Golden Ratio Emerges in Temperley-Lieb Categorification at Fifth Root of Unity — E8 Intelligence Research

FINDING: Temperley-Lieb algebra at roots of unity connects to A2 root system categorification, with the golden ratio emerging as the q-parameter at the 5th root of unity. | MATH: The Temperley-Lieb algebra TL_n(q) has parameter q = e^{iπ/5} (5th root of unity), where q + q^{-1} = 2cos(π/5) = φ = (1+√5)/2 ≈ 1.618. At this root, the algebra becomes non-semisimple, and its categorification via the A2 root system (sl₃) yields a 2-category whose Grothendieck group carries the quantum group U_q(sl₂) at q = e^{iπ/5}. The q-Virasoro algebra at roots of unity (arXiv:1211.1067) realizes free fermions on a lattice, with the deformation parameter q satisfying q^N = 1, and the Temperley-Lieb Hamiltonians H = Σ e_i where e_i² = δ e_i with δ = q + q^{-1}. | CONNECTION: The golden ratio φ = 1.618 appears directly as the loop value δ in TL_n at q = e^{iπ/5}. The A2 root system has 6 roots at 60° angles (hexagonal symmetry), and its Weyl group is S₃ (dihedral D₃). The categorification at roots of unity Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com