Lean statement

Counterexample to metric-entropy duality

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.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers

Lean statement file: MetricEntropyDuality.leanShow on the mapFamily 329. OpenAI lists this as a main result in its Lean catalogue.

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_entropy_duality_counterexample_with_covers    (a b : ℝ) (ha : 1 ≤ a) (hb : 1 ≤ b) :    ∃ n : ℕ, 0 < n ∧ ∃ K : Set (RealSpace (Fin n)),      IsSymmetricConvexBody K ∧      Coverable K (cube (Fin n)) ∧      Coverable (polar (cube (Fin n))) (a⁻¹ • polar K) ∧      b * Real.log (coveringNumber (polar (cube (Fin n))) (a⁻¹ • polar K) : ℝ) <        Real.log (coveringNumber K (cube (Fin n)) : ℝ) := 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.MetricEntropyDuality.RealSpace line 17
```
abbrev RealSpace (ι : Type u) := ι → ℝ
```

definition OAI.MetricEntropyDuality.pairing line 19

def pairing {ι : Type u} [Fintype ι] (x y : RealSpace ι) : ℝ :=  ∑ i, x i * y i

definition OAI.MetricEntropyDuality.cube line 22

def cube (ι : Type u) : Set (RealSpace ι) :=  {x | ∀ i, |x i| ≤ 1}

definition OAI.MetricEntropyDuality.polar line 25

def polar {ι : Type u} [Fintype ι] (K : Set (RealSpace ι)) : Set (RealSpace ι) :=  {y | ∀ x ∈ K, pairing x y ≤ 1}

definition OAI.MetricEntropyDuality.Covers line 28

def Covers {ι : Type u} {M : ℕ} (A B : Set (RealSpace ι))    (centers : Fin M → RealSpace ι) : Prop :=  ∀ x ∈ A, ∃ j, x - centers j ∈ B

definition OAI.MetricEntropyDuality.Coverable line 32

def Coverable {ι : Type u} (A B : Set (RealSpace ι)) : Prop :=  ∃ M : ℕ, ∃ centers : Fin M → RealSpace ι, Covers A B centers

definition OAI.MetricEntropyDuality.coveringNumber line 35

def coveringNumber {ι : Type u} (A B : Set (RealSpace ι)) : ℕ :=  sInf {M : ℕ | ∃ centers : Fin M → RealSpace ι, Covers A B centers}

structure OAI.MetricEntropyDuality.IsSymmetricConvexBody line 38

structure IsSymmetricConvexBody {ι : Type u} [Fintype ι]    (K : Set (RealSpace ι)) : Prop where  isCompact : IsCompact K  convex : Convex ℝ K  symmetric : ∀ x, x ∈ K ↔ -x ∈ K  interior_nonempty : (interior K).Nonempty

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: convex, symmetric, body, log and integer. Found in the paper’s title, which its main theorem states: counterexamples, entropy and duality. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: circ, positive and origin.

Theorem 1.1 of Counterexamples to the duality conjecture for metric entropy

Theorem

For every a,b\ge1, there are a positive integer n and an origin-symmetric convex body K\subset\R^n such that, with L=[-1,1]^n,

\log N(K,L)>b\log N(L^\circ,a^{-1}K^\circ).

Section 1, Introduction Read it in the paper TeX source, line 30

Show the TeX

For every $a,b\ge1$, there are a positive integer $n$ and an
origin-symmetric convex body $K\subset\R^n$ such that, with
$L=[-1,1]^n$,
\[
 \log N(K,L)>b\log N(L^\circ,a^{-1}K^\circ).
\]

Other possible paper matches, closest wording first

Corollary 5.1 of Counterexamples to the duality conjecture for metric entropy. Words of the Lean statement, its names split into words, that also appear in the theorem: convex, symmetric, body, log and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: origin.
Proposition 3.2 of Counterexamples to the duality conjecture for metric entropy. Words of the Lean statement, its names split into words, that also appear in the theorem: real, finite and log.
Proposition 4.3 of Counterexamples to the duality conjecture for metric entropy. Words of the Lean statement, its names split into words, that also appear in the theorem: pair.
Proposition 4.2 of Counterexamples to the duality conjecture for metric entropy. Words of the Lean statement, its names split into words, that also appear in the theorem: integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
Corollary 3.3 of Counterexamples to the duality conjecture for metric entropy. Words of the Lean statement, its names split into words, that also appear in the theorem: nonempty.

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 105 s; peak memory 2.5 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 329’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: For all a, b ≥ 1 there are n > 0 and an origin-symmetric convex body K ⊆ ℝⁿ (compact, convex, symmetric, nonempty interior) with b·log N(B_∞°, a⁻¹K°) < log N(K, B_∞), where N is the least number of translates covering (coveringNumber, ℕ-valued sInf), and Coverable hypotheses make both numbers finite and positive; this is the violation of every universal-constant comparison with the cube as covering body.
Definitions the group asks readers to compare with the paper: coveringNumber A B := sInf {M | ∃ centers : Fin M → ℝⁿ, Covers A B centers} and polar K := {y | ∀ x ∈ K, ⟨x,y⟩ ≤ 1} are hand-built (ℝⁿ as ι → ℝ with the sum pairing), standard; sInf ∅ = 0 is excluded by the stated Coverable clauses
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 formalized result disproves such a comparison: for every $a,b\ge1$, there is an origin-symmetric convex body $K$ in a positive finite dimension with $\log N(K,B_\infty)>b\log N(B_\infty^\circ,a^{-1}K^\circ)$, where $N$ is the least number of translates in a finite cover. A further construction makes the dual-to-primal logarithmic entropy ratio tend to zero as the dimension grows.”
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

59 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: 28 to its own declarations, 25 to Mathlib’s documentation. 9 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_entropy_duality_counterexample_with_covers 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 8 other declarations of the file; with the theorem itself they span 29 lines. 6 of them are named in the statement itself.

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

abbrev RealSpaceline 17in the statementlibrary
def pairingline 19library
def cubeline 22in the statementlibrary
def polarline 25in the statementlibrary
def Coversline 28library
def Coverableline 32in the statementlibrary
def coveringNumberline 35in the statementlibrary
structure IsSymmetricConvexBodyline 38in the statementlibrary
theorem exists_entropy_duality_counterexample_with_coversline 45selected theoremlibrary

import Mathlib.Analysis.Convex.Topologyimport Mathlib.Analysis.Normed.Group.Constructionsimport Mathlib.Order.Lattice.Natimport Mathlib.Analysis.SpecialFunctions.Log.Basicnamespace OAIuniverse unoncomputable sectionnamespace MetricEntropyDualityopen Filter Topologyopen scoped BigOperators Pointwiseabbrev RealSpace (ι : Type u) := ι → ℝdef pairing {ι : Type u} [Fintype ι] (x y : RealSpace ι) : ℝ :=  ∑ i, x i * y idef cube (ι : Type u) : Set (RealSpace ι) :=  {x | ∀ i, |x i| ≤ 1}def polar {ι : Type u} [Fintype ι] (K : Set (RealSpace ι)) : Set (RealSpace ι) :=  {y | ∀ x ∈ K, pairing x y ≤ 1}def Covers {ι : Type u} {M : ℕ} (A B : Set (RealSpace ι))    (centers : Fin M → RealSpace ι) : Prop :=  ∀ x ∈ A, ∃ j, x - centers j ∈ Bdef Coverable {ι : Type u} (A B : Set (RealSpace ι)) : Prop :=  ∃ M : ℕ, ∃ centers : Fin M → RealSpace ι, Covers A B centersdef coveringNumber {ι : Type u} (A B : Set (RealSpace ι)) : ℕ :=  sInf {M : ℕ | ∃ centers : Fin M → RealSpace ι, Covers A B centers}structure IsSymmetricConvexBody {ι : Type u} [Fintype ι]    (K : Set (RealSpace ι)) : Prop where  isCompact : IsCompact K  convex : Convex ℝ K  symmetric : ∀ x, x ∈ K ↔ -x ∈ K  interior_nonempty : (interior K).Nonemptytheorem exists_entropy_duality_counterexample_with_covers    (a b : ℝ) (ha : 1 ≤ a) (hb : 1 ≤ b) :    ∃ n : ℕ, 0 < n ∧ ∃ K : Set (RealSpace (Fin n)),      IsSymmetricConvexBody K ∧      Coverable K (cube (Fin n)) ∧      Coverable (polar (cube (Fin n))) (a⁻¹ • polar K) ∧      b * Real.log (coveringNumber (polar (cube (Fin n))) (a⁻¹ • polar K) : ℝ) <        Real.log (coveringNumber K (cube (Fin n)) : ℝ) := by  sorryend MetricEntropyDualityendend 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.

Counterexamples to the duality conjecture for metric entropy

Theorem

For every a,b\ge1, there are a positive integer n and an origin-symmetric convex body K\subset\R^n such that, with L=[-1,1]^n,

\log N(K,L)>b\log N(L^\circ,a^{-1}K^\circ).

Section 1, Introduction Read it in the paper TeX source, line 30

Show the TeX

For every $a,b\ge1$, there are a positive integer $n$ and an
origin-symmetric convex body $K\subset\R^n$ such that, with
$L=[-1,1]^n$,
\[
 \log N(K,L)>b\log N(L^\circ,a^{-1}K^\circ).
\]

Configuration and file facts

File: lean/ComparatorChallenges/MetricEntropyDuality.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/MetricEntropy/Main.lean
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.Analysis.Convex.Topology, Mathlib.Analysis.Normed.Group.Constructions, Mathlib.Order.Lattice.Nat, Mathlib.Analysis.SpecialFunctions.Log.Basic
Imports from outside Mathlib: None
Local vocabulary: 8 local vocabulary declarations
Lines of Lean, without comments or blank lines: 29
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