A SAPZ Threshold Criterion and Gate-Based Closure Interface for 3D Navier–Stokes Regularity (Main + Auxiliary Proofs) — v4.1

# Overview This record releases **v4. 1** of a two-document set for 3D incompressible Navier–Stokes, written in a **criterion / proof-interface** style. Files: - **Main paper (PDF + TeX): ** *A SAPZ Threshold Criterion and Gate-Based Closure Interface for 3D Navier–Stokes Regularity (v4. 1) *- **Companion (PDF + TeX): ** *Auxiliary Proofs (v4. 1) *- **Bundle (ZIP): ** packaged archive containing both TeX/PDF sources. **Positioning (important): **- This release does **not** claim unconditional global regularity for arbitrary data. - It provides (i) a **sharp threshold-type regularity criterion** (necessity + sufficiency under strict subcriticality), and (ii) a **referee-facing closure interface** organized into modular “gates”. - Any numerical experiments are **illustrative only** and are **not** proof inputs. # Canonical density (definition discipline) Fix a nonnegative unit-mass mollifier \ (_\) (canonical caloric bump). Define the convolution envelope\_ (t): = \| (| u (, t) |² * _) \, _ \|₋^ (), (t): = ₀_ (t), standard boundary cutoffs \ (_\) on bounded domains (and \ (_ 1\) on \ (R³\), \ (T³\) ). This convolution-based definition is the **official** object at the Leray–Hopf level. Any operator/trace interpretations appear only as **optional remarks** in smooth regimes and are not used in the core chain. # Main statement (criterion form) Let \ (c\) denote the critical threshold induced by the RNF/RZ constants (explicit in the papers). The Main paper establishes a **threshold criterion**: - **Necessity: ** if a singularity occurs at time \ (T\), then \ (ₓ ₓ (t) c\). - **Sufficiency (strict subcriticality): ** if a strict subcritical margin holds in a uniform-scale form (as stated in the paper), then the solution is regular on the interval and admits continuation. The Companion provides the supporting modules used in the sufficiency closure, including the rigorous time-slice handling needed at the Leray–Hopf level. # Primary proof interface (Gate A → Gate B) Sufficiency is organized as follows: 1) **RNF/RZ module. ** An ODE-type barrier for a renormalized density: \ Dₜ^+ _ (t) -a\, _ (t) ² + b\, _ (t) + c + R_ (t), \ with explicit constants and a locally integrable remainder. 2) **Gate A (Approximate-Identity \ (L^\) Identification). ** A deterministic identification principle converting finiteness of \ (₀\|f*_\|₋^\) into \ (f L^\) (a. e. in time), with boundary cutoffs and measurability/representative issues handled explicitly. 3) **Reverse concentration (kinematic contradiction). ** Microscopic CKN-type concentration forces an explosive lower bound for a boundary-normalized local density on a positive-measure set of times (averaging + covering + mollifier positivity). Gate A supplies a universal upper bound outside a null set, yielding a contradiction at sufficiently small scales. 4) **Gate B (CKN \ (\) -regularity). ** Standard \ (\) -regularity and continuation exclude singular points under the strict subcriticality hypothesis. Optional robustness modules (e. g. conditional kernel-independence under Besov control) are clearly labeled **Conditional / Not used in the primary chain**. # Program closure target (from criterion to unconditional regularity) To upgrade the criterion into unconditional global regularity, an additional a priori ingredient is required. Typical forward targets include: - **CT1 (Automatic strict subcriticality): ** prove that the strict subcriticality hypothesis holds automatically for arbitrary Leray–Hopf data. - **CT3 (Average-to-pointwise strict margin): ** provide a PDE mechanism that produces a short-window averaged strict margin and upgrades it to pointwise strict subcriticality. A candidate “no-free-saturation” route is developed as a structured target, including pressure/CZ and high-frequency anisotropy components. The present release supplies the **criterion + closure interface**; these closure targets are stated as forward goals rather than claims. # Files in this record - `SAPZSingularityPrincipleNavier-Stokesᵥ4. 1. pdf`- `SAPZSingularityPrincipleNavier-Stokesᵥ4. 1. tex`- `AuxProofᵥ4. 1. pdf`- `AuxProofᵥ4. 1. tex`- `NSSAPZᵥ4. 1bundle. zip` # Reproducibility / verification The TeX sources are included. The “gate” organization is designed for referee verification: each module is stated with explicit hypotheses, and the dependency ledger indicates where each input is used. # Citation **Lee Byoungwoo**, *A SAPZ Threshold Criterion and Gate-Based Closure Interface for 3D Navier–Stokes Regularity (Main + Auxiliary Proofs) — v4. 1*, Zenodo (2026). (Replace with the DOI assigned by Zenodo after upload. )