The free uniform spanning forest is a factor of IID
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID. (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 the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
There is one Borel, root-independent, equivariant rule
\Phi(G,U,e)\in\{0,1\}
such that, for every infinite connected locally finite simple undirected unweighted graph G and independent \operatorname{Uniform}[0,1] vertex labels U=(U_v)_{v\in V(G)}, the entire edge set
\{e\in E(G):\Phi(G,U,e)=1\}
has law \FUSF_G.
Three more possible matches
Corollary 1.4(Determinantal processes on countable groups)in The free uniform spanning forest is a factor of IID, 25 September 2026.
Corollary 1.2in The free uniform spanning forest is a factor of IID, 25 September 2026.
Theorem 1.3(Strongly Rayleigh processes on countable groups)in The free uniform spanning forest is a factor of IID, 25 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 that the free uniform spanning forest is a factor of IID on every infinite connected locally finite simple unweighted graph. One Borel rule works for all such graphs and uses no root, answering affirmatively the general factor question for unimodular random graphs. We also show that every translation-invariant strongly Rayleigh process indexed by a countable group is a factor of IID, including invariant determinantal processes with Hermitian positive-contraction kernels. This group-action conclusion requires no amenability.
Two Lean statements in this family
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There is one Borel, root-independent, equivariant rule
\Phi(G,U,e)\in\{0,1\}
such that, for every infinite connected locally finite simple undirected unweighted graph G and independent \operatorname{Uniform}[0,1] vertex labels U=(U_v)_{v\in V(G)}, the entire edge set
\{e\in E(G):\Phi(G,U,e)=1\}
has law \FUSF_G.
The words they share: vertex, equivariant, edge, fusf, connected, locally, rule, phi, finite, borel, graph, uniform, law, spanning, iid, factor, unweighted, infinite, one, root, simple, independent.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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 \Gamma be a countable group and let K be a self-adjoint positive contraction on \ell^2(\Gamma), so 0\le K\le I. If the determinantal probability law with kernel K is invariant under the regular left action, then it is a factor of IID under that action. In particular, this holds whenever K commutes with the left regular representation of \Gamma.
The words they share: kernel, determinantal, positive, contraction, groups, invariant, iid, probability, factor, countable, law, processes.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
StronglyRayleighDPP.lean, the release lists it as a supporting statement.
theoremfinite_strongly_rayleigh_response : ∀ (F : Type u_1) [Fintype F] [Nonempty F] (μ : ProbabilityMeasure (F → Bool)), StronglyRayleighFinite μ → (∀ (i j : F) (h : F → ℝ), HasDerivAt (fun r : ℝ => tiltMean μ (coordinatePerturb h j r) i) (tiltCovariance μ h i j) 0) ∧ (∀ (i : F) (h : F → ℝ), (∑ j : F, |tiltCovariance μ h i j|) ≤2 * tiltMean μ h i * (1 - tiltMean μ h i) ∧2 * tiltMean μ h i * (1 - tiltMean μ h i) ≤ (1 : ℝ) / 2) ∧ (∀ (i : F) (h h' : F → ℝ) (D : ℝ),0 ≤ D → (∀ j : F, |h j - h' j| ≤ D) → |tiltMean μ h i - tiltMean μ h' i| ≤ D / 2) := 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.
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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:fusf_is_factor_iid : ∃ Phi : FactorRule, RuleBorel Phi ∧ RuleEquivariant Phi ∧ ∀ G : GraphCode, WorksOnGraph Phi G (one rule for all connected locally finite simple graphs on ℕ, equivariant under all relabelings, no root; HasFUSFLaw = cylinder probabilities are limits of uniform-spanning-tree cylinder ratios along every connected exhaustion) and StronglyRayleighDPP (strongly_rayleigh_group_factor, invariant_positive_contraction_dpp_factor, translation_invariant_kernel_dpp_factor, DPP existence/uniqueness for positive-contraction kernels on countable sets) cover both summary claims. Note: external all in tmp/import_cones.json is a parser artifact of import all Mathlib... in JointProcessLaw.lean; the cone is Mathlib-only.
“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 that the free uniform spanning forest is a factor of independent identically distributed vertex labels on every infinite connected locally finite simple unweighted graph. One Borel equivariant rule works for all such graphs and uses no distinguished root.
The linked supplements also give equivariant IID sampling for invariant strongly Rayleigh laws on countable groups and existence, uniqueness, and IID-factor results for determinantal laws with Hermitian positive-contraction kernels on countable index sets. These group results require neither amenability nor finite generation, and the kernels need not be trace class.
AI-drafted comparisons, with no recorded human review
A universal IID factor for the free uniform spanning forestFreeUniformSpanningForest
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: one Borel root-independent equivariant rule makes the free uniform spanning forest a factor of IID on every infinite connected locally finite graph.
Strongly Rayleigh and determinantal-process IID factorsStronglyRayleighDPP
Unreviewed AI draft: states a supporting result. States IID factors for invariant strongly Rayleigh laws on countable groups and the determinantal results, which the paper states as a theorem and corollary in its introduction and the note calls supplements to the spanning-forest theorem (FreeUniformSpanningForest).
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A universal IID factor for the free uniform spanning forestFreeUniformSpanningForest
Two reports, all at the release’s recorded version.
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Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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Family 231The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
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Why this passage is shown
Theorem 1.1
In The free uniform spanning forest is a factor of IID, 25 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
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…
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 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.