{
  "abstract": "FINDING: Tao's 2019 logarithmic-density result proves almost all Collatz orbits reach values arbitrarily close to bounded — the first unconditional statistical breakthrough on the 3x+1 problem. | MATH: Collatz map T(n) = n/2 if n even, (3n+1)/2 if n odd. Tao's theorem: for almost all n (logarithmic density), inf_{k≥0} T^k(n) < f(n) where f(n) → ∞ arbitrarily slowly (e.g., log log log n). Key tool: \"Tao general difference\" (TGD) — a novel discrete difference operator generalizing finite differences to multidimensional arrays, formalized in arXiv:2401.15287. No exact constants appear; the proof uses probabilistic stopping-time bounds and dyadic decomposition. | CONNECTION: No direct geometric ratio (0.382, 0.618, 1.618) appears. However, the Collatz map's structure encodes a 2-adic symmetry — the dynamics are conjugate to a shift on the 2-adic integers, which is a self-similar fractal (the \"Collatz graph\" has a tree-like lattice with branching ratio 3:1 for odd steps vs 1:2 for even step\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
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  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
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  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
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  ],
  "bibtex": "https://research.draconitic.com/paper/4haenb4s821n9.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23526271",
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  "id": "4haenb4s821n9",
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  "language": "eng",
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  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4haenb4s821n9.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T12:21:50Z",
      "native_id": "23526271",
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  "subjects": [],
  "title": "Tao's Logarithmic-Density Breakthrough on Collatz Orbits — E8 Intelligence Research",
  "title_text": "Tao's Logarithmic-Density Breakthrough on Collatz Orbits — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-04T12:21:50Z",
  "url": "https://research.draconitic.com/paper/4haenb4s821n9",
  "venue": "Zenodo",
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