This is a statement regarding the paper On sinks and mirrors, and the statement by Josie Hlavinka about her paper Functorial Homological Mirror Symmetry for Acyclic Cluster Varieties. For the timeline please look at the email correspondence posted on Hlavinka's website. Here are some things that I want to clarify, and I can only stand for myself.

First, I want to clarify that eventually, the overlap between the papers is essentially only the Theorem A.3 ((2) in the emails) in our Appendix A (compare to Hlavinka's Theorem 1.7), where we proved the microlocal sheaf on a skeleton is equivalent to the coherent sheaves on an acyclic cluster variety. In Theorem A.2 we prove that this skeleton is the skeleton of another cluster variety, which is not claimed by Hlavinka. In particular, our example 3.12 shows that her Ansatz(*) and Cor 1.9 do not hold in general. In the main part of our paper, we develop a completely different method, which proved a much stronger result: a homological mirror symmetry statement for all braid varieties, and for Fock-Goncharov spaces (irregular character varieties) of SL_2/PGL_2 on any compact surface with boundary. For more details about what we have proved, the readers are encouraged to read the papers.

Second, I want to clarify that all the results were obtained by us (and AI) independently. I was not aware of the talk before I got the result. Later AI pointed this video to me, but since in the talk Hlavinka did not announce her results, we were not able to determine what she precisely proved (so Vivek sent the first email to ask.) I attached the AI document (the essential results there are eventually written in the Appendix A of our paper) that mentioned in Hlavinka's statement, and also in the email correspondence. It only cited the talk to clarify that it was "not used as an input". The readers are welcome to look at the file and the talk. I was not aware of the conversation between Vivek and Hlavinka in Berkeley. In the proposal posted, she conjectured that positroid Grassmannians are self-mirror. However, mirror symmetry for open Grassmannians, Braid varieties and cluster varieties has been expected for a long time (see e.g. Rietsch--Williams, CGGLSS, Prop. 6.14, Argüz-Bousseau Rmk 5.24, Fock--Goncharov, and GHKK). Both Vivek and I have been working in this direction for many years. In particular, the cases of Gr(2, 5) and Gr(2, 6), and some other irregular character varieties, were proved in our paper with Beimler and Olsen. So this project is a natural continuation of our previous work.

Finally, I want to clarify that our paper is not just an AI paper. Even for Appendix A, in the email Vivek said it was proved by AI "basically by itself", but I do not think this is quite correct (since I am the one talking to AI). I have been using AI to compute different Weinstein skeleta for months (see, for example, FS-FLTZ, which was originally my undergraduate project 10 years ago). On Oct 17, I talked about this with Vivek in a cafe in Odense. So we started to use AI to compute the skeleton of open Grassmannians (prompted with my previous calculations and some of Vivek's ideas), and eventually it was able to get the skeleton of acyclic cluster varieties. But during the week of Oct 19, i.e. before Vivek sent the email on Oct 26, we prompted many of our own ideas and did a lot of work, after which we were finally able to prove the main part of our paper, using a completely different method. Both Vivek and I think this is a very exciting result, since it solves some of many questions (or maybe of this field) for many years. For more details about the use of AI, please see the AI statement in our paper, which is very honest.

AI disclosure: this statement is purely written by human.

The AI document mentioned in the email correspondence.