Golden Ratio as Loop Value in Jones-Temperley-Lieb Algebras at k=3 — E8 Intelligence Research

FINDING: Jones-Temperley-Lieb (TL) algebras are the algebraic backbone of the Jones polynomial; their loop value δ = 2cos(π/(k+2)) equals the golden ratio φ = 1.618... exactly when k=3 (i.e., at the Jones/SU(2)₃ level), making φ the quantum dimension of the fundamental representation and the topological entanglement (loop) weight. MATH: - TL algebra generators eᵢ (i=1..n−1) satisfy: eᵢ² = δ·eᵢ, eᵢeᵢ₊₁eᵢ = eᵢ, eᵢeⱼ = eⱼeᵢ (|i−j|≥2) - Loop value: δ = −(A² + A⁻²) = 2cos(π/(k+2)) for SU(2)_k - At k=3: δ = 2cos(π/5) = φ = (1+√5)/2 = 1.6180339887... - Quantum dimension of spin-1/2 rep at level k: d = sin(π/(k+2))/sin(π/(k+2))... more precisely d1/2 = [2]₍q₎ = q + q⁻¹ with q = eiπ/(k+2); at k=3, d = φ. - Jones polynomial V_L(t) obtained via Markov trace on TL_n(δ) with t = A⁻⁴; braid group representation ρ: B_n → TL_n. - Fibonacci anyon (k=3) fusion rule: τ ⊗ τ = 1 ⊕ τ, with quantum dimensions satisfying d_τ² = 1 + d_τ → d_τ = φ. - Temperley-Lieb recoupling / F-matrix entries at k= Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com