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Bell's theorem - for or against Hidden Variables?
Gill: "As Hess and Philipp themselves admit, the distribution of the hidden variable in Alice's measurement station at any time has to depend on Bob's setting at the same time."

The admission is a feature, not a bug.  

Suppose Bob and Alice are equivalent to two simultaneously tossed fair coins.  Conventional probability tells us that there is a 1/4 chance of HH or TT.  However, there is such a thing as collective probability (von Mises), in which simultaneous tosses have probability 1/2 for each result.

“Realism” is effectively interpreted in your terms as “locality” absent of both time and measure space, such that all events observed locally are products of random observer choice of measurement criteria in an arbitrary space, independent of the time at which any choice is made.

Not so in the time-dependent Hess-Philipp schema.  Eliminating the time variable also eliminates continuity – in contradiction of Minkowski space and special relativity. Hess and Philipp explicitly showed that – just as Einstein relativity allows that every observer carries her own clock – every 3-dimension event implies a 4-dimension outcome, i.e., a Timelike Correlated Parameter (TLCP) in which space is not independent of time.

So with probability 1, there is a 1/3 chance each for HH, TT, and HT.  (This brings to mind the Monty Hall problem.)

Andrei Khrennikov asked “What is really ‘quantum’ in Quantum Theory?” He took to task conventional quantum probability models—the Kolmogorov measure-theoretic model and the Hilbert space probabilistic model—by citing Richard von Mises, regarding collectives, “ ... first the collective, and then the probability.”

This eliminates the possibility of Alice and Bob choosing a variable and the negation of that variable at the same time.  Gill, et al, purport to refute the Hess-Philipp result with the conclusion: “Time is not an issue in the proof of Bell’s theorem. What is crucial is the freedom of the experimenter to choose either of two settings at the same time. Hess and Philipp’s hidden variables model is nonlocal.”

Deceptive in this conclusion is the assumption -- not of Alice's and Bob's free will choice -- the experimenter chooses for them.  The experimenter does not have the freedom to choose either a setting or the negation of that setting at the same time. The choices taken one at a time are not equally likely.  

When we do not exclude the middle value, Bob is identified with outcome HH and Alice with TT.  Equally likely with probability 1/2.  When we add the possible outcome HT or TH, we have maximum 3/4.  The null result HT or TH, however, is never even a part of the collective -- HH and TT exist simultaneously with probability 1, without any interference by the experimenter.

Khrennikov notes in his paper:   “R. von Mises strongly criticized the conventional notion of independence, namely, event independence. He presented numerous examples in which conventional independence was represented as just a meaningless game with numbers – to obtain factorization of probability into the product of probabilities. In the frequency theory we study independence of collectives (in Khrennikov’s terminology – contexts).”
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RE: Bell's theorem - for or against Hidden Variables? - by Thomas Ray - 07-25-2016, 02:40 PM

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