I think the bigger problem is not randomness vs. pseudorandomness, but rather the question of whether uncountable probability spaces actually exist in physical situations.

I believe they do for the same reasons I take seriously the existence of other Everett branches. In fact the mapping is rather straightforward: I can’t observe or directly interact with them in full generality, but the laws governing them and what I can observe are so very much simpler than laws that excise the unobservable ones. Whether I can actually exhibit most real numbers is besides the point.

Is there a demonstration that a physics based on the computables is more complex than a physics based on the reals?

This is a complicated question. In practice, it is difficult in this particular context to measure what we mean by more or less complicated. A Blum-Shub-Smale machine which is essentially the equivalent of a Turing machine but for real numbers can do anything a regular Turing machine can do. This would suggest that physics based on the real is in general capable of doing more. But in terms of describing rules, it seems that physics based on the reals is simpler. For example, trying to talk about points in space is a lot easier when one can have any real coordinate rather than any computable coordinate. If one wants to prove something about some sort of space that only has computable coordinates the easiest thing is generally to embed it in the corresponding real manifold or the like.

I think the bigger problem is not randomness vs. pseudorandomness, but rather the question of whether uncountable probability spaces actually exist in physical situations.

I believe they do for the same reasons I take seriously the existence of other Everett branches. In fact the mapping is rather straightforward: I can’t observe or directly interact with them in full generality, but the laws governing them and what I can observe are so very much simpler than laws that excise the unobservable ones. Whether I can actually exhibit most real numbers is besides the point.

Is there a demonstration that a physics based on the computables is more complex than a physics based on the reals?

This is a complicated question. In practice, it is difficult in this particular context to measure what we mean by more or less complicated. A Blum-Shub-Smale machine which is essentially the equivalent of a Turing machine but for real numbers can do anything a regular Turing machine can do. This would suggest that physics based on the real is in general capable of doing more. But in terms of describing rules, it seems that physics based on the reals is simpler. For example, trying to talk about points in space is a lot easier when one can have any real coordinate rather than any computable coordinate. If one wants to prove something about some sort of space that only has computable coordinates the easiest thing is generally to embed it in the corresponding real manifold or the like.