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    "title": "OpenAI: A claimed torsion-free group-algebra counterexample to direct finiteness",
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    "area": "Algebra and group theory",
    "topic": "Mathematics",
    "year": 2026,
    "lane": "AI",
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            "citation": "OpenAI, A Torsion-Free Group Algebra That Is Not Directly Finite",
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          "citation": "OpenAI, A Torsion-Free Group Algebra That Is Not Directly Finite",
          "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf"
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      "statement": "# OpenAI: A claimed torsion-free group-algebra counterexample to direct finiteness\n\nThe manuscript claims a finitely presented torsion-free group G and elements a,b,c in F2[G] with ab=1, ac=0 and c nonzero. These identities would force ba!=1, disproving direct finiteness in this positive-characteristic setting. 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 known characteristic-zero theorem is unchanged. The available family scope note describes other characteristic-two and odd-characteristic constructions, not this selected torsion-free paper; that Lean coverage cannot be transferred to this target. The August nonsofic-group package needs explicit overlap review. 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, A Torsion-Free Group Algebra That Is Not Directly Finite](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf)\n",
      "summary": "The manuscript claims a finitely presented torsion-free group G and elements a,b,c in F2[G] with ab=1, ac=0 and c nonzero. These identities would force ba!=1, disproving direct finiteness in this positive-characteristic setting. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.",
      "title": "OpenAI: A claimed torsion-free group-algebra counterexample to direct finiteness"
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