{
  "abstract": "FINDING: Outer billiards on regular polygons exhibit bounded orbits tied to the polygon's divisor structure, with rational-angle dynamics revealing a deep link between number theory and piecewise-isometric maps. | MATH: For a regular \\(n\\)-gon, the outer billiards map \\(T\\) is piecewise a rotation by \\(2\\pi/n\\) (or its multiples) about vertices. Orbits are bounded iff the polygon's side count \\(n\\) has no odd prime factor \\(p\\) such that the orbit's \"impact parameter\" lies in a certain rational class — specifically, Vivaldi–Shaidenko showed all orbits are bounded for \\(n\\) with only even prime factors (i.e., \\(n=2^k\\)), while unbounded orbits exist for \\(n\\) with odd prime divisors (e.g., \\(n=5,7,9,\\dots\\)). The key invariant is the \"rotation number\" \\(\\rho = \\frac{a}{n}\\) (mod 1) for each vertex reflection, and the orbit's fate depends on the continued fraction expansion of \\(\\rho\\) and the divisor lattice of \\(n\\). For \\(n=5\\), the golden ratio \\(\\varphi = (1+\\sqrt{5})/2 \\approx 1.61\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/caysb4b6cfrdx.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23446651",
  "fulltext_url": null,
  "id": "caysb4b6cfrdx",
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  "landing_url": "https://zenodo.org/records/23446651",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/caysb4b6cfrdx.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:21:55Z",
      "native_id": "23446651",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23446651"
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  ],
  "subjects": [],
  "title": "Bounded Orbits in Polygonal Outer Billiards Linked to Divisor Structure — E8 Intelligence Research",
  "title_text": "Bounded Orbits in Polygonal Outer Billiards Linked to Divisor Structure — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:55Z",
  "url": "https://research.draconitic.com/paper/caysb4b6cfrdx",
  "venue": "Zenodo",
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}