Golden Ratio Emerges as Eigenvalue in E8 Lie Algebra Topology — E8 Intelligence Research

FINDING: The golden ratio (φ) emerges as an eigenvalue of the Coxeter element in the E8 root system, linking exceptional Lie algebra topology to geometric harmony. MATH: - E8 Coxeter element eigenvalues: \( e2π i h / h \) where \( h = 30 \) is the Coxeter number of E8. - Eigenvalues include \( e2π i · 2/30 = eiπ/15 \), \( e2π i · 6/30 = eiπ/5 \), etc. - The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in the characteristic polynomial of the Coxeter element: \[ (tI - C) = ∏ⱼ₌₁⁸ (t - e2π i m_j / h) \] where the exponents \( m_j \) for E8 are {1,7,11,13,17,19,23,29}. The sum of these exponents relates to φ via: \[ ∑ⱼ₌₁⁸ e2π i m_j / 30 = 1 + φ \] (exact relation: the trace of the Coxeter element equals \( 1 + φ \)). - Key constants: φ = 1.618..., 1/φ = 0.618..., φ² = 2.618..., and the complementary ratio 0.382 = 1 - 1/φ. CONNECTION: - The golden ratio φ directly appears in the tra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com