Bethe Ansatz Unlocks Hofstadter Butterfly via Golden-Mean Scaling — E8 Intelligence Research

Abstract

FINDING: Bethe Ansatz provides exact solution to Hofstadter butterfly, linking its recursive spectrum to continued-fraction scaling and gap-labeling via the golden mean. | MATH: Hofstadter spectrum E(ϕ) for Harper equation ψ_n+1+ψ_n−1+2cos(2π nϕ+θ)ψ_n=Eψ_n; gap labeling: each gap indexed by integer k via kϕ bmod 1; for ϕ = golden mean 1/τ (τ=(1+√5)/2), scaling exponent of central gaps governed by continued-fraction convergents p_n/q_n with q_n = Fibonacci numbers; Bethe Ansatz yields exact eigenstates as nested Bethe roots satisfying algebraic equations — no closed elementary form, but integrable structure emerges. | CONNECTION: Golden mean τ=1.618, inverse 1/τ=0.618, τ−1=0.618, τ⁻²=0.382, τ⁻³=0.236 — these appear as scaling ratios of butterfly sub-bands; continued fraction [1;1,1,…] = τ — the most irrational number, maximizing gap stability; Hofsta

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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BibTeX

@misc{caldin2026bethe,
  title = {{Bethe Ansatz Unlocks Hofstadter Butterfly via Golden-Mean Scaling — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23658193},
  url = {https://zenodo.org/records/23658193},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Bethe Ansatz Unlocks Hofstadter Butterfly via Golden-Mean Scaling — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/26
PB  - Zenodo
DO  - 10.5281/zenodo.23658193
UR  - https://zenodo.org/records/23658193
AB  - FINDING: Bethe Ansatz provides exact solution to Hofstadter butterfly, linking its recursive spectrum to continued-fraction scaling and gap-labeling via the golden mean. | MATH: Hofstadter spectrum E(ϕ) for Harper equation ψ_n+1+ψ_n−1+2cos(2π nϕ+θ)ψ_n=Eψ_n; gap labeling: each gap indexed by integer k via kϕ bmod 1; for ϕ = golden mean 1/τ (τ=(1+√5)/2), scaling exponent of central gaps governed by continued-fraction convergents p_n/q_n with q_n = Fibonacci numbers; Bethe Ansatz yields exact eigenstates as nested Bethe roots satisfying algebraic equations — no closed elementary form, but integrable structure emerges. | CONNECTION: Golden mean τ=1.618, inverse 1/τ=0.618, τ−1=0.618, τ^−2=0.382, τ^−3=0.236 — these appear as scaling ratios of butterfly sub-bands; continued fraction [1;1,1,…] = τ — the most irrational number, maximizing gap stability; Hofsta  Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER  -

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