Classification of Non-Symmetric Rank 2 Hyperbolic Root Systems — E8 Intelligence Research

FINDING: Classification of rank 2 hyperbolic root systems with non-symmetric root length ratio √(a/b) for integers a,b with ab ≥ 5. | MATH: Generalized Cartan matrix H(a,b) = [[2, -b], [-a, 2]]; a,b ∈ ℤ, ab ≥ 5; root length ratio = √(a/b); non-symmetric case when a ≠ b, generated by one long and one short simple root. | CONNECTION: Root length ratios √(a/b) for ab ≥ 5 include values like √(5/1)=2.236, √(6/1)=2.449, √(7/1)=2.646, etc. — these are not the classical golden ratio or its powers, but the condition ab ≥ 5 ensures the hyperbolic (indefinite) type, distinct from finite (elliptic) and affine (parabolic) classifications. No direct link to 0.382, 0.618, 1.618, 2.618, or base-60 found. However, the structure is crystallographic in the sense of root systems (integer Cartan integers), and the non-symmetric case breaks the usual A₂, B₂, G₂ symmetry. | DEPTH: 7 — This is a precise algebraic classification extending root system theory to hyperbolic Kac-Moody algebras, but it does not di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com