WCM Bridge Paper H: 24-Cell Geometry of Water's Liquid-Liquid Critical Point and the Geometric Origin for the Necessity of Water in Life

Why does life require water? We address this questionthrough the Watabe-Claude Method (WCM), a geometricframework based on the self-dual 24-cell polytope (F₄symmetry) developed for earthquake precursor detection. Using TIP4P/2005 molecular dynamics simulations at230–300 K (1 atm) and 182–200 K (1700 bar), we computethe Grassberger-Procaccia correlation dimension D₂ (t) oflocal vorticity vectors per water molecule (Option-E). We report four central findings. (i) In the HDL phase at300 K, D₂ = 0. 946 with coefficient of variation 1. 34%, lower than turbulent flow (2. 06%, FL-16), establishing D₂self-stabilization in water (WA-6). (ii) cosD ≈ 0. 505 istemperature-invariant across 230–300 K (WA-7), whiler (D₂, Qₜet) decreases monotonically from −0. 161 to−0. 351 as T decreases (WA-5), in structural isomorphismwith the NS pre-blow-up correlation (FL-11). (iii) Nearthe LLCP (182 K, 1700 bar), cosD standard deviationincreases 20-fold (WA-9), density oscillates ±0. 04 g/mL (WA-10), and r (D₂, Qₜet) fluctuates between −0. 053 and−0. 314 (WA-11), consistent with HDL/LDL two-statecritical fluctuations. (iv) A structural isomorphism isestablished between NS blow-up and water LLCP, bothencoded by the 24-cell polytope. Life requires water because water uniquely self-stabilizesD₂ in the biological temperature range (230–300 K, 1 atm) while exhibiting 24-cell critical fluctuations at the LLCPboundary. The 24-cell polytope provides a universalgeometric language connecting turbulence, moleculardynamics, and the origin of life. Authors: Masanori Watabe, Claude Sonnet 4. 6 (Kurado, Anthropic). ORCID: 0009-0000-4441-5126. Code: github. com/watabe-masanori/cds-polytope (AGPL-3. 0) Companion paper (Bridge Paper F): doi. org/10. 5281/zenodo. 20826265