Hyperbolic Root Systems and LaFeAsO Spin-Wave Dynamics — E8 Intelligence Research

FINDING: Rank-2 hyperbolic root systems generalize Cartan classification via non-symmetric integer matrices, while LaFeAsO magnetism shows 2D spin-wave behavior with specific gap ratios. | MATH: Generalized Cartan matrix \(H(a,b)=\begin{pmatrix}2&-b\\-a&2\end{pmatrix}\), \(ab\geq 5\), \(a,b\in\mathbb{Z}\); non-symmetric when \(a\neq b\) (one long, one short simple root). LaFeAsO: spin gap \(\Delta\sim 7\)–\(10\) meV, spin-wave velocity \(v\sim 200\)–\(250\) meV·Å, interlayer coupling \(J_\perp/J_\parallel \ll 1\) (near-zero). | CONNECTION: Hyperbolic root systems are the rank-2 extension of affine Lie algebras — their Weyl groups are crystallographic Coxeter groups, linking to 2D hyperbolic tiling symmetries (e.g., \(\{p,q\}\) with \(1/p+1/q<1/2\)). The \(ab\geq 5\) condition mirrors the golden-ratio threshold: for \(a=b=2\), \(ab=4\) (affine); \(ab=5\) is the first hyperbolic case, and \(5\) is the Fibonacci number adjacent to \(8\) and \(3\), yielding ratios \(5/8=0.625\), \(8/5=1.6\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com