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):** *Aux_Proof* (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\[δ_ε(t) := x∈Ω∫_Ω |∇ u(y,t)|^2\,φ_ε(x-y)\,dy,δ(t) := ε↓ 0δ_ε(t),\]and a **canonical barrier threshold**\[δ_c := ν^2 y_+,_+ := {b+√b^2+4ac}{2a},\]arising from a Riccati-type comparison inequality with \(ε\)-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 \(δ(t)\) yields \(|∇ u(·,t)|^2∈ L^∞\) on a.e. time-slice.- **Kinematic CKN exclusion:** a purely kinematic \(r^4\)-type estimate excludes CKN-scale parabolic concentration on sufficiently small cylinders.- **Gate B (standard \(ε\)-regularity + continuation):** once CKN concentration is excluded, classical \(ε\)-regularity yields continuation and rules out blow-up under uniform-scale strict subcriticality. Necessity is formulated contrapositive-style:if \(t→ T^-δ(t) < δ_c\), then blow-up at time \(T\) cannot occur; equivalently, blow-up forces \(t→ T^-δ(t)≥ δ_c\). ## 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 \( Aᵛᵉˡε_(t_)≥ a_\). - **(T2) Pivot:** high-pass witness \(⇒\) transport defect \( Tε_\). - **(T3) Extraction:** transport defect \(⇒\) positive transport residual \((rᵗʳε_)_+\).- 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)