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 \(j_1 \otimes j_2 = \oplus_j |j_1-j_2|^j\) are given by Racah formula: \(\langle j_1 m_1 j_2 m_2 | J M \rangle = \delta_{M,m_1+m_2} \sqrt{2J+1} \begin{pmatrix} j_1 & j_2 & J \\ m_1 & m_2 & -M \end{pmatrix}\), where the 3-j symbol is a rational function of factorials. For \(j_1=j_2=1/2\): coefficients are \(\pm 1/\sqrt{2}\) (exactly 0.7071, not 0.7071… golden-related). Regge symmetry: 72-element symmetry group (tetrahedral/octahedral permutations) on the 3-j symbol; rational orthogonality \(\sum_{m_1,m_2} \langle j_1 m_1 j_2 m_2 | J M \rangle \langle j_1 m_1 j_2 m_2 | J' M' \rangle = \delta_{JJ'}\delta_{MM'}\). | CONNECTION: Root system A1 (SU(2)) has only one positive root, so no golden-ratio structure emerges from its Cartan matrix \(\begin{pmatrix} 2 \end{pmatrix}\). The 7 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com