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.
Theorem selected for verification:OAI.RyserCoveringCounterexample.exists_prime_eventually_counterexampleRank
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremexists_prime_eventually_counterexampleRank : ∃ p : ℕ, p.Prime ∧ 5 < p ∧ p % 3 = 2 ∧ ∃ N : ℕ, ∀ n : ℕ, N ≤ n → n.Prime → 2 < n → CounterexampleRank (p ^ n + 1) := bysorry
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.
defCounterexampleRank (r : ℕ) : Prop := ∃ (V : Type) (fin : Fintype V) (dec : DecidableEq V),letI : Fintype V := finletI : DecidableEq V := dec ∃ H : Hypergraph V,0 < r ∧ H.Nonempty ∧ RPartiteUniform r H ∧ Intersecting H ∧ (∃ C : Finset V, IsCover H C) ∧ matchingNumber H = 1 ∧ coverNumber H = r
No confident match with a paper statement
No paper theorem is paired with this Lean statement.
Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements in Balanced counterexamples to Ryser’s conjecture at prime orders come first because the release’s name for this Lean statement shares words with that paper’s title.
Other possible paper matches, closest wording first
Theorem 1.1 of Balanced counterexamples to Ryser’s conjecture at prime orders. Words of the Lean statement, its names split into words, that also appear in the theorem: partite, hypergraph, intersecting, finite, uniform and prime. Found in the paper’s title, which its main theorem states: counterexamples.
Theorem 1.1 of A counterexample to Ryser’s covering conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: eq, partite, hypergraph, intersecting, integer, finite and two more. Found in the paper’s title, which its main theorem states: counterexample and covering. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: pmod, equiv, ryser and odd.
Proposition 2.1 of Balanced counterexamples to Ryser’s conjecture at prime orders. Words of the Lean statement, its names split into words, that also appear in the theorem: part, eq, number, partite, hypergraph, intersecting and three more.
Corollary 1.2 of A counterexample to Ryser’s covering conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: number, integer, finite, cover and prime. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: pmod, equiv and odd.
Corollary 1.2 of Balanced counterexamples to Ryser’s conjecture at prime orders. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, cover and prime. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: large and sufficiently.
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; 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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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 162’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.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: BalancedRyser.main_result : ∃ q₀, ∀ q ≥ q₀, q.Prime → ∃ H : PartiteHypergraph (q+1) (q+1), Intersecting ∧ Nonisolated ∧ HasCoverNumber (q+1) is the first claim verbatim (edges are functions Fin r → Fin n, so r-partite r-uniform by construction). RyserCovering/RyserOddExtensions add intersecting r-partite r-uniform hypergraphs with ν = 1, τ = r for ranks r = p^n + 1, but no challenge mentions edge-colorings of K_n or monochromatic tree covers, so the Gyárfás disproof (strictly stronger than a Ryser counterexample, since Ryser implies Gyárfás) is not stated.
Definitions the group asks readers to compare with the paper: matchings/matchingNumber/coverNumber in RyserCovering and RyserOddExtensions are custom Finset encodings (attribute [-instance] instDecidablePairwiseCoeFinsetOfDecidableEqOfDecidableRel removes a decidability instance to avoid a diamond); read as standard ν and τ ; CounterexampleRank r := ∃ (V : Type) (fin : Fintype V) (dec : DecidableEq V), ... quantifies the Fintype/DecidableEq instances as data (harmless)
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: Disproof of Gyárfás's monochromatic tree-cover conjecture (an r-edge-coloured complete graph needing r monochromatic trees) is claimed but not stated; only the hypergraph counterexamples it is derived from are stated.
OpenAI’s own note on what this Lean statement covers
“The ranks have the form $r=p^n+1$: one fixed prime $p\equiv2\pmod3$ works for all sufficiently large prime degrees $n$, and a second result covers every sufficiently large such prime $p$ and every sufficiently large odd $n$, with the degree threshold allowed to depend on $p$. Infinitely many ranks occur. Equal part sizes and numerical thresholds are not asserted.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a weaker form of a paper’s theorem”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
59 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: 20 to its own declarations, 18 to Mathlib’s documentation. 11 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
There exists $q_0$ such that, for every prime $q\ge q_0$, there is a
finite intersecting $(q+1)$-partite $(q+1)$-uniform hypergraph $H$ with
\[
|V_1|=\cdots=|V_{q+1}|=q+1,
\qquad \tau(H)=q+1.
\]
Every vertex belongs to an edge.
Corollary
For every sufficiently large prime q, there is a finite complete graph with a (q+1)-edge-coloring whose vertices require exactly q+1 monochromatic trees to cover. The same minimum holds for connected monochromatic subgraphs.
For every sufficiently large prime $q$, there is a finite complete
graph with a $(q+1)$-edge-coloring whose vertices require exactly
$q+1$ monochromatic trees to cover. The same minimum holds for
connected monochromatic subgraphs.
There exists s_0 such that, for every prime s\ge s_0 with s\equiv2\pmod3, there is an integer n_0(s)\ge3 with the following property. For every odd integer n\ge n_0(s), put q=s^n. There is a finite intersecting (q+1)-partite (q+1)-uniform hypergraph H satisfying
\tau(H)=q+1.
In particular, the intersecting case of (eq:ryser) is false.
There exists $s_0$ such that, for every prime $s\ge s_0$ with
$s\equiv2\pmod3$, there is an integer $n_0(s)\ge3$ with the following
property. For every odd integer $n\ge n_0(s)$, put $q=s^n$.
There is a finite intersecting $(q+1)$-partite $(q+1)$-uniform
hypergraph $H$ satisfying
\[
\tau(H)=q+1.
\]
In particular, the intersecting case of~\eqref{eq:ryser} is false.
Corollary
Let s_0 and n_0(s) be as in Theorem thm:main. For every prime s\ge s_0 with s\equiv2\pmod3 and every odd integer n\ge n_0(s), put r=s^n+1. There is a finite simple complete graph with an r-edge-coloring for which the minimum number of monochromatic trees needed to cover its vertices is exactly r. The same minimum holds with connected monochromatic subgraphs or maximal monochromatic components in place of trees.
Let $s_0$ and $n_0(s)$ be as in Theorem~\ref{thm:main}. For every prime
$s\ge s_0$ with $s\equiv2\pmod3$ and every odd integer $n\ge n_0(s)$,
put $r=s^n+1$. There is a finite simple complete graph with an
$r$-edge-coloring for which the minimum number of monochromatic trees
needed to cover its vertices is exactly $r$. The same minimum holds with
connected monochromatic subgraphs or maximal monochromatic components in
place of trees.
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
10 local vocabulary declarations
Lines of Lean, without comments or blank lines
29
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Fixed-prime Ryser counterexamples
Family 162: Counterexamples to Ryser’s covering conjecture
No paper statement selected
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
The Lean statement says
theoremexists_prime_eventually_counterexampleRank : ∃ p : ℕ, p.Prime ∧ 5 < p ∧ p % 3 = 2 ∧ ∃ N : ℕ, ∀ n : ℕ, N ≤ n → n.Prime → 2 < n → CounterexampleRank (p ^ n + 1) := bysorry
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.