{
  "abstract": "FINDING: Golden angle derives from golden ratio; Fibonacci convergence to φ is linear with rate 1/φ ≈ 0.618; nested radicals and continued fractions converge at different rates, with continued fractions exhibiting optimal (slowest algebraic) convergence. | MATH: Golden angle = 2π(1 − 1/φ) = 2π/φ² ≈ 137.5077° (or equivalently 2π − 2π/φ). Fibonacci ratio convergence: Fₙ₊₁/Fₙ → φ, error term ~ (−1/φ)ⁿ, giving asymptotic error ratio |eₙ₊₁/eₙ| = 1/φ ≈ 0.618. Continued fraction for φ: φ = 1 + 1/(1 + 1/(1 + …)) — all partial quotients = 1, making it the *slowest* converging continued fraction (worst-case for rational approximation). Nested radical: φ = √(1 + √(1 + √(1 + …))) converges faster — error decays doubly exponentially (rate ~ 1/φ² per nesting level squared). | CONNECTION: The convergence rate 1/φ = 0.618 is the *complement* of φ (φ − 1 = 1/φ). This is the same ratio appearing in phyllotaxis (golden angle = 2π/φ² ≈ 137.5°), which optimizes seed packing via the continued fraction of φ\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/6ss5nhjz2266j.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23553555",
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  "id": "6ss5nhjz2266j",
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  "landing_url": "https://zenodo.org/records/23553555",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ss5nhjz2266j.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:21:21Z",
      "native_id": "23553555",
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      "url": "https://zenodo.org/records/23553555"
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  "subjects": [],
  "title": "Convergence Rates of Golden Ratio Representations and the Golden Angle — E8 Intelligence Research",
  "title_text": "Convergence Rates of Golden Ratio Representations and the Golden Angle — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-05T12:21:21Z",
  "url": "https://research.draconitic.com/paper/6ss5nhjz2266j",
  "venue": "Zenodo",
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}