Lean statement

Cotype–cotype equivalence with approximation property

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.Cotype.mainTarget_proved

Lean statement file: Cotype.leanShow on the mapFamily 326. 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 mainTarget_proved [Nontrivial X] : MainTarget X := 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.Cotype.Cube line 12
```
abbrev Cube (n : ℕ) := Fin n → Bool
```

definition OAI.Cotype.sign line 14

def sign (b : Bool) : ℝ := if b then -1 else 1

definition OAI.Cotype.average line 16

def average {ι : Type u} [Fintype ι] {E : Type v} [AddCommMonoid E] [Module ℝ E]    (f : ι → E) : E := (Fintype.card ι : ℝ)⁻¹ • ∑ i, f i

definition OAI.Cotype.cubeL2 line 21

def cubeL2 {n : ℕ} (f : Cube n → E) : ℝ :=  Real.sqrt (average (fun ε => ‖f ε‖ ^ 2))

definition OAI.Cotype.rademacherSum line 24

def rademacherSum {n : ℕ} (x : Fin n → E) (ε : Cube n) : E :=  ∑ i, sign (ε i) • x i

definition OAI.Cotype.moment line 27

def moment {n : ℕ} (f : Cube n → E) (i : Fin n) : E :=  average (fun ε => sign (ε i) • f ε)

definition OAI.Cotype.radProjection line 30

def radProjection {n : ℕ} (f : Cube n → E) : Cube n → E :=  rademacherSum E (moment E f)

definition OAI.Cotype.KConvex line 33

def KConvex : Prop := ∃ K : ℝ, 0 ≤ K ∧ ∀ (n : ℕ) (f : Cube n → E),  cubeL2 E (radProjection E f) ≤ K * cubeL2 E f

definition OAI.Cotype.HasCotype line 36

def HasCotype (q : ℝ) : Prop := 2 ≤ q ∧ ∃ C : ℝ, 0 ≤ C ∧  ∀ (n : ℕ) (x : Fin n → E),    (∑ i, ‖x i‖ ^ q) ^ (1 / q) ≤ C * cubeL2 E (rademacherSum E x)

definition OAI.Cotype.FiniteCotype line 40

def FiniteCotype : Prop := ∃ q : ℝ, HasCotype E q

definition OAI.Cotype.ApproximationProperty line 42

def ApproximationProperty : Prop := ∀ (M : Set E), IsCompact M →  ∀ δ : ℝ, 0 < δ → ∃ S : E →L[ℝ] E,    FiniteDimensional ℝ (LinearMap.range S.toLinearMap) ∧    ∀ x ∈ M, ‖S x - x‖ < δ

definition OAI.Cotype.MainTarget line 47

def MainTarget [CompleteSpace E] [Nontrivial E] : Prop :=  ApproximationProperty E →    (KConvex E ↔ FiniteCotype E ∧ FiniteCotype (E →L[ℝ] ℝ))

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: property, real, convex, space and approximation. Found in the paper’s title, which its main theorem states: cotype. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonzero and banach.

Theorem 1.1 of The cotype–cotype conjecture under the approximation property

Theorem

Let X be a nonzero real Banach space with the ordinary approximation property. Then X is K-convex if and only if

C_q(X)<\infty \quad\hbox{and}\quad C_r(X^*)<\infty \qquad\text{for some }q,r\in[2,\infty).

The exponents q and r may differ. The forward implication holds for every real Banach space, without an approximation hypothesis.

Section 1, The characterization Read it in the paper TeX source, line 43

Show the TeX

Let \(X\) be a nonzero real Banach space with the ordinary approximation
property. Then \(X\) is \(K\)-convex if and only if
\[
C_q(X)<\infty
\quad\hbox{and}\quad
C_r(X^*)<\infty
\qquad\text{for some }q,r\in[2,\infty).
\]
The exponents \(q\) and \(r\) may differ. The forward implication holds
for every real Banach space, without an approximation hypothesis.

Other possible paper matches, closest wording first

Corollary 1.2 of The cotype–cotype conjecture under the approximation property. Words of the Lean statement, its names split into words, that also appear in the theorem: real, finite, cotype, convex and space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonzero and banach.
Theorem 4.3 of The cotype–cotype conjecture under the approximation property. Words of the Lean statement, its names split into words, that also appear in the theorem: maps, moment, integer and finite.
Corollary 6.1 of The cotype–cotype conjecture under the approximation property. Words of the Lean statement, its names split into words, that also appear in the theorem: dimensional, property and approximation.

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 326’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 disagree 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: mainTarget_proved : [CompleteSpace X] [Nontrivial X], ApproximationProperty X → (KConvex X ↔ FiniteCotype X ∧ FiniteCotype (X →L[ℝ] ℝ)) is the headline for every nonzero real Banach space with AP; the summary's 'equivalently, these cotype assumptions force nontrivial type' is the classical K-convex ⇔ nontrivial-type reformulation (Pisier), which is not stated in Lean.
Definitions the group asks readers to compare with the paper: KConvex E := ∃ K ≥ 0, ∀ n (f : Cube n → E), cubeL2 (radProjection f) ≤ K * cubeL2 f (uniform L²-boundedness of the Rademacher projection, the standard characterization) and HasCotype q := 2 ≤ q ∧ ∃ C ≥ 0, ∀ n (x : Fin n → E), (∑ ‖x i‖^q)^(1/q) ≤ C * cubeL2 (rademacherSum x): hand-built, standard (L² average in place of the L^q average, equivalent by Kahane)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: Only the 'Equivalently … nontrivial Rademacher type' sentence, a reformulation of the stated equivalence through Pisier's K-convexity theorem; no notion of type appears in the challenge.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result establishes this equivalence for every nonzero real Banach space with the approximation property: $X$ is $K$-convex exactly when both $X$ and $X^*$ have finite cotype.”
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

60 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: 23 to its own declarations, 23 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 mainTarget_proved 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 27 lines. 1 of them is named in the statement itself.

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

abbrev Cubeline 12library
def signline 14library
def averageline 16library
def cubeL2line 21library
def rademacherSumline 24library
def momentline 27library
def radProjectionline 30library
def KConvexline 33library
def HasCotypeline 36library
def FiniteCotypeline 40library
def ApproximationPropertyline 42library
def MainTargetline 47in the statementlibrary
theorem mainTarget_provedline 53selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsuniverse u vnamespace Cotypeabbrev Cube (n : ℕ) := Fin n → Booldef sign (b : Bool) : ℝ := if b then -1 else 1def average {ι : Type u} [Fintype ι] {E : Type v} [AddCommMonoid E] [Module ℝ E]    (f : ι → E) : E := (Fintype.card ι : ℝ)⁻¹ • ∑ i, f ivariable (E : Type u) [NormedAddCommGroup E] [NormedSpace ℝ E]def cubeL2 {n : ℕ} (f : Cube n → E) : ℝ :=  Real.sqrt (average (fun ε => ‖f ε‖ ^ 2))def rademacherSum {n : ℕ} (x : Fin n → E) (ε : Cube n) : E :=  ∑ i, sign (ε i) • x idef moment {n : ℕ} (f : Cube n → E) (i : Fin n) : E :=  average (fun ε => sign (ε i) • f ε)def radProjection {n : ℕ} (f : Cube n → E) : Cube n → E :=  rademacherSum E (moment E f)def KConvex : Prop := ∃ K : ℝ, 0 ≤ K ∧ ∀ (n : ℕ) (f : Cube n → E),  cubeL2 E (radProjection E f) ≤ K * cubeL2 E fdef HasCotype (q : ℝ) : Prop := 2 ≤ q ∧ ∃ C : ℝ, 0 ≤ C ∧  ∀ (n : ℕ) (x : Fin n → E),    (∑ i, ‖x i‖ ^ q) ^ (1 / q) ≤ C * cubeL2 E (rademacherSum E x)def FiniteCotype : Prop := ∃ q : ℝ, HasCotype E qdef ApproximationProperty : Prop := ∀ (M : Set E), IsCompact M →  ∀ δ : ℝ, 0 < δ → ∃ S : E →L[ℝ] E,    FiniteDimensional ℝ (LinearMap.range S.toLinearMap) ∧    ∀ x ∈ M, ‖S x - x‖ < δdef MainTarget [CompleteSpace E] [Nontrivial E] : Prop :=  ApproximationProperty E →    (KConvex E ↔ FiniteCotype E ∧ FiniteCotype (E →L[ℝ] ℝ))variable {X : Type u} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]theorem mainTarget_proved [Nontrivial X] : MainTarget X := by  sorryend Cotypeendend 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.

The cotype–cotype conjecture under the approximation property

Theorem

Let X be a nonzero real Banach space with the ordinary approximation property. Then X is K-convex if and only if

C_q(X)<\infty \quad\hbox{and}\quad C_r(X^*)<\infty \qquad\text{for some }q,r\in[2,\infty).

The exponents q and r may differ. The forward implication holds for every real Banach space, without an approximation hypothesis.

Section 1, The characterization Read it in the paper TeX source, line 43

Show the TeX

Let \(X\) be a nonzero real Banach space with the ordinary approximation
property. Then \(X\) is \(K\)-convex if and only if
\[
C_q(X)<\infty
\quad\hbox{and}\quad
C_r(X^*)<\infty
\qquad\text{for some }q,r\in[2,\infty).
\]
The exponents \(q\) and \(r\) may differ. The forward implication holds
for every real Banach space, without an approximation hypothesis.

Corollary (Uniform cotype bounds and entropy duality)

Fix q,r\in[2,\infty) and C,D\in[1,\infty). There is a finite constant \kappa=\kappa(q,r,C,D) such that every nonzero real Banach space X with AP and

C_q(X)\le C,\qquad C_r(X^*)\le D

satisfies K(X)\le\kappa. Consequently there are a,b\ge1, depending only on q,r,C,D, such that for every n\ge1, all origin-symmetric convex bodies K,L\subset\R^n, and every t>0,

\frac1b\log N(L^\circ,atK^\circ) \le \log N(K,tL) \le b\log N(L^\circ,a^{-1}tK^\circ),

provided either X_K or X_L satisfies the displayed primal and dual cotype bounds.

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

Show the TeX

Fix \(q,r\in[2,\infty)\) and \(C,D\in[1,\infty)\). There is a
finite constant \(\kappa=\kappa(q,r,C,D)\) such that every nonzero
real Banach space \(X\) with AP and
\[
C_q(X)\le C,\qquad C_r(X^*)\le D
\]
satisfies \(K(X)\le\kappa\).
Consequently there are \(a,b\ge1\), depending only on \(q,r,C,D\),
such that for every \(n\ge1\), all origin-symmetric convex bodies
\(K,L\subset\R^n\), and every \(t>0\),
\begin{equation}
\frac1b\log N(L^\circ,atK^\circ)
\le \log N(K,tL)
\le b\log N(L^\circ,a^{-1}tK^\circ),
\end{equation}
provided either \(X_K\) or \(X_L\) satisfies the displayed primal
and dual cotype bounds.

Configuration and file facts

File: lean/ComparatorChallenges/Cotype.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/Cotype/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
Imports from outside Mathlib: None
Local vocabulary: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 27
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.

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