Icosahedral Group A5, Golden Ratio Characters, and Lepton Mixing Symmetry — E8 Intelligence Research

FINDING: The rotational icosahedral group is isomorphic to A5 (alternating group on 5 elements); its irreducible representations carry the golden ratio φ in their character values, and this structure has been proposed as a family symmetry for lepton mixing. MATH: - Isomorphism: I (rotational icosahedral group) ≅ A5, order |A5| = 60 = 5!/2. - Conjugacy classes of A5: 5 classes → 5 irreducible representations. - Sizes: 1, 12, 12, 15, 20. - Dimensions: 1, 3, 3, 4, 5. - Check: 1² + 3² + 3² + 4² + 5² = 1 + 9 + 9 + 16 + 25 = 60 = |A5|. ✓ - Character table (key entries): the two 3-dimensional irreps (3 and 3′) have characters involving φ: - χ₃(class of order-5 elements) = (1 ± √5)/2 = φ or −1/φ = 1 − φ. - Specifically: χ₃(5A) = φ = (1+√5)/2 ≈ 1.618, χ₃(5B) = (1−√5)/2 ≈ −0.618. - χ₃′(5A) = (1−√5)/2, χ₃′(5B) = (1+√5)/2. - The 4-dimensional irrep χ₄(5A) = χ₄(5B) = −1. - The 5-dimensional irrep χ₅(5A) = χ₅(5B) = 0. - Golden ratio identity: φ² = φ + 1; φ − 1/φ = 1; φ · (1/φ) = 1. - √5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com