{"schema":"postcutoff/event@1","as_of":"2026-10-12T07:22:00+02:00","url":"https://postcutoff.com/e/2026-10-06-irrationality-exponent-pi-openai/","md":"https://postcutoff.com/e/2026-10-06-irrationality-exponent-pi-openai/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-06-irrationality-exponent-pi-openai","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges","deck":null,"takeaway":"Since Mahler's 1953 bound of 42, mathematicians had lowered the irrationality exponent of π only to about 7.1. OpenAI's release claims the conjectured value 2 in a 23-page paper with a Lean proof. Six days on, no expert has publicly assessed it; an outsider has already adapted the Lean proof to log 2.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Paper: The irrationality exponent of π is 2 (openai/math, 24 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"openai/math: CONTENTS.md, family 017","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Lean scope note for family 017","url":"https://github.com/openai/math/blob/main/lean/docs/017.md","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"Comparator challenge: PiExponent.lean","url":"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/PiExponent.lean","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"openai/math README (procedure and exceptions)","url":"https://github.com/openai/math/blob/main/README.md","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"openai/math history.md (withdrawals and fixes, 7 Oct 2026)","url":"https://github.com/openai/math/blob/main/history.md","type":"code","group":"primary","domain":"github.com"},{"n":7,"title":"'Dan Brown': The irrationality measure of the logarithm of a rational number is 2 (Lean)","url":"https://jdb19937.github.io/log-irrationality-measure/","type":"code","group":"primary","domain":"jdb19937.github.io"},{"n":8,"title":"Hacker News: The logarithms of rational numbers have irrationality exponent 2","url":"https://news.ycombinator.com/item?id=50027884","type":"discussion","group":"reactions","domain":"news.ycombinator.com"}],"official":7,"filed":"2026-10-12","updated":"2026-10-12","orgs":["OpenAI"],"title":"OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges","summary":"Family 017 of OpenAI's 6 Oct 2026 math release is \"The irrationality exponent of π is 2\" (manuscript dated 24 Sept 2026, 23 pages, author \"OpenAI\"). It claims that for every ν > 2 and all large enough q, |π − p/q| ≥ q^(−ν) for every integer p, so π is approximated by fractions no better than a typical number. The best previous bounds were 7.1032 (Zeilberger–Zudilin, 2019) and 7.1019 (Bai, Sept 2026). A corollary is that the Flint–Hills series Σ 1/(n³ sin² n) converges. OpenAI's Lean library formalizes the exponent statement against Mathlib's Real.pi. No public expert assessment was found by 12 Oct 2026.","key_facts":["Catalogue text (openai/math CONTENTS.md, family 017): 'Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational p/q satisfies |π − p/q| ≥ q^(−2−ε). This also proves convergence of the Flint–Hills series Σ 1/(n³ sin² n), with angles in radians.'","Paper: 'The irrationality exponent of π is 2', author 'OpenAI', dated 24 Sept 2026, 23 pages. Theorem 1.1 is ineffective: 'The threshold may depend on ν, and the argument does not give it effectively.' The paper stresses that exponent 2 does not mean bounded continued-fraction partial quotients","History given in the paper: Mahler (1953) exponent 42; Mignotte 20; Hata 8.0160…; Salikhov 7.6063…; Zeilberger–Zudilin 7.103205334137…; Bai (Sept 2026) 7.101862832357. 'These quantitative bounds are context, not inputs to our proof.'","Method (paper's outline): 'a multivariable interpolation theorem to construct a nonzero determinant', then two incompatible estimates for its absolute value; related to Laurent's interpolation determinants and to Roth's many-variable method, with an interpolation statement 'adapted to logarithmic curves'","Flint–Hills: Alekseyev showed convergence requires μ(π) ≤ 5/2 and Meiburg that μ(π) < 5/2 suffices; Theorem 1.1 supplies this. More generally Σ 1/(n^a |sin n|^b) converges if and only if a > max{1, b} (Corollary 5.2)","Earlier claims (paper): a convergence claim by Mantzakouras and López Zapata 'was withdrawn in September 2026'; Carella's claim of exponent two with bounded partial quotients 'contains a sign obstruction'. Neither is used","Lean: the scope note says 'The formalization proves that the irrationality exponent of π is exactly two'; the Comparator challenge PiExponent.lean states it for Mathlib's Real.pi, including the supremum characterization. 'The paper's convergence consequence for the Flint–Hills series is outside this selected statement.'","Status in OpenAI's own log: history.md (7 Oct 2026) lists three withdrawn manuscripts and 14 repaired ones; family 017 is in neither list","Follow-up by an outsider (9 Oct): a pseudonymous 'Dan Brown' (GitHub jdb19937) published 'The logarithms of rational numbers have irrationality exponent 2', claiming μ(log α) = 2 for every positive rational α ≠ 1 (so μ(log 2) = 2), with a Lean proof described as 'a module-by-module modification of OpenAI's formalization of μ(π) = 2'. Hacker News, 22 points; the author's comment: 'This is a Lean-verified variation of the OpenAI proof for pi.' Not peer-reviewed; the author's identity is unconfirmed","Not found as of 12 Oct 2026 (web and Hacker News search; X not searched): public assessments by specialists such as Wadim Zudilin, Frank Calegari or Terence Tao, or an independent re-run of this Lean proof"],"key_numbers":[],"tags":["math","number-theory","diophantine-approximation","pi","irrationality-measure","flint-hills","lean","openai","unverified"],"science":{"field":"mathematics","subfield":"Diophantine approximation","problem":"Irrationality exponent (measure) of π, conjectured to be 2; convergence of the Flint–Hills series","result":"Claimed: μ(π) = 2 (ineffective); hence Σ 1/(n³ sin² n) converges, and Σ 1/(n^a |sin n|^b) converges iff a > max{1, b}.","open_since":"1953","ai_system":["OpenAI internal model (unreleased)"],"human_role":"Autonomous per OpenAI's README: 'The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model' (about three hours of ChatGPT Pro thinking compute per result on average); family 017 is not among the listed exceptions; no human author is named","verification":{"key":"lean","label":"Lean-verified","text":"Lean formalization by OpenAI of the exponent statement (Comparator challenge published); not independently re-checked or expert-reviewed as far as found"},"status":"pending","shock":"Seventy years of work had moved the bound from 42 to 7.1; the claim jumps to the optimal value 2 in one step.","short":"Irrationality exponent of π"},"body_md":"## What happened\n\nThe irrationality exponent of a number measures how closely fractions can approximate it. Almost every real number has exponent 2,\nand π was expected to, but proofs were far from it: the record stood at about 7.10 after decades of refinements of integral\nconstructions. Among the 722 manuscripts OpenAI published on 6 Oct 2026, family 017 claims the exact value 2 by a different route, an\ninterpolation-determinant argument in many variables. The result is ineffective, meaning it gives no explicit point after which\nthe inequality holds.\n\nThe claim settles a question that became popular outside research mathematics: whether the Flint–Hills series converges. That\ndepends on how often sin n is very small, which is the same as how well π is approximated by fractions.\n\nOpenAI's Lean library contains a proof of the exponent statement, written against Mathlib's definition of π. Three days after the\nrelease, a pseudonymous programmer published a modification of that Lean proof which, he says, gives exponent 2 for the logarithm\nof every rational number, including log 2. If both hold, the technique is general.\n\n## Why it matters\n\nAmong the release's claims this is one of the easiest to state and one with a long, well-known history of small improvements,\nincluding an AI-assisted one in September 2026. The Lean statement is short and uses only standard Mathlib definitions, so a\nsuccessful independent kernel check would leave little room for doubt about the theorem itself. As of 12 Oct no such check and\nno specialist's assessment had been published, so the entry's status is pending.","disputed":[{"label":"Verification","text":"Lean formalization by OpenAI of the exponent statement (Comparator challenge published); not independently re-checked or expert-reviewed as far as found","sources":[]}],"related":[{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":74,"official":13,"filed":"2026-10-07","updated":"2026-10-12","orgs":["OpenAI"]},{"id":"2026-10-06-quasi-riemann-hypothesis-openai","url":"https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the quasi-Riemann hypothesis","deck":"No zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)","takeaway":"If correct, this is the largest advance on the zeros of ζ since the 19th century, and it would remove the Siegel-zero obstruction that runs through modern analytic number theory.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-07","updated":"2026-10-09","orgs":["OpenAI"]},{"id":"2026-09-25-zeta2-irrationality-measure-claude-lean","url":"https://postcutoff.com/e/2026-09-25-zeta2-irrationality-measure-claude-lean/","date":"2026-09-25","date_precision":"day","short_title":"New record bound on the irrationality measure of ζ(2), μ ≤ 5.0495243, released as a 92k-line Lean proof written by Claude; beaten a week later by a Claude-assisted paper","deck":null,"takeaway":"It is a new way to publish: a record in analytic number theory released as a fully kernel-checked, AI-written formal proof rather than a paper.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":5,"filed":"2026-10-05","updated":"2026-10-11","orgs":["Anthropic"]},{"id":"2026-09-10-pi-irrationality-measure-bai-gpt","url":"https://postcutoff.com/e/2026-09-10-pi-irrationality-measure-bai-gpt/","date":"2026-09-10","date_precision":"day","short_title":"Irrationality measure of π lowered to 7.101862832357, the first improvement on Zeilberger–Zudilin (2019), with GPT-5.6 Sol and GPT-6 Astra; four weeks later OpenAI claimed the exact value 2","deck":null,"takeaway":"The best known bound on how well π can be approximated by fractions moved for the first time since 2019, in a preprint whose author says OpenAI models helped with parameter search, proofs and computation. Claude-assisted papers then moved the ζ(3) and ζ(2) records the same way. OpenAI's 6 Oct release goes much further, claiming the conjectured exact value μ(π) = 2 with a Lean proof, which would make all these upper bounds obsolete.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":8,"official":7,"filed":"2026-10-11","updated":"2026-10-12","orgs":["OpenAI","Anthropic"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-12","type":"filed","text":"Created from the openai/math repository (paper, Lean scope note and challenge) and the Hacker News thread"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The openai/math repository (paper, Lean scope note and challenge) and the Hacker News thread","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-irrationality-exponent-pi"}