Contour Integral Computation of SU(2)_k Quantum Dimensions and Modular S-Matrix — E8 Intelligence Research

FINDING: Chern-Simons SU(2)_k quantum dimensions and modular S-matrix are computable via contour integrals of Feigin-Fuchs type, with braiding experiments proposed to measure k | MATH: SU(2)_k quantum dimensions d_j = sin(π(2j+1)/(k+2))/sin(π/(k+2)); Fibonacci anyons at k=3: d_1 = φ = (1+√5)/2 ≈ 1.618; Ising at k=2: d_σ = √2, d_ε = 1; modular S-matrix S_{jj'} = √(2/(k+2)) sin(π(2j+1)(2j'+1)/(k+2)); Verlinde formula N^k_{ij} = Σ_l S_{il}S_{jl}S*_{kl}/S_{0l} | CONNECTION: Fibonacci quantum dimension d_1 = φ = 1.618 (golden ratio) at k=3; Ising d_σ = √2 ≈ 1.414; quantum dimension ratios d_1/d_0 = φ, d_2/d_1 = φ at k=3; S-matrix entries involve sin ratios producing 0.618, 0.382 via sin(π/5)/sin(2π/5) = 1/φ = 0.618; k+2 = 5 for Fibonacci → pentagon symmetry (icosahedral/dodecahedral root system H₃); k+2 = 4 for Ising → square/tetrahedral symmetry | DEPTH: 9 FINDING: Equivariant spectral triples on quantum SU(2) have minimum dimension 3, with K-homology classification | MATH: Spectral tripl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com