So I've got one question about the simulation: can you run it, on two different computers, one of which gets Alice's inputs, while the other gets Bob's, such that the simulated measurement outcomes that are produced violate a Bell inequality?

If so, then I would be impressed, because it seems a trivial matter to prove that no such computer program can exist, and we don't need to think about any topological subtelties to do so, because deep down, Bell inequalities are just conditions that must hold in order for there to be a joint probability distribution that marginalizes to the correct (i.e. experimentally observed) probability distributions for each pair of observables.

So suppose we have four random variables

,

,

and

taking values in

. There exists a joint probability distribution

which can be sampled in order to produce the experimental outcomes given the measurement directions. So our program must, given the freely chosen input, produce an outcome according to

. Such outcomes will never violate a Bell inequality, for the following simple reason.

For each pair of variables, their joint probability distribution can be obtained by marginalizing the PD for all variables, e.g.

. Every correlator between two variables can then be written as

. Thus we can form the following expression:

But clearly,

, since all of those values are either

or

. But then, with

, we get:

.

This inequality, which the careful observer will realize is the Bell-CHSH inequality, must hold for every program, if we give each copy access only to some (arbitrary)

---no matter how this is arrived at---and draw values for

and

on one, and values for

and

on the other. It should be noted that this is not really a new discovery: George Boole already in the 1860s derived such inequalities (or equivalent ones) as 'conditions of possible experience', reasoning that only if some experimentally obtained quantities obeyed these inequalities, then they have a simultaneous probabilistic model and thus, could actually be obtained in an experiment. Note again that this derivation doesn't make any of the allegedly questionable assumptions Bell made---the only assumption is the existence of a joint probability distribution, and the only way this assumption can fail is via some form of influence of the outcome of one measurement on that of another.

So, where do you propose this goes wrong? How do you build a program---whose outputs can after all always be described as a joint probability distribution in the above way---that violates the inequality without any covert influences?