James, you ask: "A part of your argument ... appears to be that the computer simulations cannot be confirmation of Joy's model [...] so long as their results are expressed 'equivalent' to S^2 terms?"
I am not saying that. Quite the opposite.
I'm saying that the computer simulations prove that there are some errors somewhere in Joy's work, but whether they are harmless or fundamental, is another question.
The situation is as follows.
Joy himself describes two ways to simulate his model for the EPR-B correlations. Each simulation is a computer implementation of a classical probability model - a description of the joint probability distribution of some random variables. In the simulation we observe the average of many realizations of some function of those random variables. That's just the Monte Carlo approach for computing an expectation value, ie for calculating some integral. Starting from his S^4 based model, Joy *deduces* these two simulation models. In other words, he claims that the integrals which are implicitly defined by the two probability models in question are both identically equal to the EPR-B correlation.
The version implemented by Michel Vodje is "event-based". You pick two settings a and b, generate a hidden state lambda, and compute measurement outcomes A(a, lambda) and B(b, lambda), both equal to +/- 1. (lambda has several components - it includes a uniform random point on S^2 as well as some auxiliary randomizations).
Repeat many times and calculate the correlation.
The version implemented by Chantal Roth has a kind of short-cut: there is an expression for A(a, lambda) times B(b, lambda) equal to +/- 1. The rest is the same. Of course, by an extra coin-toss (an extra component of lambda), we could decide what A and B separately should be, given the value of their product.
Both implementations involve picking uniformly at random a point on S^2 and doing calculations in S^2. The measurement settings, by the way, are by definition, points in S^2 (directions in R^3).
Joy's model involves S^3 but S^3 is locally equivalent to S^1 x S^2 so you can imagine that specific computations coming out of his model could be rephrased in S^2 terms. And as I already remarked, certain elements in the model (the measurement settings) already belong there.
I show that the simulation results do *not* reprduce the (negative) cosine of theory - both of quantum theory, and of Joy's theory. *Both* of the simulation models fail to do their job.
Therefore either Joy's derivation of these simulation models from his original theory is wrong, or the original theory was already wrong.
The result of a computer experiment is not necessarily a mathematical proof, of course. There is the possibility of numerical error, and in a Monte Carlo experiment, there is statistical error too. However by the standards of physics or of law ("beyond reasonable doubt"), I have given conclusive proof.
