Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with \(p_c\lt 1\): at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ3. (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 Critical bond and site percolation on the cubic lattice
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
For nearest-neighbor Bernoulli bond percolation on \mathbb Z^3, there is almost surely no infinite open cluster at the bond critical parameter. The same assertion holds for nearest-neighbor Bernoulli site percolation at the site critical parameter.
In No percolation at criticality on quasi-transitive graphs
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Let G be an infinite connected locally finite quasi-transitive undirected graph. If \pc(G)<1, then
Equivalently, \PP_{\pc(G)}(|C_x|=\infty)=0 for every vertex x.
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 closer match to the description’s wording first. The atlas has not reviewed either choice.
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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.
CriticalZ3The paper states
For nearest-neighbor Bernoulli bond percolation on \mathbb Z^3, there is almost surely no infinite open cluster at the bond critical parameter. The same assertion holds for nearest-neighbor Bernoulli site percolation at the site critical parameter.
The words they share: nearest, neighbor, site, integer, parameter, critical, bernoulli, bond, cluster, infinite, surely, almost, percolation.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CriticalZ3.lean, the release lists it as a main result.
theorem critical_no_infinite : (∀ᵐ ω ∂bondLaw bondCritical, ∀ x : Vertex, (bondCluster ω x).Finite) ∧ (∀ᵐ ω ∂siteLaw siteCritical, ∀ x : Vertex, (siteCluster ω x).Finite) := 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
CriticalPercolationThe paper states
For nearest-neighbor Bernoulli bond percolation on \mathbb Z^3, there is almost surely no infinite open cluster at the bond critical parameter. The same assertion holds for nearest-neighbor Bernoulli site percolation at the site critical parameter.
The words they share: neighbor, parameter, critical, bernoulli, bond, cluster, percolation, infinite.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CriticalPercolation.lean, the release lists it as a supporting statement.
theorem no_percolation_at_criticality [Infinite V] (hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive) (hpc : G.criticalProbability < 1) : G.law G.criticalProbability G.percolates = 0 := 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.
LocallyFinite Model.lean, line 24QuasiTransitive Model.lean, line 33NearestNeighbor Model.lean, line 16BondOpen Model.lean, line 19SiteOpen Model.lean, line 23BondConnected Model.lean, line 26SiteConnected Model.lean, line 29MathVet 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: no_percolation_at_criticality [Infinite V] (hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive) (hpc : G.criticalProbability < 1) : G.law G.criticalProbability G.percolates = 0 on BondGraph V E (multigraphs, law p = setBernoulli univ p, criticalProbability = sInf {p | 0 < P_p(∃ infinite cluster)}), plus CriticalZ3.critical_no_infinite: ∀ᵐ ω ∂bondLaw bondCritical, ∀ x, (bondCluster ω x).Finite and the same for site percolation, with bondCritical = sInf {p | 0 < P_p(origin cluster infinite)}. Both summary claims are stated.
“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 critical-percolation question asks whether an infinite cluster can remain at the threshold. The formalized result proves that, at their respective critical probabilities, nearest-neighbor bond and site percolation on $\mathbb Z^3$ almost surely have no infinite cluster. Equivalently, almost surely every vertex belongs to a finite cluster in each model.
The formalization proves absence of an infinite cluster at criticality for Bernoulli bond percolation on every infinite connected locally finite quasi-transitive graph with critical probability $p_c<1$. At $p=p_c$, the probability that any infinite cluster exists is zero. The graph model permits bond multiplicities.
No infinite critical clusters on $\mathbb Z^3$ CriticalZ3
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: no infinite cluster at criticality for nearest-neighbour bond and site percolation on Z³.
No infinite cluster at the critical probability CriticalPercolation
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the quasi-transitive paper: no infinite cluster at p_c for bond percolation on every infinite connected locally finite quasi-transitive graph with p_c < 1.
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.
CriticalZ3Three 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 92 s; peak memory 6.3 GiB). | 11 October 2026 | Original report |
CriticalPercolationTwo 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 |
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
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
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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Offer to write an explanationCompare one Lean statement with its paper: CriticalZ3, CriticalPercolationSuggest 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 proof and rates its significance, if it holds up, as “Would win a Fields Medal”. 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 Critical bond and site percolation on the cubic lattice
Why this passage is shown
Theorem 1.1
In No percolation at criticality on quasi-transitive graphs
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, CriticalZ3
Proof-verification report
MathVet, CriticalZ3
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, CriticalZ3
Proof-verification report
cert-machine, CriticalPercolation
Proof-verification report
MathVet, CriticalPercolation
Lean statement
CriticalPercolation
Lean statement
CriticalZ3