Spectral Gap of Coxeter Elements Tied to Coxeter Number via Cartan Matrices — E8 Intelligence Research

FINDING: The spectral gap of a Coxeter element in an irreducible root system is governed by the eigenvalue \(4\cos^2(\pi/h)\), where \(h\) is the Coxeter number, linking Cartan matrices to spectral geometry. | MATH: For a Coxeter element \(c\) with order \(h\), its eigenvalues are \(e^{2\pi i k/h}\) for \(k\) in the exponents. The largest eigenvalue of the adjacency-like Cartan matrix \(C\) (or its symmetrized form) yields \(\lambda_{\max} = 4\cos^2(\pi/h)\) for the \(A_n\) series (e.g., \(h=n+1\)). For \(E_8\), \(h=30\), giving \(4\cos^2(\pi/30) = 2 + \sqrt{3} + \sqrt{5} + \sqrt{15}/2 \approx 3.956\). The spectral zeta function of the Coxeter graph satisfies \(\zeta(2) = \pi^2/6\) via the trace of \(C^{-1}\). | CONNECTION: \(h\) values for exceptional groups: \(E_6:12\), \(E_7:18\), \(E_8:30\). The ratio \(4\cos^2(\pi/h)\) for \(h=5\) (A_4) equals \(4\cos^2(36^\circ) = \phi^2 = 2.618\) (golden ratio squared). For \(h=10\) (H_4), \(4\cos^2(18^\circ) = 2 + \sqrt{5} \approx 4.236\), rela Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com