Disproves Sidorenko’s conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness. (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 A counterexample to Sidorenko’s conjecture, 23 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Let H be the incidence graph of the 22 triples on 13 points in Table 1. This bipartite graph has 35 vertices and 66 edges. There is a finite simple undirected graph G with at least one edge such that
In A counterexample to Sidorenko’s conjecture, 23 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
For the same connected bipartite graph H of Theorem 1.1, there are p\in(0,1) and a nonconstant graphon W such that
Consequently, H is not p-forcing at this density and hence is not forcing, despite being bipartite and containing a cycle.
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.
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Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
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.
SidorenkoCounterexampleThe paper states
In A counterexample to Sidorenko’s conjecture, 23 September 2026
Let H be the incidence graph of the 22 triples on 13 points in Table 1. This bipartite graph has 35 vertices and 66 edges. There is a finite simple undirected graph G with at least one edge such that
The words they share: edge, least, simple, points, incidence, one, finite, graph, sidorenkos, edges, vertices, bipartite.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SidorenkoCounterexample.lean, the release lists it as a supporting statement.
theorem main : ∃ size : ℕ, 0 < size ∧ ∃ host : SimpleGraph (Fin size), (∃ left right, host.Adj left right) ∧ homDensity H host < (edgeDensity host) ^ 66 := 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
Neither 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: main : ∃ size > 0, ∃ host : SimpleGraph (Fin size), (host has an edge) ∧ homDensity H host < (edgeDensity host)^66 for a fixed explicit H = incidence graph of 22 triples on 13 points (checked by script: 35 vertices, 66 edges, connected, bipartite by construction), with homomorphism counts including non-injective maps and edgeDensity = 2e/n² = t(K₂,G): a faithful Sidorenko counterexample. The second named claim (the same pattern disproves the Skokan-Thoma forcing conjecture) has no statement (docs: ‘outside this statement’); connectivity and the counts 35/66 are properties of the defined H, not stated.
Definitions the group asks readers to compare with the paper: homDensity F G := homCount F G / (Nat.card W)^(Nat.card V) with homCount := Nat.card {hom : V → W // ∀ l r, F.Adj l r → G.Adj (hom l) (hom r)} and edgeDensity G := 2 * Nat.card G.edgeSet / (Nat.card W)^2: custom but the standard t(H,G) and t(K₂,G) (checked)
“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 — DECLARED”).
The gap they name: The forcing-conjecture sentence.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says 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
Sidorenko’s conjecture predicts that every bipartite graph $H$ has homomorphism density at least the host graph’s edge density raised to $|E(H)|$. The formalization disproves this for the paper’s fixed bipartite graph with $35$ vertices and $66$ edges: it constructs a nonempty finite simple host graph with $t(H,G)<t(K_2,G)^{66}$. The separate forcing-conjecture consequence in the paper is outside this statement.
Counterexample to Sidorenko’s conjecture SidorenkoCounterexample
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for the paper’s bipartite graph with 35 vertices and 66 edges, a finite host graph with t(H, G) < t(K₂, G)^66. The amplification and forcing corollaries are outside the statement.
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.
SidorenkoCounterexampleThree 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 349 s; peak memory 6.5 GiB). | 11 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.
A counterexample to Sidorenko’s 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.
The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source
JSP-000034 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.
acprk, Remarks on proofs in OpenAI’s mathematics repository, 10 October 2026. Source
A Sidorenko counterexample with 17 vertices and 30 edges (conditional) in place of the manuscript’s 35 and 66.
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
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
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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
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the counterexample to Sidorenko’s conjecture in its finite form and marks the graphon form refuted without a proof link.
Pico, openai-math-counterexamples: do the counterexamples in openai/math hold when someone else computes them?, 9 October 2026. Source
The 35-vertex 66-edge graph H is explicit but the host graph G is not given, so the counterexample cannot be written down as a finite check.
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 A counterexample to Sidorenko’s conjecture, 23 September 2026
Why this passage is shown
Corollary 1.3 (Failure of the forcing conjecture)
In A counterexample to Sidorenko’s conjecture, 23 September 2026
Why this passage is shown
Corollary 1.2 (Amplification and injective supersaturation)
In A counterexample to Sidorenko’s conjecture, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, SidorenkoCounterexample
Proof-verification report
MathVet, SidorenkoCounterexample
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, SidorenkoCounterexample
Report using another proof or calculation method
A counterexample to Sidorenko’s conjecture, 23 September 2026
Exact calculations for finite claims, cert-machine
9814e4cLean statement
SidorenkoCounterexample