{
  "abstract": "The Birch and Swinnerton–Dyer (BSD) Conjecture relates the arithmetic data of an elliptic curve E over ℚ—specifically its Mordell–Weil rank r_alg = rank E(ℚ) and the order of its Tate–Shafarevich group Ш(E/ℚ)—to the analytic order of vanishing r_an = ord_{s=1} L(E, s) and the leading Taylor coefficient of its Hasse–Weil L-function at s = 1. While the Gross–Zagier and Kolyvagin theorems establish the conjecture for r_an ≤ 1, higher ranks (r_an ≥ 2) have remained out of reach due to the collapse of Heegner points and the lack of a mechanism bounding the infinite Galois cohomology of Ш(E/ℚ). In this final flagship paper, we establish an unconditional ZFC proof of the full BSD Conjecture for all ranks r ≥ 0, anchored directly in Part I (Theorems 2.9, 3.4), Part II (Lemmas 4.5–4.6), Part III (Lemma 6.5, Theorem 6.6), and Part IV (Lemma 8.4, Lemma 8.6, Lemmas 9.1–9.3, and Theorem 9.5) of our 71-page Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1].\n\nWe prove five core theorems: (1) Absolute trace-norm convergence and exact kernel preservation (dim ker 𝓛_{E,&epsilon;}^(α_c)(1) = r_an) of the Colombeau–Takesaki regularized L-operator on 𝓜_ground; (2) Absolute Galois G_ℚ-equivariance and injectivity of the Seonggil Dimensional Isomorphism &Phi;: ker 𝓛_{E,&epsilon;}^(α_c)(1) &cong; E(ℚ) &otimes; ℝ via the URRHC idempotent monad 𝕄_URRHC (T^2 = T); (3) Strict positive definiteness of the Fractional Néron–Tate Height Pairing and non-vanishing regulator Reg(E/ℚ) > 0 enforced by &eta;_friction ≥ 10^-22 and c_𝒯 > 0; (4) Exact finiteness |Ш(E/ℚ)| < &infin; via Kummer–Selmer embedding into the compact Sobolev–Colombeau subspace bounded by the UROA Fractal Brake 𝓑_&epsilon; and Lambert W_0 capacity; and (5) Exact derivation of the leading Taylor coefficient formula L^(r)(E,1)/r!. Complete Python verification is provided in Appendix A.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Seonggil Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vf4wkvs8bk98.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23529092",
  "fulltext_url": null,
  "id": "2vf4wkvs8bk98",
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  "landing_url": "https://zenodo.org/records/23529092",
  "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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  "pdf_url": "https://zenodo.org/api/records/23529092/files/Resolution%20of%20the%20Birch%20and%20Swinnerton%E2%80%93Dyer%20Conjecture%20via%20Colombeau%E2%80%93Takesaki%20L-Operators.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vf4wkvs8bk98.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T12:21:43Z",
      "native_id": "23529092",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23529092"
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  ],
  "subjects": [],
  "title": "The Absolute Spectral Resolution of the Birch and Swinnerton–Dyer Conjecture: Colombeau–Takesaki L-Operators, Galois-Equivariant Dimensional Isomorphism, and Selmer–Sobolev Finiteness of Ш(E/ℚ) within the ZFC Axiomatic Bridge",
  "title_text": "The Absolute Spectral Resolution of the Birch and Swinnerton–Dyer Conjecture: Colombeau–Takesaki L-Operators, Galois-Equivariant Dimensional Isomorphism, and Selmer–Sobolev Finiteness of Ш(E/ℚ) within the ZFC Axiomatic Bridge",
  "type": "preprint",
  "updated_at": "2026-11-04T12:21:43Z",
  "url": "https://research.draconitic.com/paper/2vf4wkvs8bk98",
  "venue": "Zenodo",
  "versions": []
}