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 **AuxProof 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\_ (t): =\|\, | u (, t) |² * _\, \|₋^䂲, (t): = ₀_ (t), a canonical Riccati-equilibrium threshold \ (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 \ ₓ (₀, ₓ) ₀< 䃐^BN_ (t) (1-) c \ implies regularity/continuation on \ ( (0, T]\). - **Necessity (contrapositive): ** any finite-time loss of regularity forces threshold reach: \ ₓ ₓ^- (t) 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 (AuxProof v4. 2r21) provide a closed finite-window package for: - RNF inequality for \ (_\) with \ (\) -independent coefficients on finite windows. - Residual decomposition into transport/pressure/boundary channels with vanishing \ (L¹\) -mass as \ (0\) (finite-window). - Boundary normalization (bounded no-slip domains) as a separate module; in \ (R³\) or \ (T³\) 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 \ ( (t) <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): *SAPZSingularityPrincipleNavier-Stokes v4. 2r21*- Companion modules (PDF + TeX): *AuxProof 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.