A5 Icosahedral Symmetry and Golden Ratio in Tribimaximal Lepton Mixing — E8 Intelligence Research

FINDING: The A5 icosahedral rotation group is simple and its irreducible representations embed the golden ratio φ, providing the group-theoretic scaffold for tribimaximal lepton mixing in the PMNS matrix. | MATH: A5 = ⟨r, s | r⁵ = s² = (rs)³ = 1⟩, |A5| = 60 = 5!/2; conjugacy classes: 1, 15, 20, 12, 12 (sizes 1, 15, 20, 12, 12); irreducible reps: 1, 3, 3′, 4, 5; character values involve φ = (1+√5)/2 = 1.618… and φ⁻¹ = 0.618…; the 3-dimensional irreps 3 and 3′ are complex conjugates with entries in ℤ[φ]; the 5-dim rep is the icosahedral "vector" rep. | CONNECTION: φ = 2cos(π/5) = 1.618…, φ⁻¹ = 2cos(2π/5) = 0.618…, √(φ+2) = 2cos(π/10) ≈ 1.902; pentagonal rotation angle 2π/5 = 72°; the 5-fold axis of the icosahedron generates the cyclic subgroup Z5 ⊂ A5; the golden ratio appears as the ratio of diagonal to side in the regular pentagon, and as the eigenvalue structure of the icosahedral adjacency matrix. The tribimaximal PMNS matrix U_TB has entries |U_e2| = 1/√3 ≈ 0.577, |U_μ2| = |U_τ2| = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com