{
  "abstract": "FINDING: Hofstadter butterfly spectrum exhibits exact self-similarity under rational flux transformations, with scaling governed by the golden ratio's reciprocal powers in the dominant hierarchy. | MATH: The butterfly spectrum \\(E(\\phi)\\) for \\(\\phi = p/q\\) (rational magnetic flux per plaquette) splits into \\(q\\) bands. Self-similarity emerges under the map \\(\\phi \\to \\phi'\\) where the dominant hierarchy scales as \\(E \\to E/\\alpha\\) with \\(\\alpha = \\phi^2\\) or \\(\\alpha = \\phi^{-2}\\) for specific flux intervals. The arXiv paper (1408.1006) isolates the most dominant hierarchy via continued fraction expansion of \\(\\phi\\), where the scaling factor between levels is \\(1/\\phi^2 \\approx 0.381966\\) (the complementary golden ratio squared) or \\(\\phi^2 \\approx 2.618034\\), depending on the branch. The spectrum's topological Chern numbers also obey the same recursive scaling. | CONNECTION: Direct geometric harmony link — \\(1/\\phi^2 = 2 - \\phi = 0.381966\\) and \\(\\phi^2 = \\phi + 1 = 2.618034\\). The\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/2cfe8qzz72m99.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635017",
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  "id": "2cfe8qzz72m99",
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  "language": "eng",
  "licence": "CC-BY-4.0",
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  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2cfe8qzz72m99.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:22:00Z",
      "native_id": "23635017",
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  "title": "Golden-Ratio Scaling Governs Self-Similar Hofstadter Butterfly Spectrum — E8 Intelligence Research",
  "title_text": "Golden-Ratio Scaling Governs Self-Similar Hofstadter Butterfly Spectrum — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:22:00Z",
  "url": "https://research.draconitic.com/paper/2cfe8qzz72m99",
  "venue": "Zenodo",
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