{
  "abstract": "FINDING: The golden ratio φ = [1;1,1,...] is the canonical continued fraction, and its nested radical forms connect to quasicrystal inflation via the self-similarity of its convergents (Fibonacci ratios). | MATH: φ = (1+√5)/2 = 1.6180339…; continued fraction φ = 1 + 1/(1 + 1/(1 + …)); nested radical φ = √(1+√(1+√(1+…))); convergents F(n+1)/F(n) → φ; Rogers-Ramanujan continued fraction R(q) = q^(1/5)/(1 + q/(1 + q²/(1 + q³/(…)))) — for q = e^(2πiτ), R(τ) solves quintics and is modular. | CONNECTION: φ² = φ + 1 ⇒ φ = 1 + 1/φ — the *inflation rule* for Penrose tilings (substitution: L→L+S, S→L) has eigenvalue φ. The ratio of tile frequencies in a Penrose tiling is φ:1, and the Fourier transform shows 5-fold (icosahedral) rotational symmetry — a crystallographically forbidden order, but allowed in quasicrystals. The continued fraction's period-1 structure mirrors the self-similar inflation of the tiling. Also, φ⁻¹ = 0.618, φ⁻² = 0.382 — the two fundamental length ratios in the Penrose rhom\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/4h2td282nvssa.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23431132",
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  "id": "4h2td282nvssa",
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  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4h2td282nvssa.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T12:22:03Z",
      "native_id": "23431132",
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      "url": "https://zenodo.org/records/23431132"
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  "title": "Golden Ratio as Canonical Continued Fraction Linking Quasicrystal Inflation and Modular Forms — E8 Intelligence Research",
  "title_text": "Golden Ratio as Canonical Continued Fraction Linking Quasicrystal Inflation and Modular Forms — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-31T12:22:03Z",
  "url": "https://research.draconitic.com/paper/4h2td282nvssa",
  "venue": "Zenodo",
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}