Breakthrough Tracker record
Claimed counterexamples to Rota’s matroid-flat unimodality conjecture
The authors construct matroids whose numbers of flats by rank fall and then rise, contradicting the unimodality conjecture attributed to Rota in 1970. Their mechanism applies q-lifts to matroids with sufficiently strong failures of flat-count log-concavity.
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- Stable ID
math-rota-matroid-flat-unimodality-counterexample-2026- Revision
math-rota-matroid-flat-unimodality-counterexample-2026.v1- Field
- Mathematics · Combinatorics and matroid theory
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed counterexamples
- Last checked
AI role
The authors say ChatGPT 5.6 Pro suggested a q-lift example during a search for a weaker non-convexity counterexample. They generalized that idea into the proof and state that the paper was written entirely by the authors.
Record details
- Problem or result
- Rota’s conjecture that the Whitney numbers of the second kind are unimodal for every matroid
- Authors
- Alexander Divoux, Chayim Lowen and Shouda Wang
- Institutions
- Princeton University
- Result date
- Preprint submitted July 24, 2026
Why it matters
The conjecture is a long-standing structural prediction about matroids. The construction also clarifies how a failure of the stronger Mason log-concavity conjecture can be amplified into a failure of unimodality itself.
Limits
This remains a first-version preprint without conventional peer review. A specialist MathOverflow assessment now describes the result as a disproof of Rota’s conjecture; that is meaningful independent reception, but it is not a line-by-line review of the manuscript’s infinite construction.
Sources
- Primary: Divoux, Lowen and Wang, Matroid flat counts are not unimodal
- Independent specialist assessment: Luis Ferroni’s MathOverflow assessment of the matroid-flat counterexamples
Correction and revision history
- 2026-07-27 — Added as a provisional claimed disproof after full-text review, with the construction, AI discovery role and absent independent assessment recorded.
- 2026-07-29 — Added an independent specialist assessment that treats the construction as a disproof; conventional peer review is still absent, so provisional status is retained.
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