The rules are presumably also gleanable from looking at the small speck of dust.

I don't know why you want to know T_zero. I assume T means time here, not temperature or so?

If you have eg randomness as your non-determinism, you can still build probability distributions of the rest of the universe and the rest of time. (And honestly, even if you have determinism in the laws, you always have measurement errors. Even in classical mechanics.)