Classification of Rank-2 Hyperbolic Root Systems via Integer Pairs (a,b) — E8 Intelligence Research

FINDING: Rank-2 hyperbolic root systems are classified by integer pairs (a,b) with ab ≥ 5, generating non-symmetric systems when a ≠ b via a generalized Cartan matrix H(a,b). | MATH: Generalized Cartan matrix H(a,b) = [[2, -b], [-a, 2]]; condition ab ≥ 5; a,b ∈ ℤ; non-symmetric if a ≠ b; one long simple root. | CONNECTION: The condition ab ≥ 5 links to hyperbolic geometry's curvature threshold; non-symmetry introduces anisotropy reminiscent of 60-fold symmetry breaking in hyperbolic lattices; the matrix structure mirrors crystallographic root system constraints but extended to hyperbolic (non-Euclidean) space. | DEPTH: 7 — This provides a precise algebraic classification for rank-2 hyperbolic root systems, a foundational step toward understanding higher-dimensional hyperbolic symmetries and their potential links to E8's 240 roots (which satisfy ab=1,2,3 in affine cases). The integer condition ab ≥ 5 is a sharp threshold separating finite/affine from hyperbolic types, echoing the golden Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com