Lean statement

A $C^1$ counterexample to the entropy conjecture

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.Problem340.exists_C1_entropy_counterexample

Lean statement file: C1EntropyCounterexample.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 151 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 exists_C1_entropy_counterexample :    ∃ q : ℕ, 0 < q ∧ ∃ f : CounterexampleManifold q → CounterexampleManifold q, ∃ hf : IsC1SelfMap q f, Dynamics.coverEntropy f (Set.univ : Set (CounterexampleManifold q)) = (0 : EReal) ∧ (∃ v : RealSingularHomology 2 (CounterexampleManifold q), v ≠ 0 ∧ realHomologyMap 2 f hf.1 v = (2 : ℝ) • v) ∧ 0 < Real.log (2 : ℝ) ∧ Real.log (2 : ℝ) ≤ Real.log (totalRealHomologySpectralRadius f hf.1) ∧ ¬ Function.Bijective f ∧ 5 ≤ 2 * q + 3 := 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.Problem340.ClockAmbient line 11

abbrev ClockAmbient := EuclideanSpace ℝ (Fin 2)

abbreviation OAI.Problem340.SphereAmbient line 13

abbrev SphereAmbient := EuclideanSpace ℝ (Fin 3)

abbreviation OAI.Problem340.StandardClock line 15

abbrev StandardClock := Metric.sphere (0 : ClockAmbient) (1 : ℝ)

abbreviation OAI.Problem340.StandardSphere2 line 17

abbrev StandardSphere2 := Metric.sphere (0 : SphereAmbient) (1 : ℝ)

abbreviation OAI.Problem340.CounterexampleManifold line 19

abbrev CounterexampleManifold (q : ℕ) :=  StandardClock × (Fin (q + 1) → StandardSphere2)

abbreviation OAI.Problem340.CounterexampleAmbient line 22

abbrev CounterexampleAmbient (q : ℕ) :=  ClockAmbient × (Fin (q + 1) → SphereAmbient)

definition OAI.Problem340.standardEmbedding line 25

def standardEmbedding (q : ℕ) :    CounterexampleManifold q → CounterexampleAmbient q :=  fun x => (x.1.1, fun i => (x.2 i).1)

definition OAI.Problem340.IsC1SelfMap line 29

def IsC1SelfMap (q : ℕ) (f : CounterexampleManifold q → CounterexampleManifold q) : Prop :=  Continuous f ∧    ∀ x : CounterexampleManifold q,      ∃ U : Set (CounterexampleAmbient q),        IsOpen U ∧ standardEmbedding q x ∈ U ∧          ∃ F : CounterexampleAmbient q → CounterexampleAmbient q,            ContDiffOn ℝ 1 F U ∧              ∀ y : CounterexampleManifold q,                standardEmbedding q y ∈ U →                  standardEmbedding q (f y) = F (standardEmbedding q y)

definition OAI.Problem340.continuousMapOf line 40

def continuousMapOf {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y]    (f : X → Y) (hf : Continuous f) : ContinuousMap X Y :=  ⟨f, hf⟩

abbreviation OAI.Problem340.RealSingularHomology line 44

abbrev RealSingularHomology (n : ℕ) (X : Type) [TopologicalSpace X] :=  (((AlgebraicTopology.singularHomologyFunctor (ModuleCat ℝ) n).obj    (ModuleCat.of ℝ ℝ)).obj (TopCat.of X))

definition OAI.Problem340.realHomologyMap line 48

def realHomologyMap {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y]    (n : ℕ) (f : X → Y) (hf : Continuous f) :    RealSingularHomology n X →ₗ[ℝ] RealSingularHomology n Y :=  (((AlgebraicTopology.singularHomologyFunctor (ModuleCat ℝ) n).obj    (ModuleCat.of ℝ ℝ)).map      (TopCat.ofHom (continuousMapOf f hf))).hom

definition OAI.Problem340.totalRealHomologySpectralRadius line 55

def totalRealHomologySpectralRadius {X : Type} [TopologicalSpace X]    (f : X → X) (hf : Continuous f) : ℝ :=  sSup {r : ℝ |    ∃ n : ℕ, ∃ a b : ℝ,      ∃ u v : RealSingularHomology n X,        (u ≠ 0 ∨ v ≠ 0) ∧        realHomologyMap n f hf u = a • u - b • v ∧        realHomologyMap n f hf v = b • u + a • v ∧        r = Real.sqrt (a ^ 2 + b ^ 2)}

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: log, manifold, integer, self and map. Found in the paper’s title, which its main theorem states: entropy and counterexample. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive, compact, smooth and eigenvalue.

Theorem 1.1 of A C^1 Counterexample to the Entropy Conjecture

Theorem

There are a positive integer q and a C^1 self-map f of the compact smooth manifold

M=(\R/100\Z)\times(S^2)^{q+1}

such that \htop(f)=0 and f_*:H_2(M;\R)\to H_2(M;\R) has eigenvalue 2. In particular,

\htop(f)=0<\log 2\le \log\rho(f_*).

Section 1, Introduction Read it in the paper TeX source, line 98

Show the TeX

There are a positive integer \(q\) and a \(C^1\) self-map \(f\) of the
compact smooth manifold
\[
 M=(\R/100\Z)\times(S^2)^{q+1}
\]
such that \(\htop(f)=0\) and \(f_*:H_2(M;\R)\to H_2(M;\R)\) has
eigenvalue \(2\). In particular,
\[
 \htop(f)=0<\log 2\le \log\rho(f_*).
\]

Other possible paper matches, closest wording first

Proposition 2.1 of A C^1 Counterexample to the Entropy Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: real, manifold, integer, self and map. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: compact and smooth.
Proposition 3.1 of A C^1 Counterexample to the Entropy Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: clock, metric and map. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounded and zero.
Proposition 5.1 of A C^1 Counterexample to the Entropy Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: map. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: action and eigenvalue.
Proposition 4.3 of A C^1 Counterexample to the Entropy Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: top and 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

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 151’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: Statement: ∃ q > 0 and a non-bijective C¹ self-map f of S¹ × (S²)^(q+1) (C¹ = locally the restriction of an ambient C¹ map) with Dynamics.coverEntropy f univ = 0, a nonzero v ∈ H₂(M;ℝ) with f_* v = 2v, and log 2 ≤ log(total real-homology spectral radius): exactly the summary's counterexample. Mathlib's coverEntropy is the topological entropy on compact metric spaces.
Definitions the group asks readers to compare with the paper: totalRealHomologySpectralRadius f := sSup {r | ∃ n a b u v, (u ≠ 0 ∨ v ≠ 0) ∧ f_* u = a•u − b•v ∧ f_* v = b•u + a•v ∧ r = √(a²+b²)} (custom; moduli of complex eigenvalues over all degrees; if the set were unbounded sSup = 0 and the clause log 2 ≤ log radius would fail, so it cannot trivialise)
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 gives a counterexample to the general $C^1$ self-map version: a noninvertible $C^1$ map on a compact smooth manifold without boundary has topological entropy zero, while its action on second real homology has a nonzero eigenvector with eigenvalue $2$. Thus the homological lower bound is strictly positive and fails for this map.”
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

73 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: 33 to its own declarations, 40 to Mathlib’s documentation. 13 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 exists_C1_entropy_counterexample 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 12 other declarations of the file; with the theorem itself they span 45 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.Problem340 are listed without it.

abbrev ClockAmbientline 11library
abbrev SphereAmbientline 13library
abbrev StandardClockline 15library
abbrev StandardSphere2line 17library
abbrev CounterexampleManifoldline 19in the statementlibrary
abbrev CounterexampleAmbientline 22library
def standardEmbeddingline 25library
def IsC1SelfMapline 29in the statementlibrary
def continuousMapOfline 40library
abbrev RealSingularHomologyline 44in the statementlibrary
def realHomologyMapline 48in the statementlibrary
def totalRealHomologySpectralRadiusline 55in the statementlibrary
theorem exists_C1_entropy_counterexampleline 65selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen CategoryTheorynamespace Problem340abbrev ClockAmbient := EuclideanSpace ℝ (Fin 2)abbrev SphereAmbient := EuclideanSpace ℝ (Fin 3)abbrev StandardClock := Metric.sphere (0 : ClockAmbient) (1 : ℝ)abbrev StandardSphere2 := Metric.sphere (0 : SphereAmbient) (1 : ℝ)abbrev CounterexampleManifold (q : ℕ) :=  StandardClock × (Fin (q + 1) → StandardSphere2)abbrev CounterexampleAmbient (q : ℕ) :=  ClockAmbient × (Fin (q + 1) → SphereAmbient)def standardEmbedding (q : ℕ) :    CounterexampleManifold q → CounterexampleAmbient q :=  fun x => (x.1.1, fun i => (x.2 i).1)def IsC1SelfMap (q : ℕ) (f : CounterexampleManifold q → CounterexampleManifold q) : Prop :=  Continuous f ∧    ∀ x : CounterexampleManifold q,      ∃ U : Set (CounterexampleAmbient q),        IsOpen U ∧ standardEmbedding q x ∈ U ∧          ∃ F : CounterexampleAmbient q → CounterexampleAmbient q,            ContDiffOn ℝ 1 F U ∧              ∀ y : CounterexampleManifold q,                standardEmbedding q y ∈ U →                  standardEmbedding q (f y) = F (standardEmbedding q y)def continuousMapOf {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y]    (f : X → Y) (hf : Continuous f) : ContinuousMap X Y :=  ⟨f, hf⟩abbrev RealSingularHomology (n : ℕ) (X : Type) [TopologicalSpace X] :=  (((AlgebraicTopology.singularHomologyFunctor (ModuleCat ℝ) n).obj    (ModuleCat.of ℝ ℝ)).obj (TopCat.of X))def realHomologyMap {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y]    (n : ℕ) (f : X → Y) (hf : Continuous f) :    RealSingularHomology n X →ₗ[ℝ] RealSingularHomology n Y :=  (((AlgebraicTopology.singularHomologyFunctor (ModuleCat ℝ) n).obj    (ModuleCat.of ℝ ℝ)).map      (TopCat.ofHom (continuousMapOf f hf))).homdef totalRealHomologySpectralRadius {X : Type} [TopologicalSpace X]    (f : X → X) (hf : Continuous f) : ℝ :=  sSup {r : ℝ |    ∃ n : ℕ, ∃ a b : ℝ,      ∃ u v : RealSingularHomology n X,        (u ≠ 0 ∨ v ≠ 0) ∧        realHomologyMap n f hf u = a • u - b • v ∧        realHomologyMap n f hf v = b • u + a • v ∧        r = Real.sqrt (a ^ 2 + b ^ 2)}theorem exists_C1_entropy_counterexample :    ∃ q : ℕ, 0 < q ∧ ∃ f : CounterexampleManifold q → CounterexampleManifold q, ∃ hf : IsC1SelfMap q f, Dynamics.coverEntropy f (Set.univ : Set (CounterexampleManifold q)) = (0 : EReal) ∧ (∃ v : RealSingularHomology 2 (CounterexampleManifold q), v ≠ 0 ∧ realHomologyMap 2 f hf.1 v = (2 : ℝ) • v) ∧ 0 < Real.log (2 : ℝ) ∧ Real.log (2 : ℝ) ≤ Real.log (totalRealHomologySpectralRadius f hf.1) ∧ ¬ Function.Bijective f ∧ 5 ≤ 2 * q + 3 := by  sorryend Problem340endend 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.