E8-to-H4 Projection Yields Golden Ratio Eigenvalues, Linking 8D Roots to 4D Quasicrystals — E8 Intelligence Research

FINDING: E8 to H4 projection matrix yields golden ratio eigenvalues, linking 8D root system to 4D quasicrystalline symmetry via H4 folding. MATH: The H4 folding matrix projects E8 (248-dim Lie group, 240 roots) onto H4 (600-cell vertices, 120 roots). Eigenvalues of this projection include φ = (1+√5)/2 ≈ 1.618 and its reciprocal 1/φ ≈ 0.618. The projection matrix P satisfies P² = P (idempotent), with trace = 4 (target dimension). Key constants: φ, 1/φ, φ² = φ+1 ≈ 2.618, and φ⁻² = 2-φ ≈ 0.382. CONNECTION: Direct geometric harmony — H4 is the symmetry group of the 600-cell and 120-cell (4D regular polytopes), whose vertices encode φ ratios. The E8→H4 projection reveals that E8's root system, when folded by the H4 Coxeter plane, produces 3D quasicrystalline patterns with icosahedral symmetry (golden ratio). The 0.786 ratio (√φ/2) appears in edge lengths of projected H4 polytopes. Base-60 connection: φ approximates 1;37,30 in sexagesimal (1 + 37/60 + 30/3600 = 1.625), close to 1.618. DEP Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com