Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Irrationality of Catalan's constant
Theorem selected for verification:OAI.InternalCatalan.catalan_irrational
Problem reported 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. Piotr Hajlasz, OpenAI Mathematics Preprints: Searchable Catalogue, 2026-10-09Source
The Lean statement
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: irrational and catalans. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.
Other possible paper matches, closest wording first
Proposition 5.3 of Catalan’s constant is irrational. Words of the Lean statement, its names split into words, that also appear in the theorem: real.
Proposition 2.1 of Catalan’s constant is irrational. Words of the Lean statement, its names split into words, that also appear in the theorem: integer.
Proposition 2.2 of Catalan’s constant is irrational. Words of the Lean statement, its names split into words, that also appear in the theorem: integer.
Proposition 3.4 of Catalan’s constant is irrational. Words of the Lean statement, its names split into words, that also appear in the theorem: integers.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
One independent group reports that Comparator accepted a run for this Lean statement file: MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…
No verdict reachedTheir term: “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.)).
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 005’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
OpenAI’s own note on what this Lean statement 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.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
13 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 0 to its own declarations, 1 to Mathlib’s documentation. 1 declared name also links to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.