{
  "abstract": "FINDING: The Busy Beaver function BB(n) defines the maximum runtime of halting n-state Turing machines; BB(5) was recently resolved via collaborative proof, and higher-order extensions via oracle machines link decidability to number-theoretic truth. | MATH: BB(5) = 47,176,870 (known value, proven 2024); BB(n) is non-computable (grows faster than any computable function); higher-order BB functions BB^α(n) based on Turing oracle machines satisfy: BB^α(n) is computable iff α is a sufficiently strong oracle, with decidability of Π₁ sentences tied to BB^α growth rates (arXiv:2507.20321). | CONNECTION: No direct geometric ratio (0.382, 0.618, 0.786, 1.618, 2.618) appears. However, the structure of BB(n) values for n=1..5 (1, 6, 21, 107, 47,176,870) shows no simple ratio pattern — but the *growth* is super-exponential, reminiscent of the divergence between rational approximations to φ (1.618) and the golden ratio's algebraic irrationality. The oracle hierarchy mirrors the arithmetical hierarc\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": [
    {
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        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0tvc83kjjcxkc.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23490766",
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  "id": "0tvc83kjjcxkc",
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  "language": "eng",
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  "published": "2026-10-20",
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  "ris": "https://research.draconitic.com/paper/0tvc83kjjcxkc.ris",
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    {
      "first_seen": "2026-11-03T12:21:25Z",
      "native_id": "23490766",
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  "title": "The Busy Beaver Function: From BB(5) to Oracle Machines and Decidability — E8 Intelligence Research",
  "title_text": "The Busy Beaver Function: From BB(5) to Oracle Machines and Decidability — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:25Z",
  "url": "https://research.draconitic.com/paper/0tvc83kjjcxkc",
  "venue": "Zenodo",
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