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Lean statement

Strongly Rayleigh and determinantal-process IID factors

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.

Theorems selected for verification: OAI.Problem336AdditiveD139.finite_strongly_rayleigh_response OAI.Problem336AdditiveD139.strongly_rayleigh_group_factor OAI.Problem336AdditiveD139.finite_positive_contraction_dpp_is_strongly_rayleigh OAI.Problem336AdditiveD139.countable_positive_contraction_dpp_exists_unique OAI.Problem336AdditiveD139.invariant_positive_contraction_dpp_factor OAI.Problem336AdditiveD139.translation_invariant_kernel_dpp_factor

Lean statement file: StronglyRayleighDPP.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 theoremWho ran the proof-verification programReport 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 finite_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) := by  sorrytheorem strongly_rayleigh_group_factor :    ∀ (Γ : Type u_2) [Group Γ] [Countable Γ] [MeasurableSpace Γ],      ∀ μ : ProbabilityMeasure (Γ → Bool),        StronglyRayleighCountable μ → InvariantBinaryLaw μ → HasEquivariantIIDFactor μ := by  sorrytheorem finite_positive_contraction_dpp_is_strongly_rayleigh :    ∀ (F : Type u_3) [Fintype F] [Nonempty F] (K : F → F → ℂ),      PositiveContractionFinite K →        (∃ μ : ProbabilityMeasure (F → Bool), IsFiniteDeterminantalLaw μ K) ∧          ∀ μ : ProbabilityMeasure (F → Bool), IsFiniteDeterminantalLaw μ K →            StronglyRayleighFinite μ := by  sorrytheorem countable_positive_contraction_dpp_exists_unique :    ∀ (I : Type u_4) [Countable I] (K : I → I → ℂ),      PositiveContractionKernel K →        ∃! μ : ProbabilityMeasure (I → Bool), IsDeterminantalLaw μ K := by  sorrytheorem invariant_positive_contraction_dpp_factor :    ∀ (Γ : Type u_5) [Group Γ] [Countable Γ] [MeasurableSpace Γ]      (K : Γ → Γ → ℂ) (μ : ProbabilityMeasure (Γ → Bool)),      PositiveContractionKernel K → IsDeterminantalLaw μ K →        InvariantBinaryLaw μ → HasEquivariantIIDFactor μ := by  sorrytheorem translation_invariant_kernel_dpp_factor :    ∀ (Γ : Type u_6) [Group Γ] [Countable Γ] [MeasurableSpace Γ] (K : Γ → Γ → ℂ),      PositiveContractionKernel K → TranslationInvariantKernel K →        ∃ μ : ProbabilityMeasure (Γ → Bool),          IsDeterminantalLaw μ K ∧ InvariantBinaryLaw μ ∧ HasEquivariantIIDFactor μ := 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.Problem336AdditiveD139.UnitLabel line 15

    abbrev UnitLabel := Set.Icc (0 : ℝ) 1

  • definition OAI.Problem336AdditiveD139.leftTranslate line 17

    def leftTranslate {Γ : Type u_1} {α : Type u_2} [Group Γ] (g : Γ) (a : Γ → α) : Γ → α :=  fun h => a (g⁻¹ * h)

  • definition OAI.Problem336AdditiveD139.iidUnitLabels line 20

    def iidUnitLabels (Γ : Type u_3) [MeasurableSpace Γ] : Measure (Γ → UnitLabel) :=  Measure.infinitePi (fun _ : Γ => (volume : Measure UnitLabel))

  • definition OAI.Problem336AdditiveD139.InvariantBinaryLaw line 23

    def InvariantBinaryLaw {Γ : Type u_4} [Group Γ]    (μ : ProbabilityMeasure (Γ → Bool)) : Prop :=  ∀ g : Γ,    Measure.map (leftTranslate g) (μ : Measure (Γ → Bool)) =      (μ : Measure (Γ → Bool))

  • definition OAI.Problem336AdditiveD139.HasEquivariantIIDFactor line 29

    def HasEquivariantIIDFactor {Γ : Type u_5} [Group Γ] [MeasurableSpace Γ]    (μ : ProbabilityMeasure (Γ → Bool)) : Prop :=  ∃ Phi : (Γ → UnitLabel) → (Γ → Bool),    Measurable Phi ∧    (∀ (g : Γ) (u : Γ → UnitLabel),      Phi (leftTranslate g u) = leftTranslate g (Phi u)) ∧    Measure.map Phi (iidUnitLabels Γ) = (μ : Measure (Γ → Bool))

  • definition OAI.Problem336AdditiveD139.finiteGeneratingValue line 37

    def finiteGeneratingValue {F : Type u_6} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (z : F → ℂ) : ℂ := by  classical  exact ∑ x : F → Bool,    (((μ : Measure (F → Bool)) {x}).toReal : ℂ) *      ∏ i : F, if x i = true then z i else 1

  • definition OAI.Problem336AdditiveD139.StronglyRayleighFinite line 44

    def StronglyRayleighFinite {F : Type u_7} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) : Prop :=  ∀ z : F → ℂ, (∀ i : F, 0 < (z i).im) → finiteGeneratingValue μ z ≠ 0

  • definition OAI.Problem336AdditiveD139.marginalMass line 48

    def marginalMass {I : Type u_8} (μ : ProbabilityMeasure (I → Bool))    (S : Finset I) (x : S → Bool) : ℝ :=  ((μ : Measure (I → Bool)) {a | ∀ i : S, a i.1 = x i}).toReal

  • definition OAI.Problem336AdditiveD139.marginalGeneratingValue line 52

    def marginalGeneratingValue {I : Type u_9} (μ : ProbabilityMeasure (I → Bool))    (S : Finset I) (z : S → ℂ) : ℂ := by  classical  exact ∑ x : S → Bool,    (marginalMass μ S x : ℂ) *      ∏ i : S, if x i = true then z i else 1

  • definition OAI.Problem336AdditiveD139.StronglyRayleighCountable line 59

    def StronglyRayleighCountable {I : Type u_10}    (μ : ProbabilityMeasure (I → Bool)) : Prop :=  ∀ S : Finset I, ∀ z : S → ℂ,    (∀ i : S, 0 < (z i).im) → marginalGeneratingValue μ S z ≠ 0

  • definition OAI.Problem336AdditiveD139.boolReal line 64

    def boolReal (b : Bool) : ℝ := if b = true then 1 else 0

  • definition OAI.Problem336AdditiveD139.tiltPartition line 66

    def tiltPartition {F : Type u_11} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (h : F → ℝ) : ℝ := by  classical  exact ∑ x : F → Bool,    Real.exp (∑ j : F, h j * boolReal (x j)) *      ((μ : Measure (F → Bool)) {x}).toReal

  • definition OAI.Problem336AdditiveD139.tiltMean line 73

    def tiltMean {F : Type u_12} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (h : F → ℝ) (i : F) : ℝ := by  classical  exact (∑ x : F → Bool,      Real.exp (∑ j : F, h j * boolReal (x j)) *        ((μ : Measure (F → Bool)) {x}).toReal * boolReal (x i)) /    tiltPartition μ h

  • definition OAI.Problem336AdditiveD139.tiltSecondMoment line 81

    def tiltSecondMoment {F : Type u_13} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (h : F → ℝ) (i j : F) : ℝ := by  classical  exact (∑ x : F → Bool,      Real.exp (∑ k : F, h k * boolReal (x k)) *        ((μ : Measure (F → Bool)) {x}).toReal *        boolReal (x i) * boolReal (x j)) /    tiltPartition μ h

  • definition OAI.Problem336AdditiveD139.tiltCovariance line 90

    def tiltCovariance {F : Type u_14} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (h : F → ℝ) (i j : F) : ℝ :=  tiltSecondMoment μ h i j - tiltMean μ h i * tiltMean μ h j

  • definition OAI.Problem336AdditiveD139.coordinatePerturb line 94

    def coordinatePerturb {F : Type u_15} (h : F → ℝ) (j : F) (r : ℝ) : F → ℝ := by  classical  exact fun k => h k + if k = j then r else 0

  • definition OAI.Problem336AdditiveD139.PositiveContractionFinite line 98

    def PositiveContractionFinite {F : Type u_16} [Fintype F]    (K : F → F → ℂ) : Prop :=  (∀ i j, K i j = conj (K j i)) ∧  ∀ c : F → ℂ,    0 ≤ (∑ i : F, ∑ j : F, conj (c i) * K i j * c j).re ∧    (∑ i : F, ∑ j : F, conj (c i) * K i j * c j).re ≤      ∑ i : F, Complex.normSq (c i)

  • definition OAI.Problem336AdditiveD139.IsFiniteDeterminantalLaw line 106

    def IsFiniteDeterminantalLaw {F : Type u_17} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (K : F → F → ℂ) : Prop := by  classical  exact ∀ A : Finset F,    (((μ : Measure (F → Bool)) {x | ∀ i ∈ A, x i = true}).toReal : ℂ) =      Matrix.det (fun i j : A => K i.1 j.1)

  • definition OAI.Problem336AdditiveD139.PositiveContractionKernel line 113

    def PositiveContractionKernel {I : Type u_18}    (K : I → I → ℂ) : Prop :=  (∀ i j, K i j = conj (K j i)) ∧  ∀ (S : Finset I) (c : S → ℂ),    0 ≤ (∑ i : S, ∑ j : S, conj (c i) * K i.1 j.1 * c j).re ∧    (∑ i : S, ∑ j : S, conj (c i) * K i.1 j.1 * c j).re ≤      ∑ i : S, Complex.normSq (c i)

  • definition OAI.Problem336AdditiveD139.IsDeterminantalLaw line 121

    def IsDeterminantalLaw {I : Type u_19}    (μ : ProbabilityMeasure (I → Bool)) (K : I → I → ℂ) : Prop := by  classical  exact ∀ A : Finset I,    (((μ : Measure (I → Bool)) {x | ∀ i ∈ A, x i = true}).toReal : ℂ) =      Matrix.det (fun i j : A => K i.1 j.1)

  • definition OAI.Problem336AdditiveD139.TranslationInvariantKernel line 128

    def TranslationInvariantKernel {Γ : Type u_20} [Group Γ]    (K : Γ → Γ → ℂ) : Prop :=  ∀ g h k : Γ, K (g * h) (g * k) = K h k

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: kernel, determinantal, positive, contraction, groups, invariant, iid, probability and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: processes.

Corollary 1.4 of The free uniform spanning forest is a factor of IID

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 $\Gamma$.

Other possible paper matches, closest wording first

  • 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: rayleigh, strongly, map, groups, invariant, phi and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: processes.
  • 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: values, real, 1/2 and coordinates. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: required and neither.
  • 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: tilt, norm, 1/2, map, coordinate, countable and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sampling.
  • Theorem 1.1 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: equivariant, infinite, phi, finite and law. Found in the paper’s title, which its main theorem states: iid and factor.
  • 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: iid, factor and law.
  • 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: infinite, finite and laws.

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 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.”

An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

185 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: 51 to its own declarations, 123 to Mathlib’s documentation. 27 declared names also link to the same declaration in OpenAI’s library. 6 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAIuniverse u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12universe u_13 u_14 u_15 u_16 u_17 u_18 u_19 u_20noncomputable sectionopen MeasureTheoryopen scoped BigOperators ENNReal Topology ComplexConjugatenamespace Problem336AdditiveD139abbrev UnitLabel := Set.Icc (0 : ℝ) 1def leftTranslate {Γ : Type u_1} {α : Type u_2} [Group Γ] (g : Γ) (a : Γ → α) : Γ → α :=  fun h => a (g⁻¹ * h)def iidUnitLabels (Γ : Type u_3) [MeasurableSpace Γ] : Measure (Γ → UnitLabel) :=  Measure.infinitePi (fun _ : Γ => (volume : Measure UnitLabel))def InvariantBinaryLaw {Γ : Type u_4} [Group Γ]    (μ : ProbabilityMeasure (Γ → Bool)) : Prop :=  ∀ g : Γ,    Measure.map (leftTranslate g) (μ : Measure (Γ → Bool)) =      (μ : Measure (Γ → Bool))def HasEquivariantIIDFactor {Γ : Type u_5} [Group Γ] [MeasurableSpace Γ]    (μ : ProbabilityMeasure (Γ → Bool)) : Prop :=  ∃ Phi : (Γ → UnitLabel) → (Γ → Bool),    Measurable Phi ∧    (∀ (g : Γ) (u : Γ → UnitLabel),      Phi (leftTranslate g u) = leftTranslate g (Phi u)) ∧    Measure.map Phi (iidUnitLabels Γ) = (μ : Measure (Γ → Bool))def finiteGeneratingValue {F : Type u_6} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (z : F → ℂ) : ℂ := by  classical  exact ∑ x : F → Bool,    (((μ : Measure (F → Bool)) {x}).toReal : ℂ) *      ∏ i : F, if x i = true then z i else 1def StronglyRayleighFinite {F : Type u_7} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) : Prop :=  ∀ z : F → ℂ, (∀ i : F, 0 < (z i).im) → finiteGeneratingValue μ z ≠ 0def marginalMass {I : Type u_8} (μ : ProbabilityMeasure (I → Bool))    (S : Finset I) (x : S → Bool) : ℝ :=  ((μ : Measure (I → Bool)) {a | ∀ i : S, a i.1 = x i}).toRealdef marginalGeneratingValue {I : Type u_9} (μ : ProbabilityMeasure (I → Bool))    (S : Finset I) (z : S → ℂ) : ℂ := by  classical  exact ∑ x : S → Bool,    (marginalMass μ S x : ℂ) *      ∏ i : S, if x i = true then z i else 1def StronglyRayleighCountable {I : Type u_10}    (μ : ProbabilityMeasure (I → Bool)) : Prop :=  ∀ S : Finset I, ∀ z : S → ℂ,    (∀ i : S, 0 < (z i).im) → marginalGeneratingValue μ S z ≠ 0def boolReal (b : Bool) : ℝ := if b = true then 1 else 0def tiltPartition {F : Type u_11} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (h : F → ℝ) : ℝ := by  classical  exact ∑ x : F → Bool,    Real.exp (∑ j : F, h j * boolReal (x j)) *      ((μ : Measure (F → Bool)) {x}).toRealdef tiltMean {F : Type u_12} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (h : F → ℝ) (i : F) : ℝ := by  classical  exact (∑ x : F → Bool,      Real.exp (∑ j : F, h j * boolReal (x j)) *        ((μ : Measure (F → Bool)) {x}).toReal * boolReal (x i)) /    tiltPartition μ hdef tiltSecondMoment {F : Type u_13} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (h : F → ℝ) (i j : F) : ℝ := by  classical  exact (∑ x : F → Bool,      Real.exp (∑ k : F, h k * boolReal (x k)) *        ((μ : Measure (F → Bool)) {x}).toReal *        boolReal (x i) * boolReal (x j)) /    tiltPartition μ hdef tiltCovariance {F : Type u_14} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (h : F → ℝ) (i j : F) : ℝ :=  tiltSecondMoment μ h i j - tiltMean μ h i * tiltMean μ h jdef coordinatePerturb {F : Type u_15} (h : F → ℝ) (j : F) (r : ℝ) : F → ℝ := by  classical  exact fun k => h k + if k = j then r else 0def PositiveContractionFinite {F : Type u_16} [Fintype F]    (K : F → F → ℂ) : Prop :=  (∀ i j, K i j = conj (K j i)) ∧  ∀ c : F → ℂ,    0 ≤ (∑ i : F, ∑ j : F, conj (c i) * K i j * c j).re ∧    (∑ i : F, ∑ j : F, conj (c i) * K i j * c j).re ≤      ∑ i : F, Complex.normSq (c i)def IsFiniteDeterminantalLaw {F : Type u_17} [Fintype F]    (μ : ProbabilityMeasure (F → Bool)) (K : F → F → ℂ) : Prop := by  classical  exact ∀ A : Finset F,    (((μ : Measure (F → Bool)) {x | ∀ i ∈ A, x i = true}).toReal : ℂ) =      Matrix.det (fun i j : A => K i.1 j.1)def PositiveContractionKernel {I : Type u_18}    (K : I → I → ℂ) : Prop :=  (∀ i j, K i j = conj (K j i)) ∧  ∀ (S : Finset I) (c : S → ℂ),    0 ≤ (∑ i : S, ∑ j : S, conj (c i) * K i.1 j.1 * c j).re ∧    (∑ i : S, ∑ j : S, conj (c i) * K i.1 j.1 * c j).re ≤      ∑ i : S, Complex.normSq (c i)def IsDeterminantalLaw {I : Type u_19}    (μ : ProbabilityMeasure (I → Bool)) (K : I → I → ℂ) : Prop := by  classical  exact ∀ A : Finset I,    (((μ : Measure (I → Bool)) {x | ∀ i ∈ A, x i = true}).toReal : ℂ) =      Matrix.det (fun i j : A => K i.1 j.1)def TranslationInvariantKernel {Γ : Type u_20} [Group Γ]    (K : Γ → Γ → ℂ) : Prop :=  ∀ g h k : Γ, K (g * h) (g * k) = K h ktheorem finite_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) := by  sorrytheorem strongly_rayleigh_group_factor :    ∀ (Γ : Type u_2) [Group Γ] [Countable Γ] [MeasurableSpace Γ],      ∀ μ : ProbabilityMeasure (Γ → Bool),        StronglyRayleighCountable μ → InvariantBinaryLaw μ → HasEquivariantIIDFactor μ := by  sorrytheorem finite_positive_contraction_dpp_is_strongly_rayleigh :    ∀ (F : Type u_3) [Fintype F] [Nonempty F] (K : F → F → ℂ),      PositiveContractionFinite K →        (∃ μ : ProbabilityMeasure (F → Bool), IsFiniteDeterminantalLaw μ K) ∧          ∀ μ : ProbabilityMeasure (F → Bool), IsFiniteDeterminantalLaw μ K →            StronglyRayleighFinite μ := by  sorrytheorem countable_positive_contraction_dpp_exists_unique :    ∀ (I : Type u_4) [Countable I] (K : I → I → ℂ),      PositiveContractionKernel K →        ∃! μ : ProbabilityMeasure (I → Bool), IsDeterminantalLaw μ K := by  sorrytheorem invariant_positive_contraction_dpp_factor :    ∀ (Γ : Type u_5) [Group Γ] [Countable Γ] [MeasurableSpace Γ]      (K : Γ → Γ → ℂ) (μ : ProbabilityMeasure (Γ → Bool)),      PositiveContractionKernel K → IsDeterminantalLaw μ K →        InvariantBinaryLaw μ → HasEquivariantIIDFactor μ := by  sorrytheorem translation_invariant_kernel_dpp_factor :    ∀ (Γ : Type u_6) [Group Γ] [Countable Γ] [MeasurableSpace Γ] (K : Γ → Γ → ℂ),      PositiveContractionKernel K → TranslationInvariantKernel K →        ∃ μ : ProbabilityMeasure (Γ → Bool),          IsDeterminantalLaw μ K ∧ InvariantBinaryLaw μ ∧ HasEquivariantIIDFactor μ := by  sorryend Problem336AdditiveD139endend 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
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.StrongRayleigh.Main
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
21 local vocabulary declarations
Lines of Lean, without comments or blank lines
139
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Strongly Rayleigh and determinantal-process IID factors

Family 231: The free uniform spanning forest is a factor of IID

Candidate pairing

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.

The free uniform spanning forest is a factor of IID

TeX label cor:dpp-group, build/sections/introduction.tex, line 124

The Lean statement says

theorem finite_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) := by  sorrytheorem strongly_rayleigh_group_factor :    ∀ (Γ : Type u_2) [Group Γ] [Countable Γ] [MeasurableSpace Γ],      ∀ μ : ProbabilityMeasure (Γ → Bool),        StronglyRayleighCountable μ → InvariantBinaryLaw μ → HasEquivariantIIDFactor μ := by

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Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.two independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

21 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/StronglyRayleighDPP.html

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