Joy.
I have looked at your paper, and at Richard Gill's refutation. I have also downloaded your rebuttal of Gill. It will take a while to go through all of the arguments.
In the meantime, I have constructed a somewhat pedantic, but pretty explicit derivation of the Clauser-Horne-Shimony-Holt version of Bell's inequality. This is the core of Bell's theorem. Could you explain, in simple terms, either what is wrong with it or why it does not apply to the analysis of the entangled states that Bell considered.
Assume that there are 2 systems, labelled 1 and 2. Assume that there are 2 properties that system 1 might or might not possess, labelled A and A'. Assume that there are 2 properties that system 2 might or might not possess, labelled B and B'. These properties, A, A', B, B', can bear any logical relationship whatsoever to one another. They might be the same. They might be opposite. They might be independent, or they might be correlated (positively or negatively). Define quantities, a, a', b, b' as follows: a = 1 if system 1 possesses property A; a = -1 if system 1 does not possess property A. Define a', b, and b' analogously.
Now consider the quantity constructed by multiplying the quantities from different systems in pairs and adding or subtracting them as follows:
ab + ab' + a'b - a'b'
There are 16 possible combinations of values of a, a', b, and b', resulting in 8 possible combinations for ab, ab', a'b, a'b' :
a a' b b' ab + ab' + a'b - a'b'
+1 +1 +1 +1 +1 +1 +1 -1 = +2
+1 +1 +1 -1 +1 -1 +1 +1 = +2
+1 +1 -1 +1 -1 +1 -1 -1 = -2
+1 +1 -1 -1 -1 -1 -1 +1 = -2
+1 -1 +1 +1 +1 +1 -1 +1 = +2
+1 -1 +1 -1 +1 -1 -1 -1 = -2
+1 -1 -1 +1 -1 +1 +1 +1 = +2
+1 -1 -1 -1 -1 -1 +1 -1 = -2
-1 +1 +1 +1 -1 -1 +1 -1 = -2
-1 +1 +1 -1 -1 +1 +1 +1 = +2
-1 +1 -1 +1 +1 -1 -1 -1 = -2
-1 +1 -1 -1 +1 +1 -1 +1 = +2
-1 -1 +1 +1 -1 -1 -1 +1 = -2
-1 -1 +1 -1 -1 +1 -1 -1 = -2
-1 -1 -1 +1 +1 -1 +1 +1 = +2
-1 -1 -1 -1 +1 +1 +1 -1 = +2
In every case, the quantity, ab + ab' + a'b - a'b', is either 2 or -2.
Each of the 16 cases can occur with some probability between 0 and 1.
These 16 cases are mutually exclusive and logically exhaustive, so
the sum of any set of them must be less than or equal to 1 (and, of
course, it must be greater than or equal to 0). The expectation value
of the quantity can be computed by multiplying the probability of each case
by the value of the quantity in that case (either 2 or -2). The maximum
value is +2, which occurs when the probabilities of all of the cases with
negative values are zero. The minimum value is -2, which occurs when the
probabilities of all of the cases with positive values are zero. It cannot
be less than -2 or greater than +2.
Thanks,
Ed