{
  "abstract": "FINDING: Vieta jumping is the elementary face of an affine Weyl group orbit structure on integer solutions of quadratic forms; the arXiv paper formalizes the infinite reduced word geometry underlying this. MATH: Vieta jumping on \\(x^2 + y^2 = kxy + c\\) (or \\(ax^2 + bxy + cy^2 = n\\)) corresponds to the action of the simple reflection \\(s_i\\) on the root lattice: \\((x,y) \\mapsto (kx - y, x)\\) for \\(k = \\frac{x^2+y^2-c}{xy}\\). This is precisely the Weyl group \\(W(\\widetilde{A}_1)\\) (or higher \\(\\widetilde{A}_n\\)) acting on the integer points of the quadratic form's level set. The affine Weyl group \\(\\widetilde{A}_n\\) has Coxeter generators \\(s_0, s_1, \\dots, s_n\\) with \\(s_i^2=1\\), \\((s_i s_{i+1})^3=1\\), and the orbit of a solution under these reflections is the full integer solution set. The paper's \"minimal infinite reduced words\" encode the infinite descent/ascent chains of Vieta jumps — each jump is a simple reflection, and the sequence of jumps is a reduced word in the affine Weyl\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/avr70228fp7ad.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23409733",
  "fulltext_url": null,
  "id": "avr70228fp7ad",
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  "landing_url": "https://zenodo.org/records/23409733",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/avr70228fp7ad.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:27Z",
      "native_id": "23409733",
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      "url": "https://zenodo.org/records/23409733"
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  "subjects": [],
  "title": "Vieta Jumping as Affine Weyl Group Orbits on Quadratic Forms — E8 Intelligence Research",
  "title_text": "Vieta Jumping as Affine Weyl Group Orbits on Quadratic Forms — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:24Z",
  "url": "https://research.draconitic.com/paper/avr70228fp7ad",
  "venue": "Zenodo",
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}