Lean statement

Mixing of every finite order for one mixing transformation

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.Rokhlin.mixing_all_finite_orders

Lean statement file: Rokhlin.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 145 link it.

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

Problems reported

Problem reported Listed as broken in the same way, without further detail in the post. Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 2026-10-08 Source

The Lean statement

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

theorem mixing_all_finite_orders (μ : Measure Ω) [IsProbabilityMeasure μ]    (T : Ω ≃ᵐ Ω) (h_pres : MeasurePreserving T μ μ) (h_mix : IsMixing μ T) :    ∀ k : ℕ, 3 ≤ k → MixingOfOrder μ T k := 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.Rokhlin.timeMap line 12

def timeMap (T : Ω ≃ᵐ Ω) (n : ℤ) : Ω → Ω :=  ⇑((T.toEquiv : Equiv.Perm Ω) ^ n)

definition OAI.Rokhlin.IsMixing line 15

def IsMixing (μ : Measure Ω) (T : Ω ≃ᵐ Ω) : Prop :=  ∀ A B : Set Ω, MeasurableSet A → MeasurableSet B →    Tendsto (fun n : ℤ => μ (A ∩ timeMap T n ⁻¹' B))      (comap Int.natAbs atTop) (𝓝 (μ A * μ B))

definition OAI.Rokhlin.layoutTime line 20

def layoutTime {k : ℕ} (gaps : Fin (k - 1) → ℕ+) (i : Fin k) : ℕ :=  ∑ j : Fin (k - 1), if j.val < i.val then (gaps j : ℕ) else 0

definition OAI.Rokhlin.MixingOfOrder line 23

def MixingOfOrder (μ : Measure Ω) (T : Ω ≃ᵐ Ω) (k : ℕ) : Prop :=  ∀ A : Fin k → Set Ω, (∀ i, MeasurableSet (A i)) →    Tendsto      (fun gaps : Fin (k - 1) → ℕ+ =>        μ (⋂ i : Fin k, timeMap T (layoutTime gaps i : ℤ) ⁻¹' A i))      atTop (𝓝 (∏ i : Fin k, μ (A i)))

Definitions taken from OpenAI’s library 4

Comparator compares only the name and type of each definition listed here. When a theorem is configured, its proof uses the definition in OpenAI’s solution library. Read that definition as well as the one in this file when comparing the statement with the paper.

Comparator checks the name and type of timeMap; its definition is in OpenAI’s library, in the modules that OAI.Dynamics.MultipleMixing.Main imports.
Comparator checks the name and type of IsMixing; its definition is in OpenAI’s library, in the modules that OAI.Dynamics.MultipleMixing.Main imports.
Comparator checks the name and type of layoutTime; its definition is in OpenAI’s library, in the modules that OAI.Dynamics.MultipleMixing.Main imports.
Comparator checks the name and type of MixingOfOrder; its definition is in OpenAI’s library, in the modules that OAI.Dynamics.MultipleMixing.Main imports.

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: finite, invertible, mixing, order, preserving, probability and transformation.

Theorem 1.1 of Rokhlin’s multiple-mixing problem for one transformation

Theorem

Every invertible mixing probability-preserving transformation is mixing of every finite order. More explicitly, under the assumptions above, for every k\ge3 and every A_1,\ldots,A_k\in\mathcal F,

\mu\bigl(A_1\cap T^{-n_1}A_2\cap\cdots\cap T^{-(n_1+\cdots+n_{k-1})}A_k\bigr) \longrightarrow\prod_{i=1}^k\mu(A_i)

as \min(n_1,\ldots,n_{k-1})\to\infty through positive integers. No standardness or countable-generation assumption is imposed on the original probability space.

Section 1, Introduction Read it in the paper TeX source, line 21

Show the TeX

Every invertible mixing probability-preserving transformation is mixing
of every finite order. More explicitly, under the assumptions above, for
every $k\ge3$ and every $A_1,\ldots,A_k\in\mathcal F$,
\[
 \mu\bigl(A_1\cap T^{-n_1}A_2\cap\cdots\cap
             T^{-(n_1+\cdots+n_{k-1})}A_k\bigr)
 \longrightarrow\prod_{i=1}^k\mu(A_i)
\]
as $\min(n_1,\ldots,n_{k-1})\to\infty$ through positive integers.
No standardness or countable-generation assumption is imposed on the
original probability space.

Other possible paper matches, closest wording first

Proposition 2.1 of Rokhlin’s multiple-mixing problem for one transformation. Words of the Lean statement, its names split into words, that also appear in the theorem: layouts, order, integer, measure, mixing, finite and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: transformation and one.
Corollary 8.1 of Rokhlin’s multiple-mixing problem for one transformation. Words of the Lean statement, its names split into words, that also appear in the theorem: gap, preserves, integer, measurable and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: infinity, successive, tends, product and transformations.
Proposition 5.5 of Rokhlin’s multiple-mixing problem for one transformation. Words of the Lean statement, its names split into words, that also appear in the theorem: time, map, measurable, mixing, finite and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: product and one.
Proposition 6.1 of Rokhlin’s multiple-mixing problem for one transformation. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, mixing and probability. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Proposition 5.2 of Rokhlin’s multiple-mixing problem for one transformation. Words of the Lean statement, its names split into words, that also appear in the theorem: time and measurable. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Proposition 3.2 of Rokhlin’s multiple-mixing problem for one transformation. Words of the Lean statement, its names split into words, that also appear in the theorem: map.

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 405 s; peak memory 6.9 GiB).
Reported 10 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.

Definitions compared by name and type only (4): timeMap, IsMixing, layoutTime, MixingOfOrder. Their bodies come from OpenAI’s solution library.

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 145’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
Definitions the group asks readers to compare with the paper: Hand-written timeMap, IsMixing, layoutTime, MixingOfOrder (all four are required definition_names); all-pairwise-gaps-diverge is encoded as atTop on Fin (k-1) → ℕ+ (product order), standard
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 this implication. For each $k\ge3$ and every $k$ measurable sets, the measure of their translated intersection tends to the product of their measures as all successive time gaps tend to infinity.”
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

37 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: 5 to its own declarations, 25 to Mathlib’s documentation. 5 declared names also link to the same declaration in OpenAI’s library. 2 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 mixing_all_finite_orders 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. For a definition marked “name and type only”, Comparator does not compare the definition’s body with the one here. When a theorem is configured, it verifies the proof using the definition in OpenAI’s library. Follow the library link to compare the two definitions.

It reaches all 4 other declarations of the file; with the theorem itself they span 18 lines. 2 of them are named in the statement itself.

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

def timeMapline 12name and type onlylibrary
def IsMixingline 15in the statementname and type onlylibrary
def layoutTimeline 20name and type onlylibrary
def MixingOfOrderline 23in the statementname and type onlylibrary
theorem mixing_all_finite_ordersline 30selected theoremlibrary

import Mathlibnamespace OAIopen MeasureTheory Filteropen scoped BigOperators Topologynamespace Rokhlinvariable {Ω : Type*} [MeasurableSpace Ω]def timeMap (T : Ω ≃ᵐ Ω) (n : ℤ) : Ω → Ω :=  ⇑((T.toEquiv : Equiv.Perm Ω) ^ n)def IsMixing (μ : Measure Ω) (T : Ω ≃ᵐ Ω) : Prop :=  ∀ A B : Set Ω, MeasurableSet A → MeasurableSet B →    Tendsto (fun n : ℤ => μ (A ∩ timeMap T n ⁻¹' B))      (comap Int.natAbs atTop) (𝓝 (μ A * μ B))def layoutTime {k : ℕ} (gaps : Fin (k - 1) → ℕ+) (i : Fin k) : ℕ :=  ∑ j : Fin (k - 1), if j.val < i.val then (gaps j : ℕ) else 0def MixingOfOrder (μ : Measure Ω) (T : Ω ≃ᵐ Ω) (k : ℕ) : Prop :=  ∀ A : Fin k → Set Ω, (∀ i, MeasurableSet (A i)) →    Tendsto      (fun gaps : Fin (k - 1) → ℕ+ =>        μ (⋂ i : Fin k, timeMap T (layoutTime gaps i : ℤ) ⁻¹' A i))      atTop (𝓝 (∏ i : Fin k, μ (A i)))theorem mixing_all_finite_orders (μ : Measure Ω) [IsProbabilityMeasure μ]    (T : Ω ≃ᵐ Ω) (h_pres : MeasurePreserving T μ μ) (h_mix : IsMixing μ T) :    ∀ k : ℕ, 3 ≤ k → MixingOfOrder μ T k := by  sorryend Rokhlinend 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.

Rokhlin’s multiple-mixing problem for one transformation

Theorem

Every invertible mixing probability-preserving transformation is mixing of every finite order. More explicitly, under the assumptions above, for every k\ge3 and every A_1,\ldots,A_k\in\mathcal F,

\mu\bigl(A_1\cap T^{-n_1}A_2\cap\cdots\cap T^{-(n_1+\cdots+n_{k-1})}A_k\bigr) \longrightarrow\prod_{i=1}^k\mu(A_i)

as \min(n_1,\ldots,n_{k-1})\to\infty through positive integers. No standardness or countable-generation assumption is imposed on the original probability space.

Section 1, Introduction Read it in the paper TeX source, line 21

Show the TeX

Every invertible mixing probability-preserving transformation is mixing
of every finite order. More explicitly, under the assumptions above, for
every $k\ge3$ and every $A_1,\ldots,A_k\in\mathcal F$,
\[
 \mu\bigl(A_1\cap T^{-n_1}A_2\cap\cdots\cap
             T^{-(n_1+\cdots+n_{k-1})}A_k\bigr)
 \longrightarrow\prod_{i=1}^k\mu(A_i)
\]
as $\min(n_1,\ldots,n_{k-1})\to\infty$ through positive integers.
No standardness or countable-generation assumption is imposed on the
original probability space.

Configuration and file facts

File: lean/ComparatorChallenges/Rokhlin.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 145 link it.
Solution module: OAI.Dynamics.MultipleMixing.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: 4 local vocabulary declarations
Lines of Lean, without comments or blank lines: 18
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