I don’t think Turing-completeness implies omniperiodicity. I’m imagining a cellular automaton which follows Game of Life rules on even-numbered generations and does nothing on odd-numbered generations, which is trivially Turing-complete because it’s just Conway’s Game of Life if you ignore every other generation, but also trivially has no odd-period oscillators.
I don’t think Turing-completeness implies omniperiodicity. I’m imagining a cellular automaton which follows Game of Life rules on even-numbered generations and does nothing on odd-numbered generations, which is trivially Turing-complete because it’s just Conway’s Game of Life if you ignore every other generation, but also trivially has no odd-period oscillators.