Disproves Ryser’s covering conjecture by constructing intersecting \((q+1)\)-partite, \((q+1)\)-uniform hypergraphs with covering number \(q+1\), rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás’s monochromatic tree-cover conjecture. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
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
Every vertex belongs to an edge.
No Lean statement in the release is paired with this theorem.
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
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.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.
Choose one to see it beside the paper theorem it is paired with.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
RyserCoveringThe paper states
No paper theorem is paired with this Lean statement.
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements it shares words with:
The Lean statement
No paper theorem is paired with it.
RyserCovering.lean, the release lists it as a main result.
theorem exists_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) := 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
BalancedRyserThe paper states
No paper theorem is paired with this Lean statement.
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements it shares words with:
The Lean statement
No paper theorem is paired with it.
BalancedRyser.lean, the release lists it as a supporting statement.
def MainResult : Prop := ∃ q₀ : ℕ, ∀ q : ℕ, q₀ ≤ q → q.Prime → BalancedCounterexample q theorem main_result : MainResult := 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
RyserOddExtensionsThe paper states
In A counterexample to Ryser’s covering conjecture, 23 September 2026
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
In particular, the intersecting case of (1) is false.
The words they share: eq, ryser, odd, partite, hypergraph, intersecting, integer, finite, uniform, prime, counterexample, covering, pmod, equiv.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
RyserOddExtensions.lean, the release lists it as a main result.
theorem eventualOddFailures_and_infinite : (∃ primeThreshold : ℕ, ∀ prime : ℕ, primeThreshold ≤ prime → prime.Prime → prime % 3 = 2 → ∃ degreeThreshold : ℕ, 3 ≤ degreeThreshold ∧ ∀ degree : ℕ, degreeThreshold ≤ degree → Odd degree → Specification.ExplicitFailureAt (prime ^ degree + 1)) ∧ Set.Infinite {rank : ℕ | Specification.ExplicitFailureAt rank} := 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Covers Model.lean, line 17Intersecting Model.lean, line 20Nonisolated Model.lean, line 23HasCoverNumber Model.lean, line 26BalancedCounterexample Model.lean, line 32MainResult Model.lean, line 36Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 part of the advertised result (their term: “partial”).
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)
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
The gap they name: 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.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
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
Ryser’s covering conjecture predicts that an intersecting $r$-partite hypergraph has a vertex cover of size at most $r-1$. The formalization proves that every sufficiently large prime $q$ has a finite intersecting $(q+1)$-partite, $(q+1)$-uniform hypergraph with covering number $q+1$ and exactly $q+1$ nonisolated vertices in each part. Thus the conjecture fails even with equal part sizes. The prime threshold is existential.
Ryser’s covering conjecture predicts $\tau\le(r-1)\nu$ for an $r$-partite hypergraph, where $\tau$ and $\nu$ are its covering and matching numbers. The formalized constructions give finite intersecting $r$-partite $r$-uniform hypergraphs with $\nu=1$ and $\tau=r$, contradicting the bound. 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.
Fixed-prime Ryser counterexamples RyserCovering
Unreviewed AI draft: states a weaker form of a paper’s theorem. States counterexamples of rank p^n + 1 for one fixed prime p ≡ 2 (mod 3) and all large prime degrees n, whereas the main theorem covers every large such prime and every large odd degree; the latter is RyserOddExtensions.
Balanced Ryser counterexamples at prime orders BalancedRyser
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the balanced paper: for every large prime q, an intersecting (q + 1)-partite (q + 1)-uniform hypergraph with equal parts and covering number q + 1.
Counterexamples in odd extension degrees RyserOddExtensions
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for every sufficiently large prime p ≡ 2 (mod 3) and every sufficiently large odd n, an intersecting (p^n + 1)-partite hypergraph with ν = 1 and τ = r, giving infinitely many ranks.
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
RyserCoveringTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
BalancedRyserTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
RyserOddExtensionsTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
Other people’s accounts of running proof-verification software, with links to their reports.
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.
Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
A counterexample to Ryser’s covering conjecture, 23 September 2026
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.
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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Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Offer to write an explanationCompare one Lean statement with its paper: BalancedRyser, RyserCovering, RyserOddExtensionsSuggest a source attributionSuggest a change
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 counterexample 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
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 Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Corollary 1.2
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Corollary 1.2
In A counterexample to Ryser’s covering conjecture, 23 September 2026
Why this passage is shown
Theorem 1.1
In A counterexample to Ryser’s covering conjecture, 23 September 2026
Why this passage is shown
Theorem 1.1
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Theorem 1.1
In A counterexample to Ryser’s covering conjecture, 23 September 2026
Why this passage is shown
Proposition 2.1 (Deterministic reduction)
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Corollary 1.2
In A counterexample to Ryser’s covering conjecture, 23 September 2026
Why this passage is shown
Corollary 1.2
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Proposition 5.1 (Selection with bounded cylinder probabilities)
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Proposition 7.1
In A counterexample to Ryser’s covering conjecture, 23 September 2026
Why this passage is shown
Corollary 3.4 (Concentration of core covers)
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Theorem 1.1
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Proposition 2.1 (Deterministic reduction)
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Theorem 1.1
In A counterexample to Ryser’s covering conjecture, 23 September 2026
Why this passage is shown
Corollary 1.2
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Proposition 5.1 (Selection with bounded cylinder probabilities)
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Why this passage is shown
Corollary 1.2
In A counterexample to Ryser’s covering conjecture, 23 September 2026
Why this passage is shown
Proposition 7.1
In A counterexample to Ryser’s covering conjecture, 23 September 2026
Why this passage is shown
Proposition 4.1
In Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, RyserCovering
Proof-verification report
MathVet, RyserCovering
Proof-verification report
cert-machine, BalancedRyser
Proof-verification report
MathVet, BalancedRyser
Proof-verification report
cert-machine, RyserOddExtensions
Proof-verification report
MathVet, RyserOddExtensions
Report using another proof or calculation method
Balanced counterexamples to Ryser’s conjecture at prime orders, 27 September 2026
Exact calculations for finite claims, cert-machine
9814e4cReport using another proof or calculation method
A counterexample to Ryser’s covering conjecture, 23 September 2026
Exact calculations for finite claims, cert-machine
9814e4cLean statement
BalancedRyser
Lean statement
RyserCovering
Lean statement
RyserOddExtensions