You say:

“If you like: a=a(x,w) is a function of x and of the information w available before the game started, but is not a function of y.”

Does that mean she can’t flip a coin to decide a if x = 0? Of course not. Does it mean she must determine the value of a without the use of her lab? Then what purpose is her lab? The rest of your comment on realism is mere flapdoodle.

The proof that they can win with at most probability 3/4 requires the existence of all four of (a|0), (a|1), (b|0), (b|1). However a given trial observes only two of them, e.g. (a|1) and (b|0). Where do the other two come from? They come from the assumption of realism (= determinism = hidden variables = counterfactual definiteness in this frame work). Realism says that a is determined and may depend on whether Alice receives x = 0 or 1, i.e. there is a classical algorithm (perhaps based on the state of reality) that gives the values (a|0) and (a|1), e.g. (a|0) = (a|1) = 0 (or perhaps a is determined by the flip of a fair coin). Same for Bob. Mind you, in some cases only God knows the algorithm.

Under the assumption no information is passed (locality) the QM violation of the “at most 3/4” shows that realism is false.

]]>So why not count that a loophole remaining?

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