Source authenticated

The unrestricted infinite matroid intersection and packing/covering conjectures

Theorem 1.1: in ZFC there are a countably infinite set $E$ and two self-dual infinite matroids $M_0,M_1$ on $E$ with $I_0\cup I_1\ne E$ for all independent $I_0,I_1$ ; the pair has no packing/covering partition, so the infinite matroid packing/covering conjecture is false. Corollary 1.2: the same partitional pair fails intersection, refuting the unrestricted infinite Matroid Intersection Conjecture and answering Joo's Question 1.5 negatively. The examples are neither finitary nor cofinitary, so Nash-Williams' original conjecture for finitary matroids and the cofinitary cases are untouched. The local building block also answers the Bowler-Geschke ZFC question on self-dual uniform matroids (separate entry).

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The unrestricted infinite matroid intersection and packing/covering conjectures · Infinite matroid intersection

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The unrestricted infinite matroid intersection and packing/covering conjectures

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against the conjectures as the paper quotes them from Bowler-Carmesin and Joo. formalization.yaml lists ComparatorChallenges/InfiniteMatroid.json with declaration OAI.InfiniteMatroidCounterexample.main in OAI/Combinatorics/InfiniteMatroid/Main.lean (file exists at the pinned commit). Its statement, read here, gives two Mathlib matroids (which allow infinite ground sets with the maximality axiom) on a countable infinite type, both self-dual, with no independent cover $I_0\cup I_1$ of the ground set and no packing/covering partition: the headline disproof. lean/docs/185.md also links InfiniteMatroidCorollaries.json (module OAI.Combinatorics.InfiniteMatroid.Corollaries, file exists, not in the formalization catalogue), whose statement adds that both matroids are partitional and fail intersection, and that the separate covering and packing conjectures fail. Not rebuilt here.

Source ↗

Lean corollaries (partitional intersection, covering, packing)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Bowler-Carmesin, Matroid intersection, base packing and base covering for infinite matroids

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

The unrestricted infinite matroid intersection and packing/covering conjectures

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean proof (OAI.InfiniteMatroidCounterexample.main)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean corollaries (partitional intersection, covering, packing)

lean artifact · pending

Artifact ↗

Lean proof (OAI.InfiniteMatroidCounterexample.main)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

The unrestricted infinite matroid intersection and packing/covering conjectures parent of this event

This event attributed to Unreleased internal OpenAI model

VibeMathed record: The unrestricted infinite matroid intersection and packing/covering conjectures evidence for this event

The unrestricted infinite matroid intersection and packing/covering conjectures evidence for this event

Bowler-Carmesin, Matroid intersection, base packing and base covering for infinite matroids evidence for this event

Lean proof (OAI.InfiniteMatroidCounterexample.main) evidence for this event

Lean corollaries (partitional intersection, covering, packing) evidence for this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph