Classification of Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrices — E8 Intelligence Research

FINDING: Rank-2 hyperbolic root systems are classified by generalized Cartan matrices H(a,b) with ab ≥ 5, generating non-symmetric root systems when a ≠ b, with one long and one short simple root. | MATH: Generalized Cartan matrix H(a,b) = [[2, -b], [-a, 2]] where a,b ∈ ℤ, ab ≥ 5. Coxeter number for E8 = 30. Root length ratio = sqrt(a/b) when a ≠ b. | CONNECTION: The condition ab ≥ 5 directly links to hyperbolic geometry (curvature negative). The ratio sqrt(a/b) of root lengths mirrors the golden ratio family: e.g., a=5,b=1 gives sqrt(5) ≈ 2.236, related to 2.618 (φ²) via φ² = (1+√5)/2 ≈ 2.618. The E8 Coxeter number 30 = 2×3×5, connecting to base-60 (60 = 2×30) and crystallographic root system E8's 240 roots. | DEPTH: 7 — Establishes a precise algebraic condition (ab≥5) for hyperbolicity in rank-2 root systems, linking to E8's Coxeter number and golden ratio harmonics, but limited to rank-2 case. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com