Comment by UncombedCoconut

Mostly no: we did find some non-halting TMs that required new proofs, but none of those had the flavor of new math, per se. Indeed, we found that all but 30 of them could be proved by finite automata methods, meaning the TM's state/tape at any step could be reduced to one of finitely many states and we'd still know all we needed to know about future steps. I would argue that such a non-halting proof can't have much mathematical content. (Maybe a bit, in about the same way that an integer equation is sometimes proved unsolvable by considering it modulo n and checking every case.) Also, I learned some math I wasn't personally familiar with from the analysis of a particular machine: https://www.sligocki.com/2023/03/14/skelet-10.html (Zeckendorf's Theorem).