SAPZ Singularity Principle for the 3D Incompressible Navier-Stokes Equations (v5.6r1): Spectral-Entropy Threshold, Gate A, and Route-T Discharge

# OverviewThis record releases **v5.6r1** of a *journal-cut* two-paper set (plus a short verification note) developing the **SAPZ**(Spectral–Averaged Parabolic Zone) threshold framework for the **3D incompressible Navier–Stokes equations**. **Concept DOI (all versions / series DOI):** 10.5281/zenodo.15846588 Package roles (read this as a criterion-plus-discharge interface, not a single-paper black-box claim):- **Main paper (CRIT):** defines the convolution-first SAPZ envelope \(δ(t)\), fixes a canonical Riccati-equilibrium threshold \(δ_c\), and states the finite-horizon criterion + necessity (threshold reach).- **Companion (DISCHARGE + modules):** supplies theorem-level analytic modules (TCE/RNF/RZ/BN) and discharges the strict-subcriticality condition on each finite horizon via **Route–T (transport-bypass)**, then realizes the endpoint closure via **Gate A \(→\) Gate B**.- **Minimal Verification Note (formal):** referee-facing; isolates a small trusted core (**TCB = 3**) for independent checking and adds **no new proof inputs**. :contentReference[oaicite:3]{index=3} > **Scope note (important).** This record is a *journal-cut* proof package. Numerical protocols, figures, and visualization material are **not** used as proof inputs.> Expository/visual content (when provided) is handled as a separate supplement/record. --- ## Files in this record (PDF-only)- Main paper (PDF): `Main_v5.6r1.pdf`- Companion (PDF): `Aux_v5.6r1.pdf`- Minimal Verification Note (PDF): `Verify_v5.6r1.pdf` (referee-facing; no new proof inputs) --- ## Core functional and canonical threshold (Main)Fix a canonical mollifier family \(φ_ε\). For a (weak) solution \(u\), define\[δ_ε(t):=\|\, |∇ u(·,t)|^2 * φ_ε \,\|L^∞_x,δ(t):=ε↓ 0δ_ε(t).\]The SAPZ mechanism yields an \(ε\)-independent **Riccati normal form (RNF)** with universal coefficientsand a canonical equilibrium threshold\[δ_c=ν^2 y_+,_+={b + √b^2 + 4ac}{2a},\]with exact normalization and coefficient provenance fixed in the companion modules. **Envelope convention.** In the Leray–Hopf regime we do **not** assume \(δ_ε(t)\) converges as \(ε↓ 0\).All threshold statements use only the envelope \(δ(t)=ε↓0δ_ε(t)\) (or a truncated \(0<ε≤ε_0\)). :contentReference[oaicite:4]{index=4} --- ## Main theorem and proof interface (high-level)The main paper presents a two-direction structure. ### (1) Sufficiency (criterion-level)On any finite horizon \((0,T)\), **strict SAPZ subcriticality** implies smoothness and continuation.At the package level, this is realized as:\[DISCHARGE (Route–T on (0,T))\ \ 0<t<Tδ(t)≤ (1-η_T)δ_c\ \ CONTINUE (Gate A \(→\) Gate B).\]Gate B is classical (CKN \(ε\)-regularity/continuation); the genuinely nonstandard work is isolated in the Gate A / persistence / Route–T discharge package. :contentReference[oaicite:5]{index=5} ### (2) Necessity (contrapositive; threshold reach)Any finite-time loss of regularity forces **threshold reach**\[t→ T^-δ(t)\ ≥\ δ_c.\]This is framed as a contrapositive/reach statement rather than a trivial “\(δ=∞\)” narrative. --- ## Companion modules (what is proved where)The companion provides theorem-level modules used by the main paper:- **TCE** (trace–convolution equivalence), **RNF** (Riccati normal form), **RZ** (residual-zero reduction and constant independence), **BN** (boundary normalization), and the sufficiency interface (**Gate A \(→\) Gate B**). :contentReference[oaicite:6]{index=6} Primary endpoint realization (label-first; numbering secondary):- **Gate A:** approximate-identity \(L^∞\) identification (Aux, Theorem 12.7). :contentReference[oaicite:7]{index=7}- **Kinematic exclusion of CKN-scale concentration:** (Aux, Theorem 19.2).- **Gate B (standard):** CKN \(ε\)-regularity/continuation closure (Aux, Section 20: finite-window smoothness + threshold-to-criterion closure). > **Optional/historical modules.** The legacy quantitative injection engine (pressure/cutoff/commutator/BN ledger) is retained only for robustness and bookkeeping cross-checks,> and is explicitly labeled as optional (not used on the primary Gate A\(→\)B route). :contentReference[oaicite:8]{index=8} --- ## Solution-class contract (no hidden regularity)Base class: **Leray–Hopf weak solutions** (global energy inequality).Whenever CKN-scale endpoint regularity is invoked, the setting is **suitable weak solutions**(Leray–Hopf plus the local energy inequality, LEI).Distributional commutator identities are justified via standard approximation(Galerkin / mollification / Steklov-in-time) and then passed to the limit. The main paper includes Proposition 1.6 (“Standard-approximation Leray–Hopf solutions are suitable”) with a referee-facing checklist proof. :contentReference[oaicite:9]{index=9} :contentReference[oaicite:10]{index=10} --- ## Minimal Verification Note (formal): TCB = 3 checkpointsThe verification note isolates a small trusted core (TCB) for independent checking, with an explicit object dictionary to prevent a common failure mode (mixing distinct residual objects): :contentReference[oaicite:11]{index=11} 1) **Gate A** (Aux Theorem 12.7), 2) **CT3 persistence / scale-last selection** (Aux Lemma 1.8, supported by Lemma 1.7), 3) **Route–T transport extraction** (Aux Lemma 1.52; TR1–TR3 sealed; kernel lower bound as a separate micro-lemma). After these cores, the endpoint step is standard CKN \(ε\)-regularity/continuation. --- ## Proof vs evidence (discipline)- Any figures and numerical material are **illustrative only** and are **not** used as proof inputs.- Engineering/visual content, when provided, is maintained as a separate supplement/record. --- ## Recommended citationLee Byoungwoo, “**A Spectral–Entropy Threshold Framework for Regularity and Blow-up in the Navier–Stokes Equations: The SAPZ Principle**” (Version v5.6r1), with companion “**Auxiliary Proof Modules for the SAPZ Singularity Principle**” (Version v5.6r1) and “**SAPZ Navier–Stokes v5.6r1: Minimal Verification Note (formal)**” (Version v5.6r1), Zenodo, March 5, 2026. DOI: 10.5281/zenodo.15846588 ========================= Author: Lee Byoungwoo(이병우) E-mail: leeclinic@protonmail.com