A Spectral–Entropy Threshold Framework for Regularity and Blow-up in 3D Navier–Stokes: The SAPZ Principle (v4.2r20)

# Overview This record releases **The SAPZ Principle (v4. 2r20) ** — a *threshold-and-proof-interface* framework for the 3D incompressible Navier–Stokes regularity problem. The core object is a scalar diagnostic built from the mollified trace-energy density\_ (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. This is a **journal-cut** version: expository figures, FAQs, and numerical visualizations are moved to a separate Visual Guide / Zenodo supplement and are **not** used as proof inputs. # Contents - **Main paper (PDF/TeX): ** *The SAPZ Principle for Navier–Stokes Regularity: Threshold Architecture and Proof Interface* (v4. 2r20) - **Companion (PDF/TeX): ** *AuxProof* (v4. 2r20), providing theorem-level analytic modules and the closure interface- **Visual/Zenodo supplement (PDF/TeX): ** *SAPZNSVisualGuide* (v4. 2r20) — exposition-only; not used in proofs- **Bundle ZIP: ** journal-cut bundle for archival convenience # What is proved in v4. 2r20 The papers are organized as a “module chain + gate interface”. ## Closed analytic modules (theorem-level, in the companion) - **TCE: ** trace–convolution equivalence for the SAPZ diagnostic- **RNF: ** Riccati normal form (distributional at Leray–Hopf level) - **RZ: ** residual-zero reduction (commutator and pressure residuals vanish in \ (L¹₋₎₂\) ) - **BN: ** boundary normalization for bounded no-slip domains (with \ (₁ () \) absorbed into normalization) ## Primary sufficiency route (Gate A → Gate B) - **Gate A (approximate-identity \ (L^\) identification): ** finiteness of \ ( (t) \) upgrades to \ (| u (, t) |² L^\) on a. e. time-slice. - **Kinematic CKN exclusion: ** a purely kinematic \ (r⁴\) -type bound excludes CKN-scale parabolic concentration. - **Gate B (standard CKN \ (\) -regularity + continuation): ** once CKN concentration is excluded on sufficiently small cylinders, standard \ (\) -regularity yields continuation and rules out blow-up under uniform-scale strict subcriticality. ## Necessity (contrapositive form) All necessity statements are formulated contrapositive-style: if \ (ₓ ₓ^- (t) < c\), then blow-up at time \ (T\) cannot occur. Equivalently, any finite-time blow-up must satisfy \ (ₓ ₓ^- (t) c\). # Program status and the v4. 3 trigger This release is a **proof-interface consolidation** version. The remaining Clay-level completion target is isolated as a single referee-facing acceptance test: - **CT3- (A3): ** a short-window averaged strict-margin condition for the normalized diagnostic. **v4. 3 trigger: ** the first version that proves CT3- (A3) as a PDE estimate (rather than recording it as an explicit target). # Optional robustness material For robustness only, the companion retains an independent “legacy injection-engine” module producing a reverse-concentration theorem. This module is explicitly **optional** and not required for the primary Gate A → Gate B route. # Proof vs evidence (numerics) Any numerical experiments and fitted validation thresholds are provided for reproducibility and visualization only. They are **not** used as proof inputs. # Author and version - Author: Lee Byoungwoo- Version: v4. 2r20 (March 1, 2026)