A Spectral-Entropy Threshold Framework for Regularity and Blow-up in the Navier–Stokes Equations: The SAPZ Principle (v5.3r2A)

TitleSAPZ Singularity Principle for the 3D Navier–Stokes Equations: A Spectral–Entropy Threshold Criterion with Route–T Discharge (v5. 3r2A) KeywordsNavier–Stokes; 3D incompressible flow; global regularity; finite-time blow-up; continuation criterion; epsilon-regularity; CKN; Leray–Hopf weak solutions; energy concentration; Littlewood–Paley; mollification; spectral methods; entropy methods; transport defect; boundary effects # Overview This record releases **v5. 3r2A** of a two-paper set developing the **SAPZ (Spectral–Averaged Parabolic Zone) principle** for the 3D incompressible Navier–Stokes equations. The program is organized around a **verifiable, scale-uniform threshold criterion** for continuation, together with a companion paper that discharges the internal “Route–T” targets used in the proof interface. The SAPZ envelope is defined in a convolution-first manner by\[_ (t): = \| \, | u (, t) |² * _ \, \|₋^䂲, (t): = _{00 \) is constructed from fixed analytic profiles (mollifier / cutoffs / normalization) and the viscosity. # Record contents - Main paper (PDF): SAPZSingularityPrincipleNavier-Stokes (v5. 3r2A) - Companion paper (PDF): AuxProof (v5. 3r2) # Main statements (high-level) ## Continuation criterion (finite-horizon) For a Leray–Hopf weak solution \ (u\), if on a given horizon \ ( (0, T) \) one has uniform-scale subcriticality below the universal threshold, then \ (u\) is smooth up to time \ (T\) and continues beyond \ (T\). Conversely, any finite-time singularity forces threshold reach in the quantitative “necessity” sense formulated in the main paper. ## Route–T / Gate A / Gate B closure interface (companion) The companion paper supplies theorem-level modules that implement the “Route–T” discharge chain and the Gate A/B interfaces: - Gate A: approximate-identity \ (L^\) identification at the declared solution class;- Route–T (transport-bypass extraction): defect \ (\) strictly positive transport residual;- Gate B: standard CKN \ (\) -regularity closure. # What is new in v5. 3r2A (this record) v5. 3r2A aligns the **top-level conclusion strength** with the criterion architecture by adding a **global corollary** in the main paper: - Combining the main finite-horizon criterion with the companion Route–T discharge on each finite horizon yields \ (T^=\) (global continuation), hence global regularity. In addition, a one-page **proof map** is inserted in the main paper to make the dependency chain immediately auditable. # Scope 2) CT3 persistence / scale selection without assumption leakage;3) Route–T transport extraction (defect \ (\) positive residual with explicit constant dependencies). # Recommended citation Lee Byoungwoo, "SAPZ Singularity Principle for the 3D Navier–Stokes Equations: A Spectral–Entropy Threshold Criterion with Route–T Discharge" (Version v5. 3r2A), with companion "Auxiliary Proof Modules for the SAPZ Singularity Principle" (Version v5. 3r2), Zenodo, 2026.