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

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

# Overview
This 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
\[
\delta_\varepsilon(t):=\sup_{x\in\Omega}\int_\Omega |\nabla u(y,t)|^2\,\varphi_\varepsilon(x-y)\,dy,
\qquad
\delta(t):=\limsup_{\varepsilon\downarrow 0}\delta_\varepsilon(t),
\]
and a Riccati-type normal form with \(\varepsilon\)-independent coefficients that yields a canonical critical threshold
\[
\delta_c=\nu^2 y_+,\qquad
y_+ = \frac{b+\sqrt{b^2+4ac}}{2a}.
\]

# 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
  \(\limsup_{t\to T^-}\delta(t)\ge \delta_c\).

- **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 \(\sup_{t\ge t_0}\delta(t)\le \tfrac12\delta_c\) 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

# Keywords
Navier–Stokes; global regularity; blow-up; Leray–Hopf solutions; Caffarelli–Kohn–Nirenberg; ε-regularity;
Riccati inequality; Littlewood–Paley; commutators; threshold criterion; spectral entropy.

# Author
Lee Byoungwoo