Lean statement

Bi-Lipschitz absorption of $c_0$

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.C0Absorption.main_result

Lean statement file: C0Absorption.leanShow on the mapFamily 324. OpenAI lists this as a main result in its Lean catalogue.

See the closest candidates 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.

def MainConclusion : Prop :=  ∃ (Z : Type) (_ : NormedAddCommGroup Z) (_ : NormedSpace ℝ Z),    CompleteSpace Z ∧ TopologicalSpace.SeparableSpace Z ∧ NoLinearC0 Z ∧    (∃ F : Z × C0 → Z, Function.Surjective F ∧ BiLip F) ∧    (∃ f : C0 → Z, BiLip f) ∧ MetricUniversal Z ∧    ¬ Nonempty ((Z × C0) ≃L[ℝ] Z)⋮theorem main_result : MainConclusion := 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.C0Absorption.C0 line 9
```
abbrev C0 := C₀(ℕ, ℝ)
```

definition OAI.C0Absorption.BiLip line 11

def BiLip {X Y : Type*} [MetricSpace X] [MetricSpace Y] (f : X → Y) : Prop :=  ∃ c C : ℝ, 0 < c ∧ c ≤ C ∧    ∀ x y, c * dist x y ≤ dist (f x) (f y) ∧ dist (f x) (f y) ≤ C * dist x y

definition OAI.C0Absorption.NoLinearC0 line 15

def NoLinearC0 (Z : Type*) [NormedAddCommGroup Z] [NormedSpace ℝ Z] : Prop :=  ∀ T : C0 →L[ℝ] Z, ¬ ∃ c : ℝ, 0 < c ∧ ∀ x, c * ‖x‖ ≤ ‖T x‖

definition OAI.C0Absorption.MetricUniversal line 18

def MetricUniversal (Z : Type*) [MetricSpace Z] : Prop :=  ∀ (M : Type) [MetricSpace M] [TopologicalSpace.SeparableSpace M],    ∃ f : M → Z, BiLip f

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements in Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0 come first because the release’s name for this Lean statement shares words with that paper’s title.

Other possible paper matches, closest wording first

Corollary 8.1 of Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0. Words of the Lean statement, its names split into words, that also appear in the theorem: linear, metric, bi, separable, real and space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equivalent, copy, linearly, isomorphic, contains, lipschitz and one more.
Theorem 1.1 of Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0. Words of the Lean statement, its names split into words, that also appear in the theorem: linear, bi, separable, real and space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: containing, isomorphic, lipschitz, closed, subspace and banach.
Proposition 4.3 of Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0. Words of the Lean statement, its names split into words, that also appear in the theorem: bi. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonlinear and lipschitz.
Proposition 7.5 of Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0. Words of the Lean statement, its names split into words, that also appear in the theorem: lip and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Proposition 7.6 of Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0. Words of the Lean statement, its names split into words, that also appear in the theorem: space. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: linearly, copy, contains, isomorphic and banach.

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 309 s; peak memory 6.3 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 324’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: LipschitzEquivalence.main: ∃ separable real Banach spaces X, Y and a bijection Ψ with (4/21)‖s-t‖ ≤ ‖Ψ s - Ψ t‖ ≤ (76/25)‖s-t‖, IsEmpty (X ≃L[ℝ] Y), X ⊇ isometric c₀(ℓ₂), Y has no bounded-below linear image of c₀(ℓ₂): exactly the summary's bi-Lipschitz-equivalent but not linearly isomorphic pair. C0Absorption adds the second paper's single space Z (no linear c₀, Z × c₀ bi-Lipschitz onto Z, metrically universal, ¬ Nonempty ((Z × c₀) ≃L Z)).
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 constructs one separable real Banach space $X$ that is bi-Lipschitz equivalent to $X\times c_0$ but contains no closed linear subspace isomorphic to $c_0$. The same space contains a bi-Lipschitz copy of $c_0$, is bi-Lipschitz universal for separable metric spaces, and is not linearly isomorphic to $X\times c_0$. Thus nonlinear absorption of $c_0$ does not force a linear copy of it.”
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

33 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: 10 to its own declarations, 17 to Mathlib’s documentation. 6 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 main_result 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 5 other declarations of the file; with the theorem itself they span 16 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.C0Absorption are listed without it.

abbrev C0line 9library
def BiLipline 11library
def NoLinearC0line 15library
def MetricUniversalline 18library
def MainConclusionline 22in the statementlibrary
theorem main_resultline 29selected theoremlibrary

import Mathlibnamespace OAInamespace C0Absorptionopen scoped ZeroAtInftyabbrev C0 := C₀(ℕ, ℝ)def BiLip {X Y : Type*} [MetricSpace X] [MetricSpace Y] (f : X → Y) : Prop :=  ∃ c C : ℝ, 0 < c ∧ c ≤ C ∧    ∀ x y, c * dist x y ≤ dist (f x) (f y) ∧ dist (f x) (f y) ≤ C * dist x ydef NoLinearC0 (Z : Type*) [NormedAddCommGroup Z] [NormedSpace ℝ Z] : Prop :=  ∀ T : C0 →L[ℝ] Z, ¬ ∃ c : ℝ, 0 < c ∧ ∀ x, c * ‖x‖ ≤ ‖T x‖def MetricUniversal (Z : Type*) [MetricSpace Z] : Prop :=  ∀ (M : Type) [MetricSpace M] [TopologicalSpace.SeparableSpace M],    ∃ f : M → Z, BiLip fdef MainConclusion : Prop :=  ∃ (Z : Type) (_ : NormedAddCommGroup Z) (_ : NormedSpace ℝ Z),    CompleteSpace Z ∧ TopologicalSpace.SeparableSpace Z ∧ NoLinearC0 Z ∧    (∃ F : Z × C0 → Z, Function.Surjective F ∧ BiLip F) ∧    (∃ f : C0 → Z, BiLip f) ∧ MetricUniversal Z ∧    ¬ Nonempty ((Z × C0) ≃L[ℝ] Z)theorem main_result : MainConclusion := by sorryend C0Absorptionend 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.

Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic

Theorem

There are separable real Banach spaces X,Y and a bijection \Psi:X\to Y such that, for every s,t\in X,

\frac4{21}\|s-t\|_X \le \|\Psi(s)-\Psi(t)\|_Y \le \frac{76}{25}\|s-t\|_X.

The space X contains a linearly isometric copy of c_0(\ell_2), while Y contains no closed subspace linearly isomorphic to c_0(\ell_2). In particular, X and Y are not linearly isomorphic.

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

Show the TeX

There are separable real Banach spaces \(X,Y\) and a bijection
\(\Psi:X\to Y\) such that, for every \(s,t\in X\),
\begin{equation}

 \frac4{21}\|s-t\|_X
 \le \|\Psi(s)-\Psi(t)\|_Y
 \le \frac{76}{25}\|s-t\|_X.
\end{equation}
The space \(X\) contains a linearly isometric copy of \(c_0(\ell_2)\), while
\(Y\) contains no closed subspace linearly isomorphic to
\(c_0(\ell_2)\). In particular, \(X\) and \(Y\) are not linearly
isomorphic.

Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0

Theorem

There is a separable real Banach space Z containing no closed linear subspace isomorphic to c_0, and an onto bi-Lipschitz map

F:Z\oplus_\infty c_0\longrightarrow Z.

Section 1, Introduction Read it in the paper TeX source, line 51

Show the TeX

There is a separable real Banach space \(Z\) containing no closed linear
subspace isomorphic to \(c_0\), and an onto bi-Lipschitz map
\[
                 F:Z\oplus_\infty c_0\longrightarrow Z.
\]

Configuration and file facts

File: lean/ComparatorChallenges/C0Absorption.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/C0Absorption/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: 5 local vocabulary declarations
Lines of Lean, without comments or blank lines: 16
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

Bi-Lipschitz absorption of $c_0$

Family 324: Lipschitz equivalent Banach spaces need not be linearly isomorphic

No paper statement selected

no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

The Lean statement says

def MainConclusion : Prop :=  ∃ (Z : Type) (_ : NormedAddCommGroup Z) (_ : NormedSpace ℝ Z),    CompleteSpace Z ∧ TopologicalSpace.SeparableSpace Z ∧ NoLinearC0 Z ∧    (∃ F : Z × C0 → Z, Function.Surjective F ∧ BiLip F) ∧    (∃ f : C0 → Z, BiLip f) ∧ MetricUniversal Z ∧    ¬ Nonempty ((Z × C0) ≃L[ℝ] Z)⋮theorem main_result : MainConclusion := by sorry

Recorded comparisons with the paper

Atlas records no contributed comparison of this statement with the paper.

Contributed Comparator run reports

Atlas records no contributed report of running Comparator for this statement file.

Compare this statement with the paper Share a Comparator run report Suggest a change

The Lean statement

Definitions needed to read it

No confident match with a paper statement

Other possible paper matches, closest wording first

Reported Comparator runs

Carlos Toledo’s cert-machine

MathVet

Neal McBurnett’s openai-math-lean-tooling

Do family 324’s Lean statements express the advertised result?

Supporting material

The complete Lean statement file

Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic

Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0

Statement comparison

Recorded comparisons with the paper

Contributed Comparator run reports