Busy beaver for Fractran
https://en.wikipedia.org/wiki/FRACTRAN
https://wiki.bbchallenge.org/wiki/Fractran
Work more on register machines (see register/).
Write a decider that provides Lean proofs.
For example, see oned_progress.py+oned_progress_to_lean.py and nd_progress.py+nd_progress_to_lean.py.
Try understanding the AI-written deciders (1DPC, NDPC).