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  "volume_id": "bsd",
  "title": "The Hidden Music of Elliptic Curves",
  "subtitle": "A guided journey to the Birch and Swinnerton–Dyer conjecture",
  "status": "Open Millennium Prize Problem; documentary exposition; no proof claim",
  "claim_boundary": "Browser-native, source-normalized exposition of rational points, the Mordell–Weil group, the Hasse–Weil L-function, low-rank theorem terrain, and the strong BSD formula. Numerical agreement, parity, Selmer bounds, family results, one-prime theorems, and p-adic analogues are not promoted to the universal complex conjecture.",
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      "title": "The Hidden Music of Elliptic Curves",
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      "label": "Plate I",
      "title": "The ancient question of rational solutions",
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      "alt": "A paired arithmetic and analytic panorama links the Mordell–Weil group of rational points to the vanishing of an elliptic-curve L-function.",
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      "label": "Plate III",
      "title": "From prime counts to the central point",
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      "alt": "A decorative ledger surrounds the leading-term formula with period, regulator, Tamagawa, torsion, and Tate–Shafarevich contributions.",
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      "label": "Plate V",
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      "alt": "A map distinguishes established modularity and low-rank results from the unresolved higher-rank and universal leading-term frontier.",
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      "id": "central-bridge",
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  "appendices": [
    {
      "id": "appendix-curves",
      "title": "Elliptic Curves and the Mordell–Weil Group",
      "source_heading": "Elliptic Curves and the Mordell--Weil Group",
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      "id": "appendix-lfunction",
      "title": "The L-Function of an Elliptic Curve",
      "source_heading": "The $L$-Function of an Elliptic Curve",
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    {
      "id": "appendix-strong",
      "title": "The Strong BSD Formula",
      "source_heading": "The Strong BSD Formula",
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    },
    {
      "id": "appendix-selmer",
      "title": "Selmer Groups, Descent, and the Hidden Term",
      "source_heading": "Selmer Groups, Descent, and the Hidden Term",
      "claim_classes": [
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        "guardrail"
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    },
    {
      "id": "appendix-trust",
      "title": "Known Results and Claim-Level Trust Matrix",
      "source_heading": "Known Results and Logical Boundaries; MATH-PROGRAMME Trust Matrix",
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  "sources": [
    "https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/",
    "https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf",
    "https://doi.org/10.1515/crll.1965.218.79",
    "https://doi.org/10.1090/S0894-0347-01-00370-8",
    "https://doi.org/10.1007/BF01388809",
    "https://www.mathnet.ru/eng/aa47"
  ]
}