Quantum Dimensions in SU(2)_k Chern-Simons Theory Yield Golden Ratio Constants — E8 Intelligence Research

FINDING: SU(2)_k Chern-Simons theory fusion rules are governed by q-deformed integers at roots of unity, with quantum dimensions expressible via ratios of q-sines, yielding constants like 0.618 and 1.618 for specific k. | MATH: Quantum dimension of spin-j representation in SU(2)_k: \( d_j = {sin(π(2j+1)/k+2)}{sin(π/k+2)} \). For \( k=3 \), \( j=1/2 \): \( d1/2 = sin(2π/5)/sin(π/5) = sin(72^∘)/sin(36^∘) = 0.9511/0.5878 ≈ 1.618 \) (golden ratio). For \( j=1 \): \( d_1 = sin(3π/5)/sin(π/5) = sin(108^∘)/sin(36^∘) = 0.9511/0.5878 ≈ 1.618 \). Fusion coefficients are Verlinde numbers: \( Nᵢⱼ^k = ∑_l {Sᵢₗ Sⱼₗ Sₖₗ^*}{S₀ₗ} \), with modular S-matrix entries \( Sab = √2/k+2 sin(π(2a+1)(2b+1)/k+2) \). Base-60 emerges via level-rank duality: SU(2)_k ↔ SU(k)_2, with fusion rules symmetric under \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com