Lean statement

Failure of the separable quotient assertion under the continuum hypothesis

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.SeparableQuotient.negative_main

Lean statement file: SeparableQuotientNegative.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 323 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 negative_main (hCH : CH) : ¬ SQ.{u} ℝ ∧ ¬ SQ.{u} ℂ := 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.

definition OAI.SeparableQuotient.CH line 12

def CH : Prop := Cardinal.mk ℝ = Cardinal.aleph 1

definition OAI.SeparableQuotient.HasSeparableQuotient line 15

def HasSeparableQuotient (𝕜 : Type) [RCLike 𝕜]    (X : Type u) [NormedAddCommGroup X] [NormedSpace 𝕜 X] : Prop :=  ∃ (Y : Type u) (_ : NormedAddCommGroup Y) (_ : NormedSpace 𝕜 Y),    CompleteSpace Y ∧ TopologicalSpace.SeparableSpace Y ∧      (¬ FiniteDimensional 𝕜 Y) ∧ ∃ T : X →L[𝕜] Y, Function.Surjective T

definition OAI.SeparableQuotient.SQ line 22

def SQ (𝕜 : Type) [RCLike 𝕜] : Prop :=  ∀ (X : Type u) [NormedAddCommGroup X] [NormedSpace 𝕜 X] [CompleteSpace X],    (¬ FiniteDimensional 𝕜 X) → HasSeparableQuotient 𝕜 X

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: onto, surjection, linear, bounded, norm, assertion, complex, banach and five more.

Proposition 2.2 of Relative independence of the separable quotient problem

Proposition (Evaluation criterion)

For an infinite-dimensional real or complex Banach space X, the following are equivalent:

1.
X admits a bounded linear surjection onto a separable infinite-dimensional Banach space.
2.
There is a closed infinite-dimensional subspace F\subset X^* for which R_F(X) is norm separable.

In the second assertion, F may be required to be separable.

Section 2, Separable evaluation images Read it in the paper TeX source, line 81

Show the TeX

For an infinite-dimensional real or complex Banach space $X$, the following
are equivalent:
\begin{enumerate}
\item $X$ admits a bounded linear surjection onto a separable
      infinite-dimensional Banach space.
\item There is a closed infinite-dimensional subspace $F\subset X^*$
      for which $R_F(X)$ is norm separable.
\end{enumerate}
In the second assertion, $F$ may be required to be separable.

Other possible paper matches, closest wording first

Theorem 1.1 of Relative independence of the separable quotient problem. Words of the Lean statement, its names split into words, that also appear in the theorem: hypothesis, continuum, sq, cardinal and real. Found in the paper’s title, which its main theorem states: quotient and separable. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: consistency and failure.
Theorem 9.4 of Relative independence of the separable quotient problem. Words of the Lean statement, its names split into words, that also appear in the theorem: ch, quotient, banach, space, dimensional, separable and two more.
Corollary 4.3 of Relative independence of the separable quotient problem. Words of the Lean statement, its names split into words, that also appear in the theorem: assertions, complex, banach, quotient, space, dimensional and three more.
Proposition 4.1 of Relative independence of the separable quotient problem. Words of the Lean statement, its names split into words, that also appear in the theorem: complex, banach, quotient, space, dimensional, separable and two more.
Corollary 9.5 of Relative independence of the separable quotient problem. Words of the Lean statement, its names split into words, that also appear in the theorem: norm, ch, aleph and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: counterexample.
Theorem 3.5 of Relative independence of the separable quotient problem. Words of the Lean statement, its names split into words, that also appear in the theorem: space, banach, quotient, dimensional, separable, infinite and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: full.

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 348 s; peak memory 6.4 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 323’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: negative_main (hCH : CH) : ¬ SQ.{u} ℝ ∧ ¬ SQ.{u} ℂ (CH := Cardinal.mk ℝ = aleph 1; SQ = every complete infinite-dimensional normed space has a bounded linear surjection onto an infinite-dimensional separable Banach space) is the continuum-hypothesis counterexample half over both fields. The other half of the independence claim, consistency of the positive assertion relative to a measurable cardinal, has no statement (docs: 'the positive consistency direction ... outside the selected statement').
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The positive consistency direction (from a measurable cardinal), hence the independence statement as a whole.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“This is the conditional counterexample direction. The positive consistency direction and the paper's full relative-independence conclusions are outside the selected statement.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

31 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: 4 to its own declarations, 16 to Mathlib’s documentation. 4 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 negative_main 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 3 other declarations of the file; with the theorem itself they span 11 lines. 2 of them are named in the statement itself.

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

def CHline 12in the statementlibrary
def HasSeparableQuotientline 15library
def SQline 22in the statementlibrary
theorem negative_mainline 26selected theoremlibrary

import Mathlibnamespace OAInamespace SeparableQuotientopen MeasureTheory Cardinaluniverse u/-- The continuum hypothesis. -/def CH : Prop := Cardinal.mk ℝ = Cardinal.aleph 1/-- A bounded linear surjection onto an infinite-dimensional separable Banach space. -/def HasSeparableQuotient (𝕜 : Type) [RCLike 𝕜]    (X : Type u) [NormedAddCommGroup X] [NormedSpace 𝕜 X] : Prop :=  ∃ (Y : Type u) (_ : NormedAddCommGroup Y) (_ : NormedSpace 𝕜 Y),    CompleteSpace Y ∧ TopologicalSpace.SeparableSpace Y ∧      (¬ FiniteDimensional 𝕜 Y) ∧ ∃ T : X →L[𝕜] Y, Function.Surjective T/-- The separable quotient assertion at an arbitrary ambient universe. -/def SQ (𝕜 : Type) [RCLike 𝕜] : Prop :=  ∀ (X : Type u) [NormedAddCommGroup X] [NormedSpace 𝕜 X] [CompleteSpace X],    (¬ FiniteDimensional 𝕜 X) → HasSeparableQuotient 𝕜 Xtheorem negative_main (hCH : CH) : ¬ SQ.{u} ℝ ∧ ¬ SQ.{u} ℂ := by  sorryend SeparableQuotientend 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.

Relative independence of the separable quotient problem

Theorem

For each \mathbb F\in\{\R,\C\}, ZFC proves the following implications.

(i)
If \mathfrak c is real-valued measurable, then \SQ_{\mathbb F} holds.
(ii)
If the continuum hypothesis holds, then \SQ_{\mathbb F} fails.

Consequently, if ZFC with a measurable cardinal is consistent, then \SQ_{\mathbb F} is independent of ZFC. Consistency of its failure requires only consistency of ZFC.

Section 1, Introduction Read it in the paper TeX source, line 34

Show the TeX

For each $\mathbb F\in\{\R,\C\}$, ZFC proves the following implications.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item If $\mathfrak c$ is real-valued measurable, then $\SQ_{\mathbb F}$
 holds.
 \item If the continuum hypothesis holds, then $\SQ_{\mathbb F}$ fails.
\end{enumerate}
Consequently, if ZFC with a measurable cardinal is consistent, then
$\SQ_{\mathbb F}$ is independent of ZFC. Consistency of its failure
requires only consistency of ZFC.

Corollary

If ZFC is consistent, then \SQ^{\aleph_1}_{\mathbb F} is independent of ZFC for each \mathbb F\in\{\R,\C\}.

Section 1, Introduction Read it in the paper TeX source, line 53

Show the TeX

If ZFC is consistent, then $\SQ^{\aleph_1}_{\mathbb F}$ is independent
of ZFC for each $\mathbb F\in\{\R,\C\}$.

Configuration and file facts

File: lean/ComparatorChallenges/SeparableQuotientNegative.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 323 link it.
Solution module: OAI.Analysis.SeparableQuotients.Main
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: 3 local vocabulary declarations
Lines of Lean, without comments or blank lines: 11
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