Joy Christian wrote:They are obsessed about the number of pre-states used in the simulation and how many of these pre-states are actually selected in a given experiment for a given pair (a, b) of settings. But who cares about the pre-ensemble and the pre-states in R^3? The pre-ensemble does not exist in Nature. Only S^3 and the real states (u, s) exist in Nature. Pre-ensemble M and pre-states are like the wooden frame, and S^3 and the corresponding real states (u, s) are like the Picasso. Thus Gill and his proxy are simply barking on the wrong tree!

The problem is that in local realism the states have to be prepared and chosen without knowing a and b. Simply because they are yet undefined and unknown - at least if one excludes superdeterminism, at those places where an Einstein-local theory could define and prepare its hidden variables.

This property - independence of a and b - is the essential, key property of the local hidden variables. That's why I care about it. You may name this "obsessed", but I simply know - its trivial - that without this independence of lambda on a and b one cannot prove Bell's inequalities, thus, an example where these initial values depend on a and b is worthless and irrelevant.

Thus, the hidden variables lambda can only be those variables which are defined without knowing a and b. In the computation this property has the pre-ensemble, and the pre-ensemble only.

Once the definition of what are the "real states (u,s)" depends on a and b, they can be identified only at measurement time, because only at measurement time a and b are created by the free choices of the experimenters. The preparations should have been some element of the pre-ensemble.

What happens, if the pre-ensemble element to become a "real state" once a and b come into existence? However one names it, it will be a failure of the particular experiment. Or A(a,l) will not be +-1 but 0, or B(b,l) will be 0. This possibility is named (or equivalent to) the detector inefficiency loophole. And it is also well-known that low detector efficiency allows to obtain violations of Bell's inequalities within local realistic models.

So what I do is simply to defend Bell's theorem against "counterexamples" which are not really counterexamples but simple long ago known possibilities to obtain violations of Bell's inequalities, which in simple and obvious ways violate the conditions of Bell's theorem.

Here, again, a clear statement that one of the central points of Bell's inequality - the independence of the states lambda as well as its probability distribution rho(lambda)d lambda - is simply ignored in your "counterexample", which makes it completely worthless:

Joy Christian wrote:Therefore what is physically relevant in the simulation is the ratio of the total number N of the simultaneous events observed by Alice and Bob and the total number L of the initial states (u, s) in S^3. Whether or not the absolute numbers N and L depend on the settings (a, b) is completely irrelevant.

What you have created is a "counterexample" which could be written in the following form:

And what you have declared now is that this dependence on a and b is completely irrelevant. But it is relevant, for such an expression one cannot prove Bell's inequality.