{
  "abstract": "FINDING: The canonical height pairing on elliptic curves defines a positive-definite quadratic form whose matrix eigenvalues encode the curve's rank and lattice structure — the rank conjecture asks whether this rank is unbounded over families. MATH: - Elliptic curve: \\(E: y^2 = x^3 + ax + b\\), \\(a,b \\in \\mathbb{Q}\\) - Mordell–Weil group: \\(E(\\mathbb{Q}) \\cong \\mathbb{Z}^r \\oplus E_{\\text{tors}}\\) - Canonical height: \\(\\hat{h}(P) = \\lim_{n\\to\\infty} 4^{-n} h(2^n P)\\) - Height pairing: \\(\\langle P,Q \\rangle = \\frac{1}{2}[\\hat{h}(P+Q) - \\hat{h}(P) - \\hat{h}(Q)]\\) - Matrix \\(M_{ij} = \\langle P_i, P_j \\rangle\\) for independent points → eigenvalues \\(\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge \\lambda_r > 0\\) - Rank \\(r\\) = number of positive eigenvalues (or dimension of kernel of \\(M\\) over \\(\\mathbb{R}\\)) - Birch–Swinnerton-Dyer: \\(r = \\text{ord}_{s=1} L(E,s)\\) - Rank conjecture: \\(\\sup\\{r : \\exists E/\\mathbb{Q} \\text{ with rank } r\\} = \\infty\\) (open; current record \\(r \\ge 28\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": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bcefjg5a2681p.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635613",
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  "id": "bcefjg5a2681p",
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  "landing_url": "https://zenodo.org/records/23635613",
  "language": "eng",
  "licence": "CC-BY-4.0",
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  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bcefjg5a2681p.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:21:52Z",
      "native_id": "23635613",
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  "subjects": [],
  "title": "Eigenvalues of the Canonical Height Pairing and the Unbounded Rank Conjecture — E8 Intelligence Research",
  "title_text": "Eigenvalues of the Canonical Height Pairing and the Unbounded Rank Conjecture — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:52Z",
  "url": "https://research.draconitic.com/paper/bcefjg5a2681p",
  "venue": "Zenodo",
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}