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Even though it is hidden in plain sight, let me bring out the

sleight of hand in the proof of Bell's theorem (or in the derivation of the CHSH inequality). The sleight of hand occurs in Bell's proof in the form of a nonsensical application of Einstein's idea of realism. For convenience, I am including a figure below from one of my papers so that nothing is left to the imagination. In the present context of an EPR-Bohm type experiment, Einstein's idea of realism dictates that ALL possible measurement results such as A(a, h) about the measurement directions "a" are

real in the sense that they are at least counterfactually realizable. If, for example, Alice chooses to perform her measurement about the direction "a", then she would observe the result A(a, h), where "h" is the hidden variable or an initial state of the singlet system, which is usually denoted by "lambda." Thus all, infinitely many, measurement results A(a, h) are "real" in Einstein's sense. They all exist and are in fact predetermined for Alice, at least counterfactually. Likewise, all measurement results B(b, h) are real for Bob, and they again exist, at least counterfactually, about the freely chosen measurement directions "b". And since all such A(a, h) and B(b, h) exist, so do their products such as A(a, h)B(b, h), A(a, h)B(b', h), A(a', h)B(b, h) and A(a', h)B(b', h).

So far so good. However, the proof of Bell's theorem, or the derivation of the bound of 2 in the CHSH inequality, depends on the evaluation of the following integral:

See eq. (16) of

my paper to understand the details. As we can see, the integrand of this expression involves quantities like A(a, h) { B(b, h) + B(b', h) }. The immediate question then is: Are the quantities like A(a, h) { B(b, h) + B(b', h) } "real" in Einstein's sense? Are they at least counterfactually realizable? Note that it would be quite deceitful to claim that these are just intermediate mathematical quantities. They are not. They are THE quantities that are averaged over to obtain the bound of 2 in the CHSH inequality. Thus it is quite pertinent to ask whether these quantities are real in Einstein's sense. If they are real in Einstein's sense, then Bell's theorem goes through. But if they are not real in Einstein's sense, then Bell inequalities are simply mathematical curiosities without any physical significance. Now it is very easy to see that quantities like A(a, h) { B(b, h) + B(b', h) } are entirely fictitious. They are NOT realizable in ANY possible world. Indeed, they are analogous to the events like

While Jack reaches Los Angeles and buys apple juice, Jill reaches New York and buys orange juice and Jill reaches Miami and buys apple juice at exactly the same time.

Thus the fictitious quantities such as A(a, h) { B(b, h) + B(b', h) } appearing in the above integral are NOT real in Einstein's sense, and therefore

Bell's theorem fails.

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