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 \ ( (t) \), fixes a canonical Riccati-equilibrium threshold \ (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**.: contentReferenceoaicite: 3index=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): `Mainᵥ5. 6r1. pdf`- Companion (PDF): `Auxᵥ5. 6r1. pdf`- Minimal Verification Note (PDF): `Verifyᵥ5. 6r1. pdf` (referee-facing; no new proof inputs) --- ## Core functional and canonical threshold (Main) Fix a canonical mollifier family \ (_\). For a (weak) solution \ (u\), define\_ (t): =\|\, | u (, t) |² * _ \, \|₋^䂲, (t): = ₀_ (t). SAPZ mechanism yields an \ (\) -independent **Riccati normal form (RNF) ** with universal coefficientsand a canonical equilibrium threshold\c=² y_+, _+=b + b² + 4ac2a, exact normalization and coefficient provenance fixed in the companion modules. **Envelope convention. ** In the Leray–Hopf regime we do **not** assume \ (_ (t) \) converges as \ (0\). All threshold statements use only the envelope \ ( (t) =₀_ (t) \) (or a truncated \ (₀ **₎₏ₓ₈₎₍₀₋/₇₈ₒₓ₎ₑ₈₂₀₋ ₌₎₃ₔ₋₄ₒ. ** ₓ₇₄ ₋₄₆₀₂ₘ ₐₔ₀₍ₓ₈ₓ₀ₓ₈ₕ₄ ₈₍₉₄₂ₓ₈₎₍ ₄₍₆₈₍₄ (₏ₑ₄ₒₒₔₑ₄/₂ₔₓ₎₅₅/₂₎₌₌ₔₓ₀ₓ₎ₑ/₁₍ ₋₄₃₆₄ₑ) ₈ₒ ₑ₄ₓ₀₈₍₄₃ ₎₍₋ₘ ₅₎ₑ ₑ₎₁ₔₒₓ₍₄ₒₒ ₀₍₃ ₁₎₎₊₊₄₄₏₈₍₆ ₂ₑ₎ₒₒ-₂₇₄₂₊ₒ, > ₀₍₃ ₈ₒ ₄ₗ₏₋₈₂₈ₓ₋ₘ ₋₀₁₄₋₄₃ ₀ₒ ₎₏ₓ₈₎₍₀₋ (₍₎ₓ ₔₒ₄₃ ₎₍ ₓ₇₄ ₏ₑ₈₌₀ₑₘ ₆₀ₓ₄ ₀\ (\) ₁ ₑ₎ₔₓ₄).: ₂₎₍ₓ₄₍ₓₑ₄₅₄ₑ₄₍₂₄₎₀₈₂₈ₓ₄: ₈₈₍₃₄ₗ=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.: contentReferenceoaicite: 9{index=9: contentReferenceoaicite: 10index=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): : contentReferenceoaicite: 11index=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 \ (\) -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