Even simpler, you're just trying to figure out if it halts or not, not the exact length of the cycle, so you can do it with only a single counter for the total number of states.

Say you have a machine with 128 bits of state, if it halts in 2^128 (roughly 340 undecillion) steps, that means it halts. If it's still going after 2^128 steps, then it'll keep going forever.

So, to see if a 128 bit machine halts, you only need an extra 128 bits to count up 1-by-1 to 2^128.

Sure, you also need a few thousand times the lifetime of the known universe just to count that high, but you know, It's O(2^128) instructions which means it's O(1).

Good luck doing that on any machine with any real amount of memory of course. Really easy to write "just count to 2^128", but not so easy to do it.