Hi Ed,
I will tell you both what is wrong with the Bell-CHSH argument and why it does not apply to the analysis of the entangled states considered by Bell.
As stated, there is nothing wrong with your or Bell's argument. The inequality you and Bell-CHSH derived is a completely straightforward and mathematically valid inequality.
But here is a problem: Bell-CHSH inequality is not respected by Nature. It is routinely violated in the actual experiments. So, clearly, at least one physical assumption that has gone into the derivation of the inequality must be wrong, or at least unjustified. The question is: Which assumption?
Bell of course thought that it was the assumption of local causality that was unjustified. But I think that he was unduly influenced by his fondness of Bohm's theory to think that.
Suppose we did not know anything about quantum theory or Bohm's theory. We could of course still derive Bell-CHSH inequality, as Bool did before Bell. Without the knowledge of quantum theory do you think we would blame the violations on non-locality? Not unless we are completely mad.
So what is going on behind the violations? Well, to begin with not all alternatives, ab, ab', a'b, and a'b', can be simultaneously realized in any actual experiment. Only one pair can be realized at a time. So there is clearly an assumption of counterfactual definiteness of the joint outcomes ab, ab', a'b, and a'b' that has gone into the derivation of the inequality. But this can be eliminated by considering a single pair, say ab, for the sake of argument, because even a single pair produces stronger-than-classical correlation. So, counterfactual definiteness cannot be the real culprit behind the violations.
What other assumption, then, has gone into the derivation that could be wrong. Well, I claim that it is the assumption of wrong topology of the co-domain of the measurement functions Bell considered. Bell assumed measurement functions of the form
A(a, L) = +1 or -1
in the very first equation of his famous paper. But one cannot write a function like this without specifying its co-domain. Usually one assumes the co-domain to be just {-1, +1}. But I have proved that that makes the above prescription of Bell incomplete. With {-1, +1} it cannot satisfy the completeness criterion of EPR. The only way to satisfy the completeness criterion is by taking the co-domain to be a parallelized 3-sphere.
Here is where things get a bit technical. To understand why what I am saying is true, I invite you to read the attached paper of mine (which is the first chapter of my book). Please read at least up to page 4 to understand my argument. Further details can be found in several other chapters of my book.
I hope this helps,
JoyAttachment #1: 11_Origins.pdf
