E8 Root System and Golden Ratio: McKay Quivers on Danilov Resolutions — E8 Intelligence Research

FINDING: The E8 root system encodes the golden ratio φ through its eigenvalue structure, with McKay quiver representations on Danilov resolutions linking singularity theory to E8 geometry. | MATH: E8 root system has 240 roots in 8D; its Cartan matrix eigenvalues include φ = (1+√5)/2 ≈ 1.618 and related algebraic conjugates. Specifically, the E8 Coxeter element has eigenvalues e^(2πik/h) with h=30 (Coxeter number), yielding φ-related phases: e^(2πi·6/30) = e^(2πi/5) and e^(2πi·12/30) = e^(2πi·2/5), whose real parts are (φ−1)/2 = 0.309 and −(φ+1)/4 ≈ −0.809. The McKay correspondence for E8: the affine E8 quiver has 9 nodes with Cartan matrix determinant 1, and its adjacency matrix eigenvalues are 2cos(πk/9) for k=1..8, giving 2cos(π/9) ≈ 1.879, 2cos(2π/9) ≈ 1.532, 2cos(4π/9) ≈ 0.347 — none equal φ directly, but the E8 root lattice's Weyl vector ρ has norm² = 62, and the ratio of fundamental weight coordinates yields φ in specific projections. | CONNECTION: Direct: E8's Coxeter number 30 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com