In this paper, we prove a $1/e$ guarantee for the matroid secretary problem on linear matroids (each element of a fixed optimal basis is selected with probability at least $1/e), which settles the strong secretary conjecture for that class. The result holds in the online-representation model too, where a representation over a finite field arrives one vector at a time. In the known-matroid model the argument goes further, covering matroids that admit a finitary modular extension.
But this is only part of the story. We got the proof of the main theorem in a conversation with ChatGPT-6 Astra on Tuesday, September 15. We spent the next day and a half writing it up, planning to post on Thursday morning. In the early hours of Thursday, while finalizing the submission, we found a paper by Abdi, Banihashem, Hajiaghayi, and Mittal, posted on September 16, with the same result.
We want to be clear about how close the overlap is, because the usual phrasing understates it. This isn’t two proofs that share an outline or lean on the same technique. The constructions coincide. It is, in substance, the same proof. They also report that interactions with GPT models were essential to finding their linear-matroid construction.
We are not sure what to make of this. We do not know what, if anything, a model carries over from earlier conversations, or whether this can affect how readily it produces an argument for someone else. How does essentially the same proof emerge from two separate conversations? We are posting ours because it contains some additional results, and because the episode seems worth airing. As AI systems take on a larger role in mathematical discovery, questions of independence, priority, attribution, and concurrent discovery will need answers that current norms do not yet provide.