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, Q_tet) 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, Q_tet) 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