A counterexample of the form "X cycles to X after N steps" is easy to check. A counterexample of the form "starting with X we keep going up forever" is hard to check in finite time.

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IsTom 2 hours ago | parent [-]

And still that requires X to not be particularly large. It could conceivably be in ballpark of BB(40).

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AnimalMuppet an hour ago | parent [-]

You, sir, have a very different definition of "not particularly large" than I do!

	▲	Sharlin 22 minutes ago \| parent [-]
		To be fair, BB(40) is smaller than almost all positive integers.