Lean statement

A universal IID factor for the free uniform spanning forest

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.Problem336.fusf_is_factor_iid

Lean statement file: FreeUniformSpanningForest.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 231 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem fusf_is_factor_iid :    ∃ Phi : FactorRule, RuleBorel Phi ∧ RuleEquivariant Phi ∧ ∀ G : GraphCode, WorksOnGraph Phi G := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.Problem336.GraphCode line 12

abbrev GraphCode :=  {a : ℕ → ℕ → Bool //    (∀ u, a u u = false) ∧ (∀ u v, a u v = a v u)}

abbreviation OAI.Problem336.EdgeCode line 16

abbrev EdgeCode := {p : ℕ × ℕ // p.1 < p.2}

abbreviation OAI.Problem336.EdgeConfiguration line 18
```
abbrev EdgeConfiguration := EdgeCode → Bool
```

abbreviation OAI.Problem336.FactorRule line 20

abbrev FactorRule := GraphCode → (ℕ → ℝ) → EdgeCode → Bool

abbreviation OAI.Problem336.FiniteSubgraphCode line 22

abbrev FiniteSubgraphCode := Finset ℕ × Finset (ℕ × ℕ)

definition OAI.Problem336.Adj line 24

def Adj (G : GraphCode) (u v : ℕ) : Prop := G.1 u v = true

definition OAI.Problem336.GraphConnected line 26

def GraphConnected (G : GraphCode) : Prop :=  ∀ u v, Relation.ReflTransGen (fun x y => Adj G x y) u v

definition OAI.Problem336.GraphLocallyFinite line 29

def GraphLocallyFinite (G : GraphCode) : Prop :=  ∀ u, Set.Finite {v : ℕ | Adj G u v}

definition OAI.Problem336.mkEdge line 32

def mkEdge (u v : ℕ) (h : u ≠ v) : EdgeCode := by  by_cases huv : u < v  · exact ⟨(u, v), huv⟩  · exact ⟨(v, u), lt_of_le_of_ne (Nat.le_of_not_gt huv) (Ne.symm h)⟩

definition OAI.Problem336.rawPair line 37

def rawPair (u v : ℕ) : ℕ × ℕ :=  if u < v then (u, v) else (v, u)

definition OAI.Problem336.relabelGraph line 40

def relabelGraph (σ : Equiv ℕ ℕ) (G : GraphCode) : GraphCode :=  ⟨fun u v => G.1 (σ.symm u) (σ.symm v), by    constructor    · intro u      exact G.2.1 (σ.symm u)    · intro u v      exact G.2.2 (σ.symm u) (σ.symm v)⟩

definition OAI.Problem336.relabelLabels line 48

def relabelLabels (σ : Equiv ℕ ℕ) (U : ℕ → ℝ) : ℕ → ℝ :=  fun u => U (σ.symm u)

definition OAI.Problem336.mapEdge line 51

def mapEdge (σ : Equiv ℕ ℕ) (e : EdgeCode) : EdgeCode :=  mkEdge (σ e.1.1) (σ e.1.2) (by    intro h    apply Nat.ne_of_lt e.2    exact σ.injective h)

definition OAI.Problem336.RuleBorel line 57

def RuleBorel (Phi : FactorRule) : Prop :=  Measurable    (fun q : GraphCode × ((ℕ → ℝ) × EdgeCode) =>      Phi q.1 q.2.1 q.2.2)

definition OAI.Problem336.RuleEquivariant line 62

def RuleEquivariant (Phi : FactorRule) : Prop :=  ∀ (σ : Equiv ℕ ℕ) (G : GraphCode) (U : ℕ → ℝ) (e : EdgeCode),    Phi (relabelGraph σ G) (relabelLabels σ U) (mapEdge σ e) =      Phi G U e

definition OAI.Problem336.TreeAdj line 67

def TreeAdj (T : Finset (ℕ × ℕ)) (u v : ℕ) : Prop :=  rawPair u v ∈ T

definition OAI.Problem336.SubgraphAdj line 70

def SubgraphAdj (K : FiniteSubgraphCode) (u v : ℕ) : Prop :=  u ∈ K.1 ∧ v ∈ K.1 ∧ rawPair u v ∈ K.2

definition OAI.Problem336.TreeOn line 73

def TreeOn (K : FiniteSubgraphCode)    (T : Finset (ℕ × ℕ)) : Prop :=  T ⊆ K.2 ∧  T.card + 1 = K.1.card ∧  ∀ u ∈ K.1, ∀ v ∈ K.1,    Relation.ReflTransGen (TreeAdj T) u v

definition OAI.Problem336.spanningTrees line 80

def spanningTrees (K : FiniteSubgraphCode) :    Finset (Finset (ℕ × ℕ)) := by  classical  exact K.2.powerset.filter (TreeOn K)

definition OAI.Problem336.ustCylinder line 85

def ustCylinder (K : FiniteSubgraphCode)    (A B : Finset EdgeCode) : ℝ≥0∞ := by  classical  let all := spanningTrees K  let good := all.filter (fun T =>    (∀ e ∈ A, e.1 ∈ T) ∧ (∀ e ∈ B, e.1 ∉ T))  exact (good.card : ℝ≥0∞) / (all.card : ℝ≥0∞)

definition OAI.Problem336.IsFiniteConnectedSubgraph line 93

def IsFiniteConnectedSubgraph    (G : GraphCode) (K : FiniteSubgraphCode) : Prop :=  K.1.Nonempty ∧  (∀ p ∈ K.2,    p.1 ∈ K.1 ∧ p.2 ∈ K.1 ∧ p.1 < p.2 ∧ Adj G p.1 p.2) ∧  ∀ u ∈ K.1, ∀ v ∈ K.1,    Relation.ReflTransGen (SubgraphAdj K) u v

definition OAI.Problem336.IsConnectedExhaustion line 101

def IsConnectedExhaustion    (G : GraphCode) (H : ℕ → FiniteSubgraphCode) : Prop :=  (∀ n, IsFiniteConnectedSubgraph G (H n)) ∧  (∀ n, (H n).1 ⊆ (H (n + 1)).1 ∧    (H n).2 ⊆ (H (n + 1)).2) ∧  (∀ u v, u < v → Adj G u v →    ∃ n, (u, v) ∈ (H n).2)

definition OAI.Problem336.uniform01 line 109

def uniform01 : Measure ℝ :=  volume.restrict (Set.Icc (0 : ℝ) 1)

definition OAI.Problem336.IsVertexIID line 112

def IsVertexIID {Ω : Type} [MeasurableSpace Ω]    (P : Measure Ω) (U : ℕ → Ω → ℝ) : Prop :=  (∀ i, Measurable (U i)) ∧  ∀ (s : Finset ℕ) (A : ℕ → Set ℝ),    (∀ i ∈ s, MeasurableSet (A i)) →    P {ω | ∀ i ∈ s, U i ω ∈ A i} =      Finset.prod s (fun i => uniform01 (A i))

2 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: vertex, equivariant, edge, fusf, connected, locally, rule, phi and five more. Found in the paper’s title, which its main theorem states: spanning, iid and factor. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unweighted, infinite, one, root, simple and independent.

Theorem 1.1 of The free uniform spanning forest is a factor of IID

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 54

Show the TeX

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 $\FUSF_G$.

Other possible paper matches, closest wording first

Proposition 2.2 of The free uniform spanning forest is a factor of IID. Words of the Lean statement, its names split into words, that also appear in the theorem: exhaustion, tree, subgraphs, fusf, connected, locally and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: infinite, simple, free, forests and independent.
Corollary 1.2 of The free uniform spanning forest is a factor of IID. Words of the Lean statement, its names split into words, that also appear in the theorem: works, rule, iid, spanning, factor, graph and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: free and forest.
Theorem 1.3 of The free uniform spanning forest is a factor of IID. Words of the Lean statement, its names split into words, that also appear in the theorem: map, phi, borel, uniform and law. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: independent.
Proposition 4.2 of The free uniform spanning forest is a factor of IID. Words of the Lean statement, its names split into words, that also appear in the theorem: relabeling, map, finite, borel and law. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one and independent.
Proposition 2.4 of The free uniform spanning forest is a factor of IID. Words of the Lean statement, its names split into words, that also appear in the theorem: real, borel and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: root.
Corollary 1.4 of The free uniform spanning forest is a factor of IID. Words of the Lean statement, its names split into words, that also appear in the theorem: iid, factor and law.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

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 (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 231’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

149 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 59 to its own declarations, 46 to Mathlib’s documentation. 27 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves fusf_is_factor_iid exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 26 other declarations of the file; with the theorem itself they span 106 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.Problem336 are listed without it.

abbrev GraphCodeline 12in the statementlibrary
abbrev EdgeCodeline 16library
abbrev EdgeConfigurationline 18library
abbrev FactorRuleline 20in the statementlibrary
abbrev FiniteSubgraphCodeline 22library
def Adjline 24library
def GraphConnectedline 26library
def GraphLocallyFiniteline 29library
def mkEdgeline 32library
def rawPairline 37library
def relabelGraphline 40library
def relabelLabelsline 48library
def mapEdgeline 51library
def RuleBorelline 57in the statementlibrary
def RuleEquivariantline 62in the statementlibrary
def TreeAdjline 67library
def SubgraphAdjline 70library
def TreeOnline 73library
def spanningTreesline 80library
def ustCylinderline 85library
def IsFiniteConnectedSubgraphline 93library
def IsConnectedExhaustionline 101library
def uniform01line 109library
def IsVertexIIDline 112library
def HasFUSFLawline 120library
def WorksOnGraphline 133in the statementlibrary
theorem fusf_is_factor_iidline 141selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheoryopen scoped BigOperators ENNReal Topologynamespace Problem336abbrev GraphCode :=  {a : ℕ → ℕ → Bool //    (∀ u, a u u = false) ∧ (∀ u v, a u v = a v u)}abbrev EdgeCode := {p : ℕ × ℕ // p.1 < p.2}abbrev EdgeConfiguration := EdgeCode → Boolabbrev FactorRule := GraphCode → (ℕ → ℝ) → EdgeCode → Boolabbrev FiniteSubgraphCode := Finset ℕ × Finset (ℕ × ℕ)def Adj (G : GraphCode) (u v : ℕ) : Prop := G.1 u v = truedef GraphConnected (G : GraphCode) : Prop :=  ∀ u v, Relation.ReflTransGen (fun x y => Adj G x y) u vdef GraphLocallyFinite (G : GraphCode) : Prop :=  ∀ u, Set.Finite {v : ℕ | Adj G u v}def mkEdge (u v : ℕ) (h : u ≠ v) : EdgeCode := by  by_cases huv : u < v  · exact ⟨(u, v), huv⟩  · exact ⟨(v, u), lt_of_le_of_ne (Nat.le_of_not_gt huv) (Ne.symm h)⟩def rawPair (u v : ℕ) : ℕ × ℕ :=  if u < v then (u, v) else (v, u)def relabelGraph (σ : Equiv ℕ ℕ) (G : GraphCode) : GraphCode :=  ⟨fun u v => G.1 (σ.symm u) (σ.symm v), by    constructor    · intro u      exact G.2.1 (σ.symm u)    · intro u v      exact G.2.2 (σ.symm u) (σ.symm v)⟩def relabelLabels (σ : Equiv ℕ ℕ) (U : ℕ → ℝ) : ℕ → ℝ :=  fun u => U (σ.symm u)def mapEdge (σ : Equiv ℕ ℕ) (e : EdgeCode) : EdgeCode :=  mkEdge (σ e.1.1) (σ e.1.2) (by    intro h    apply Nat.ne_of_lt e.2    exact σ.injective h)def RuleBorel (Phi : FactorRule) : Prop :=  Measurable    (fun q : GraphCode × ((ℕ → ℝ) × EdgeCode) =>      Phi q.1 q.2.1 q.2.2)def RuleEquivariant (Phi : FactorRule) : Prop :=  ∀ (σ : Equiv ℕ ℕ) (G : GraphCode) (U : ℕ → ℝ) (e : EdgeCode),    Phi (relabelGraph σ G) (relabelLabels σ U) (mapEdge σ e) =      Phi G U edef TreeAdj (T : Finset (ℕ × ℕ)) (u v : ℕ) : Prop :=  rawPair u v ∈ Tdef SubgraphAdj (K : FiniteSubgraphCode) (u v : ℕ) : Prop :=  u ∈ K.1 ∧ v ∈ K.1 ∧ rawPair u v ∈ K.2def TreeOn (K : FiniteSubgraphCode)    (T : Finset (ℕ × ℕ)) : Prop :=  T ⊆ K.2 ∧  T.card + 1 = K.1.card ∧  ∀ u ∈ K.1, ∀ v ∈ K.1,    Relation.ReflTransGen (TreeAdj T) u vdef spanningTrees (K : FiniteSubgraphCode) :    Finset (Finset (ℕ × ℕ)) := by  classical  exact K.2.powerset.filter (TreeOn K)def ustCylinder (K : FiniteSubgraphCode)    (A B : Finset EdgeCode) : ℝ≥0∞ := by  classical  let all := spanningTrees K  let good := all.filter (fun T =>    (∀ e ∈ A, e.1 ∈ T) ∧ (∀ e ∈ B, e.1 ∉ T))  exact (good.card : ℝ≥0∞) / (all.card : ℝ≥0∞)def IsFiniteConnectedSubgraph    (G : GraphCode) (K : FiniteSubgraphCode) : Prop :=  K.1.Nonempty ∧  (∀ p ∈ K.2,    p.1 ∈ K.1 ∧ p.2 ∈ K.1 ∧ p.1 < p.2 ∧ Adj G p.1 p.2) ∧  ∀ u ∈ K.1, ∀ v ∈ K.1,    Relation.ReflTransGen (SubgraphAdj K) u vdef IsConnectedExhaustion    (G : GraphCode) (H : ℕ → FiniteSubgraphCode) : Prop :=  (∀ n, IsFiniteConnectedSubgraph G (H n)) ∧  (∀ n, (H n).1 ⊆ (H (n + 1)).1 ∧    (H n).2 ⊆ (H (n + 1)).2) ∧  (∀ u v, u < v → Adj G u v →    ∃ n, (u, v) ∈ (H n).2)def uniform01 : Measure ℝ :=  volume.restrict (Set.Icc (0 : ℝ) 1)def IsVertexIID {Ω : Type} [MeasurableSpace Ω]    (P : Measure Ω) (U : ℕ → Ω → ℝ) : Prop :=  (∀ i, Measurable (U i)) ∧  ∀ (s : Finset ℕ) (A : ℕ → Set ℝ),    (∀ i ∈ s, MeasurableSet (A i)) →    P {ω | ∀ i ∈ s, U i ω ∈ A i} =      Finset.prod s (fun i => uniform01 (A i))def HasFUSFLaw {Ω : Type} [MeasurableSpace Ω]    (G : GraphCode) (P : Measure Ω)    (X : Ω → EdgeConfiguration) : Prop :=  (∀ e, Measurable (fun ω => X ω e)) ∧  ∀ (H : ℕ → FiniteSubgraphCode), IsConnectedExhaustion G H →    ∀ (A B : Finset EdgeCode),      Filter.Tendsto        (fun n => ustCylinder (H n) A B)        Filter.atTop        (nhds (P {ω |          (∀ e ∈ A, X ω e = true) ∧          (∀ e ∈ B, X ω e = false)}))def WorksOnGraph (Phi : FactorRule) (G : GraphCode) : Prop :=  GraphConnected G → GraphLocallyFinite G →  ∀ (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω)      (U : ℕ → Ω → ℝ),    IsVertexIID P U →    HasFUSFLaw G P      (fun ω e => Phi G (fun v => U v ω) e)theorem fusf_is_factor_iid :    ∃ Phi : FactorRule, RuleBorel Phi ∧ RuleEquivariant Phi ∧ ∀ G : GraphCode, WorksOnGraph Phi G := by  sorryend Problem336endend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

The free uniform spanning forest is a factor of IID

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 54

Show the TeX

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 $\FUSF_G$.

Corollary

The free uniform spanning forest is a graph factor of IID under every unimodular law supported on the stated class of graphs. The same rule works for every such law.

Section 1, Introduction Read it in the paper TeX source, line 69

Show the TeX

The free uniform spanning forest is a graph factor of IID under every
unimodular law supported on the stated class of graphs. The same rule works
for every such law.

Theorem (Strongly Rayleigh processes on countable groups)

Let \Gamma be a countable group and let \nu be a strongly Rayleigh probability law on \{0,1\}^{\Gamma} invariant under every \tau_g. There is a Borel map

\Phi_\nu:[0,1]^{\Gamma}\longrightarrow\{0,1\}^{\Gamma}

such that \Phi_\nu(\tau_g u)=\tau_g\Phi_\nu(u) for every g\in\Gamma and every u\in[0,1]^{\Gamma}, and such that \Phi_\nu(U) has law \nu when the coordinates of U are independent \operatorname{Uniform}[0,1] variables.

Section 1, Introduction Read it in the paper TeX source, line 102

Show the TeX

Let $\Gamma$ be a countable group and let $\nu$ be a strongly Rayleigh
probability law on $\{0,1\}^{\Gamma}$ invariant under every $\tau_g$.
There is a Borel map
\[
 \Phi_\nu:[0,1]^{\Gamma}\longrightarrow\{0,1\}^{\Gamma}
\]
such that $\Phi_\nu(\tau_g u)=\tau_g\Phi_\nu(u)$ for every $g\in\Gamma$
and every $u\in[0,1]^{\Gamma}$, and such that $\Phi_\nu(U)$ has law $\nu$
when the coordinates of $U$ are independent $\operatorname{Uniform}[0,1]$
variables.

Corollary (Determinantal processes on countable groups)

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.

Section 1, Introduction Read it in the paper TeX source, line 124

Show the TeX

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 $\Gamma$.

Configuration and file facts

File: lean/ComparatorChallenges/FreeUniformSpanningForest.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 231 link it.
Solution module: OAI.Probability.SpanningForest.Factor
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 26 local vocabulary declarations
Lines of Lean, without comments or blank lines: 106
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas