I have no original insight into this question but I think it’s really interesting and I just want to highlight two relevant posts on the forum being hinted at in this thread since I recently did a read-through. I don’t mean to intrude with information you already know. I’m just connecting the dots for myself and maybe some others.
Tao had some thoughts on this
Proving Collatz is Baker Hard - This post details Tao’s statement that creating exponential separation between powers is necessary to prove Collatz, utilizing a specific inequality.
Proving the opposite, that 2^a - 3^b > b/2 can be false, would even be sufficient to prove Collatz false, but we know it’s always true.
A preliminary study of sequences with a significantly redundant parity vector led us to promising advances earlier on this forum, but it is far from sufficient.
Longest-Repeated-Substring - This argument rules out integer cycles with sufficiently long repeating substrings. Using Ellison’s bounds, the repeated substrings have to be at least half the length of the cycle to rule out an integer solution, and the theoretical limit with current techniques still requires them to be more than a third of the cycle length.
Is it possible to combine this with the combinatorics fact (also mentioned in the linked thread) that the longest repeated substring in a necklace must be at least something like log_2(k) (still extremely small) to give the kind of statement we’re looking for here, even if the bound would be absurd?
Edit: log_2(k) not log_2(k + x); shortcut steps means k is the whole length. While I’m at it, let me just answer my own question. No, because for longest repeated substring r, we’re looking for 3^{k - r} < 2^k - 3^x and r = log_2(k) is not going to bring 3^{k - r} below 3^x.
even if we could make it at Roth’s level, such inequality is not sufficient to rule out cycles with a lot of local minima, whose worst case is the high cycle
Similar question: Is there a level at which it would be sufficient for all m-cycles? Even if it required some crazy huge effective bound for the cycle length? I’m assuming not since you also said
the question of whether there is a bound of this kind that could be sufficient for a complete proof remains largely open
By the way I’m not trying to suggest it would be practical. I’m just honing in on the question of the thread.
To the idea that
if you’re working on cycles or whatever you just waste your time without any real progress
I would imagine even if there were such a bound, it wouldn’t be both sufficient and necessary. Like, there’s a weak necessary bound and maybe a strong sufficient bound, but no bound that is identical to no nontrivial cycles. There’s always the possibility that the sufficient bound turns out to be false and you would have to figure out cycles some other way.
Last thing, and this might be even more of a reach, but maybe there is a concern that such a bound could be used to answer questions which should be undecidable about general Collatz systems.