Justo wrote:Since he answered instead of me before, I answer this one: the result is not independent of the particles. It also depends on their state. They are supposed to be in the singlet state; otherwise, the prediction is not valid.
Are you sure that Is that all? Gill already knew that we were talking about particles in the singlet state when he made the statement, and Bell's theorem is dealing with particles in the singlet state. So perhaps he has something else in mind when he says "All the particle pairs would be created in the same way". Please be absolutely sure that is the only thing that is meant because it would be a pity if you have to backtrack to amend this point later.
Justo wrote:minkwe wrote:In other words, say we have 4 random researchers scattered around the world each measuring just one of the terms above in their lab. How similar will their particles have to be to each other before they can confirm the relationship

, and for combined their results to produce:

Tthe same answer as before. All particles should be prepared in the singlet state.
So let us be clear then, if the 4 random researchers scattered around the world each prepare their sets of particles in the singlet state, you would expect their results to obey the relationship:

Again you believe "All the particle pairs would be created in the same way" means ONLY that the particles are they are prepared in the singlet state.
Now let us generalize a little bit. Let us keep everything just agreed above for the singlet state and instead let us relax the singlet state requirement and say all that the researchers have to do is prepare their particles anyway they like. They can use the singlet state if they like, but they can also use any other type of state. In fact, we don't even restrict the type of particle they are to use. All they have to do is perform a similar experiment with any set of pairs of particles (no restrictions) and record the correlations. Once they bring their results together, the maximum value they can observe for the following expression is 4.
 - E\left(a_1, b_2\right) + E\left(a_2, b_1\right) + E\left(a_2, b_2\right) \le 4)
Note that experimentally,
 = \left \langle AB \right \rangle)
Were

is the outcome for measurement

and

is the outcome of measurement

If you disagree explain why. If you agree then we can proceed. Note, the only thing I've removed is Gill's condition that "all the particles be prepared in exactly the same way", or as you say I've removed the condition that the particles are prepared in a singlet state.