Lean statement

Finite-entropy system without a smooth positive-volume model

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.SmoothObstruction.main

Lean statement file: SmoothObstruction.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 152 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 main :    ∃ (X : Type) (_ : MeasurableSpace X) (μ : Measure X)      (_ : IsProbabilityMeasure μ) (T : X ≃ᵐ X),      StandardNonatomic μ ∧ Ergodic T μ ∧ FiniteKSEntropy μ T ∧        ¬ HasSmoothPositiveVolumeModel μ T := 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.

definition OAI.SmoothObstruction.entropyTerm line 12

noncomputable def entropyTerm (p : ℝ) : ℝ := -(p * Real.log p) / Real.log 2

definition OAI.SmoothObstruction.blockEntropy line 15

noncomputable def blockEntropy {X : Type*} [MeasurableSpace X]    (μ : Measure X) (T : X → X) {q : ℕ} (P : X → Fin q) (n : ℕ) : ℝ :=  ∑ w : Fin n → Fin q,    entropyTerm ((μ {x | ∀ j : Fin n, P ((T^[j.val]) x) = w j}).toReal)

definition OAI.SmoothObstruction.FiniteKSEntropy line 23

def FiniteKSEntropy {X : Type*} [MeasurableSpace X]    (μ : Measure X) (T : X → X) : Prop :=  ∃ C : ℝ, ∀ (q : ℕ) (P : X → Fin (q + 1)), Measurable P →    ∃ h : ℝ, h ≤ C ∧      Tendsto (fun n : ℕ => blockEntropy μ T P n / (n : ℝ)) atTop (𝓝 h)

definition OAI.SmoothObstruction.ConullIsomorphic line 30

def ConullIsomorphic {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]    (μ : Measure X) (ν : Measure Y) : Prop :=  ∃ (A : Set X) (B : Set Y), MeasurableSet A ∧ MeasurableSet B ∧    μ Aᶜ = 0 ∧ ν Bᶜ = 0 ∧    ∃ e : A ≃ᵐ B,      MeasurePreserving e (μ.comap (Subtype.val : A → X))        (ν.comap (Subtype.val : B → Y))

definition OAI.SmoothObstruction.StandardNonatomic line 39

def StandardNonatomic {X : Type*} [MeasurableSpace X] (μ : Measure X) : Prop :=  ConullIsomorphic μ    (volume.comap (Subtype.val : Set.Icc (0 : ℝ) 1 → ℝ))

definition OAI.SmoothObstruction.ConullConjugate line 45

def ConullConjugate {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]    (μ : Measure X) (T : X → X) (ν : Measure Y) (S : Y → Y) : Prop :=  ∃ (A : Set X) (B : Set Y), MeasurableSet A ∧ MeasurableSet B ∧    μ Aᶜ = 0 ∧ ν Bᶜ = 0 ∧ T ⁻¹' A = A ∧ S ⁻¹' B = B ∧    ∃ e : A ≃ᵐ B,      MeasurePreserving e (μ.comap (Subtype.val : A → X))        (ν.comap (Subtype.val : B → Y)) ∧      ∀ (x y : A), T x = y → (e y : Y) = S (e x)

definition OAI.SmoothObstruction.SmoothPositiveDensity line 57

def SmoothPositiveDensity {d : ℕ} {H : Type*} [TopologicalSpace H]    (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H)    {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [MeasurableSpace M]    (ν : Measure M) : Prop :=  ∀ x : M, ∃ f : EuclideanSpace ℝ (Fin d) → ℝ,    ContDiffOn ℝ ∞ f (extChartAt I x).target ∧    (∀ y ∈ (extChartAt I x).target, 0 < f y) ∧    Measure.map (extChartAt I x) (ν.restrict (extChartAt I x).source) =      (volume.withDensity (fun y => ENNReal.ofReal (f y))).restrict        (extChartAt I x).target

definition OAI.SmoothObstruction.HasModelWith line 70

def HasModelWith {X : Type*} [MeasurableSpace X]    (μ : Measure X) (T : X → X) {d : ℕ} {H : Type} [TopologicalSpace H]    (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) : Prop :=  ∃ (M : Type) (_ : TopologicalSpace M) (_ : MeasurableSpace M)      (_ : BorelSpace M) (_ : T2Space M) (_ : SecondCountableTopology M)      (_ : CompactSpace M) (_ : ChartedSpace H M) (_ : IsManifold I ∞ M)      (ν : Measure M) (_ : IsProbabilityMeasure ν)      (S : Diffeomorph I I M M ∞),    SmoothPositiveDensity I ν ∧ MeasurePreserving S ν ν ∧      ConullConjugate μ T ν S

definition OAI.SmoothObstruction.HasSmoothPositiveVolumeModel line 84

def HasSmoothPositiveVolumeModel {X : Type*} [MeasurableSpace X]    (μ : Measure X) (T : X → X) : Prop :=  (∃ d : ℕ, HasModelWith μ T (𝓡 d)) ∨    (∃ d : ℕ, ∃ _ : NeZero d, HasModelWith μ T (𝓡∂ d))

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: conjugate, conull, orientable, smooth, nonatomic, standard, boundary, preserving and ten more. Found in the paper’s title, which its main theorem states: system, model, without, volume, entropy and zero.

Theorem 1.1 of A zero-entropy system without a smooth positive-volume model

Theorem

There exists an ergodic invertible measure-preserving transformation (X,\mu,T) of a standard nonatomic probability space such that h_\mu(T)=0 and (X,\mu,T) is not measurably conjugate, on invariant conull sets, to any C^\infty diffeomorphism of a compact finite-dimensional manifold preserving a strictly positive smooth probability density. The manifold may have smooth boundary and need not be orientable.

Section 1, Introduction Read it in the paper TeX source, line 91

Show the TeX

There exists an ergodic invertible measure-preserving transformation
$(X,\mu,T)$ of a standard nonatomic probability space such that
$h_\mu(T)=0$ and $(X,\mu,T)$ is not measurably conjugate, on invariant conull sets, to
any $C^\infty$ diffeomorphism of a compact finite-dimensional manifold
preserving a strictly positive smooth probability density. The manifold
may have smooth boundary and need not be orientable.

Other possible paper matches, closest wording first

Theorem 4.1 of A zero-entropy system without a smooth positive-volume model. Words of the Lean statement, its names split into words, that also appear in the theorem: map, partition, dimension, sense, measurable, boundary and ten more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: conclusions.
Proposition 6.1 of A zero-entropy system without a smooth positive-volume model. Words of the Lean statement, its names split into words, that also appear in the theorem: borel, modulo, null, nonatomic, standard, ergodic and four more.
Proposition 3.1 of A zero-entropy system without a smooth positive-volume model. Words of the Lean statement, its names split into words, that also appear in the theorem: real, upper, second, factor, rates, empty and six more.
Proposition 2.2 of A zero-entropy system without a smooth positive-volume model. Words of the Lean statement, its names split into words, that also appear in the theorem: sequence, integer, compact, positive and finite.
Proposition 5.1 of A zero-entropy system without a smooth positive-volume model. Words of the Lean statement, its names split into words, that also appear in the theorem: choose, allowed, finite and probability.
Proposition 6.3 of A zero-entropy system without a smooth positive-volume model. Words of the Lean statement, its names split into words, that also appear in the theorem: log, integer, invariant and probability.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 238 s; peak memory 6.6 GiB).
Reported 11 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 152’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: main gives an ergodic invertible T on a standard nonatomic probability space with FiniteKSEntropy (all finite-partition entropy rates exist and share one real bound) and ¬ HasSmoothPositiveVolumeModel (no C^∞ positive-density model on any compact manifold, closed of every dimension or with boundary); the summary's zero-entropy requirement is not stated (docs: 'the paper's stronger zero-entropy conclusion is outside this statement'), so the Lean is implied by the headline but does not imply it.
Definitions the group asks readers to compare with the paper: def FiniteKSEntropy μ T := ∃ C : ℝ, ∀ q (P : X → Fin (q+1)), Measurable P → ∃ h ≤ C, Tendsto (fun n => blockEntropy μ T P n / n) atTop (𝓝 h): custom; says sup of partition entropies is finite, not zero ; HasModelWith/SmoothPositiveDensity/ConullConjugate: custom manifold-model encoding (density of Measure.map (extChartAt I x) against Lebesgue in every chart, conjugacy on invariant conull sets); read as standard
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: Zero Kolmogorov–Sinai entropy is claimed; the statement gives only finite entropy.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The linked statement gives finite entropy; the paper's stronger zero-entropy conclusion is outside this statement.”
Quoted from OpenAI’s Lean note for this family

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

Supporting material

The complete Lean statement file

99 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: 10 to its own declarations, 85 to Mathlib’s documentation. 8 declared names also link to the same declaration in OpenAI’s library. 3 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main 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 9 other declarations of the file; with the theorem itself they span 58 lines. 3 of them are named in the statement itself.

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

def entropyTermline 12library
def blockEntropyline 15
def FiniteKSEntropyline 23in the statement
def ConullIsomorphicline 30library
def StandardNonatomicline 39in the statementlibrary
def ConullConjugateline 45library
def SmoothPositiveDensityline 57library
def HasModelWithline 70library
def HasSmoothPositiveVolumeModelline 84in the statementlibrary
theorem mainline 90selected theoremlibrary

import Mathlibnamespace OAIopen MeasureTheory Set Filteropen scoped ENNReal Topology Manifold ContDiffnamespace SmoothObstruction/-- Shannon entropy in bits; the value at mass zero is zero. -/noncomputable def entropyTerm (p : ℝ) : ℝ := -(p * Real.log p) / Real.log 2/-- The entropy of the length-`n` join of the partition with labels in `Fin q`. -/noncomputable def blockEntropy {X : Type*} [MeasurableSpace X]    (μ : Measure X) (T : X → X) {q : ℕ} (P : X → Fin q) (n : ℕ) : ℝ :=  ∑ w : Fin n → Fin q,    entropyTerm ((μ {x | ∀ j : Fin n, P ((T^[j.val]) x) = w j}).toReal)/-- All finite-partition entropy rates exist and have one finite real upperbound. This spells out `sup_P lim_n H(join_{j<n} T^{-j}P)/n < ∞` withoutchoosing a value for limits of nonconvergent sequences. Empty atoms are allowed. -/def FiniteKSEntropy {X : Type*} [MeasurableSpace X]    (μ : Measure X) (T : X → X) : Prop :=  ∃ C : ℝ, ∀ (q : ℕ) (P : X → Fin (q + 1)), Measurable P →    ∃ h : ℝ, h ≤ C ∧      Tendsto (fun n : ℕ => blockEntropy μ T P n / (n : ℝ)) atTop (𝓝 h)/-- Measure-space isomorphism after removing measurable null sets. -/def ConullIsomorphic {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]    (μ : Measure X) (ν : Measure Y) : Prop :=  ∃ (A : Set X) (B : Set Y), MeasurableSet A ∧ MeasurableSet B ∧    μ Aᶜ = 0 ∧ ν Bᶜ = 0 ∧    ∃ e : A ≃ᵐ B,      MeasurePreserving e (μ.comap (Subtype.val : A → X))        (ν.comap (Subtype.val : B → Y))/-- Standard nonatomic in the source's explicit modulo-null-set sense. -/def StandardNonatomic {X : Type*} [MeasurableSpace X] (μ : Measure X) : Prop :=  ConullIsomorphic μ    (volume.comap (Subtype.val : Set.Icc (0 : ℝ) 1 → ℝ))/-- Measurable conjugacy on invariant conull measurable subsets, not acontinuous conjugacy and not merely a factor map. -/def ConullConjugate {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]    (μ : Measure X) (T : X → X) (ν : Measure Y) (S : Y → Y) : Prop :=  ∃ (A : Set X) (B : Set Y), MeasurableSet A ∧ MeasurableSet B ∧    μ Aᶜ = 0 ∧ ν Bᶜ = 0 ∧ T ⁻¹' A = A ∧ S ⁻¹' B = B ∧    ∃ e : A ≃ᵐ B,      MeasurePreserving e (μ.comap (Subtype.val : A → X))        (ν.comap (Subtype.val : B → Y)) ∧      ∀ (x y : A), T x = y → (e y : Y) = S (e x)/-- Positive smooth local coordinate densities. In a half-space chart,`ContDiffOn` includes smoothness up to the boundary. Absolute Lebesgue densityis used; no orientation or volume form is part of the data. -/def SmoothPositiveDensity {d : ℕ} {H : Type*} [TopologicalSpace H]    (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H)    {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [MeasurableSpace M]    (ν : Measure M) : Prop :=  ∀ x : M, ∃ f : EuclideanSpace ℝ (Fin d) → ℝ,    ContDiffOn ℝ ∞ f (extChartAt I x).target ∧    (∀ y ∈ (extChartAt I x).target, 0 < f y) ∧    Measure.map (extChartAt I x) (ν.restrict (extChartAt I x).source) =      (volume.withDensity (fun y => ENNReal.ofReal (f y))).restrict        (extChartAt I x).target/-- A smooth positive-volume model with a specified Euclidean or half-spacemanifold model. All model dimensions will be quantified below. -/def HasModelWith {X : Type*} [MeasurableSpace X]    (μ : Measure X) (T : X → X) {d : ℕ} {H : Type} [TopologicalSpace H]    (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) : Prop :=  ∃ (M : Type) (_ : TopologicalSpace M) (_ : MeasurableSpace M)      (_ : BorelSpace M) (_ : T2Space M) (_ : SecondCountableTopology M)      (_ : CompactSpace M) (_ : ChartedSpace H M) (_ : IsManifold I ∞ M)      (ν : Measure M) (_ : IsProbabilityMeasure ν)      (S : Diffeomorph I I M M ∞),    SmoothPositiveDensity I ν ∧ MeasurePreserving S ν ν ∧      ConullConjugate μ T ν S/-- Closed manifolds of all dimensions (including zero), and smooth manifoldswith boundary of every positive finite dimension. Nonorientable manifoldsare included: no orientability hypothesis occurs. -/def HasSmoothPositiveVolumeModel {X : Type*} [MeasurableSpace X]    (μ : Measure X) (T : X → X) : Prop :=  (∃ d : ℕ, HasModelWith μ T (𝓡 d)) ∨    (∃ d : ℕ, ∃ _ : NeZero d, HasModelWith μ T (𝓡∂ d))/-- A finite-entropy standard nonatomic ergodic system with no smooth positive-volume model. -/theorem main :    ∃ (X : Type) (_ : MeasurableSpace X) (μ : Measure X)      (_ : IsProbabilityMeasure μ) (T : X ≃ᵐ X),      StandardNonatomic μ ∧ Ergodic T μ ∧ FiniteKSEntropy μ T ∧        ¬ HasSmoothPositiveVolumeModel μ T := by  sorryend SmoothObstructionend 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.

A zero-entropy system without a smooth positive-volume model

Theorem

Section 1, Introduction Read it in the paper TeX source, line 91

Show the TeX

There exists an ergodic invertible measure-preserving transformation
$(X,\mu,T)$ of a standard nonatomic probability space such that
$h_\mu(T)=0$ and $(X,\mu,T)$ is not measurably conjugate, on invariant conull sets, to
any $C^\infty$ diffeomorphism of a compact finite-dimensional manifold
preserving a strictly positive smooth probability density. The manifold
may have smooth boundary and need not be orientable.

Configuration and file facts

File: lean/ComparatorChallenges/SmoothObstruction.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 152 link it.
Solution module: OAI.Dynamics.SmoothObstruction.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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 58
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