SAPZ Singularity Principle for the 3D Navier--Stokes Equations: A Spectral--Entropy Threshold Criterion with Route--T Discharge (v5.4r7)

TITLESAPZ Singularity Principle for the 3D Navier--Stokes Equations: A Spectral--Entropy Threshold Criterion with Route--T Discharge (v5. 4r7) KEYWORDS (comma-separated) Navier--Stokes, 3D incompressible flow, global regularity, finite-time blow-up, continuation criterion, epsilon-regularity, CKN, Leray--Hopf weak solutions, suitable weak solutions, local energy inequality, energy concentration, Littlewood--Paley, mollification, spectral methods, entropy methods, transport defect, commutator estimates, high-frequency filtering, boundary effects DESCRIPTION # Overview This record releases **v5. 4r7** of a two-paper set developing the SAPZ (Spectral--Averaged Parabolic Zone) principle for the 3D incompressible Navier--Stokes equations. **Files included in this record (PDF only): **- Main paper (PDF): `SAPZSingularityPrincipleNavier-Stokesᵥ5. 4r7. pdf`- Companion paper (PDF): `AuxProofᵥ5. 4r7. pdf` The program is organized around a **verifiable, scale-uniform threshold envelope** built from a convolution-first energy-density observable: \_ (t): = \|\, | u (, t) |² * _ \, \|₋^䂲, (t): = _{00\) determined by fixed analytic profiles (mollifier / cutoffs / normalization) and the viscosity. # Main statements (high-level) ## Continuation criterion (finite horizon) For a Leray--Hopf weak solution \ (u\), the main paper establishes a continuation criterion of the following form: - If a uniform-scale subcriticality bound below the universal threshold holds on a given horizon \ ( (0, T) \), then \ (u\) is smooth on \ ( (0, T\) and continues beyond \ (T\). - Conversely, any finite-time singularity forces threshold reach in the quantitative necessity sense formulated in the main paper. ## Companion closure interface (Route--T / Gate A / Gate B) The companion paper supplies theorem-level modules implementing the proof interface: - **Gate A: ** approximate-identity \ (L^\) identification on the declared solution class. - **Route--T (transport-bypass extraction): ** defect \ (\) strictly positive transport residual, using a high-frequency filtering / commutator-extraction chain. - **Gate B: ** standard CKN \ (\) -regularity closure. (The endpoint closure reduces to classical \ (\) -regularity once the Gate A/B interfaces and Route--T discharge are in place. ) # Cognitive defense (reader-facing) This release adds reviewer-friendly navigation and anti-misreading defenses: - A parameter/constant hierarchy is made explicit to rule out circular dependence of scales and thresholds. - A proof map and “verification focal points” identify the trusted core components that independent verification naturally concentrates on. - A scope disclaimer clarifies why the Route--T bypass avoids the typical failure modes of pointwise pressure/nonlocal control. - A suitability/LEI reminder isolates the framework from “wild solution” pathologies outside the declared class. # Scope The framework targets standard 3D incompressible Navier--Stokes settings, including whole-space/periodic geometries and bounded no-slip domains (handled via boundary-normalized variants in the companion). The writing is modular: the main paper isolates the criterion statement, while the companion isolates analytic modules and proof interfaces. # Recommended citation Lee Byoungwoo, "SAPZ Singularity Principle for the 3D Navier--Stokes Equations: A Spectral--Entropy Threshold Criterion with Route--T Discharge" (Version v5. 4r7), with companion "Auxiliary Proof Modules for the SAPZ Singularity Principle" (Version v5. 4r7), Zenodo, 2026.