OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves \(p_c\lt p_u\) for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition. (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
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
Let \Lambda be a nonamenable finitely generated group, and let S be any finite symmetric generating set with the identity removed. For Bernoulli bond percolation on the simple undirected Cayley graph with edges \{x,xs\},x\in\Lambda,s\in S, the critical connection operator is bounded on counting-measure \ell^2(\Lambda) and
\pc<\ptwo\le\pu.
The simultaneous nonuniqueness interval in Theorem 1.1 exists for every such fixed generating set. At every fixed p\in(\pc,\pu), there are almost surely infinitely many infinite open clusters.
There exist \pc<p_1<p_2<1 such that, in the coupling obtained from independent uniform edge labels, almost surely there are infinitely many infinite open clusters simultaneously for every p\in[p_1,p_2].
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.
One more possible match
Corollary 2(Critical laws and exponential connectivity)in Nonuniqueness of percolation on nonamenable quasi-transitive graphs, 24 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft’s stronger operator-threshold conjecture, establishing \(p_c\lt p_{2\to2}\le p_u\).
Two Lean statements in this family
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.
Let \Lambda be a nonamenable finitely generated group, and let S be any finite symmetric generating set with the identity removed. For Bernoulli bond percolation on the simple undirected Cayley graph with edges \{x,xs\},x\in\Lambda,s\in S, the critical connection operator is bounded on counting-measure \ell^2(\Lambda) and
\pc<\ptwo\le\pu.
The simultaneous nonuniqueness interval in Theorem 1.1 exists for every such fixed generating set. At every fixed p\in(\pc,\pu), there are almost surely infinitely many infinite open clusters.
The words they share: almost, bernoulli, bond, cayley, clusters, connection, critical, finite, finitely, generated, generating, graphs, group, infinite, infinitely, interval, main, many, nonamenable, nonuniqueness, operator, percolation, simultaneous, surely, symmetric.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
BenjaminiSchramm.lean, the release lists it as a supporting statement.
theoremfull_main {V : Type u} {E : Type v} [Infinite V] (G : BondGraph V E) (hc : Connected G) (hl : LocallyFinite G) (hq : QuasiTransitive G) (hh : 0 < hV G) : MainConclusion G := bysorry
This 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.
Let \Lambda be a nonamenable finitely generated group, and let S be any finite symmetric generating set with the identity removed. For Bernoulli bond percolation on the simple undirected Cayley graph with edges \{x,xs\},x\in\Lambda,s\in S, the critical connection operator is bounded on counting-measure \ell^2(\Lambda) and
\pc<\ptwo\le\pu.
The simultaneous nonuniqueness interval in Theorem 1.1 exists for every such fixed generating set. At every fixed p\in(\pc,\pu), there are almost surely infinitely many infinite open clusters.
The words they share: group, measure, arbitrary, cayley, edges, interval, bernoulli, pu, operator, bond, graph, clusters, ptwo, infinite, bounded, pc, symmetric, generating, finitely, nonamenable, nonuniqueness, finite.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CayleyPercolation.lean, the release lists it as a supporting statement.
theoremarbitrary_cayley {Λ : Type u} [Group Λ] [Group.FG Λ] (hna : ¬ AmenableGroup Λ) (S : Finset Λ) (hsym : ∀ s ∈ S, s⁻¹ ∈ S) (hgen : Subgroup.closure (S : Set Λ) = ⊤) (h1 : 1 ∉ S) : OperatorBounded (cayleyGraph S) (pc (cayleyGraph S)) ∧ pc (cayleyGraph S) < ptwo (cayleyGraph S) ∧ ptwo (cayleyGraph S) ≤ pu (cayleyGraph S) ∧ (∃ p₁ p₂ : unitInterval, pc (cayleyGraph S) < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧ ∀ᵐ labels ∂(labelLaw : Measure (CayleyEdge S → unitInterval)), ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters (cayleyGraph S) (openEdges labels p)).Infinite) ∧ (∀ p : unitInterval, pc (cayleyGraph S) < p → p < pu (cayleyGraph S) → ∀ᵐ ω ∂law p, (infiniteClusters (cayleyGraph S) ω).Infinite) := bysorry
This 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
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
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:full_main (infinite, connected, locally finite, quasi-transitive multigraph with 0 < hV G) concludes MainConclusion G including pc G < ptwo G ∧ ptwo G ≤ pu G and a coupled interval [p₁,p₂] above pc with infinitely many infinite clusters a.s.; critical_laws gives the finite triangle diagram triangleDiagram G (pc G) x ≤ operatorNorm^3 ∧ < ⊤ plus mean-field laws; cayley_main/arbitrary_cayley handle every finite symmetric generating set of a non-Folner-amenable group for every p ∈ (pc, pu). Nonamenability is encoded as positive vertex-isoperimetric constant (graphs) or failure of the Folner condition (groups).
Definitions the group asks readers to compare with the paper:Custom ‘amenable’: def FolnerAmenable (Λ) : Prop := ∀ K : Finset Λ, ∀ ε > 0, ∃ A : Finset Λ, A.Nonempty ∧ ∀ k ∈ K, ((A.image (· * k)) \ A).card < ε * A.card (and the identical AmenableGroup in CayleyPercolation) is the standard right-Folner criterion, not Mathlib’s; for general graphs nonamenability is 0 < hV G with hV G := sInf {|∂A| / |A| : A finite nonempty} (outer vertex boundary). Standard but hand-written.
“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.
The formalization proves the nonuniqueness phase conjecture for Bernoulli bond percolation on infinite connected locally finite nonamenable quasi-transitive graphs. It establishes $p_c<p_{2\to2}\le p_u$ and a common nonempty coupled interval with infinitely many infinite clusters almost surely. For Cayley graphs, the result holds for every finite symmetric generating set of a nonamenable finitely generated group and every $p\in(p_c,p_u)$.
The selected critical estimates include a finite triangle diagram, susceptibility of order $(p_c-p)^{-1}$, percolation probability of order $p-p_c$, cluster-volume tail of order $n^{-1/2}$, and intrinsic and extrinsic radius tails of order $n^{-1}$. Connection probabilities decay exponentially below $p_{2\to2}$.
AI-drafted comparisons, with no recorded human review
Nonuniqueness phase and critical laws on quasi-transitive graphsBenjaminiSchramm
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (p_c < p_{2→2} ≤ p_u with infinitely many infinite clusters on a common interval) together with the Cayley-graph and critical-law corollaries.
Nonuniqueness for every nonamenable Cayley graphCayleyPercolation
Unreviewed AI draft: states a consequence of a paper’s theorem. States the paper’s corollary for Cayley graphs: nonuniqueness for every finite symmetric generating set of a nonamenable group and every p in (p_c, p_u).
Who ran the proof-verification program?
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.
Nonuniqueness phase and critical laws on quasi-transitive graphsBenjaminiSchramm
Two 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.
What these papers cite in the release, and what cites them
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.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite 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
Explain a result, compare statements or propose a correction
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.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
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 “Major math journal”. The page says that nothing on it is verified.
Tasmin Chu says the OpenAI pc < pu preprint builds on the approach their advisor Tom Hutchcroft laid out a year ago.
Questions awaiting review
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 214No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Address a question raised in an independent assessment
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Independent assessment
MathVet
Their term
“full”
What their term means
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
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; 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…
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 170 s).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.