Resolves the Harary–Hill conjecture and Turán’s brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
In The crossing number of complete bipartite graphs, 23 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For all positive integers m,n,
In The crossing number of complete graphs, 23 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every integer n\ge3,
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
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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.
BipartiteCrossingThe paper states
In The crossing number of complete bipartite graphs, 23 September 2026
For all positive integers m,n,
The words they share: integers, bipartite, crossing, positive.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
BipartiteCrossing.lean, the release lists it as a main result.
def MainTarget : Prop := ∀ m n : ℕ, 0 < m → 0 < n → (∃ D : AdmissibleDrawing m n, crossingCount D = axisPairs m * axisPairs n) ∧ (∀ D : AdmissibleDrawing m n, axisPairs m * axisPairs n ≤ crossingCount D)theorem mainTarget_proof : MainTarget := 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
CompleteCrossingThe 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.
CompleteCrossing.lean, the release lists it as a main result.
theorem complete_graph_crossing_number (order : ℕ) (at_least_three : 3 ≤ order) : ordinaryCrossingNumber (completeGraph order) = hill order := 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. 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: complete_graph_crossing_number : 3 ≤ n → ordinaryCrossingNumber (completeGraph n) = hill n (Hill’s formula; the ℕ-division by 4 is exact because the product of the four floors is divisible by 4) and MainTarget : ∀ m n > 0, (∃ admissible drawing with Z(m)Z(n) crossings) ∧ (∀ admissible drawings, Z(m)Z(n) ≤ crossings) state both formulas for all sizes, for drawings by arbitrary continuous simple arcs with crossings counted as points. The minimum ranges only over the custom AdmissibleDrawing class (finitely many crossing points, each a topologically transversal crossing, no triple points), which is the usual crossing-number class.
Definitions the group asks readers to compare with the paper: structure AdmissibleDrawing... extends ContinuousDrawing with crossings_finite, crossings_proper : ProperCrossing... (local chart straightening the two arcs to the axes) and no_triples; ordinaryCrossingNumber := sInf (Set.range crossingCount) (sInf ∅ = 0, but a nonempty class is forced for n ≥ 5 since hill n > 0); the two files carry separate copies of the definitions (Zarankiewicz vs Paper170 namespaces)
“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 formalized result proves Hill’s proposed formula for the crossing number of the complete graph: for every $n\ge3$, $\mathrm{cr}(K_n)=\frac14\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor\lfloor(n-2)/2\rfloor\lfloor(n-3)/2\rfloor$. It proves the lower bound and constructs a matching two-page drawing. Crossings are spatial points, so repeated crossings of one edge pair count separately.
Zarankiewicz’s crossing-number conjecture predicts $\mathrm{cr}(K_{m,n})=\lfloor m/2\rfloor\lfloor(m-1)/2\rfloor\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor$. The formalized result proves this equality for every positive $m,n$ and constructs a drawing attaining it. Crossings are counted as spatial points for continuous simple edge paths, including repeated crossings between the same pair of edges.
Crossing number of complete bipartite graphs BipartiteCrossing
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: Zarankiewicz’s formula for the crossing number of K_{m,n} for all positive m, n, with a drawing attaining it.
Crossing number of complete graphs CompleteCrossing
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: Hill’s formula for the crossing number of K_n for every n ≥ 3, with the lower bound and a matching drawing.
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.
BipartiteCrossingTwo 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 |
CompleteCrossingThree 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 |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 64 s). | 10 October 2026 | Original report |
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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Commentary on this family by others, dated and credited. It is commentary, not a finding.
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Why this passage is shown
Theorem 1.1
In The crossing number of complete bipartite graphs, 23 September 2026
Why this passage is shown
Theorem 1.1
In The crossing number of complete graphs, 23 September 2026
Why this passage is shown
Theorem 1.1
In The crossing number of complete graphs, 23 September 2026
Why this passage is shown
Theorem 1.1
In The crossing number of complete bipartite graphs, 23 September 2026
Why this passage is shown
Proposition 4.3 (Odd lower bound)
In The crossing number of complete graphs, 23 September 2026
Why this passage is shown
Proposition 5.1
In The crossing number of complete graphs, 23 September 2026
Why this passage is shown
Corollary 2.3 (Rank sum for differences)
In The crossing number of complete bipartite graphs, 23 September 2026
Why this passage is shown
Corollary 4.4
In The crossing number of complete graphs, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, BipartiteCrossing
Proof-verification report
MathVet, BipartiteCrossing
Proof-verification report
cert-machine, CompleteCrossing
Proof-verification report
MathVet, CompleteCrossing
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, CompleteCrossing
Lean statement
BipartiteCrossing
Lean statement
CompleteCrossing