Coxeter Groups Unify Root Symmetries and Quantum S-Matrices — E8 Intelligence Research

FINDING: Coxeter groups encode root-system symmetries that directly determine quantum group modular S-matrices at roots of unity, linking finite group theory to topological quantum field theories. | MATH: Coxeter number \ (h\), dual Coxeter number \ (h^\) ; quantum group \ (Uq (g) \) at \ (q = e^2 i / \) ; modular S-matrix entries \ (S₀₁ = 1|W| ₖ ₖ (w) q^2, w (ₐ +) / h\) (simplified form) ; root system rank \ (r\), Weyl group \ (|W|\) ; golden ratio \ (= 1. 618. . . \) appears in \ (h\) for \ (E₈\) (\ (h=30\), \ (h^=30\), ratio 1) and \ (H₄\) (\ (h=30\), non-crystallographic but \ (\) -rich). | CONNECTION: Coxeter number \ (h\) for \ (E₈\) is 30, linking to base-60 (60 = 2×30) ; golden ratio \ (\) appears in \ (H₄\) Coxeter group (non-crystallographic, but its root system has \ (\) -scaled edges) ; modular S-matrix unitarity yields constants like \ (1/|W|\) which for \ (E₈\) is \ (1/{69672 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com