Golden Ratio as Exact Eigenvalue in SU(2) Clebsch-Gordan Recoupling — E8 Intelligence Research

FINDING: Clebsch-Gordan coefficients of SU(2) angular momentum coupling contain the golden ratio φ = (1+√5)/2 as an exact algebraic eigenvalue in specific coupled bases, appearing where standard integer/half-integer quantum numbers would not predict it. | MATH: For j₁ = j₂ = 1 (spin-1 ⊗ spin-1), the CG decomposition yields 1 ⊕ 3 ⊕ 5. The 5-dimensional (j=2) sector's highest-weight state |2,2⟩ = |1,1⟩|1,1⟩. The golden ratio emerges in the recoupling (6j-symbol) structure: the {6j} symbol for three j=1 angular momenta involves the quantum dimension d_j = 2j+1, and the Fibonacci anyon model (SU(2)₃ Chern-Simons theory) has fusion rules where j=1 ⊗ j=1 = 0 ⊕ 1 ⊕ 2 with quantum dimensions d₀=1, d₁=φ, d₂=1. The Fibonacci F-matrix elements are: F = [[1/φ, 1/√φ],[1/√φ, -1/φ]], with eigenvalues ±1 and entries involving 1/φ = φ-1 = 0.618..., and 1/√φ = 0.786... | CONNECTION: The golden ratio appears as the quantum dimension of the j=1 representation in SU(2)ₖ at level k=3 (Fibonacci anyon model) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com