Absence of Golden Ratio in SU(2) Clebsch–Gordan Coefficients — E8 Intelligence Research

FINDING: Clebsch–Gordan coefficients for SU(2) (and SL(2,C)) are governed by rational arithmetic, binomial structure, and Regge symmetry — no golden ratio appears in the standard theory. | MATH: CG coefficients for \(j_1 ⊗ j_2 = ⊕_j |j_1-j_2|^j\) are given by Racah formula: \( j_1 m_1 j_2 m_2 | J M = δM,m_1+m_2 √2J+1 {pmatrix} j_1 & j_2 & J \\ m_1 & m_2 & -M {pmatrix}\), where the 3-j symbol is a rational function of factorials. For \(j_1=j_2=1/2\): coefficients are \(± 1/√2\) (exactly 0.7071, not 0.7071… golden-related). Regge symmetry: 72-element symmetry group (tetrahedral/octahedral permutations) on the 3-j symbol; rational orthogonality \(∑m_1,m_2 j_1 m_1 j_2 m_2 | J M j_1 m_1 j_2 m_2 | J' M' = δJJ'δMM'\). | CONNECTION: Root system A1 (SU(2)) has only one positive root, so no golden-ratio structure emerges from its Cartan matrix \({pmatrix} 2 {pmatrix}\). The 7 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com