A Spectral–Entropy Threshold Framework for Regularity and Blow-up in the Navier–Stokes Equations: The SAPZ Principle (v4.3r1)

# 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\[δ_ε(t):=x∈Ω∫_Ω |∇ u(y,t)|^2\,φ_ε(x-y)\,dy,δ(t):=ε↓ 0δ_ε(t),\]and a Riccati-type normal form with \(ε\)-independent coefficients that yields a canonical critical threshold\[δ_c=ν^2 y_+,_+ = {b+√b^2+4ac}{2a}.\] # 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 \(t→ T^-δ(t)≥ δ_c\). - **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 \(t≥ t_0δ(t)≤ 12δ_c\) 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