Ben6993 wrote:If I read it correctly, Joy's recent post on FQXi is a stunner. He doesn't need your, nor anybody's, computer programs. He probably doesn't even need a computer, spreadsheet or calculator. He just needs the back of an envelope to record the four numbers in categories 00, 01, 10 and 11. Just to be calculated for one pair of a and b values. He is probably correct!
Dear Ben
Yes it is a stunner. Well, we do need computer programs, but they are so simple there is no need to discuss coding issues now.
Speaking metaphorically, I need the back of four envelopes. Joy will only need one, to check my finding.
The experiment (N exploding balls, and analysis of a a lot of video footage of those explosions) generates two computer files each containing N directions of angular momentum. Spherical coordinates theta (azimuth), phi (zenith).
Let's call the directions of angular momentum in Alice's file u_k, k=1, ..., N, and in Bob's file v_k, k = 1, ..., N
In theory one would have v_k = - u_k, but in practice that might not be exactly the case.
If I pick measurement directions a and b, then according to Joy's experimental paper the outcomes left and right are
A_k = sign(a . u_k) and
B_k = sign(b . v_k),
and the estimated (observed, sample, experimental ...) correlation is
E(a, b) = 1/N sum_k A_k B_k
= ( N(++) + N(--) - N(+-) - N(-+) ) / ( N(++) + N(--) + N(+-) + N(-+) )
in the obvious notation.
Joy predicts the theoretical (population, large N limit, ensemble) correlation rho(a, b) = - a . b = - cos(angle between a and b)
Now we are going to look at two possible directions for Alice and two for Bob. They are all in the equatorial plane so they can be described just by azimuthal angles alpha = 0 and 90 for Alice and beta = 45 and 135 for Bob.
Joy predicts
E(0, 45) = - 0.7071...,
E(0, 135) = + 0.7071...,
E(90, 45) = - 0.7071...,
E(90, 135) = - 0.7071....
I will win my bet if I show him that one of these four predictions has failed, by an amount 0.2 or more.
He will give me two computer files "AliceDirections.txt" and "BobDirections.txt"
I'll claim that I'll be able to show him that one of his predictions has failed by a large amount (0.2 in either direction, or more).
I am actually certain that I will win. Notice that I even generously went to 0.2 whereas in my old proposal (when we randomly sampled subsets of the runs) I only went to 0.1