Non-Symmetric Lattices from Rank-2 Hyperbolic Root Systems with ab ≥ 5 — E8 Intelligence Research

FINDING: Rank-2 hyperbolic root systems with generalized Cartan matrices \( H(a,b) \) where \( ab \geq 5 \) and \( a \neq b \) produce non-symmetric lattices generated by one long and one short simple root. | MATH: Generalized Cartan matrix \( \begin{pmatrix} 2 & -b \ -a & 2 \end{pmatrix} \), condition \( ab \geq 5 \), \( a,b \in \mathbb{Z} \). Eigenvalue relation of this matrix: characteristic polynomial \( \lambda^2 - 4\lambda + (4 - ab) = 0 \). For \( ab = 5 \), eigenvalues are \( \lambda = 2 \pm \sqrt{ab} = 2 \pm \sqrt{5} \). Ratio of eigenvalues: \( (2 + \sqrt{5})/(2 - \sqrt{5}) = (2 + \sqrt{5})^2 / (4 - 5) = -(2 + \sqrt{5})^2 \). The golden ratio \( \phi = (1 + \sqrt{5})/2 \approx 1.618 \) appears: \( 2 + \sqrt{5} = \phi^3 \approx 4.236 \), \( 2 - \sqrt{5} = -\phi^{-3} \approx -0.236 \). | CONNECTION: Eigenvalue ratio \( \phi^6 \) (since \( \phi^3 / (-\phi^{-3}) = -\phi^6 \)). The condition \( ab \geq 5 \) is the hyperbolic threshold; \( ab = 5 \) gives the golden ratio eigenval Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com