Golden Ratio Bound on Topological Entanglement Entropy in Fibonacci Anyon Loops — E8 Intelligence Research

FINDING: Temperley–Lieb loop model with loop value √5 (Fibonacci anyon sector) yields a topological entanglement entropy correction that is non-integer, tied to the golden ratio, and a universal lower bound on topological entanglement entropy exists from anyonic quantum dimensions. | MATH: Temperley–Lieb algebra: generators \(e_i\) satisfy \(e_i^2 = \delta e_i\), \(e_i e_{i\pm1} e_i = e_i\), with loop value \(\delta = \sqrt{5}\) (Fibonacci anyons: quantum dimension \(\phi = (1+\sqrt{5})/2 \approx 1.618\), and \(\delta = \phi^2 - 1 = \phi\)? No — for Fibonacci, \(\delta = \phi^2 = \phi+1 \approx 2.618\) in some normalizations, but the loop value for the golden chain is \(\delta = \phi \approx 1.618\) or \(\delta = \sqrt{5}\) depending on the representation. The topological entanglement entropy \(S_{\text{topo}} = -\ln \mathcal{D}\), where \(\mathcal{D} = \sqrt{\sum_a d_a^2}\) is the total quantum dimension. For Fibonacci: \(\mathcal{D} = \sqrt{1 + \phi^2} = \sqrt{2+\phi} = \sqrt{2.618} Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com