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

Title
SAPZ Singularity Principle for the 3D Navier–Stokes Equations: A Spectral–Entropy Threshold Criterion with Route–T Discharge (v5.3r2A)

Keywords
Navier–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
\[
\delta_\varepsilon(t)
:= \left\| \, |\nabla u(\cdot,t)|^2 * \phi_\varepsilon \, \right\|_{L^\infty_x},
\qquad
\delta(t) := \sup_{0<\varepsilon\le \varepsilon_0} \delta_\varepsilon(t),
\]
and a universal threshold level \( \delta_c>0 \) is constructed from fixed analytic profiles (mollifier / cutoffs / normalization) and the viscosity.

# Record contents

- Main paper (PDF): SAPZ_Singularity_Principle_Navier-Stokes (v5.3r2A)
- Companion paper (PDF): Aux_Proof (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 \(L^\infty\) identification at the declared solution class;
- Route–T (transport-bypass extraction): defect \(\Rightarrow\) strictly positive transport residual;
- Gate B: standard CKN \( \varepsilon \)-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 \(T^\ast=\infty\) (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 & non-toy status

The framework targets standard 3D incompressible Navier–Stokes settings (including periodic and whole-space geometries, and bounded no-slip domains via boundary-normalized variants in the companion). The papers are written to separate (i) a clean criterion statement from (ii) modular proof components, so independent verification can focus on the trusted core.

# Verification focal points (TCB-style)

Independent verification naturally concentrates on:
1) Gate A (approximate-identity \(L^\infty\) identification);
2) CT3 persistence / scale selection without assumption leakage;
3) Route–T transport extraction (defect \(\Rightarrow\) 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.