Gill’s theorem (no-go to classical, distributed, simulation of a Bell-CHSH type experiment with externally generated random and unpredictable settings) was published in arXiv papers in 2001 and 2003. Those papers were also published in conference proceedings, both in 2003. My results and methods have been much improved, most recently by Stephanie Wehner and David Elkouss in Delft. They are used, and a full proof given, in the Nature paper of the 2015 Bell experiment, Hensen er al (supplementary material). Here is what I write in Section 3.2 newly written my newest paper (latest version: 9 November)
https://www.math.leidenuniv.nl/~gill/gull.pdf, where you can find full references.
Consider such an experiment and let us say that the nth trial results in a success, if and only if the two outcomes are equal and the settings are not both the setting with label “2”, or the two outcomes are opposite and the settings are both the setting with label “2”. (Recall that measurement outcomes take the values ±1; the settings will be labelled “1” and “2”, and these labels correspond to certain choices of measurement directions, in each of the two wings of the experiment). The quantum engineering is set up so as to ensure a large positive correlation between the outcomes for setting pairs 11, 12 and 21, but a large negative correlation for setting pair 22. Let us denote the total number of successes in a fixed number, N, of trials, by SN .
Then Hensen et al. (2015a, b), and see also Bierhorst (2015) and Elkouss and Wehner (2016) for further generalisations, show that, for all x,
P(S_N ≥ x) ≤ P(Bin(N,3/4) ≥ x),
where Bin(N, p) denotes a binomally distributed random variable with parameters N, the number of trials, and success probability, per independent trial, p.
Above we wrote “under the assumption of local realism”. Those are physics concepts. The important point here is that a network of two classical PC’s both performing a completely deterministic computation, and allowed to communicate over a classical wired connection between every trial and the next, does satisfy those assumptions. The theorem applies to a classical distributed computer simulation of the usual quantum optics lab experiment. Time trends and time jumps in the simulated physics, and correlations (dependency) due to use of memory of past settings (even of the past settings in the other wing in the experiment) do not destroy the theorem. It is driven solely by the random choice anew, trial after trial, of one of the four pairs of settings, and such that each computer is only fed its own setting, not that given to the other computer.
Take for instance N = 10000. Take a critical level of x = 0.8N. Local realism says that S_N is stochastically smaller (in the right tail) than the Bin(N, 0.75) distribution. According to quantum mechanics, and using the optimal pairs of settings and the optimal quantum state, S_N has approximately the Bin(N,0.85) distribution. Under those two distributions, the probabilities of outcomes respectively larger and smaller than 0.80N are about 10^−30 and 10^−40 respectively. These probabilities give an excellent basis for making bets.
You really are completely clueless. It's a quantum mechanics calculation. Got nothing to do with Gill's theory.