What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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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People and systems
OpenAI (organizational manuscript byline); individual human roles not fully disclosed The recorded author list is incomplete.
OpenAI 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 attribution; provider attestation is unavailable.
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English catalogue narrative
OpenAI: A claimed irrationality exponent of two for pi
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.
Scientific scope
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.
Reading status
This 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.
People and systems
OpenAI (organizational manuscript byline); individual human roles not fully disclosed
OpenAI 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).