Very neat, thanks for working this out. (Why would it not be sufficient to start with just five symbols?)

The solution is like a restricted version of Hilbert's Hotel, we're still in a group of invertible maps on binary sequences {0,1}^N but we aren't allowed to do anything non-local.

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...