{"schema":"postcutoff/event@1","as_of":"2026-10-20T07:28:00+02:00","url":"https://postcutoff.com/e/2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt/","md":"https://postcutoff.com/e/2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt","date":"2026-10-19","date_precision":"day","short_title":"Gregory Chudnovsky, with ChatGPT, claims the lemniscate period has irrationality exponent 2, with a Lean 4 proof built on OpenAI's π-exponent library","deck":null,"takeaway":"After OpenAI's claim that π has irrationality exponent 2, a well-known transcendence theorist carries the method to the lemniscate period Ω = 2∫₀¹(1−t⁴)^(−1/2) dt. He says ChatGPT helped with the exposition and the formalization, and he ships a complete Lean 4 proof built on OpenAI's library. Like the π claim, it has not been reviewed independently.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.26539: The irrationality exponent of the lemniscate period (Chudnovsky)","url":"https://arxiv.org/abs/2610.26539","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"openai/math (π irrationality-exponent family 017 and Lean library)","url":"https://github.com/openai/math","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI"],"title":"Gregory Chudnovsky, with ChatGPT, claims the lemniscate period has irrationality exponent 2, with a Lean 4 proof built on OpenAI's π-exponent library","summary":"arXiv 2610.26539 (19 Oct 2026), \"The irrationality exponent of the lemniscate period: A determinant proof with a Lean 4 formalization\" by Gregory V. Chudnovsky, proves that for every ν > 2 and all large q, |Ω − p/q| > q^(−ν), so Ω = 2∫₀¹(1−t⁴)^(−1/2) dt has irrationality exponent 2. The proof uses elliptic separated-weight interpolation on the curve y² = x³ − x and a determinant bound on a complex-multiplication grid, an elliptic analogue of the method in OpenAI's π-exponent manuscript. The title-page footnote says the paper was \"Developed in collaboration with OpenAI ChatGPT, including mathematical exposition and formalization assistance\". The ancillary Lean 4 sources are said to prove the endpoint \"without additional hypotheses\" and use a vendored copy of OpenAI's library (namespace OAI…PiExponent). Unrefereed.","key_facts":["Paper: arXiv 2610.26539 [math.NT], Gregory V. Chudnovsky, dated 10 Oct 2026, submitted 19 Oct; 33 pages; comment: 'Lean 4 proof sources and reproduction instructions included as ancillary files'","Theorem (abstract): 'for every real ν > 2, there is Q ≥ 2 such that q^(−ν) < |Ω − p/q| for every integer p and every integer q ≥ Q. Consequently Ω is irrational and its irrationality exponent is two.'","Method: 'elliptic separated-weight interpolation, arithmetic denominator estimates, and an analytic determinant bound on a square complex-multiplication grid', using E: y² = x³ − x","AI disclosure, verbatim (title-page footnote): 'Developed in collaboration with OpenAI ChatGPT, including mathematical exposition and formalization assistance. The accompanying Lean source files contain the formal proof.' Reference [5]: 'G. V. Chudnovsky, with OpenAI ChatGPT assistance, Lemniscate-period formal proof snapshot'","Lean: 'The accompanying Lean 4 development proves the endpoint without additional hypotheses'; Appendix A says no additional axiom, sorryAx or unfinished proof is present, only propext, Classical.choice and Quot.sound. 'The generic graded-jet, cohomological, and collision-counting infrastructure is supplied by the vendored OAI library [4]' (openai/math snapshot fd4aeeb); arXiv ancillary files sit under anc/lean/OAI/NumberTheory/PiExponent/…","Status: the formal statement was written by the author with ChatGPT's help, and the shared infrastructure comes from OpenAI's unrefereed π-exponent development; no independent review found by 20 Oct"],"key_numbers":[],"tags":["math","number-theory","diophantine-approximation","irrationality-measure","transcendence","elliptic-curves","chatgpt","lean","openai-math-release"],"science":{"field":"mathematics","subfield":"Diophantine approximation / transcendence","problem":"Irrationality exponent of the lemniscate period Ω = Γ(1/4)²/(2√(2π))","result":"Claimed proof that the irrationality exponent of Ω equals 2, with a complete Lean 4 development.","open_since":"","ai_system":["ChatGPT"],"human_role":"Human-led with AI: Chudnovsky with ChatGPT for exposition and formalization, adapting OpenAI's AI-generated π method","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed arXiv preprint; complete Lean 4 proof in ancillary files, statement written by the author, built on OpenAI's unreviewed library"},"status":"pending","shock":"","short":"Irrationality exponent of the lemniscate period"},"body_md":"## What happened\n\nOpenAI's 6 October release claimed that π has irrationality exponent 2, with a Lean proof. Two follow-ups already carried the\nmethod over to logarithms of rationals. Gregory Chudnovsky, a long-standing figure in transcendence theory, now gives an\nelliptic-curve version for the lemniscate period. He worked with ChatGPT and built his Lean development on a vendored copy of\nOpenAI's π-exponent library.\n\n## Why it matters\n\nBefore 2026, exponent exactly 2 was proved for very few classical constants. A Lean proof with a short, readable statement,\nfrom a well-known specialist, is the strongest signal yet that OpenAI's interpolation-determinant method works and carries over.\nIt is still not an independent check of OpenAI's shared infrastructure.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint; complete Lean 4 proof in ancillary files, statement written by the author, built on OpenAI's unreviewed library","sources":[]}],"related":[{"id":"2026-10-20-ai-assisted-results-october-20-listing","url":"https://postcutoff.com/e/2026-10-20-ai-assisted-results-october-20-listing/","date":"2026-10-20","date_precision":"day","short_title":"AI-assisted results in the 20 Oct 2026 arXiv listing","deck":"Poset dimension proofs found by GPT-6 Astra, Hitchin domination from ChatGPT 5.6 Sol's ideas, the section conjecture for generic curves, Yau's geometric conjecture, Erdős–Gyárfás r = 5, a Claude-built Gray-code reduction, and a Lean harness solving 6/6 IMO 2026","takeaway":"Tuesday's arXiv announcement was the strongest maths batch yet. Seven results have their own entries, among them the Erdős–Szemerédi sunflower conjecture, Erdős–Hajnal for all paths and a non-sofic hyperbolic group. This page catalogues the next tier: about forty papers whose AI disclosures run from 'the proofs were found by ChatGPT-6 Astra' to proofreading.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI","Anthropic","DeepSeek","AWS","various mathematicians"]},{"id":"2026-10-06-irrationality-exponent-pi-openai","url":"https://postcutoff.com/e/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":9,"official":8,"filed":"2026-10-12","updated":"2026-10-13","orgs":["OpenAI"]},{"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":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-20","type":"filed","text":"Created (PDF read in full text; arXiv ancillary listing checked)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-20","run":null,"sources_read":"PDF read in full text; arXiv ancillary listing checked","updated":"2026-10-20","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":172,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":111,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":202,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":141,"in_training_data":false}],"short_url":null}