A Spectral–Entropy Threshold Criterion and Route–T Closure for the Three-Dimensional Navier–Stokes Equations (v6.0)

## Overview This record uploads **v6. 0** of a single-manuscript journal-cut version of the Navier–Stokes SAPZ program. The paper introduces a **Spectral–Averaged Parabolic Zone (SAPZ) ** criterion for the three-dimensional incompressible Navier–Stokes equations and organizes the argument in a single theorem-facing manuscript. The central scalar object is the convolution-first SAPZ envelope\_ (t): =\|\, | u (, t) |² * _\|₋^䂲, (t): = ₀_ (t), from a fixed canonical caloric-bump mollifier. An \ (\) -independent Riccati normal form yields a canonical barrier level\c=² y_+, _+=b+b²+4ac2a. \ The manuscript presents a single-manuscript route from this threshold criterion to finite-horizon continuation and then to the global continuation theorem. It also records the converse threshold-reach statement: any finite-time loss of regularity forces\ₓ ₓ^- (t) c. \ ## Main theorem-facing content The manuscript is organized around the following theorem-level chain. 1. A **finite-window Riccati normal form** for the dimensionless SAPZ profile, with \ (\) -independent coefficients and a measurable remainder ledger. 2. A **finite-window residual-envelope package CT2 (T) ** and the resulting **CT3 averaged residual budget**. 3. A **CT3 persistence mechanism with scale-last selection**, producing a contradiction scale \ (_\), a short backward parabolic subwindow, and a fixed-scale persistence estimate on that same subwindow. 4. A **Route–T contradiction-window discharge**, converting contradiction-window persistence into a positive transport residual contribution and then into a contradiction with the short-window RNF residual budget. 5. A **Gate A approximate-identity endpoint bridge**, upgrading SAPZ control to almost-everywhere time-slice \ (L^ₓ\) control of \ (| u|²\). 6. A **standard endpoint closure** via kinematic exclusion of Caffarelli–Kohn–Nirenberg scale concentration and the usual CKN \ (\) -regularity continuation mechanism. The resulting main forward statement is a global regularity theorem in the suitable-weak framework, written as the internal theorem-level conclusion of the present manuscript. ## What is in v6. 0 This version is the **single-manuscript journal-cut** form of the SAPZ Navier–Stokes project. Relative to the earlier multi-document/package-facing presentation, v6. 0 is organized as one theorem-facing paper with: - a unified main theorem section, - an internal proof chain through Sections 3–8, - an appendix-level object/constant ledger, - appendix-level fixed-scale support lemmas for the Route–T extraction chain, - an expanded appendix for the fixed-scale Riccati normal-form derivation, - and an expanded appendix for the witness and local-to-global commutator comparison steps used in the Route–T closure. In particular, the goal of this version is not to add numerical or expository supplement material, but to make the proof architecture readable as a single manuscript. ## Structure - **Section 2** fixes the solution class, the SAPZ envelope, and the main theorem statements. - **Section 3** develops the Riccati normal form, residual decomposition, finite-window envelopes, and the CT2 (T) package. - **Section 4** develops CT3 persistence and the scale-last contradiction-scale choice. - **Section 5** gives the Route–T contradiction-window discharge. - **Section 6** proves Gate A, the approximate-identity time-slice identification step. - **Section 7** records the endpoint concentration exclusion and standard CKN continuation step. - **Section 8** assembles the global theorem. - **Appendix A** records the threshold-reach statement near a finite maximal time. - **Appendix B** records the object dictionary and constant hierarchy. - **Appendix C** records the fixed-scale support lemmas for the Route–T extraction chain. - **Appendix D** expands the fixed-scale Riccati normal-form derivation. - **Appendix E** expands the witness-recovery and local-to-global commutator-comparison steps used in the Route–T closure. ## Scope This record should be read as a **theorem-facing single-manuscript snapshot** of the SAPZ Navier–Stokes program. The paper is written for mathematical review and verification; it is not a numerical or visual supplement, and the core claims are presented as theorem-level statements inside the manuscript itself. ## Author Lee Byoungwoo ========================= Author: Lee Byoungwoo (이병우) E-mail: leeclinic@protonmail. com