Gregory Chudnovsky, with ChatGPT, claims the lemniscate period has irrationality exponent 2, with a Lean 4 proof built on OpenAI’s π-exponent library
Awaiting review
The 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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- 3 of 5
- Last verified
- 20 October 2026
Your AI and this story
- GPT-6 Astra172 days after its cutoff
- Claude Opus 5.5111 days after its cutoff
- Gemini 4 Argon202 days after its cutoff
- Grok 4.7141 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 111 days before it.
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
What happened
OpenAI’s 6 October release claimed that π has irrationality exponent 2, with a Lean proof. Two follow-ups already carried the method over to logarithms of rationals. Gregory Chudnovsky, a long-standing figure in transcendence theory, now gives an elliptic-curve version for the lemniscate period. He worked with ChatGPT and built his Lean development on a vendored copy of OpenAI’s π-exponent library.
Why it matters
Before 2026, exponent exactly 2 was proved for very few classical constants. A Lean proof with a short, readable statement, from a well-known specialist, is the strongest signal yet that OpenAI’s interpolation-determinant method works and carries over. It is still not an independent check of OpenAI’s shared infrastructure.
What is disputed or not yet verified
| Verification | Unrefereed arXiv preprint; complete Lean 4 proof in ancillary files, statement written by the author, built on OpenAI’s unreviewed library |
|---|
Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Filed (PDF read in full text; arXiv ancillary listing checked)