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B2c3a6ac7e/S1e2cik3ejnr4i adjustable ships. Sorry my friends in China.
https://docs.google.com/spreadsheets/d/ ... sp=sharing

It can be proved that any speed with the usual constraints[a] has a ship of that speed. First, it is possible to construct a front end by overlaying a sequence of replicators which are destroyed at the front (see diagram): Next, consider the very first space where a replicator could be. If it is wrong (on when it should be off or vice versa), this can be fixed by adding a replicator in the original pattern (period) cells[b‎] away from where the error will be. If it is the same we do nothing. We then move to the next spot and do the same thing.
After moving far enough away from the front end[c], the next cell will always be a simple xor of certain cells within the last (period) cells. This sequence will be periodic for obvious reasons. Therefore, eventually it will loop back to the sequence at the front[c]. The back end can be placed at the new front and we will have a complete spaceship.

[a]: The speed is less than or equal to c/3 and the sum of the period and displacement is even. Also, nc/3n spaceships are impossible, but it should be possible to have, say, 2c/10 spaceships which aren't just c/5s.
[b‎]: This does not affect any cells beyond the wrong cell because it travels at light speed and so it takes a full cycle to move to the current spot.
[c]: While it definitely will loop back to the sequence at the front in the same phase, it is possible that it may appear earlier but phase-shifted relative to the front.

toroidalet wrote: July 29th, 2022, 3:15 pm ...Therefore, eventually it will loop back to the sequence at the front[c]. The back end can be placed at the new front and we will have a complete spaceship.

[c]: While it definitely will loop back to the sequence at the front in the same phase, it is possible that it may appear earlier but phase-shifted relative to the front.

How do you attach the back end? To me, it’s not obvious that a suitable backend can be attached to the front sequence.

pzq_alex wrote: July 29th, 2022, 9:07 pm How do you attach the back end? To me, it’s not obvious that a suitable backend can be attached to the front sequence.

The second front end in the pattern behaves identically to the one at the front, which means its replicators have to be blocked in the same positions. Therefore, the replicators right before it are blocked in positions moving to the left 1 cell at a time in the same intervals, which is equivalent to a pulling reaction. See the diagram: Note that with larger speeds, it may be possible that the back and front ends are different, but this only makes them shorter.