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It can be finite, but you have to definitively prove the non halting cases are infinite (which, being infinite, can't be computed with brute force)

Look, you're arguing the collatz cobjecture can be proven by counting up by 1 & seeing when a number hits a loop without reaching 1 (which, if there is such a case, would eventually prove via refutation, but if not, you'd never know if you were about to hit an answer or not)

Mathematical uncomputability for problems with answers is a thing. Read up on Gödel's Incompleteness Theorem

& some conjectures have been refuted by computers: https://math.stackexchange.com/questions/2638897/conjectures... but until it's done the question remains unknown