BusyBeaver(6) Is Quite Large

So what is the richest logic whose proofs can be enumerated with only a five state TM?

That entirely depends on how you want to interpret a finite binary string as an enumeration of logic proofs?!

While that question depends on what you count as an 'enumeration', there's the related question of "What's the richest logic that cannot prove the halting status of all 5-state TMs?" That is, what's the richest logic that some 5-state TM's halting status is independent of?

I've pondered that version of the question a bit, but I couldn't get very far due to my lack of expertise in first-order logic. What I do know is that Skelet #17 [0] is one of the toughest machines to prove non-halting on a mathematical level [1], so any theory sufficient to prove that Skelet #17 doesn't halt is likely sufficient to decide the rest of the 5-state machines.

[0] https://bbchallenge.org/1RB---_0LC1RE_0LD1LC_1RA1LB_0RB0RA

[1] https://arxiv.org/abs/2407.02426