{
  "abstract": "FINDING: Complete 2-descent on elliptic curves over ℚ reduces the Mordell-Weil rank problem to finite checks on quartic (degree-4) models, linking torsion structure to rational point lattices. | MATH: For E: y² = x³ + ax + b, 2-descent maps E(ℚ)/2E(ℚ) ↪ Sel⁽²⁾(E/ℚ), with quartic models C: y² = f(x) where deg(f)=4 (or genus-1 binary quartics). The 2-torsion E[2] ≅ (ℤ/2ℤ)² over ℚ determines the factorization of f into quadratics; the Selmer group bounds rank r ≤ dim₂ Sel⁽²⁾ − dim₂ E[2](ℚ). Yesin's work: divisibility by 2 on quartic models connects to rational Diophantine D(q)-quintuples — sets {a₁,…,a₅} where aᵢaⱼ + q is a square for all i≠j, with q ∈ ℚ*. | CONNECTION: The quartic model's binary form has discriminant Δ = 16·disc(f). The torsion lattice E[2] corresponds to the root system A₁³ (three orthogonal ℤ/2ℤ axes) — a crystallographic symmetry. The D(q)-quintuple condition aᵢaⱼ + q = square is structurally identical to the condition for rational points on a genus-1 curve, whose j-i\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,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
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      ],
      "name": "Andrew Stewart Caldin"
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  ],
  "bibtex": "https://research.draconitic.com/paper/71q1f6er7e8t4.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635153",
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  "id": "71q1f6er7e8t4",
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  "language": "eng",
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  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71q1f6er7e8t4.ris",
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    {
      "first_seen": "2026-11-08T12:21:58Z",
      "native_id": "23635153",
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  "title": "2-Descent via Quartic Models: Torsion Structure Bounds Elliptic Curve Ranks — E8 Intelligence Research",
  "title_text": "2-Descent via Quartic Models: Torsion Structure Bounds Elliptic Curve Ranks — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:58Z",
  "url": "https://research.draconitic.com/paper/71q1f6er7e8t4",
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