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.
Theorems selected for verification:OAI.CubicFirstMoment.patterson_firstMomentOAI.CubicFirstMoment.patterson_angularComparisonOAI.CubicFirstMoment.patterson_angularCancellation
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.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: first, moment, primary and primes.
Theorem 1.1 of An unconditional first moment for cubic Gauss sums
As $X\longrightarrow\infty$,
\begin{equation}
\sum_{\substack{N\pi\le X\\\pi\equiv1\pmod3\ \mathrm{prime}}}G(\pi)
=\frac65c_*\frac{X^{5/6}}{\log X}
+o\!\left(\frac{X^{5/6}}{\log X}\right).
\end{equation}
The sum is over all primary primes in $\mathcal O$.
Other possible paper matches, closest wording first
Theorem A.1 of An unconditional first moment for cubic Gauss sums. Words of the Lean statement, its names split into words, that also appear in the theorem: nonzero, mode, integer, pi, comparison, theta and five more.
Theorem 2.1 of An unconditional first moment for cubic Gauss sums. Words of the Lean statement, its names split into words, that also appear in the theorem: 2/3, cubic, cutoffs and primary.
Theorem 2.4 of An unconditional first moment for cubic Gauss sums. Words of the Lean statement, its names split into words, that also appear in the theorem: star, unit, 2/3, theta, pi, gamma and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: term.
Proposition A.5 of An unconditional first moment for cubic Gauss sums. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, moments and angular.
Proposition 6.1 of An unconditional first moment for cubic Gauss sums. Words of the Lean statement, its names split into words, that also appear in the theorem: scales, cutoff, sqrt, norm, log, moments and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: terms.
Proposition 4.1 of An unconditional first moment for cubic Gauss sums. Words of the Lean statement, its names split into words, that also appear in the theorem: norm, log, moments, primary and prime.
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
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; Neal McBurnett’s openai-math-lean-tooling. 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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 488 s).
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 023’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.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: None; custom Gauss-sum/cubic-symbol definitions match the paper's displayed definition (checked in paper.tex).
Definitions the group asks readers to compare with the paper: Gauss sum, cubic residue symbol and primary-prime predicate are all custom (cubicSymbolAtPrime uses Classical.choose and a fallback value 0; residueRepresentative := Classical.choose ...; attribute [local instance] Classical.propDecidable), but they agree with the paper's definition and the summand is independent of the representative
OpenAI’s own note on what this Lean statement covers
“The formalization proves the sharp-cutoff asymptotic with main term $\frac65c_*X^{5/6}/\log X$, where $c_*=(2\pi)^{2/3}/(3\Gamma(2/3))$. It also proves the corresponding angular comparison for every integer Fourier mode and cancellation of order $o(X^{5/6}/\log X)$ for each fixed nonzero mode. All primary Eisenstein primes are included, without an extra conjectural premise.”
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
99 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: 58 to its own declarations, 35 to Mathlib’s documentation. 24 declared names also link 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.
As $X\longrightarrow\infty$,
\begin{equation}
\sum_{\substack{N\pi\le X\\\pi\equiv1\pmod3\ \mathrm{prime}}}G(\pi)
=\frac65c_*\frac{X^{5/6}}{\log X}
+o\!\left(\frac{X^{5/6}}{\log X}\right).
\end{equation}
The sum is over all primary primes in $\mathcal O$.
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.