: The SACO entry inequality (near + far) can be closed using an average directional coherence bound derived from a localized interface dissipation budget on stopping-time selected active parabolic boxes, removing the need for pointwise coherence.How to verify: See Theorems 3.1–3.3 in the main note and the complete technical proofs in the attached Supplement (angle-energy estimates, fractional bookkeeping, revised near-field bounds, and patched unlock lemma near/far). STATUSMain paper + technical supplement (complete proofs). WHAT THIS WORK DOES NOT CLAIMThis volume is a modular component; it does not by itself establish unconditional global regularity of 3D Navier–Stokes. The remaining dynamical task is to establish the assumed local interface dissipation budget (and Morrey-calibrated threshold construction) from Navier–Stokes evolution without circularity.

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