Can we make the hardest statement in transcendental theory to prove collatz?

[mathkook]

@AcidicJello These questions are great! Tao showed “(Weak) Collatz implies (Weak) Baker”, but is there anything that would imply (Weak) Collatz? Reducing X to Y is really clarifying sometimes.

@AcidicJello

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.

Right on the money! Except … when r is small, constructing the 3^{k-r} worst case isn’t actually possible, because you wind up with (0|1)^r 1^{k-r-1} 0, which no longer has an LRS of r = \log(k). So it’s pretty interesting. I’ve been meaning to post about this for a while, hopefully I can get some time. [Edit: found some time.]