{
  "abstract": "FINDING: Nested radicals of the form √(1+√(1+√(1+...))) converge to φ with doubly exponential error decay, revealing a self-similar recursive structure tied to the golden ratio's algebraic minimal polynomial. | MATH: The canonical nested radical φ = √(1+√(1+√(1+...))) satisfies φ² = 1+φ → φ = (1+√5)/2 = 1.6180339887… Error term εₙ₊₁ ≈ εₙ²/(2φ) — quadratic convergence (doubly exponential in iteration count: εₙ ~ φ⁻²ⁿ). General form: √(a + b√(a + b√(...))) converges iff |b| < 2√a, with limit L = (b + √(b²+4a))/2. The Wolfram demonstration and video analyses confirm this rate; the \"impossible\" radical √(2√(4√(8...))) = 2^(1+1/2+1/4+...) = 2² = 4, a geometric series in exponents. | CONNECTION: The convergence rate εₙ₊₁ = εₙ²/(2φ) directly involves φ = 1.618 and its inverse 1/φ = 0.618. The nested structure mirrors the continued fraction of φ = [1;1,1,1,...] — both are infinite self-similar representations. The error contraction factor 1/(2φ) ≈ 0.309 is related to 0.382 (φ⁻²) via the identi\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/0kt7veg1gyq9r.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23616390",
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  "id": "0kt7veg1gyq9r",
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  "language": "eng",
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  "licence_label": "CC BY 4.0",
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  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0kt7veg1gyq9r.ris",
  "sources": [
    {
      "first_seen": "2026-11-07T12:20:53Z",
      "native_id": "23616390",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23616390"
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  "subjects": [],
  "title": "Nested Radicals Converge to Golden Ratio with Doubly Exponential Error — E8 Intelligence Research",
  "title_text": "Nested Radicals Converge to Golden Ratio with Doubly Exponential Error — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-07T12:20:53Z",
  "url": "https://research.draconitic.com/paper/0kt7veg1gyq9r",
  "venue": "Zenodo",
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