I'm working through trying to understand this, but the math of the original paper, and subsequent actual academic papers are a bit beyond me. So I've turned to simplified explanations, all of which are more or less the same, but for reference i'll stick with this one
However, it seems to me that the assumptions about a locally deterministic theory are trivially easy to challenge simply by actually adding an extra "hidden" variable.
As a corollary, lets say we create a pseudo random number generator, that outputs a single number 1-6. And then we create a second one with the same exact seed, such that they would output the same number, except we add 2, so if the first rolls a 6 the second will be a 2.
Then lets say the program only reports to us if its either 1-3(A) or 4-6(B). Now, individually each is perfectly unpredictable happening 50% of the time, and thus we might assume, that the results should be AA,AB,BA,BB, and we'd have agreement 50% of the time. However, because only a first roll of 1 or 4 results in an agreement, the actual result is 1/3.
So here is an example of a system that's perfectly locally deterministic, and unpredictable, while still producing a disagreement, between the "random" outcome and a correlated one.
And thus my suggestion would be that Bell's Theorem fails, in assuming that the hidden variables have to adhere to the top level random odds.
First question, why is my explanation wrong?
Second Question, After a photon goes through a polarized filter it takes on the polarization of the filter. What if we setup two filters with one rotated to always have perpendicular polarization to the first. Then, set it up so the exact orientation is unknown to us and random(excepting the relation between the two) And then fired photons through this setup, and did the Bell Theorem tests, would the predictions of QM for some reason be different, then in the case of the "entangled" photons?