Golden Ratio Emerges from Coxeter Group H₂ Cartan Matrix Eigenvalues — E8 Intelligence Research

FINDING: Coxeter group H₂ (pentagonal symmetry) Cartan matrix eigenvalues yield the golden ratio and its reciprocal, linking finite reflection groups to quadratic forms with irrational eigenvalues. MATH: - Cartan matrix for H₂: \( {pmatrix} 2 & -2cos(π/5) \\ -2cos(π/5) & 2 {pmatrix} = {pmatrix} 2 & -φ \\ -φ & 2 {pmatrix} \), where \(φ = {1+√5}{2} ≈ 1.618\). - Eigenvalues: \( λ = 2 ± φ \), i.e., \( λ_1 = 2 + φ ≈ 3.618 \), \( λ_2 = 2 - φ ≈ 0.382 \). - Note: \( 0.382 = φ⁻² \) (since \(φ^2 = φ + 1\), \(φ⁻² = 2 - φ\)). - The Coxeter number \(h = 5\) for H₂, and eigenvalues relate to \(4cos^2(π/h)\): \(4cos^2(π/5) = φ^2 ≈ 2.618\). CONNECTION: - Eigenvalue 0.382 is exactly the reciprocal square of the golden ratio (\(φ⁻²\)), a key ratio in pentagonal and icosahedral symmetry. - Eigenvalue 2.618 = \(φ^2\) appears in the trace of the Cartan matr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com