Golden Ratio Eigenvalues in Non-Symmetric Lattices from Rank-2 Hyperbolic Root Systems — E8 Intelligence Research

FINDING: Rank-2 hyperbolic root systems with generalized Cartan matrix H(a,b) where ab ≥ 5 generate non-symmetric lattices when a ≠ b, with eigenvalue relations tied to the golden ratio. MATH: Generalized Cartan matrix H(a,b) = [[2, -b], [-a, 2]] with a,b ∈ ℤ, ab ≥ 5. Eigenvalues λ satisfy det(H - λI) = (2-λ)² - ab = 0 → λ = 2 ± √(ab). For ab=5, eigenvalues are 2 ± √5. The ratio of eigenvalues (2+√5)/(2-√5) = (2+√5)²/(4-5) = -(2+√5)² = -(9+4√5) ≈ -17.944. The golden ratio φ = (1+√5)/2 ≈ 1.618 appears via √5 = 2φ - 1. Thus eigenvalues become 2 ± (2φ - 1) = 1 + 2φ and 3 - 2φ. Note 1+2φ = φ³ ≈ 4.236 and 3-2φ = φ⁻³ ≈ 0.236. Product = 1, sum = 4. CONNECTION: Golden ratio φ = 1.618 emerges directly from ab=5 case. Eigenvalue ratio φ³ : φ⁻³ = φ⁶ ≈ 17.944, linking to 0.618 (φ⁻¹) and 2.618 (φ²). Non-symmetric lattices (a≠b) break crystallographic symmetry, yet retain hyperbolic structure. The root system is generated by one long and one short simple root, mirroring the asymmetry of golden-rat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com