{
  "abstract": "FINDING: The Collatz map is reformulated as an automorphic Cayley colour graph, yielding a proof of the $3n+1$ conjecture via decidability of $an+b$ conjectures. | MATH: Collatz map $T(n) = n/2$ if $n$ even, $T(n) = 3n+1$ if $n$ odd; trivial cycle $4 \\to 2 \\to 1 \\to 4$; branching number $c=4$; automorphism group of the Cayley graph acts transitively on root paths; decidability of $an+b$ conjectures reduces to graph automorphism properties. | CONNECTION: The branching number $c=4$ and the cyclic root $4\\to2\\to1\\to4$ form a 4-cycle — a square, the simplest crystallographic lattice in 2D (square lattice symmetry $p4m$). The ratio of the cycle's edge lengths (if mapped to a geometric square) is $1:1$, but the *walk lengths* to the root from any $n$ scale logarithmically — the inverse of the golden ratio appears in the density of odd numbers in the Collatz sequence: for a random odd $n$, the expected number of divisions by 2 before next odd is $2$, giving a contraction factor of $3/4$ per o\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/48rmm0jmd3hkd.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635651",
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  "id": "48rmm0jmd3hkd",
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  "landing_url": "https://zenodo.org/records/23635651",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/48rmm0jmd3hkd.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:21:52Z",
      "native_id": "23635651",
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      "url": "https://zenodo.org/records/23635651"
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  "subjects": [],
  "title": "Cayley Graph Proof of the Collatz Conjecture via Decidable an+b Problems — E8 Intelligence Research",
  "title_text": "Cayley Graph Proof of the Collatz Conjecture via Decidable an+b Problems — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:52Z",
  "url": "https://research.draconitic.com/paper/48rmm0jmd3hkd",
  "venue": "Zenodo",
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}