Is there a way of making precise, and proving, something like this?
For any noisy dynamic system describable with differential equations, observed through a noisy digitised channel, there exists a program which produces an output stream indistinguishable from the system.
It would be good to add some notion of input too.
There are several issues with making this precise and avoiding certain problems, but I suspect all of this is already solved so it's probably not worth me going into detail here. In the unlikely event this isn't already a solved problem, I could have a go at precisely stating and proving this.
I don't completely understand what you mean (in particular, I would really like you to be more specific about what you mean by "noisy" and "indistinguishable"), but this looks like it shouldn't be true on cardinality grounds. There should be uncountably many possible distinguishable noisy behaviors of a dynamical system.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.