Quantum Braiding, Verlinde Formula, and SU(2) WZW Invariants — E8 Intelligence Research

FINDING: Quantum group U_q(sl₂) braiding yields 3-manifold invariants; Verlinde formula computes SU(2) WZW modular S-matrix from fusion rules. | MATH: S_{ij} = √(2/(k+2)) · sin(π(i+1)(j+1)/(k+2)); fusion coefficients N_{ij}^k from Verlinde: N_{ij}^k = Σ_n S_{in}S_{jn}S*_{kn}/S_{0n}; q = e^{iπ/(k+2)}; Jones polynomial V_L(q) from R-matrix of U_q(sl₂) at roots of unity. | CONNECTION: SU(2) root system A₁: two roots ±α at 180°; Weyl group order 2; S-matrix entries are sine ratios — discrete Fourier transform on Z_{k+2} ≅ cyclic group, a crystallographic 1D lattice symmetry. Quantum dimensions d_j = sin(π(j+1)/(k+2))/sin(π/(k+2)) → golden ratio φ = 1.618... at k=3, j=1 (Fibonacci anyon). | DEPTH: 9 --- FINDING: Braided monoidal structure on U_q(g) provides knot/link invariants via Reshetikhin–Turaev functor; WZW correlators = conformal blocks on Riemann surfaces. | MATH: R-matrix satisfies Yang–Baxter: (R⊗1)(1⊗R)(R⊗1) = (1⊗R)(R⊗1)(1⊗R); Ribbon element θ; invariant Z(M³) = Σ_λ d_λ² · (col Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com