A Spectral–Entropy Threshold Framework for Regularity and Blow-up in the Navier–Stokes Equations: The SAPZ Principle (v3.10.6)

# Overview This record releases **PDF-only** versions of the SAPZ Navier–Stokes paper set (Version **v3. 10. 6**, dated **2026-02-27**) by **Lee Byoungwoo**: - **Main paper (conceptual + architecture): ** *A Spectral–Entropy Threshold Framework for Regularity and Blow-up in the Navier–Stokes Equations: The SAPZ Principle*- **Companion (theorem-level analytic modules): ** *Auxiliary Proof Modules for the SAPZ Singularity Principle* (proof interfaces for TCE/RNF/RZ/BN/RCP) The project introduces a mollified trace-energy functional \ (_ (t) \) and its limiting SAPZ density \ ( (t) \), and organizes the regularity problem around a **universal Riccati-type inequality** with **\ (\) -independent coefficients**, yielding a canonical threshold\c = ² y_+, _+ = b+b²+4ac2a, \ (with the reduced-case shorthand \ (c= (b/a) ²\) only when \ (c=0\) ). # Main claims and structure (as stated in the papers) ## Main paper (SAPZ Principle) The main paper states the **threshold architecture**: - the Riccati barrier mechanism for \ (_ (t) \) / \ ( (t) \) ;- the **necessity-side signal** (in contrapositive form): under blow-up, \ (ₓ ₓ^- (t) c\) ;- the **sufficiency direction** is organized as a theorem-level dependency on a single explicit closure bridge (reverse-concentration / local-to-SAPZ injection), discharged in the companion. ## Companion (AuxProof) The companion supplies theorem-level analytic modules underlying the SAPZ mechanism: - **TCE** (Trace–Convolution Equivalence) - **RNF** (Riccati Normal Form) - **RZ** (Residual Zero + coefficient universality) - **BN** (Boundary Normalization for bounded no-slip domains) - **RCP** (Reverse-Concentration Closure Interface) A key referee-facing feature is that the **decisive remaining closure bottleneck** is isolated explicitly as a single quantitative reverse-concentration injection lemma in **Part V / Section 19**, converting **CKN-scale parabolic concentration** into **near-critical SAPZ mass** at a nearby time/scale. The error budget is tracked via an explicit ledger (pressure / cutoff / commutator / boundary-normalization), so the closure step is presented as a finite checklist of quantitative estimates rather than an informal strategy. # What is new in v3. 10. 6 (referee-facing tightening) Version v3. 10. 6 tightens the “unique bottleneck” interface in the companion: - a boxed **Sniper check (S1–S3) ** whose pointer text consists of **equation labels only**, ending in a single displayed implication of the form \ C (u;Qᵣ) ₀ \ \ (sᵣ) > (1-) c, \ with minimal proof-input pointers;- a **Sniper index** (one-line-per-pointer “30-second checklist”) mapping the assembled error ledger, BN/TCE discrepancy, and parameter ladder to precise lemma/equation locations. # Proof vs Evidence Numerical visualizations and reproducibility protocols are included **only as supporting evidence** and are **not used as proof inputs**. All “proof” claims refer strictly to analytic derivations. # Keywords Navier–Stokes; global regularity; blow-up; Leray–Hopf weak solutions; Caffarelli–Kohn–Nirenberg; \ (\) -regularity; Beale–Kato–Majda; Prodi–Serrin; trace energy; Riccati inequality; Calderón–Zygmund; commutator estimates; boundary normalization; reverse concentration; harmonic analysis; PDE thresholds