The SAPZ Principle for Navier–Stokes Regularity: Threshold Architecture and Proof Interface (v4.3r13)

# Summary (v4. 3r13, PDF-only) This record releases **v4. 3r13 (PDF-only) ** of the SAPZ Navier–Stokes program: a **threshold-and-proof-interface** framework for 3D incompressible Navier–Stokes regularity vs blow-up. The presentation is organized as a **module chain + gate interface**, with a referee-facing acceptance test that isolates the **single remaining Clay-level PDE target**. ## Files in this record (PDF-only) - **Main paper (PDF): ** *The SAPZ Principle for Navier–Stokes Regularity: Threshold Architecture and Proof Interface* (v4. 3r13) - **Companion (PDF): ** *AuxProof* (v4. 3r13) — theorem-level analytic modules and closure interface ## Core diagnostic and canonical threshold The program is centered on the mollified trace-energy diagnostic\_ (t): = ₗ_ | u (y, t) |²\, _ (x-y) \, dy, (t): = ₀_ (t), a **canonical barrier threshold**\c: = ² y_+, _+: = b+b²+4ac2a, from a Riccati-type normal form with \ (\) -independent coefficients \ (a, b, c\). ## What is proved (criterion-level closure) The closed chain is presented as **Gate A → Gate B**: - **Gate A (approximate-identity \ (L^\) identification): ** finiteness of \ ( (t) \) yields \ (| u (, t) |² L^\) on a. e. time-slice. - **Kinematic CKN exclusion: ** a purely kinematic \ (r⁴\) -type estimate excludes CKN-scale parabolic concentration on sufficiently small cylinders. - **Gate B (standard \ (\) -regularity + continuation): ** once CKN concentration is excluded, classical \ (\) -regularity yields continuation and rules out blow-up under uniform-scale strict subcriticality. Necessity is formulated contrapositive-style: if \ (ₓ ₓ^- (t) < c\), then blow-up at time \ (T\) cannot occur; equivalently, blow-up forces \ (ₓ ₓ^- (t) c\). ## What is new in v4. 3r13 (referee-facing) - **Route T (transport-bypass) is fixed as the primary closure blueprint**, explicitly avoiding delicate pressure-cancellation analysis. - The remaining Clay-level PDE difficulty is compressed to a **single non-vacuous target inequality** (“N2-core★”) expressed in terms of a **symmetrized difference-quotient / transport defect** at the selected scale. - A quantitative **dichotomy lemma** fixes the two exclusive scenarios on a contradiction window: (i) structured capture triggers the Route-T witness, or (ii) capture failure forces a positive defect lower bound. - The key implication “high-pass mass ⇒ transport defect lower bound” is recorded with proof-text (not only proof-idea). - The companion’s referee checklist is **mirrored in the main paper**, so the Step-3 target and its reduction chain are visible from the main paper alone. ## Nonvacuity (theorem-level examples) To show the acceptance test is nonempty, the companion records theorem-level certificates in standard regularity regimes (e. g. critical small-data settings; Serrin-type smallness on finite windows) where CT3- (A3) holds automatically. ## Remaining Clay-level completion target The only remaining Clay-level completion target is isolated as the averaged strict-margin condition: - **CT3- (A3): ** a short-window averaged strict-margin condition for the normalized diagnostic. ## Optional robustness module For robustness only, the companion retains an independent “legacy injection-engine” reverse-concentration theorem. This module is explicitly optional and not required on the primary Gate A → Gate B route. ## Author and version - Author: Lee Byoungwoo- Version: v4. 3r13 (March 1, 2026)