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        "title": "OpenAI, The irrationality exponent of π is 2",
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        "description": "The denominator threshold may depend on nu and is ineffective. The statement does not imply a uniform c/q^2 bound or bounded continued-fraction entries. The Flint–Hills convergence consequence is outside the selected formal statement in the scope note. 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, The irrationality exponent of π is 2",
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      "statement": "# OpenAI: A claimed irrationality exponent of two for pi\n\nThe manuscript claims that for every exponent nu>2, all sufficiently large denominators q satisfy |pi-p/q|>=q^(-nu) for every integer numerator p. This specifies how exceptionally well fractions can approximate pi, beyond merely proving that pi is irrational. 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 denominator threshold may depend on nu and is ineffective. The statement does not imply a uniform c/q^2 bound or bounded continued-fraction entries. The Flint–Hills convergence consequence is outside the selected formal statement in the scope note. 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, The irrationality exponent of π is 2](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf)\n",
      "summary": "The manuscript claims that for every exponent nu>2, all sufficiently large denominators q satisfy |pi-p/q|>=q^(-nu) for every integer numerator p. This specifies how exceptionally well fractions can approximate pi, beyond merely proving that pi is irrational. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.",
      "title": "OpenAI: A claimed irrationality exponent of two for pi"
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