Dang it -- I messed up cutting and pasting. I will repost correctly here, and hope I can get the last version deleted. Sorry.
Michael,
I am so grateful -- as I expect Joy is as well -- to be able to have meaningful dialogue on the real issues. For so long, and for Joy many years longer than I, we've been forced to respond to straw man arguments. Very debilitating and demoralizing.
One of those persistent straw men describes Joy's model as algebraic (though one has to be innocent of what "geometric algebra" really means, to think that way), when of course a topological framework can't be other than analytical. The detractor then proceeds to identify a nonexistent "algebraic error" and dismiss the whole argument.
Anyway:
I think it fruitful to approach the subject the way you're doing, because the issues do go deep into FOM as well as physics -- and actually, as you imply, have to do so -- in order to reconcile local discrete measures with globally continuous functions.
Key to the structure is orientability, that only a topological model can supply. I really only became aware of this about a year ago -- when I read a 30 year old unpublished paper by the eminent computer scientist Leslie Lamport titled "Buridan's Principle." His analysis of the Stern-Gerlach apparatus convinced me that the principle ("A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time") really does generalize, as a physical law, to all measurement functions continuous from an initial condition. I suggested to him that the paper really needed to be published, and fortunately, the medium I suggested -- Foundations of Physics -- accepted and published it in the April 2012 issue.
I've filled 3 notebooks with arguments and equations in the last year and half, and I'm itching to get it into publishable form -- back in 2 March I wrote " ... the continuous range of measurement results are recorded -- not on unit S^0 as Bell assumed by the functions A(a,l) = 1 or - 1, but on S^1, a unit 2-sphere. As Joy Christian explains, 'After all, no one has ever observed a 'click' in an experiment other than about some experimental direction a. With this simple change in the function A now takes on values in a topological 2-sphere, not the real line, thereby correctly representing the EPR elements of reality. The values of the spin components are still 1 or - 1, but they now reside on the surface of a unit ball.'" Orientability matters. It matters, though, over the whole range of parallelizable spheres, which are simply connected and therefore accommodate the flatness condition.
Like you, I have tended to translate Joy's research into my own familiar terms of complex analysis, information theory and number theory. I have tried not to do that, though without complete success. In any case, we bump up against your conclusion: "The philosophical point is that realism in terms of observational predictions is retained, but at the expense of the descriptive realism of the dynamics being compromised." And that is why, as I think you'll see is obvious, that I apply the criterion of Godel completeness rather than the incompleteness theorem. It meets Popper falsifiability, in the context of Tarski correspondence theory of truth, and it satisfies metaphysical realism. In other words, we recover the dynamics in a continuous function model of argument and value -- I characterize Joy's correlation result, E(a,b) = - a.b as the input argument to a continuous range of values, which generalize Buridan's Principle to the topological limit.
I hope you get a chance to visit my essay site, where some of these same issues are discussed in a different way.
All best,
Tom
(P.S. I trust that you got my email reply with my mailing address. Looking forward to reading your book!)