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: \( d_j = \frac{\sin\left(\frac{\pi(2j+1)}{k+2}\right)}{\sin\left(\frac{\pi}{k+2}\right)} \). For \( k=3 \), \( j=1/2 \): \( d_{1/2} = \frac{\sin(2\pi/5)}{\sin(\pi/5)} = \frac{\sin(72^\circ)}{\sin(36^\circ)} = \frac{0.9511}{0.5878} \approx 1.618 \) (golden ratio). For \( j=1 \): \( d_1 = \frac{\sin(3\pi/5)}{\sin(\pi/5)} = \frac{\sin(108^\circ)}{\sin(36^\circ)} = \frac{0.9511}{0.5878} \approx 1.618 \). Fusion coefficients are Verlinde numbers: \( N_{ij}^k = \sum_l \frac{S_{il} S_{jl} S_{kl}^*}{S_{0l}} \), with modular S-matrix entries \( S_{ab} = \sqrt{\frac{2}{k+2}} \sin\left(\frac{\pi(2a+1)(2b+1)}{k+2}\right) \). 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