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\[δ_ε(t) := x∈Ω∫_Ω |∇ u(y,t)|^2 \,φ_ε(x-y)\,dy,δ(t) := ε↓ 0δ_ε(t),\]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. 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):** *Aux_Proof* (v4.2r20), providing theorem-level analytic modules and the closure interface- **Visual/Zenodo supplement (PDF/TeX):** *SAPZ_NS_Visual_Guide* (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 \(L^1loc\))- **BN:** boundary normalization for bounded no-slip domains (with \(λ_1(Ω)\) absorbed into normalization) ## Primary sufficiency route (Gate A → Gate B) - **Gate A (approximate-identity \(L^∞\) identification):** finiteness of \(δ(t)\) upgrades to \(|∇ u(·,t)|^2∈ L^∞\) on a.e. time-slice.- **Kinematic CKN exclusion:** a purely kinematic \(r^4\)-type bound excludes CKN-scale parabolic concentration.- **Gate B (standard CKN \(ε\)-regularity + continuation):** once CKN concentration is excluded on sufficiently small cylinders, standard \(ε\)-regularity yields continuation and rules out blow-up under uniform-scale strict subcriticality. ## Necessity (contrapositive form) All necessity statements are formulated contrapositive-style:if \(t→ T^-δ(t) < δ_c\), then blow-up at time \(T\) cannot occur.Equivalently, any finite-time blow-up must satisfy \(t→ T^-δ(t)≥ δ_c\). # 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)