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

Andrew Stewart Caldin

Published Sep 28, 2026• 0 citations

Abstract

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

DOI: 10.5281/zenodo.23006999

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

Andrew Stewart Caldin

Published Sep 28, 2026• 0 citations

Publisher

Abstract

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

DOI: 10.5281/zenodo.23006999

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FINDING: No direct evidence of quantum dimension sqrt(6) or golden ratio squared 2.618 in toric code anyon fusion from provided sources. Toric code anyons (e, m, ε) have quantum dimensions 1, 1, √2 respectively; golden ratio appears in Fibonacci anyon models, not toric code. MATH: Toric code anyon quantum dimensions: d_e = 1, d_m = 1, d_ε = √2. Total quantum dimension D = √(1² + 1² + (√2)²) = √4 = 2. No sqrt(6) or 2.618 present. CONNECTION: √2 relates to square lattice symmetry (crystallographic group p4m), not golden ratio. Toric code is defined on square lattice with Z₂ gauge theory — no base-60 or pentagonal symmetry. DEPTH: 1 — The search query's hypothesized constants (sqrt(6), 2.618) are absent from toric code. The provided sources do not support the claimed connection. Toric code is a distinct topological order from Fibonacci anyon models where golden ratio appears. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: No direct evidence of quantum dimension sqrt(6) or golden ratio squared 2.618 in toric code anyon fusion from provided sources. Toric code anyons (e, m, ε) have quantum dimensions 1, 1, √2 respectively; golden ratio appears in Fibonacci anyon models, not toric code. MATH: Toric code anyon quantum dimensions: d_e = 1, d_m = 1, d_ε = √2. Total quantum dimension D = √(1² + 1² + (√2)²) = √4 = 2. No sqrt(6) or 2.618 present. CONNECTION: √2 relates to square lattice symmetry (crystallographic group p4m), not golden ratio. Toric code is defined on square lattice with Z₂ gauge theory — no base-60 or pentagonal symmetry. DEPTH: 1 — The search query's hypothesized constants (sqrt(6), 2.618) are absent from toric code. The provided sources do not support the claimed connection. Toric code is a distinct topological order from Fibonacci anyon models where golden ratio appears. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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\dot{\rho} = -i[H,\rho] + \sum_k \gamma_k (L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\})

— no golden ratio coupling appears in the cited material. - Golden ratio

\Phi = (1+\sqrt{5})/2 \approx 1.618

, conjugate

\phi = \Phi - 1 = 1/\Phi \approx 0.618

. - arXiv 2510.08934: treats

\Phi

as a stable local recurrence procedure — no Lindblad or quantum dephasing content. CONNECTION: - The golden ratio appears only in the convergence-rate video (continued fractions vs. nested radicals) and the self-application paper — neither connects to quantum dissipation or FMO complexes. - No evidence of 0.382, 0.7 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

T \approx e^{-2\kappa L}

\kappa = \sqrt{2m(V_0-E)}/\hbar

. Golden ratio:

\phi = (1+\sqrt{5})/2 \approx 1.618

\phi^{-2} = (3-\sqrt{5})/2 \approx 0.382

. No equation in the sources equates

T

\phi^{-2}

. | CONNECTION: The arXiv paper (2510.08934) uses

\Phi

as a model of stable local recurrence — operationally,

\Phi^{-1} = \phi-1 \approx 0.618

, and

\Phi^{-2} \approx 0.382

— but this is in self-referential computation, not tunneling. The Numberphile video proves infinite sequences share the golden ratio property, but again no quantum link. | DEPTH: 2 — The search failed to find the claimed deriv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\Delta E = \mu_B g_F m_F B

(linear regime) - Hyperfine Hamiltonian (contact term):

H_{\text{hf}} = A \, \mathbf{I} \cdot \mathbf{J}

- Breit-Rabi formula (intermediate field):

E_{F,m_F} = -\frac{\Delta E_{\text{hf}}}{2(2I+1)} + g_I \mu_B m_F B \pm \frac{\Delta E_{\text{hf}}}{2} \sqrt{1 + \frac{4 m_F x}{2I+1} + x^2}

, with

x = \frac{(g_J - g_I)\mu_B B}{\Delta E_{\text{hf}}}

- No occurrence of 0.382, 0.618, 0.786, 1.618, 2.618, or base-60 in any of the listed lecture notes or videos. CONNECTION: None found. The arXiv paper (2510.08934) uses the golden ratio

\Phi

as a model for stable loc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com