Dear Declan
I also have been investigating classically produced quantum correlations, in the context of Rob McEachern's one bit hypothesis. The simplified model I came up with had a similar amount of noise obscuring the cosine, but when the number of trials was increased to one billion, a residual systematic error of about 1 percent was discovered. This was verified with a purely probabilistic treatment instead of random trials. (see vixra.org/abs/1705.0377)
I have a vague interest in simulating quantum computation, and it seems to me that a systematic error would likely limit the number of qubits. Can you increase the number of trials to see if there might be some unwanted correlation? I think the real problem is to find a method that actually converges to the cosine.
Richard Gill found such a method in the work of Philip Pearle, as pointed out by Austin. I recall following (or trying to follow) the discussion mainly between Gill and Joy Christian, and then forgot about it until Austin's post. (see another of Gill's papers arxiv.org/abs/1505.04431) Having some experience now with the problem, I see that Gill does a nice job of explaining the procedure, but I find R code can be quite obscure. Here are some basic details.
The method requires three(!) random numbers R1,R2,R3 generated uniformly over the interval 0-1 for each trial. Gill stores a large set of transformed random numbers, z,x,s, to be reused in trials for any combination of settings by Alice and Bob.
The first two random numbers are transformed to cover a spherical shell, and then projected onto a plane running through the center, forming a disc. Points on the plane are taken as 2d vectors (z,x), so the distribution of their magnitude is biased towards the edge of the disc, where it is most dense.
The third random number sets the threshold, s, for detection, with another carefully crafted distribution.
z = 2 R1 - 1
x = sqrt(1 - z^2) cos(2 pi R2)
s = [ 2 / sqrt(3 R3 1) ] - 1
A unit vector (az,ax) in the z-x plane sets Alice's angle, with (bz,bx) for Bob. Projections are calculated as follows
pa = (z,x) . (az,ax) = z az x ax
pb = (z,x) . (bz,bx) = z bz x bx
A detection occurs when the absolute value of both pa and pb is greater than s. The correlation for a detected event is given by the product of the signs of their projections
C = sign(pa) sign(pb)
The average correlation converges to the cosine expectation as the number of trials is increased. Unfortunately, it is not clear to me how to measure noise for McEachern's hypothesis, but that is a problem for another day.
Thanks for entering an essay on the technique of quantum steering, which I will have to experiment with.
Cheers,
Colin