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): `SAPZ_Singularity_Principle_Navier-Stokes_v5.4r7.pdf`- Companion paper (PDF): `Aux_Proof_v5.4r7.pdf` The program is organized around a **verifiable, scale-uniform threshold envelope** built from a convolution-first energy-density observable:\[δ_ε(t):= \|\, |∇ u(·,t)|^2 * φ_ε \,\|L^∞_x,δ(t) := 0<ε≤ ε_0δ_ε(t),\]together with a universal threshold level \(δ_c>0\) 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 \(ε\)-regularity closure. (The endpoint closure reduces to classical \(ε\)-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.