Hi Edwin,
Something that bothered me early on, when I was introduced to Joy's model, is what I perceived to be a carefully buried assumption that one could perform a Bayesian type analysis on the experimental results the model generates. This is not trivial, for if true, it would destroy the foundation of the framework. This is because Bayesians cannot avoid the assumption of a definite probability on the interval [0,1]. To get this probability, one must apply a degree of personal belief dependent on previous outcomes.
I had to be convinced that Joy's statistical analysis was based purely on a frequentist interpretation -- that every measurement trial is independent of every other (i.e., Bernoullian). This is what got me into some hot exchanges with Richard Gill, over the law of large numbers -- he insists that the central limit theorem that guarantees 0.5 in the interval proves that Joy's prediction is wrong, because no matter how many trials, the upper bound is set by the middle value based on faith in the law of large numbers (this is what gets us the upper bound in Bell-Aspect). Gill misses the point. Joy's framework does not address a single interval of probability in a unified series of trials; it deals with discrete non-probabilistic outcomes on both sides of the singularity that exists in every measurement function continuous from an initial condition (Lamport).
As a consequence, discrete measurement outcomes, 1 or - 1, are not equally likely for every orientation of the measurement apparatus (the observer) choosing from a continuous range of possible measurement values of a fixed input argument (- a.b). For a run of trials in one probability interval [0,1], in some orientation that outputs some value, we're going to get a unitarily corresponding value on the oppositely oriented interval. Joy explains this in terms of trigonometric functions, though I prefer to think of it in analytical terms; i.e., angle-preserving conformal mapping to infinity. We don't need all the tools of complex analysis that the link describes, because geometric algebra simplifies and reduces the calculation to all real values. The only way this is possible, however, is by continuing the inherently 2-dimensional complex analysis to a 4-dimension domain (where the toplogy of S^3 lives). And from there, we get the sigificance of Joy's parallelizability: S^0, S^1, S^3, S^7. The topology is complete and self-limiting.
So when it comes to Alice and Bob and the measurement angles they choose, though they record their results on a 1-dimension line in a 2-dimension space, the continuous range of those values actually lives in 8 dimensions, from which Joy derives the (CHSH) upper bound 2sqrt2, which is identical to sqrt8 and gives us an integral norm. Which leads to Joy's statement in his "What really sets the upper bound ..." paper:
" ... we have proven that the upper bound of 2sqrt2 on the strength of all possible quantum correlations is derived from the maximum of parallelizing torsions within all possible norm-composing parallelizable manifolds."
Please take this for what it's worth. My support for Joy's program is based almost entirely on mathematical completeness. That opinion in turn is taken from Einstein's conviction that no probabilistic framework can lead to a mathematically complete theory of how nature works. Since your premise is the same -- there's nothing to say you can't get there by a different road.
Best,
Tom