Zero entropy does not guarantee a smooth positive-volume model
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
There 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.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s 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.
The words they share: conjugate, conull, orientable, smooth, nonatomic, standard, boundary, preserving, manifold, ergodic, measure, density, space, compact, invariant, positive, finite, probability, system, model, without, volume, entropy, zero.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SmoothObstruction.lean, the release lists it as a supporting statement.
theoremmain : ∃ (X : Type) (_ : MeasurableSpace X) (μ : Measure X) (_ : IsProbabilityMeasure μ) (T : X ≃ᵐ X), StandardNonatomic μ ∧ Ergodic T μ ∧ FiniteKSEntropy μ T ∧ ¬ HasSmoothPositiveVolumeModel μ T := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express a weaker statement than the advertised result (their term: “weaker-statement”).
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
“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
The paper asks whether a measure-preserving system can be represented by a smooth diffeomorphism preserving positive smooth volume. The formalized supporting result constructs an ergodic invertible transformation of a standard nonatomic probability space with finite Kolmogorov–Sinai entropy that has no such model on any compact finite-dimensional manifold, including manifolds with smooth boundary. The comparison is measurable conjugacy after discarding null sets.
The linked statement gives finite entropy; the paper’s stronger zero-entropy conclusion is outside this statement.
AI-drafted comparisons, with no recorded human review
Finite-entropy system without a smooth positive-volume modelSmoothObstruction
Unreviewed AI draft: states a weaker form of a paper’s theorem. States an ergodic system with finite entropy and no smooth positive-volume model, where the main theorem gives such a system with zero entropy.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Finite-entropy system without a smooth positive-volume modelSmoothObstruction
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 152The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 238 s; peak memory 6.6 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
Finite-entropy system without a smooth positive-volume model
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.