Absence of Golden-Ratio Energy Gaps in Perturbed Toric Code — E8 Intelligence Research

FINDING: No direct evidence of a golden-ratio energy-gap ratio in the toric code under anisotropic perturbation; the search results are pedagogical (toric code lectures, perturbation theory) and one unrelated paper on golden-ratio self-application. | MATH: Toric code Hamiltonian: \(H = -\sum_v A_v - \sum_f B_f\), with \(A_v = \prod_{i\in v} \sigma^x_i\), \(B_f = \prod_{i\in f} \sigma^z_i\). Perturbation theory: \(E_n^{(2)} = \sum_{m\neq n} \frac{|\langle m|V|n\rangle|^2}{E_n^{(0)}-E_m^{(0)}}\). No derived ratio from sources. | CONNECTION: Toric code is a \(\mathbb{Z}_2\) lattice gauge theory on a square lattice — crystallographic symmetry \(p4m\) (wallpaper group), but no golden-ratio or base-60 link appears in the cited material. The arXiv paper (2510.08934) uses \(\Phi = (1+\sqrt{5})/2\) as a model of stable recurrence, but it is not connected to the toric code. | DEPTH: 2 — The query's premise (anisotropic perturbation → golden ratio gap) is unsupported by the retrieved sources; onl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com