Hyperbolic Root Systems from Non-Symmetric Rank-2 Cartan Matrices — E8 Intelligence Research

FINDING: Rank-2 hyperbolic root systems are defined by Cartan matrices with integer entries a,b where ab≥5; non-symmetric cases (a≠b) generate one long and one short simple root, linking to affine limits and E8 via ab=1,2,3. MATH: Generalized Cartan matrix H(a,b) = [[2, -b], [-a, 2]] with a,b ∈ ℤ, ab ≥ 5. For ab=1,2,3, the system is affine (not hyperbolic). The E8 root system is finite, rank 8, with Cartan matrix determinant 1; its hyperbolic extensions involve rank-9 Lorentzian lattices. The affine limit ab=1,2,3 corresponds to the boundary where hyperbolic behavior degenerates into affine (infinite) root systems, which are known to embed into E8 via folding or lattice embeddings. CONNECTION: The condition ab≥5 for hyperbolicity mirrors the golden ratio condition: the largest real root of the characteristic polynomial of H(a,b) is >2 when ab≥5. For ab=5, the spectral radius approaches φ² ≈ 2.618 (since φ² = φ+1 ≈ 2.618). The affine cases ab=1,2,3 correspond to spectral radii 2, √2+√ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com