Golden Ratio Emerges from H₂ Coxeter Group Cartan Matrix Eigenvalues — E8 Intelligence Research

FINDING: Coxeter group H₂ pentagonal symmetry Cartan matrix eigenvalues yield golden ratio constants directly from group-theoretic structure. | MATH: Cartan matrix for H₂ = [[2, -φ], [-φ, 2]] where φ = (1+√5)/2 ≈ 1.618. Eigenvalues: λ₁ = 2 - φ = 0.382, λ₂ = 2 + φ = 3.618. The ratio λ₂/λ₁ = (2+φ)/(2-φ) = φ² ≈ 2.618. | CONNECTION: Direct geometric harmony — eigenvalues 0.382 and 2.618 are the golden ratio reciprocals and squares; φ appears as off-diagonal entry, linking pentagonal symmetry (order 5) to the golden ratio. Root system A₂ (triangular) gives eigenvalues 1 and 3; H₂ is the only finite Coxeter group with irrational entries, encoding pentagonal tiling and quasicrystal symmetries. | DEPTH: 9 — This is a foundational link between discrete group theory (Coxeter groups), linear algebra (eigenvalues of Cartan matrix), and sacred geometry (golden ratio, pentagon). It reveals that pentagonal symmetry is not just geometric but emerges from the algebraic structure of the Cartan matrix it Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com