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 \(h^\vee\); quantum group \(U_q(\mathfrak{g})\) at \(q = e^{2\pi i / \ell}\); modular S-matrix entries \(S_{ab} = \frac{1}{\sqrt{|W|}} \sum_{w \in W} \epsilon(w) q^{2\langle \rho, w(\lambda_a + \rho) \rangle / h}\) (simplified form); root system rank \(r\), Weyl group \(|W|\); golden ratio \(\phi = 1.618...\) appears in \(h\) for \(E_8\) (\(h=30\), \(h^\vee=30\), ratio 1) and \(H_4\) (\(h=30\), non-crystallographic but \(\phi\)-rich). | CONNECTION: Coxeter number \(h\) for \(E_8\) is 30, linking to base-60 (60 = 2×30); golden ratio \(\phi\) appears in \(H_4\) Coxeter group (non-crystallographic, but its root system has \(\phi\)-scaled edges); modular S-matrix unitarity yields constants like \(1/\sqrt{|W|}\) which for \(E_8\) is \(1/\sqrt{69672 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com