Hyperbolic Root Systems: Extending Cartan Classification via Asymmetric Rank-2 Matrices — E8 Intelligence Research

FINDING: The most profound mathematical structure here is the **rank-2 hyperbolic root systems** paper, which directly extends the Cartan classification into indefinite territory, revealing a discrete family parameterized by integers \(a,b\) with \(ab \geq 5\). | MATH: Generalized Cartan matrix \(H(a,b) = \begin{pmatrix} 2 & -b \\ -a & 2 \end{pmatrix}\), \(a,b \in \mathbb{Z}\), \(ab \geq 5\). Non-symmetric case (\(a \neq b\)) yields one long and one short simple root — a crystallographic asymmetry. The condition \(ab \geq 5\) is the key threshold: it separates hyperbolic (indefinite) from affine (degenerate) and finite (positive-definite) types. | CONNECTION: This is a **direct crystallographic root-system generalization**. The classical Cartan types \(A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2\) correspond to \(ab \leq 3\) (finite) and \(ab = 4\) (affine). The hyperbolic case \(ab \geq 5\) introduces new symmetries. Notably, the ratio \(5/4 = 1.25\) and the golden ratio \(\varphi = 1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com