# TitleA Spectral–Entropy Threshold Framework for Regularity and Blow-up in the Navier–Stokes Equations: The SAPZ Principle (v4.3r1) # OverviewThis record releases a two-paper set: - **Main paper (PDF):** *A Spectral–Entropy Threshold Framework for Regularity and Blow-up in the Navier–Stokes Equations: The SAPZ Principle*- **Companion (PDF):** *Auxiliary Proof Modules for the SAPZ Singularity Principle* The framework centers on the mollified trace–energy functional\[\delta_\varepsilon(t):=\sup_{x\in\Omega}\int_\Omega |\nabla u(y,t)|^2\,\varphi_\varepsilon(x-y)\,dy,\qquad\delta(t):=\limsup_{\varepsilon\downarrow 0}\delta_\varepsilon(t),\]and a Riccati-type normal form with \(\varepsilon\)-independent coefficients that yields a canonical critical threshold\[\delta_c=\nu^2 y_+,\qquady_+ = \frac{b+\sqrt{b^2+4ac}}{2a}.\] # What is proved vs. what remains (referee-facing)- **Criterion-level (proved as an interface):** Uniform-scale SAPZ subcriticality implies regularity/continuation via the companion closure chain (Gate A ⇒ kinematic CKN-exclusion ⇒ Gate B). - **Necessity (contrapositive form):** Any finite-time loss of regularity forces threshold reach \(\limsup_{t\to T^-}\delta(t)\ge \delta_c\). - **Single Clay-level PDE completion target (isolated):** The averaged strict-margin input **CT3-(A3)** is explicitly isolated as the only remaining PDE target. Route T (transport-bypass) is the preferred blueprint: it reduces CT3-(A3) to a one-page trigger statement plus standard Littlewood–Paley / spectral-gap / commutator micro-lemmas. # Nonvacuity example (theorem-level)To show the acceptance test is nonempty, the main paper includes a theorem-level example:in standard critical small-data regimes (e.g. \(L^3\) or \(BMO^{-1}\)),classical smoothing implies \(\sup_{t\ge t_0}\delta(t)\le \tfrac12\delta_c\) for sufficiently small data,hence CT3-(A3) is automatically certified on every finite horizon \([t_0,T]\). # Files in this record- SAPZ_Singularity_Principle_Navier-Stokes_v4.3r1.pdf- Aux_Proof_v4.3r1.pdf # KeywordsNavier–Stokes; global regularity; blow-up; Leray–Hopf solutions; Caffarelli–Kohn–Nirenberg; ε-regularity;Riccati inequality; Littlewood–Paley; commutators; threshold criterion; spectral entropy. # AuthorLee Byoungwoo