Lean statement

Quantitative renorming of real $\ell_1$

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.DiscreteFree.real_l1_quantitative_renorming

Lean statement file: RealL1Renorming.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 330 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.

def RealL1RenormingStatement : Prop :=  ∀ p : ℕ, 1 ≤ p → ∃ N : AddGroupNorm RealL1,    (∀ (r : ℝ) (c : RealL1), N (r • c) = |r| * N c) ∧    (∀ c : RealL1, (1 / 2 : ℝ) * ‖c‖ ≤ N c ∧ N c ≤ blockBound p * ‖c‖) ∧    (let m : Module ℝ RealL1 := inferInstance     letI : NormedAddCommGroup RealL1 := N.toNormedAddCommGroup     ∃ s : NormedSpace ℝ RealL1, s.toModule = m ∧       letI := s       CompleteSpace RealL1 ∧       HasBAP RealL1 (2 * blockBound p) ∧ ¬ HasBAP RealL1 (p : ℝ) ∧         (p = 1 → ¬ HasBAP RealL1 1))⋮theorem real_l1_quantitative_renorming : RealL1RenormingStatement := 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.DiscreteFree.HasBAP line 19

def HasBAP (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] (Λ : ℝ) : Prop :=  ∀ C : Set X, IsCompact C → ∀ ε : ℝ, 0 < ε →    ∃ T : X →L[ℝ] X, Module.Finite ℝ (LinearMap.range T.toLinearMap) ∧      ‖T‖ ≤ Λ ∧ ∃ η : ℝ, η < ε ∧ ∀ u ∈ C, ‖T u - u‖ ≤ η

definition OAI.DiscreteFree.blockBound line 24

def blockBound (p : ℕ) : ℝ := 300 * (p : ℝ)^2 + 150 * (p : ℝ) + 5

definition OAI.DiscreteFree.RealL1 line 28

def RealL1 : Type := lp (fun _ : ℕ => ℝ) (1 : ENNReal)

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: real, map, integer, 1/2, norm and bap. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equivalent, fails and ell.

Corollary 1.3 of A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

Corollary

For each integer p\ge1, real \ell_1 has an equivalent norm \norm{\cdot}_p such that

\tfrac12\norm{c}_1\le\norm{c}_p\le B_p\norm{c}_1,

and (\ell_1,\norm{\cdot}_p) has 2B_p-BAP but fails p-BAP. For p=1, this norm fails MAP.

Section 1, Introduction Read it in the paper TeX source, line 119

Show the TeX

For each integer $p\ge1$, real $\ell_1$ has an equivalent norm
$\norm{\cdot}_p$ such that
\[
 \tfrac12\norm{c}_1\le\norm{c}_p\le B_p\norm{c}_1,
\]
and $(\ell_1,\norm{\cdot}_p)$ has $2B_p$-BAP but fails $p$-BAP.
For $p=1$, this norm fails MAP.

Other possible paper matches, closest wording first

Theorem 1.1 of A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, bap and space. Found in the paper’s title, which its main theorem states: bounded. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: fails.
Theorem 1.2 of A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, bounded, integer, linear, 1/2, norm and one more.
Proposition 2.1 of A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: map, linear, 1/2, spaces, norm and bap. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ell.

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 81 s; peak memory 6.2 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 330’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: MainStatement (inside source_endpoints) gives a countable metric space M with all distinct distances ≥ 1, unbounded and not proper, such that the real Lipschitz-free space FreeSpace o has HasAP but ¬ HasBAP for every Λ ≥ 1, exactly the summary; BlockStatement (finite quantitative blocks) and the ℓ¹ renorming with 2B_p-BAP but not p-BAP are the retained earlier results.
Definitions the group asks readers to compare with the paper: FreeSpace o := closure of span {evaluation o x} ⊆ (Lip0 o →L[ℝ] ℝ) with Lip0 o normed by the slope map into ℓ^∞(M × M): hand-built Lipschitz-free space, standard; HasAP/HasBAP state approximation of the identity on every compact set by finite-rank operators (with ‖T‖ ≤ Λ for BAP)
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 earlier quantitative renorming result for real $\ell_1$ is retained: for each integer $p\ge1$, an equivalent complete norm has the $2B_p$-bounded approximation property but fails the $p$-bounded approximation property, where $B_p=300p^2+150p+5$.”
Quoted from OpenAI’s Lean note for this family

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

Supporting material

The complete Lean statement file

55 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: 18 to its own declarations, 23 to Mathlib’s documentation. 5 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. 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 real_l1_quantitative_renorming 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 4 of the file’s 6 other declarations; with the theorem itself they span 19 lines. 1 of them is named in the statement itself.

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

def HasBAPline 19library
def blockBoundline 24library
def RealL1line 28library
def RealL1RenormingStatementline 36in the statementlibrary
theorem real_l1_quantitative_renormingline 48selected theoremlibrary

Other Lean rules that may affect the statement 2

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance NormedAddCommGroup RealL1 line 30
instance NormedSpace ℝ RealL1 line 33

import Mathlib.Analysis.Normed.Lp.lpSpaceimport Mathlib.Analysis.Normed.Operator.Basicimport Mathlib.RingTheory.Finiteness.Defsnamespace OAInoncomputable section/-!# Quantitative renorming of real ℓ¹ with fixed vector-space operations-/open scoped BigOperators ENNRealnamespace DiscreteFreevariable {M : Type*} [MetricSpace M]def HasBAP (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] (Λ : ℝ) : Prop :=  ∀ C : Set X, IsCompact C → ∀ ε : ℝ, 0 < ε →    ∃ T : X →L[ℝ] X, Module.Finite ℝ (LinearMap.range T.toLinearMap) ∧      ‖T‖ ≤ Λ ∧ ∃ η : ℝ, η < ε ∧ ∀ u ∈ C, ‖T u - u‖ ≤ ηdef blockBound (p : ℕ) : ℝ := 300 * (p : ℝ)^2 + 150 * (p : ℝ) + 5open scoped BigOperators ENNRealdef RealL1 : Type := lp (fun _ : ℕ => ℝ) (1 : ENNReal)noncomputable instance : NormedAddCommGroup RealL1 :=  inferInstanceAs (NormedAddCommGroup (lp (fun _ : ℕ => ℝ) (1 : ENNReal)))noncomputable instance : NormedSpace ℝ RealL1 :=  inferInstanceAs (NormedSpace ℝ (lp (fun _ : ℕ => ℝ) (1 : ENNReal)))def RealL1RenormingStatement : Prop :=  ∀ p : ℕ, 1 ≤ p → ∃ N : AddGroupNorm RealL1,    (∀ (r : ℝ) (c : RealL1), N (r • c) = |r| * N c) ∧    (∀ c : RealL1, (1 / 2 : ℝ) * ‖c‖ ≤ N c ∧ N c ≤ blockBound p * ‖c‖) ∧    (let m : Module ℝ RealL1 := inferInstance     letI : NormedAddCommGroup RealL1 := N.toNormedAddCommGroup     ∃ s : NormedSpace ℝ RealL1, s.toModule = m ∧       letI := s       CompleteSpace RealL1 ∧       HasBAP RealL1 (2 * blockBound p) ∧ ¬ HasBAP RealL1 (p : ℝ) ∧         (p = 1 → ¬ HasBAP RealL1 1))theorem real_l1_quantitative_renorming : RealL1RenormingStatement := by  sorryend DiscreteFreeendend 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.

A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

Theorem

There is a countable, unbounded, nonproper pointed metric space (M,d,o) such that d(x,y)\ge1 for distinct points and \F(M) has AP but fails \lambda-BAP for every finite \lambda\ge1.

Section 1, Introduction Read it in the paper TeX source, line 39

Show the TeX

There is a countable, unbounded, nonproper pointed metric space $(M,d,o)$
such that $d(x,y)\ge1$ for distinct points and $\F(M)$ has AP but fails
$\lambda$-BAP for every finite $\lambda\ge1$.

Theorem

For each integer p\ge1, there are a bounded countably infinite pointed metric space (M_p,d_p,o) and a finite set A_p\subseteq M_p containing o such that

d_p(x,y)\ge1\quad(x\ne y),\qquad \operatorname{diam}M_p\le B_p:=300p^2+150p+5,

and every bounded finite-rank linear operator T_0:\F(M_p)\to\F(M_p) with \norm{T_0}\le p satisfies

\max_{a\in A_p}\norm{T_0\delta_a-\delta_a}\ge\frac12.

Section 1, Introduction Read it in the paper TeX source, line 92

Show the TeX

For each integer $p\ge1$, there are a bounded countably infinite pointed
metric space $(M_p,d_p,o)$ and a finite set $A_p\subseteq M_p$ containing
$o$ such that
\[
 d_p(x,y)\ge1\quad(x\ne y),\qquad
 \operatorname{diam}M_p\le B_p:=300p^2+150p+5,
\]
and every bounded finite-rank linear operator
$T_0:\F(M_p)\to\F(M_p)$ with $\norm{T_0}\le p$ satisfies
\[
 \max_{a\in A_p}\norm{T_0\delta_a-\delta_a}\ge\frac12.
\]

Corollary

For each integer p\ge1, real \ell_1 has an equivalent norm \norm{\cdot}_p such that

\tfrac12\norm{c}_1\le\norm{c}_p\le B_p\norm{c}_1,

and (\ell_1,\norm{\cdot}_p) has 2B_p-BAP but fails p-BAP. For p=1, this norm fails MAP.

Section 1, Introduction Read it in the paper TeX source, line 119

Show the TeX

For each integer $p\ge1$, real $\ell_1$ has an equivalent norm
$\norm{\cdot}_p$ such that
\[
 \tfrac12\norm{c}_1\le\norm{c}_p\le B_p\norm{c}_1,
\]
and $(\ell_1,\norm{\cdot}_p)$ has $2B_p$-BAP but fails $p$-BAP.
For $p=1$, this norm fails MAP.

Configuration and file facts

File: lean/ComparatorChallenges/RealL1Renorming.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 330 link it.
Solution module: OAI.Analysis.LipschitzFree.Renorming
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.Normed.Lp.lpSpace, Mathlib.Analysis.Normed.Operator.Basic, Mathlib.RingTheory.Finiteness.Defs
Imports from outside Mathlib: None
Local vocabulary: 6 local vocabulary declarations
Lines of Lean, without comments or blank lines: 23
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