This is a really exciting essay; I'm really intrigued by the connections you suggest between quantum mechanics and chaos theory, and am now keen to learn more about this area.
I did have one general question about the motivation for this approach. If I understand you correctly, the idea is that by constraining the state of the universe to evolve on some uncomputable fractal subset of state space, we get a natural way to violate statistical independence (without the denial of free will) and thus we can have violations of Bell's inequality without violations of locality.I wonder, though, why you consider it important to avoid violations of locality? As I understand it, the similarity of Schrodinger's equation to the Liouville equations leads you to consider underlying deterministic dynamics, and then since Bell's theorem rules out any underlying deterministic local dynamics, you turn to non-computability as a means of violating statistical independence and thus invalidating Bell's theorem. But an alternative possible route would have been to accept the existence of nonlocality and consider how Schrodinger's equation could arise from an underlying deterministic non-local dynamics - is there a specific reason you chose not to go down this route?
I also have some questions about the sense in which 'locality' is preserved by your model. First, consider applying the constraint of evolution on a fractal subset to an indeterministic model. Then if the evolution of the universe is constrained to remain on the fractal subset, it would seem that the (non-deterministic) evolution of the universe at any one spacetime point must depend on the (non-deterministic) evolution of the universe at all points spacelike related to that point, as if the points evolved independently and non-deterministically then it would be possible to go off the fractal subset. So constraining the universe to lie on the fractal subset does not, in the absence of determinism, seem to give us a local theory. So now consider a deterministic model as you propose. Here the evolution of the universe is fully determined by the initial state (I am assuming here that by 'deterministic' you are referring to initial-value determinism, as the term is commonly used), and so the constraint you suggest comes down to requiring that the universe has a fine-tuned initial state which ensures that its evolution always remains on the fractal subset. But surely if this sort of fine-tuning is allowed then we can quite easily explain nonlocality without needing to appeal to noncomputability or fractal subspaces - i.e. we can encode the choices of measurements and the measurement results for all Bell experiments which will ever be performed directly into the initial state, and thus produce experimental results which appear non-local even though they are in fact produced from local evolution from this fine-tuned initial state. I think most physicists are not keen to adopt this approach to eliminating nonlocality because it seems unreasonably conspiratorial and fine-tuned - do you think your fractal approach gets round this complaint in some way, and if so, how?
I was also interested in the approach you take to recovering 'free will.' The distinction you make between defining free will via counterfactuals vs defining free will as the absence of constraint clearly ties into long-standing arguments in philosophy about the nature of free will, and I think there are indeed good arguments in favour of the latter approach even before one comes to the specific theoretical model that you introduce here - indeed I would be fascinated to read a paper discussing the links between your proposal and the body of philosophical literature on this topic!
