{
  "abstract": "FINDING: Vieta jumping on quadratic Diophantine equations reveals hidden affine Weyl group orbits, with the rational equation (a+1)/b + (b+1)/a = k forcing k ∈ {3,4} and solutions tied to Fibonacci-type recurrences. | MATH: Equation: (a+1)/b + (b+1)/a = k → a² + a + b² + b = kab. Vieta jumping: if (a,b) solves, then (a, ka−b−1) also solves (root reflection). Classification: k=3 or k=4 only. For k=3, solutions follow recurrence a_{n+1} = 3a_n − a_{n−1} − 1 (Fibonacci-like with offset); for k=4, a_{n+1} = 4a_n − a_{n−1} − 1. The quadratic form a² + b² − kab + a + b = 0 has discriminant Δ = k² − 4, so k=3 → Δ=5 (golden ratio base), k=4 → Δ=12 (2√3, hexagonal lattice). | CONNECTION: k=3 gives Δ=5 → √5 appears → ratio (1+√5)/2 = φ ≈ 1.618, and its inverse 0.618. The Vieta jump is a reflection in the affine Weyl group of type A₁ (or Ã₁) — the same group underlying the Fibonacci/Lucas recurrences and the golden ratio's continued fraction [1;1,1,...]. k=4 gives Δ=12 → √12 = 2√3 → ratio √3 ≈ 1.\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/3cvhh4wfn7tny.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23447046",
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  "id": "3cvhh4wfn7tny",
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  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3cvhh4wfn7tny.ris",
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    {
      "first_seen": "2026-11-01T12:21:48Z",
      "native_id": "23447046",
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  "title": "Vieta Jumping Reveals Affine Weyl Orbits in Quadratic Diophantine Equations — E8 Intelligence Research",
  "title_text": "Vieta Jumping Reveals Affine Weyl Orbits in Quadratic Diophantine Equations — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:48Z",
  "url": "https://research.draconitic.com/paper/3cvhh4wfn7tny",
  "venue": "Zenodo",
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