would be cool to discuss what (if) invariants occur no matter what initial condition a universe starts on

There are a bunch of different formulations of "eternal recurrence"; some are merely similar, not identical.

Not all require literal repetition, either, and the recurrence is purely semantic.

Literal exact recurrence is certainly a poetic subset, though! It would fit right into "Einstein's Dreams".

I was thinking of Nietzsche's eternal recurrence:

What, if some day or night a demon were to steal after you into your loneliest loneliness and say to you: 'This life as you now live it and have lived it, you will have to live once more and innumerable times more; and there will be nothing new in it, but every pain and every joy and every thought and sigh and everything unutterably small or great in your life will have to return to you, all in the same succession and sequence' ... Would you not throw yourself down and gnash your teeth and curse the demon who spoke thus? Or have you once experienced a tremendous moment when you would have answered him: 'You are a god and never have I heard anything more divine.'

https://en.wikipedia.org/wiki/Eternal_return#Friedrich_Nietz...