{
  "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(\\phi)\\) for Harper equation \\(\\psi_{n+1}+\\psi_{n-1}+2\\cos(2\\pi n\\phi+\\theta)\\psi_n=E\\psi_n\\); gap labeling: each gap indexed by integer \\(k\\) via \\(k\\phi \\bmod 1\\); for \\(\\phi\\) = golden mean \\(1/\\tau\\) (\\(\\tau=(1+\\sqrt5)/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 \\(\\tau=1.618\\), inverse \\(1/\\tau=0.618\\), \\(\\tau-1=0.618\\), \\(\\tau^{-2}=0.382\\), \\(\\tau^{-3}=0.236\\) — these appear as scaling ratios of butterfly sub-bands; continued fraction \\([1;1,1,\\dots]\\) = \\(\\tau\\) — the most irrational number, maximizing gap stability; Hofsta\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2fe2n51f4jdfz.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23658193",
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  "id": "2fe2n51f4jdfz",
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  "landing_url": "https://zenodo.org/records/23658193",
  "language": "eng",
  "licence": "CC-BY-4.0",
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  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2fe2n51f4jdfz.ris",
  "sources": [
    {
      "first_seen": "2026-11-09T12:21:38Z",
      "native_id": "23658193",
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      "url": "https://zenodo.org/records/23658193"
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  "subjects": [],
  "title": "Bethe Ansatz Unlocks Hofstadter Butterfly via Golden-Mean Scaling — E8 Intelligence Research",
  "title_text": "Bethe Ansatz Unlocks Hofstadter Butterfly via Golden-Mean Scaling — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:38Z",
  "url": "https://research.draconitic.com/paper/2fe2n51f4jdfz",
  "venue": "Zenodo",
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}