Proves that Catalan’s constant \(G=\sum_{j\ge0}(-1)^j/(2j+1)^2\) is irrational. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
One problem has been reported by others; the atlas has not judged it.
The paper states
In Catalan’s constant is irrational, 24 September 2026
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Catalan’s constant is irrational.
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CatalanThe paper states
In Catalan’s constant is irrational, 24 September 2026
Catalan’s constant is irrational.
The words they share: irrational, catalans, constant.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
Catalan.lean, the release lists it as a supporting statement.
theorem catalan_irrational : Irrational (∑' j : ℕ, (-1 : ℝ) ^ j / ((2 * j + 1 : ℕ) : ℝ) ^ 2) := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
One independent group reports that Comparator accepted a run for this Lean statement file: MathVet. Who ran the proof-verification program
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note: None; statement is the headline verbatim with Mathlib Irrational/tsum.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The formalization proves that Catalan’s constant $G=\sum_{j=0}^{\infty}(-1)^j/(2j+1)^2$ is irrational. This is the mathematical assertion in the paper’s title.
Irrationality of Catalan’s constant Catalan
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s main theorem, the irrationality of Catalan’s constant.
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CatalanTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “REFUSED” | Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)). | 10 October 2026 | Original report |
| MathVet | “comparator-pass” | Re-run through Comparator: accepted. | 10 October 2026 | Original report |
Other people’s accounts of running proof-verification software, with links to their reports.
Chokmah LLC, chokmah-me/fragile-proof-audit: replay audits including receipts for families 237 and 005, 9 October 2026. Source
Same receipt for the Catalan irrationality family.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Catalan’s constant is irrational, 24 September 2026
Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.
Piotr Hajlasz, OpenAI Mathematics Preprints: Searchable Catalogue, 9 October 2026. Source
A reader of the catalogue thread notes that OpenAI’s formalisation catalogue does not list the Catalan paper although a formalisation exists; at the pinned commit the YAML file indeed omits it while a Comparator challenge and a scope page are present.
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Hexagon, Hexagon: a repository for research in the mathematical sciences and theoretical computer science, 10 October 2026. Source
hexagon:2610.00187v1 (Sean Litsey, produced by Claude Opus 5.5) bounds the irrationality exponent of Catalan’s constant by 61,300, building on OpenAI’s proof and extending its Lean formalisation.
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the irrationality of Catalan’s constant.
Frank Calegari, A Brief Digest of Some New OpenAI Results, 8 October 2026. Source
The Catalan’s constant paper makes unmotivated optimisations and cites two crank papers; SLMath will hold an irrationality conference in January.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.1
In Catalan’s constant is irrational, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, Catalan
Proof-verification report
MathVet, Catalan
Report using another proof or calculation method
Catalan’s constant is irrational, 24 September 2026
Exact calculations for finite claims, cert-machine
9814e4cLean statement
Catalan