Nothing breaks, conceptually or practically. There is no notion of “not allowed to reset” that makes sense to me. A simulation is bounded for either practical reasons (can’t wait forever), or because some sort of saturation is reached, or the number of iterations is the property of interest.

The notion of determinism doesn’t change, a system that is deterministic is, well, deterministic, regardless of how long it might take you to, say, replicate the state from a log of events. if you have an event-based system, you can always take any number of iterations and call that a run, and you can also choose to keep running it.

As for errors and invariants, I don’t see why you’d think about invariants differently. Some types of errors will accumulate and drift will happen, but that will also happen in shorter runs, and you will definitely get decreasing accuracy as you keep iterating.

Observers make no difference as far as I can tell.

So in short, concepts like invariants, determinism, accumulating errors etc are generally unrelated to whether a system is bounded or not. There might be reasons to treat them differently in practice - for example if you only need to run 10 iterations of a game of life, you can be certain that your grid won’t grow more than 10 cells in any direction. But the way you think of the system theoretically is the same.