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

# Summary (v4. 3) This record releases **v4. 3 (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. 3) - **Companion (PDF): ** *AuxProof* (v4. 3) — theorem-level analytic modules and the 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 comparison inequality 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. 3 (referee-facing) - **Route T (transport-bypass) is fixed as the primary closure blueprint**, explicitly avoiding delicate pressure-cancellation analysis. - The Route T interface is organized as a **three-box reduction**: - ** (T1) Trigger: ** contradiction window \ (\) high-pass witness \ (A^vel_ (t_) a_\). - ** (T2) Pivot: ** high-pass witness \ (\) transport defect \ (T_\). - ** (T3) Extraction: ** transport defect \ (\) positive transport residual \ ( (r^tr_) _+\). - Two **sealed quantitative lemmas** are recorded to fix constants and remove “proof-idea” ambiguity: - **Tail absorption** (buffered split fixed explicitly). - **Kernel lower bound for symmetrized difference-quotient weights** (explicit coercivity input). ## Program status (remaining Clay-level target) The only remaining Clay-level completion target is isolated as a referee-facing acceptance test: - **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. 3 (March 1, 2026)