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\[δ_ε(t):=\|\,|∇ u(·,t)|^2 * φ_ε\,\|L^∞_x,δ(t):=ε↓ 0δ_ε(t),\]together with a canonical RNF (Riccati normal form) equilibrium threshold \(δ_c=ν^2 y_+\) (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 \(η∈(0,1)\) and \(ε_0>0\) such that \[ t∈(0,T)0<ε≤ ε_0δBN_ε(t)≤ (1-η)\,δ_c, \] then the solution is smooth on \((0,T]\) and extends beyond \(T\). (In \(R^3\) or \(T^3\), boundary normalization is trivial.) - **Necessity (contrapositive).** Any finite-time loss of regularity forces threshold reach: \[ _{t→ T⁻}δ(t)≥ δ_c. \] # Closed modules (companion) Aux_Proof v4.9 provides a finite-window closure package for: - **RNF (Riccati normal form):** a Dini-derivative inequality for \(δ_ε\) with \(ε\)-independent coefficients on finite windows.- **Residual decomposition:** transport / pressure / boundary channels with vanishing \(L^1\)-mass as \(ε↓ 0\) (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 \(ε_\) is formulated to eliminate ambiguity between \(ε↓ 0\) 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.