gill1109 wrote:Exactly. And therefore, when you calculate each of the four correlations on a different random subset, then *on average* the upper bound won't exceed 2 either.

First of all, that is not the definition of an *upper bound*. An *upper bound* in an inequality is a mathematical statement that no value above that is possible within the assumptions used to derive it, it is not a statement about what the average value will be. When we determine the upper bound, we look at the extremes, not what the most likely value will be. Secondly, when we say *upper bound*, we do not mean that average values can not be less than that, rather we mean that when that value is exceeded, we will have to revisit our assumptions used to derive the bound or a mathematical error has been committed.

We have hereby proved Bell's inequality in its usual form (an inequality about expectation values): local realism (and no conspiracy) implies CHSH <= 2 (on average).

And you did that prove by assuming that you have a single set. This is easy to see by looking at Bell's original derivation, as well as ALL derivations including yours which include a factorization step, a step that can not be done if you do not have a single set. That inequality is valid,

for a single set.

It is easy to exhibit a single set of particles, and a particular partition of that set into four disjoint subsets

Once you divide the set up into disjoint sets, we no longer have a single set, and the inequality you derived for the single set no longer applies to a the disjoint sets, without conspiracy as I have clearly explained.

So in principle, experimental data could violate CHSH.

Yes, because of the mathematical error of using terms from 4 disjoint sets to substitute for terms which should have been calculated in a single set as assumed in the derivation.

QM predicts that in certain QM experiments, it does.

Yes, because of the mathematical error of using QM predictions, which are for completely distinct experimental arrangements, and therefore necessary for disjoint sets of particles to substitute in an expression which was derived starting with the assumption that we have a single set.

We have hereby proved Bell's theorem.

And hereby, I've proven that Bell's theorem is false.

It's just that, if local realism holds (according to which the observed data can be thought of as having arisen from a partition of a complete data set into four parts as described above), then if one would select the four subsets at random, the subsets which give such extreme values of CHSH are very rare. That's Bell's inequality in my strengthened (probabilistic) form.

Bell's theorem does not say it is rare to violate the inequality. It says it is impossible. It does not matter whether you start from a single set and partition the data into 4, you are still calculating the correlations from 4 disjoint sets and for this purpose the upper bound is 4. Probability can not save Bell's theorem here, it still fails.

There is no way that this expression <a1b1> + <a2b2′> + <a3′b3′> − <a4′b4> can have an upper bound lower than 4 is if the values in one set, impose constraints on the values another set, and the only way that can happen is if the sets are not disjoint to begin with. So what you are claiming is equivalent to the claim that

probability can cause 4 disjoint sets to not be disjoint -- a contradiction.