Rethlas helps disprove two major conjectures about Kazhdan–Lusztig polynomials of matroids
Ronnie Cheng and Shurui Liu show that, over every finite field, there exist representable matroids whose Kazhdan–Lusztig polynomials are not unimodal (arXiv:2607.24186). This disproves both the real-rootedness conjecture (Gedeon–Proudfoot–Young, 2017) and the log-concavity conjecture (Elias–Matherne–Proudfoot–Wang, 2014). These conjectures had attracted significant attention and had been established for many special classes of matroids before these counterexamples were found. The counterexamples are constructed by deleting points from finite projective geometries, a method that is interesting and inspiring in its own right.
The work was carried out using Rethlas, with GPT-5.6 Pro Sol as its base model. Tom Braden and Nick Proudfoot announced the result in their ICM 2026 talk, “Intersection cohomology without spaces.” Proudfoot was one of the mathematicians who originally formulated the real-rootedness conjecture. The result is also recorded in a MathOverflow thread on AI contributions to major mathematical developments. Jens Eberhardt independently obtained a counterexample—a self-dual rank-12 matroid—to the same conjectures.
Also this month: Cheng and Liu posted “Tangent classes of matroids and wonderful compactifications” (arXiv:2607.05835). The paper’s main mathematical content was produced autonomously by Danus, a multi-agent reasoning system built on Rethlas’s worker–verifier core and equipped with fact-graph memory. Danus solved the problem before the human-authored version (arXiv:2606.22650) became public. The experiment is documented in Appendix B.