I have a theory that all finite state machines can be monitored by a second turing machine with infinite tape to determine if the state of the first machine was repeated thus reaching the conclusion that the finite state machine must be looping. I’m fairly confident that this is not a new idea but I could not find anything relevant through all my searches and reading here. Knowing that all modern digital computer hardware has finite RAM and local storage, we can make the statement that all modern digital computers are equivalent to an arbitrary large FSM.

for example, a program that outputs all the non-negative integers {0,1,2,…}

this program will never halt if it is run on a turing machine with infinite tape. however, this program would eventually overflow and rollover to 0 on a machine with finite ram. assuming that an overflow does not halt. this machine also does not halt. however, the implementation with finite RAM can be monitored by another turing machine that records a copy of the contents of RAM in a new dictionary file on the second turing machine with infinite tape. you could in theory then run a program on the second machine that checks if any new state exists in the dictionary. if it exists in the dictionary then output a warning message that the FSM is looping. I believe the implication of this conclusion is that the halting problem must have a turing machine with infinite tape as a requirement for the halting problem to remain undecidable. This is equivalent to the statement that any modern computer hardware can in theory have a watchdog process that monitors a sandbox that has allocated a very small limited RAM for repeating patterns to confirm that a program is in fact undecidable. an example of this is hardware PRNG which always loops and the period can be determined by another process on the same machine. Is there a name for this conjecture?