SAPZ Singularity Principle for 3D Navier–Stokes (v4.2r21): A Threshold–Criterion Proof Interface (Main + Companion Modules)

# Overview This record releases **SAPZ Singularity Principle for 3D Navier–Stokes (v4.2r21)** together with a companion module paper **Aux_Proof v4.2r21**, providing a *proof-interface* architecture for a threshold–criterion approach to regularity. The core object is an energy-class, convolution-first SAPZ envelope\[δ_ε(t):=\|\,|∇ u(·,t)|^2 * φ_ε\,\|L^∞_x,δ(t):=ε↓ 0δ_ε(t),\]and a canonical Riccati-equilibrium threshold \(δ_c\) defined by an RNF (Riccati normal form) module. The main theorem is presented in a “two-direction” form: - **Sufficiency (criterion):** a uniform-scale subcriticality condition \[ t∈(0,T)0<ε≤ ε_0δBN_ε(t)≤ (1-η)δ_c \] implies regularity/continuation on \((0,T]\). - **Necessity (contrapositive):** any finite-time loss of regularity forces threshold reach: \[ _{t→ T⁻}δ(t)≥ δ_c. \] The companion paper supplies the closure modules (RNF + residual decomposition + boundary normalization + reverse concentration / closure theorems) and the proof-status ledger. # Closed results (module-level) The companion modules (Aux_Proof v4.2r21) provide a closed finite-window package for: - RNF inequality for \(δ_ε\) with \(ε\)-independent coefficients on finite windows.- Residual decomposition into transport/pressure/boundary channels with vanishing \(L^1\)-mass as \(ε↓ 0\) (finite-window).- Boundary normalization (bounded no-slip domains) as a separate module; in \(R^3\) or \(T^3\) the BN step is trivial.- Reverse-concentration and threshold-to-criterion closure modules used as proof inputs for the sufficiency direction. # What is new in v4.2r21 - **Definition hygiene:** the SAPZ functional is fixed in the energy class by the convolution-first definition; any operator-trace realization is treated as a smooth-regime interpretation only.- **Quantifier discipline:** “criterion sufficiency” and “contrapositive necessity” are cleanly separated to avoid overstatement (no claim that \(δ(t)<δ_c\) is necessary for smoothness).- **Finite-window closure ledger:** CT2(T)-type finite-window envelope closure is packaged as a proved module in the companion, with explicit dependence tracking on the fixed cutoff/weight profiles.- **Route-T localization:** a Littlewood–Paley / weighted almost-orthogonality block is isolated and bound into an explicit tail control module. # Scope & non-toy status - The framework is stated at the Leray–Hopf energy level, using only convolution envelopes and finite-window module inputs.- The result is presented as a **verifiable criterion architecture**: it does not assume the existence of singular solutions, and the necessity direction is stated in contrapositive form. # Program closure and targets This release is a journal-cut consolidation of the proof-interface. The companion ledger isolates the remaining Clay-level target as a single explicit step (CT3-(A3): a short-window averaged strict margin input in the reverse-concentration SAPZ injection chain). No claim of unconditional Clay-level resolution is made in this version. # Contents (files) - Main paper (PDF + TeX): *SAPZ_Singularity_Principle_Navier-Stokes v4.2r21*- Companion modules (PDF + TeX): *Aux_Proof v4.2r21* # Suggested citation Lee Byoungwoo, “SAPZ Singularity Principle for 3D Navier–Stokes (v4.2r21): A Threshold–Criterion Proof Interface (Main + Companion Modules),” Zenodo, 2026.