Lean statement

Infinite matroid packing/covering counterexample

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Theorem selected for verification: OAI.InfiniteMatroidCounterexample.main

Lean statement file: InfiniteMatroid.leanShow on the mapFamily 185. 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 main :    Countable E ∧ Infinite E ∧    ∃ M₀ M₁ : Matroid E,      M₀.E = Set.univ ∧ M₁.E = Set.univ ∧      M₀.dual = M₀ ∧ M₁.dual = M₁ ∧      (∀ I₀ I₁ : Set E, M₀.Indep I₀ → M₁.Indep I₁ → I₀ ∪ I₁ ≠ Set.univ) ∧      ¬ HasPackingCovering M₀ M₁ := 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.InfiniteMatroidCounterexample.D line 8

abbrev D := (m : ℕ) × ((Fin m → Bool) → Bool)

abbreviation OAI.InfiniteMatroidCounterexample.E line 9
```
abbrev E := ℤ × D
```

definition OAI.InfiniteMatroidCounterexample.contractOnto line 11

def contractOnto {α : Type*} (M : Matroid α) (C : Set α) : Matroid α :=  (M.dual ↾ C).dual

definition OAI.InfiniteMatroidCounterexample.HasPackingCovering line 14

def HasPackingCovering {α : Type*} (M₀ M₁ : Matroid α) : Prop :=  ∃ P C S₀ S₁ I₀ I₁ : Set α,    Disjoint P C ∧ P ∪ C = M₀.E ∧ M₁.E = M₀.E ∧    S₀ ⊆ P ∧ S₁ ⊆ P ∧ Disjoint S₀ S₁ ∧    (M₀ ↾ P).Spanning S₀ ∧ (M₁ ↾ P).Spanning S₁ ∧    I₀ ⊆ C ∧ I₁ ⊆ C ∧    (contractOnto M₀ C).Indep I₀ ∧    (contractOnto M₁ C).Indep I₁ ∧ I₀ ∪ I₁ = C

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: covering, packing, matroid and infinite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: countably, ground, two, partition, pair, conjecture and independent.

Theorem 1.1 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture

Theorem

In ZFC, there exist a countably infinite set E and two infinite matroids M_0,M_1 on E such that M_i^*=M_i on the same labelled ground set for i=0,1, and

I_0\cup I_1\ne E \quad\text{for every independent }I_0\text{ in }M_0 \text{ and independent }I_1\text{ in }M_1.

The pair (M_0,M_1) has no packing/covering partition. Consequently, the infinite matroid packing/covering conjecture is false.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

In ZFC, there exist a countably infinite set $E$ and two infinite matroids
$M_0,M_1$ on $E$ such that $M_i^*=M_i$ on the same labelled ground set for
$i=0,1$, and
\[
 I_0\cup I_1\ne E
 \quad\text{for every independent }I_0\text{ in }M_0
 \text{ and independent }I_1\text{ in }M_1.
\]
The pair $(M_0,M_1)$ has no packing/covering partition. Consequently, the
infinite matroid packing/covering conjecture is false.

Other possible paper matches, closest wording first

Corollary 1.2 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: countable, infinite and matroids. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: partitional, intersection, constructed, ground, two, partition and four more.
Corollary 7.1 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: covering and packing. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unrestricted and conjectures.
Theorem 5.5 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: dual and matroid. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: independent.
Proposition 5.4 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: infinite.
Proposition 4.6 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.

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 48 s; peak memory 6.1 GiB).
Reported 10 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 185’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: InfiniteMatroid.main gives countable infinite E, two self-dual matroids with ∀ I₀ I₁ indep, I₀ ∪ I₁ ≠ univ and ¬ HasPackingCovering M₀ M₁; partitional_intersection_counterexample adds IsPartitional M₀ ∧ IsPartitional M₁ ∧ ¬ HasIntersection M₀ M₁ (answering Joó's partitional question), and separate_covering_packing_counterexamples refutes the covering and packing conjectures separately. Uses Mathlib's infinite Matroid; the conjecture predicates (HasIntersection, HasPackingCovering, contractOnto M C := (M.dual ↾ C).dual) are written out by hand and look standard.
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 formalization constructs two self-dual partitional matroids on one countably infinite ground set that admit neither an independent covering nor a packing/covering partition. The same pair has no intersection witness, so it also refutes the unrestricted infinite matroid intersection conjecture.”
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

34 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: 8 to its own declarations, 17 to Mathlib’s documentation. 5 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 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 4 other declarations of the file; with the theorem itself they span 20 lines. 2 of them are named in the statement itself.

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

abbrev Dline 8library
abbrev Eline 9in the statementlibrary
def contractOntoline 11library
def HasPackingCoveringline 14in the statementlibrary
theorem mainline 23selected theoremlibrary

import Mathlibnamespace OAInamespace InfiniteMatroidCounterexampleopen Matroidabbrev D := (m : ℕ) × ((Fin m → Bool) → Bool)abbrev E := ℤ × Ddef contractOnto {α : Type*} (M : Matroid α) (C : Set α) : Matroid α :=  (M.dual ↾ C).dualdef HasPackingCovering {α : Type*} (M₀ M₁ : Matroid α) : Prop :=  ∃ P C S₀ S₁ I₀ I₁ : Set α,    Disjoint P C ∧ P ∪ C = M₀.E ∧ M₁.E = M₀.E ∧    S₀ ⊆ P ∧ S₁ ⊆ P ∧ Disjoint S₀ S₁ ∧    (M₀ ↾ P).Spanning S₀ ∧ (M₁ ↾ P).Spanning S₁ ∧    I₀ ⊆ C ∧ I₁ ⊆ C ∧    (contractOnto M₀ C).Indep I₀ ∧    (contractOnto M₁ C).Indep I₁ ∧ I₀ ∪ I₁ = Ctheorem main :    Countable E ∧ Infinite E ∧    ∃ M₀ M₁ : Matroid E,      M₀.E = Set.univ ∧ M₁.E = Set.univ ∧      M₀.dual = M₀ ∧ M₁.dual = M₁ ∧      (∀ I₀ I₁ : Set E, M₀.Indep I₀ → M₁.Indep I₁ → I₀ ∪ I₁ ≠ Set.univ) ∧      ¬ HasPackingCovering M₀ M₁ := by  sorryend InfiniteMatroidCounterexampleend 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 Counterexample to the Infinite Matroid Packing/Covering Conjecture

Theorem

In ZFC, there exist a countably infinite set E and two infinite matroids M_0,M_1 on E such that M_i^*=M_i on the same labelled ground set for i=0,1, and

I_0\cup I_1\ne E \quad\text{for every independent }I_0\text{ in }M_0 \text{ and independent }I_1\text{ in }M_1.

The pair (M_0,M_1) has no packing/covering partition. Consequently, the infinite matroid packing/covering conjecture is false.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

In ZFC, there exist a countably infinite set $E$ and two infinite matroids
$M_0,M_1$ on $E$ such that $M_i^*=M_i$ on the same labelled ground set for
$i=0,1$, and
\[
 I_0\cup I_1\ne E
 \quad\text{for every independent }I_0\text{ in }M_0
 \text{ and independent }I_1\text{ in }M_1.
\]
The pair $(M_0,M_1)$ has no packing/covering partition. Consequently, the
infinite matroid packing/covering conjecture is false.

Corollary

The same pair (M_0,M_1) constructed in Theorem thm:main consists of partitional matroids and does not satisfy intersection. More explicitly, for every set J independent in both M_0 and M_1 and every partition J=J_0\dot\cup J_1,

\cl_{M_0}(J_0)\cup\cl_{M_1}(J_1)\ne E.

Consequently, the unrestricted infinite Matroid Intersection Conjecture is false in ZFC. The pair also gives a negative ZFC answer to Joó’s Question 1.5 for two partitional matroids on a common countable ground set.

Section 1, Introduction Read it in the paper TeX source, line 103

Show the TeX

The same pair $(M_0,M_1)$ constructed in Theorem~\ref{thm:main} consists of
partitional matroids and does not satisfy intersection. More explicitly,
for every set $J$ independent in both $M_0$ and $M_1$ and every partition
$J=J_0\dot\cup J_1$,
\[
 \cl_{M_0}(J_0)\cup\cl_{M_1}(J_1)\ne E.
\]
Consequently, the unrestricted infinite Matroid Intersection Conjecture is
false in ZFC. The pair also gives a negative ZFC answer to Jo\'o's
Question~1.5 for two partitional matroids on a common countable ground set.

Configuration and file facts

File: lean/ComparatorChallenges/InfiniteMatroid.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/InfiniteMatroid/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: 4 local vocabulary declarations
Lines of Lean, without comments or blank lines: 20
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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