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    "title": "OpenAI: Exact BSD formula under Selmer corank zero or one",
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        "title": "OpenAI, Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank",
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      "scope": "The hypothesis is on the full Selmer group, not merely Mordell–Weil rank. This is not BSD in every rank. The paper imports the unrestricted converse and the two-primary formula from separate manuscripts. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean."
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            "citation": "OpenAI, Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank",
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        "description": "The hypothesis is on the full Selmer group, not merely Mordell–Weil rank. This is not BSD in every rank. The paper imports the unrestricted converse and the two-primary formula from separate manuscripts. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean.",
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        {
          "citation": "OpenAI, Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf"
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      "statement": "# OpenAI: Exact BSD formula under Selmer corank zero or one\n\nFor an elliptic curve over the rational numbers, the manuscript claims the complete Birch–Swinnerton-Dyer leading-term formula whenever its full q-power Selmer group has corank zero or one at some prime q. It includes finiteness of the Tate–Shafarevich group and all prime factors of the formula. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.\n\n## Scientific scope\n\nThe hypothesis is on the full Selmer group, not merely Mordell–Weil rank. This is not BSD in every rank. The paper imports the unrestricted converse and the two-primary formula from separate manuscripts. WRS has inspected the cited statement and source-reported formalization scope where available; it has not audited the full proof or replayed Lean.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\n\n\nOpenAI (organizational manuscript byline); individual human roles not fully disclosed\n\nOpenAI research model / OpenAI / Unreleased internal frontier model; exact identifier not disclosed: Source-reported mathematical argument generation; human preparation/editing roles retained separately where disclosed (source reported).\n\n## Original sources\n\n- [OpenAI, Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf)\n",
      "summary": "For an elliptic curve over the rational numbers, the manuscript claims the complete Birch–Swinnerton-Dyer leading-term formula whenever its full q-power Selmer group has corank zero or one at some prime q. It includes finiteness of the Tate–Shafarevich group and all prime factors of the formula. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.",
      "title": "OpenAI: Exact BSD formula under Selmer corank zero or one"
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