Jones Polynomial at 5th Roots of Unity Yields Golden-Ratio Fusion Rules — E8 Intelligence Research

FINDING: The Jones polynomial's evaluation at 5th roots of unity (the "Chern–Simons level" k=1,2,3,4) yields quantum invariants whose underlying algebraic structure — the Temperley–Lieb algebra at q = e^{2πi/5} — closes on Fibonacci-like fusion rules, directly linking knot topology to golden-ratio arithmetic. | MATH: Jones polynomial V_L(t), t = e^{2πi/r}; skein relation t⁻¹V(L₊) − tV(L₋) = (t^{1/2} − t^{−1/2})V(L₀). At r=5: quantum dimension d = −2cos(2π/5) = φ = (1+√5)/2 ≈ 1.618. Fusion: τ⊗τ = 1 ⊕ τ (Fibonacci anyon). Quantum group U_q(sl₂) at q = e^{2πi/5}, level k=3, central charge c = 1 − 6/((k+2)(k+3)) = 7/10. | CONNECTION: φ = 1.618, 1/φ = 0.618, φ² = 2.618 all appear as quantum dimensions and braiding eigenvalues in the Fibonacci anyon model. The 5th root of unity q = e^{2πi/5} generates the golden ratio via 2cos(2π/5) = 1/φ. This is the same φ governing pentagonal/quasicrystalline symmetry (icosahedral group, H₃ root system). | DEPTH: 9 --- FINDING: The Fibonacci sequence em Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com