Golden Ratio Anyons: Temperley-Lieb Algebra at SU(2)_3 Chern-Simons Level — E8 Intelligence Research

FINDING: Temperley-Lieb algebra with loop value √5 emerges at SU(2)_k Chern-Simons level k=3, where the quantum dimension of the fundamental anyon is the golden ratio φ = (1+√5)/2 ≈ 1.618. | MATH: For SU(2)_k, the loop value (quantum dimension) is d = [2]_q = q + q⁻¹ with q = e^{iπ/(k+2)}. At k=3: q = e^{iπ/5}, so d = 2cos(π/5) = φ ≈ 1.618. The Temperley-Lieb algebra TL_n(δ) has loop parameter δ = d = φ. The Fibonacci anyon (the non-trivial particle in SU(2)_3) has fusion rule τ×τ = 1 + τ, with quantum dimension φ. The associated Jones polynomial at t = e^{2πi/5} evaluates to powers of φ. | CONNECTION: Direct geometric harmony — φ = 1.618 is the golden ratio, and its inverse φ⁻¹ = 0.618. The pentagonal symmetry (5-fold) is encoded: k+2 = 5, giving the 5th root of unity. The loop value √5 appears in the quadratic relation δ² = φ² = φ+1 = 2.618 (the golden ratio squared). The Catalan numbers count basis states of TL_n(φ), and the asymptotic growth rate is 4, but the specific quantum defo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com