{
  "abstract": "FINDING: The Stern-Brocot tree, Farey sequences, Calkin-Wilf tree, and continued fractions are isomorphic representations of ℚ⁺, with the Lagrange spectrum and Markov triples emerging as the arithmetic shadow of badly-approximable irrationals — the same objects that generate the golden ratio and its metallic-mean family.\n\nMATH:\n\nCore structures (all bijective on ℚ⁺): - Stern-Brocot: mediant insertion. From 0/1, 1/0, insert (a+c)/(b+d). Every positive rational appears exactly once. - Farey F_n: fractions in [0,1] with denominator ≤ n; consecutive terms a/b, c/d satisfy bc − ad = 1. - Calkin-Wilf: breadth-first enumeration via f(a/b) = a/(a+b), (a+b)/b. - Continued fractions: [a₀; a₁, a₂, …] ↔ Stern-Brocot path (L/R runs of length aᵢ).\n\nUnimodular invariance: - det[[a, c],[b, d]] = ad − bc = ±1 for Farey neighbors → SL(2,ℤ) action. - Stern-Brocot = Cayley graph of PSL(2,ℤ) with generators L, R.\n\nLagrange spectrum: - L(α) = liminf_{q→∞} q‖qα‖. - Lagrange's theorem: α badly approximable ⟺\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/0fb03fege4hg2.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23658699",
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  "id": "0fb03fege4hg2",
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  "landing_url": "https://zenodo.org/records/23658699",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0fb03fege4hg2.ris",
  "sources": [
    {
      "first_seen": "2026-11-09T12:21:33Z",
      "native_id": "23658699",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23658699"
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  "subjects": [],
  "title": "Isomorphic Rational Trees Link Farey, Stern-Brocot, and Markov Spectra — E8 Intelligence Research",
  "title_text": "Isomorphic Rational Trees Link Farey, Stern-Brocot, and Markov Spectra — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:33Z",
  "url": "https://research.draconitic.com/paper/0fb03fege4hg2",
  "venue": "Zenodo",
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}