{
  "abstract": "This paper proves both the rank equality and the exact leading-coefficient formula of the Birch and Swinnerton-Dyer Conjecture for all elliptic curves $E/\\mathbb{Q}$ under Topological Vacuum Electrodynamics. By Milnor’s theorem and the Modularity Theorem, an elliptic curve $E/\\mathbb{Q}$ of conductor $N$ is realized on the $N$-fold congruence covering $Y_0(N)=\\Gamma_0(N)\\backslash\\mathbb{H}$ of the trefoil knot complement via a weight-2 primitive Hecke newform $f_E$, corresponding to a force-free Beltrami magnetic 1-form $\\omega_E$. Truncating the cusps at the unknot minor-radius cutoff $y=a$ with the Sommerfeld outgoing radiation condition yields a maximally dissipative Lax–Phillips generator $B_E$ whose characteristic scattering function in the $f_E$-channel is governed by the completed Hasse–Weil function $\\Lambda(E,s)$. At the central point $s=1$ (static threshold $\\lambda=s-1=0$), Green’s identity forces the boundary dissipation to vanish on $\\ker(B_E)$. Under the modular Abel–Jacobi parameterization and Eichler–Shimura isomorphism, the static zero-leakage flux cycles quantized on the Néron lattice are in bijection with the free generators of $E(\\mathbb{Q})$, giving $\\dim\\ker(B_E)=\\operatorname{rank}E(\\mathbb{Q})=r$. Because $B_E$ is maximally dissipative, $\\lambda=0$ is strictly semisimple with no Jordan blocks, proving $\\operatorname{ord}_{s=1}L(E,s)=r$. Compactness of $Y_0(N)_a$ below the ultraviolet capacity $\\alpha^{-1}=4\\pi^3+\\pi^2+\\pi$ forces the Tate–Shafarevich defect group \\Sha$(E/\\mathbb{Q})$ to be finite, and the Cheeger–Müller analytic torsion factorization at $\\lambda=0$ recovers the exact leading Taylor coefficient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Truth Vector"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b9cs3xmvz071a.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23459257",
  "fulltext_url": null,
  "id": "b9cs3xmvz071a",
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  "landing_url": "https://zenodo.org/records/23459257",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23459257/files/TVE_%20Birch%20and%20Swinnerton-Dyer%20Conjecture%2020261018.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b9cs3xmvz071a.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:20:53Z",
      "native_id": "23459257",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23459257"
    }
  ],
  "subjects": [],
  "title": "Birch and Swinnerton-Dyer Conjecture Proof via Causal Scattering and Analytic Torsion on Truncated Modular Coverings under Topological Vacuum Electrodynamics",
  "title_text": "Birch and Swinnerton-Dyer Conjecture Proof via Causal Scattering and Analytic Torsion on Truncated Modular Coverings under Topological Vacuum Electrodynamics",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:53Z",
  "url": "https://research.draconitic.com/paper/b9cs3xmvz071a",
  "venue": "Zenodo",
  "versions": []
}