A math problem open for more than 35 years now has a proof, and a chatbot helped find it.
Researchers posted a preprint resolving what's called the finite-signed uniqueness problem for the basic adjoint relationship (BAR) conjecture. The conjecture asks whether a specific mathematical relationship uniquely pins down the stationary distribution of a reflected diffusion, a model class used in queueing network analysis. The result applies within stable Harrison-Reiman data where the reflection matrix satisfies a nonsingular M-matrix condition, a setting the authors show is structurally necessary. The proof was discovered with ChatGPT's assistance, the paper states, and then independently verified by the authors.
The result matters because it closes both sides of a question posed by Dai and Dieker: uniqueness holds in the M-matrix class, but not in the broader completely-S class, where the paper exhibits an explicit four-parameter, three-dimensional counterexample. That double answer, a positive result bounded by a sharp negative one, tells researchers exactly where the BAR approach runs out of power, which is more useful than either result alone.
Whether ChatGPT's role was genuine structural insight or a mechanical nudge toward a proof the authors were already close to, the paper doesn't say, and that question will matter more than any single theorem as mathematicians decide how much to trust this kind of collaboration.