Counterexamples to infinite matroid intersection and packing/covering
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Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
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.
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
The same pair (M_0,M_1) constructed in Theorem 1.1 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.
No Lean statement in the release is paired with this theorem.
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Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We construct in ZFC two self-dual partitional matroids on a countably infinite common ground set that admit neither a packing/covering partition nor an intersection witness. This disproves the unrestricted infinite matroid packing/covering and intersection conjectures and answers Joó’s question for two partitional matroids negatively. The examples are neither finitary nor cofinitary.
Two Lean statements in this family
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For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s 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.
The words they share: covering, packing, matroid, infinite, countably, ground, two, partition, pair, conjecture, independent.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
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.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
The infinite matroid packing/covering conjecture predicts a partition of a common ground set into parts admitting the corresponding packing and covering. 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.
Separate formalized consequences give counterexamples to the individual covering and packing conjectures. The construction is in ordinary set theory with Choice and assumes no finitary restriction on the matroids.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: two self-dual partitional matroids on a countable set with neither an independent covering nor a packing/covering partition.
Partitional intersection, covering, and packing counterexamplesInfiniteMatroidCorollaries
Unreviewed AI draft: states a consequence of a paper’s theorem. States the paper’s corollaries: the same pair refutes the infinite matroid intersection conjecture, and separate counterexamples refute the covering and packing conjectures.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Other people’s accounts of running proof-verification software, with links to their reports.
VolMax-Studio, VolMax-Studio/p10-openai-math-verification-pilot: pilot verification of one family 185 theorem, 7 October 2026. Source
Comparator and a fresh Lean build accept the infinite-matroid counterexample theorem; four negative controls are rejected.
VolMax-Studio, VolMax-Studio/p10-openai-math-verification-pilot: pilot verification of one family 185 theorem, 7 October 2026. Source
Human-ratified pilot: provenance binding, an independent lake build, Comparator with the default kernel and dependency-integrity checks all passed for OAI.InfiniteMatroidCounterexample.main; a changed statement, a sorry, a wrong binding and an injected axiom were each rejected. Run at adc7f124, where this challenge file is unchanged at the atlas’s pin.
What these papers cite in the release, and what cites them
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Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
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References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
VolMax-Studio, VolMax-Studio/p10-openai-math-verification-pilot: pilot verification of one family 185 theorem, 7 October 2026. Source
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Identify the theorem the catalogue describes
Family 185The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 185Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In A Counterexample to the Infinite Matroid Packing/Covering Conjecture, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
conjecture, countably, covering, ground, infinite, matroid, matroids, packing, same
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Theorem 5.5
In A Counterexample to the Infinite Matroid Packing/Covering Conjecture, 24 September 2026
How it was found
Its words were compared with the words of the Lean statement InfiniteMatroidCorollaries.
Words they share
dual, family, matroid
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Proposition 5.4
In A Counterexample to the Infinite Matroid Packing/Covering Conjecture, 24 September 2026
How it was found
Its words were compared with the words of the Lean statement InfiniteMatroidCorollaries.
Words they share
family, infinite
Review
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Proposition 4.6 (Interval alternatives for one class)
In A Counterexample to the Infinite Matroid Packing/Covering Conjecture, 24 September 2026
How it was found
Its words were compared with the words of the Lean statement InfiniteMatroidCorollaries.
Review
Atlas records no review confirming that the two statements mean the same thing.
Independent assessment
MathVet
Their term
“full”
What their term means
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
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.
The challenge statements state the headline as OpenAI wrote it.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 48 s; peak memory 6.1 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 51 s; peak memory 6.1 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.
Partitional intersection, covering, and packing counterexamples
Role
the release lists it as a supporting statement
Paired with
No paper theorem is paired with this Lean statement.
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.