SAPZ Singularity Principle for 3D Navier–Stokes (v5.3r): Threshold–Criterion Interface with Constant-Flow Diagram (Main + Companion Modules)

# Overview This record releases **SAPZ Singularity Principle for 3D Navier–Stokes (v5.3)** together with the companion module paper **Aux_Proof v5.3**. The pair is written as a **proof interface**: the main paper states the threshold–criterion theorem and a referee-facing ledger; the companion supplies the analytic modules (RNF/RZ/BN/TCE), the sealed Route-T chain, and a **constant-flow diagram** that makes dependencies and parameter choices auditable. The central diagnostic is fixed in the energy class by a convolution-first envelope\[δ_ε(t):=\|\,|∇ u(·,t)|^2 * φ_ε\,\|L^∞_x,δ(t):=ε↓ 0δ_ε(t),\]together with the canonical RNF equilibrium threshold\[δ_c=ν^2 y_+,_+={b+√b^2+4ac}{2a}.\] # Main theorem (two-direction structure) The theorem is organized in a two-direction logical form: - **Sufficiency (criterion).** On a finite window \((0,T)\), if there exist \(η∈(0,1)\) and \(ε_0>0\) such that \[ t∈(0,T)0<ε≤ ε_0δ_εBN(t)≤ (1-η)\,δ_c, \] 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: \[ _{t→ T⁻}δ(t)≥ δ_c. \] # Companion modules (Aux_Proof v5.3) Aux_Proof v5.3 provides a finite-window closure package for: - **TCE (trace–convolution equivalence):** treated as a smooth-regime interpretation (not used to justify energy-class steps).- **RNF (Riccati normal form):** a Dini-derivative inequality for \(δ_ε\) with \(ε\)-independent coefficients on finite windows.- **RZ (residual-zero reduction):** transport/pressure/boundary residual channels with vanishing \(L^1\)-mass as \(ε↓ 0\) on finite windows.- **BN (boundary normalization):** a separate module for bounded no-slip domains.- **Route-T (transport-bypass) localization:** Littlewood–Paley / weighted almost-orthogonality and tail absorption, plus commutator-based positivity/extraction blocks. # What is new in v5.3 - **Constant-flow diagram (auditable dependencies).** A one-page diagram records the order of choices and dependencies: system constants \(→\) fixed kernel/LP/BN profiles \(→\) margin \(η\) \(→\) Route-T slack parameters \((α,β,κ)\) \(→\) contradiction scale \(ε_\). It also records how \((α,β,κ)\) propagate to a uniform witness \(a_ᵘⁿⁱᶠ\), then to a transport-defect lower bound \(ρT,ηᵗʳ\), and finally to the CT3 margin bookkeeping. - **Sharper quantitative bookkeeping.** The fixed-scale short-window persistence step and Route-T bookkeeping are written to make constant losses and scale/time selection steps easy to audit. # Scope & non-toy status - The framework is stated at the energy (Leray–Hopf / suitable weak) level using convolution envelopes.- Operator-trace viewpoints are used only as optional smooth-regime interpretations; they are not used to justify energy-class steps.- Expository figures/FAQs are moved to a separate visual supplement and are not proof inputs. # Contents (files) - Main paper (PDF + TeX): *SAPZ_Singularity_Principle_Navier-Stokes v5.3*- Companion modules (PDF + TeX): *Aux_Proof v5.3*- (Optional) Visual guide: *SAPZ_NS_Visual_Guide* (separate Zenodo record) # Suggested citation Lee Byoungwoo, “SAPZ Singularity Principle for 3D Navier–Stokes (v5.3): Threshold–Criterion Interface with Constant-Flow Diagram (Main + Companion Modules),” Zenodo, 2026.