SAPZ Singularity Principle for 3D Navier–Stokes (v4.9): Threshold–Criterion Interface with Sealed Route-T Closure (Main + Companion Modules)

# Overview

This record releases **SAPZ Singularity Principle for 3D Navier–Stokes (v4.9)** together with the companion module paper **Aux_Proof v4.9**. The pair is written as a **proof interface**: the main paper states a threshold–criterion theorem plus a referee ledger, while the companion supplies the modular closure packages and the sealed Route-T chain.

The central energy-class diagnostic is fixed by a convolution-first definition
\[
\delta_\varepsilon(t):=\bigl\|\,|\nabla u(\cdot,t)|^2 * \phi_\varepsilon\,\bigr\|_{L^\infty_x},
\qquad
\delta(t):=\limsup_{\varepsilon\downarrow 0}\delta_\varepsilon(t),
\]
together with a canonical RNF (Riccati normal form) equilibrium threshold \(\delta_c=\nu^2 y_+\) (normalization fixed in the companion).

# Main theorem (two-direction form)

The theorem is presented in a “two-direction” architecture:

- **Sufficiency (criterion).** On any finite window \((0,T)\), if there exist \(\eta\in(0,1)\) and \(\varepsilon_0>0\) such that
  \[
  \sup_{t\in(0,T)}\sup_{0<\varepsilon\le \varepsilon_0}\delta^{BN}_\varepsilon(t)\le (1-\eta)\,\delta_c,
  \]
  then the solution is smooth on \((0,T]\) and extends beyond \(T\). (In \(\mathbb{R}^3\) or \(\mathbb{T}^3\), boundary normalization is trivial.)

- **Necessity (contrapositive).** Any finite-time loss of regularity forces threshold reach:
  \[
  \limsup_{t\to T^{\ast-}}\delta(t)\ge \delta_c.
  \]

# Closed modules (companion)

Aux_Proof v4.9 provides a finite-window closure package for:

- **RNF (Riccati normal form):** a Dini-derivative inequality for \(\delta_\varepsilon\) with \(\varepsilon\)-independent coefficients on finite windows.
- **Residual decomposition:** transport / pressure / boundary channels with vanishing \(L^1\)-mass as \(\varepsilon\downarrow 0\) (finite windows).
- **Boundary normalization (BN):** a separate module for bounded no-slip domains.
- **Reverse concentration + Gate structure:** the closure chain supplying the criterion-level implication.
- **Route-T (transport-bypass) localization:** Littlewood–Paley / weighted almost-orthogonality blocks and tail absorption.
- **Route-T extraction (T3):** commutator representation, positivity, and near-maximizer localization are sealed as a proved module in v4.9.

# What is new in v4.9

- **Sealed Route-T closure:** the transport-bypass chain is organized as a three-box route (T1–T3) with T3 sealed as a proved module, enabling a referee-facing “attack–defense–combine” narrative.
- **Scale/time coherence closure:** the fixed-scale short-window persistence mechanism at \(\varepsilon_\star\) is formulated to eliminate ambiguity between \(\limsup_{\varepsilon\downarrow 0}\) diagnostics and a fixed dyadic scale used inside a contradiction window.
- **Version/ledger hygiene:** legacy internal references are aligned to v4.9 and the status snapshot is made consistent across main and companion.

# Scope & non-toy status

- The framework is stated at the energy (Leray–Hopf / suitable weak) level using convolution envelopes.
- Operator-trace viewpoints are treated only as smooth-regime interpretations; they are not used to justify energy-class steps.

# Contents (files)

- Main paper (PDF): *SAPZ_Singularity_Principle_Navier-Stokes v4.9*
- Companion modules (PDF): *Aux_Proof v4.9*

# Suggested citation

Lee Byoungwoo, “SAPZ Singularity Principle for 3D Navier–Stokes (v4.9): Threshold–Criterion Interface with Sealed Route-T Closure (Main + Companion Modules),” Zenodo, 2026.