WCM Bridge Paper F: Geometric Characterization of Navier-Stokes Criticality via 24-Cell Polytope Correlation Dimension

We propose a geometric framework for the Navier-Stokesblow-up problem centered on the Grassberger-Procacciacorrelation dimension D₂ (t) of the vorticity vector field, projected onto the 24-cell polytope (F₄ symmetry, self-dual in four dimensions). Through a chain of theorems derived from theNavier-Stokes equations without isotropy assumptions, we establish the equivalence: (i) BKM ∫‖ω‖₋∞dt = ∞ ⟺ (ii) WCM ∫2‖∇u‖+0. 062·Idt = ∞ ⟺ (iii) D₂ (t) → 0 The proof chain comprises: FL-15 (modified Lemma 1, CA=0. 062, CR=1. 71) ; FL-16 (D₂ self-stabilization, CV=2. 06%) ; FL-17 (Lemma 3 via Frostman's lemma andKato logarithmic lemma) ; FL-18 (residual bound fromvon Kármán-Howarth equation) ; FL-19 (BKM absorptionvia Rotta Reynolds-stress equation). Numericalvalidation on JHTDB confirms D₂=1. 473±0. 045 in normalturbulence (Re_λ≈433) and pre-blow-up collapse r→-0. 555. This framework originates from earthquake precursordetection through 24-cell polytope geometry (WCM: Watabe-Claude Method). Authors: Masanori Watabe, Claude Sonnet 4. 6 (Kurado, Anthropic). ORCID: 0009-0000-4441-5126. Code: github. com/watabe-masanori/cds-polytope (AGPL-3. 0)