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 \(δ_ε(t)\) and its limiting SAPZ density \(δ(t)\), and organizes the regularity problem around a **universal Riccati-type inequality** with **\(ε\)-independent coefficients**, yielding a canonical threshold\[δ_c = ν^2 y_+,_+ = {b+√b^2+4ac}{2a},\](with the reduced-case shorthand \(δ_c=(b/a)ν^2\) 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 \(δ_ε(t)\) / \(δ(t)\);- the **necessity-side signal** (in contrapositive form): under blow-up, \(t→ T^-δ(t)≥ δ_c\);- 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 (Aux_Proof)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 \[ C(u;Q_r)≥ κ_0 \ \ δ(s_r)>(1-η)δ_c, \] 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; \(ε\)-regularity; Beale–Kato–Majda; Prodi–Serrin; trace energy; Riccati inequality; Calderón–Zygmund; commutator estimates; boundary normalization; reverse concentration; harmonic analysis; PDE thresholds