Gill's R script has nothing whatsoever to do with his challenge, or my refutation of his contentions. The terms of his challenge are spelt out by him
here and
here. Evidently, his challlange is about my proposed experiment. My proposed experiment is about testing my local model for the EPRB correlation. My local model for the EPRB correlation is based on my hypothesis that we live in a parallelized 3-sphere, S^3, and our usual perception that we live in R^3 is just an illusion.
The idea behind the simulation of the experiment is thus to produce N spin vectors that are consistent with the geometry and topology of the 3-sphere. In my proposed experiment the experimenters are supposed to follow the procedure described on the page 4 of
my paper. They are supposed to observe N spin directions, record them as N points of S^2, and calculate the correlations using eq. (16) of my paper. What they will find is what I have predicted,
theoretically, in this simulation (modulo experimental errors):
http://rpubs.com/jjc/19298. A much more detailed theoretical description of my local model can be found
here.
The evidence presented in the above simulation is completely consistent with the terms of the Gill challenge. To reiterate: (1) a single set of spin directions u_k for Alice is used to calculate all four correlations, the negative of which, -u_k, being the spin directions used for Bob; (2) the same number of trials, N = 7070, is used in the calculations of all four correlations; and (3) the standard dot product, in the standard formula for the mean value, is used to calculate all four correlations. That is what Gill has been demanding. These calculations are not only fully consistent with the terms of the Gill challenge, but also with the eq. (16) of my paper.
The straight forward calculations computed in my R script give the following results:
alpha <- 0 * pi/180
beta <- 45 * pi/180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
ca <- colSums(u * a) ## Inner products of cols of 'u' with 'a'
cb <- colSums(u * b) ## Inner products of cols of 'u' with 'b'
(E_0_45 <- sum(sign(ca) * sign(-cb))/N)
## [1] -0.6993
(N)
## [1] 7070
alpha <- 0 * pi/180
beta <- 135 * pi/180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
ca <- colSums(u * a) ## Inner products of cols of 'u' with 'a'
cb <- colSums(u * b) ## Inner products of cols of 'u' with 'b'
(E_0_135 <- sum(sign(ca) * sign(-cb))/N)
## [1] 0.703
(N)
## [1] 7070
alpha <- 90 * pi/180
beta <- 45 * pi/180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
ca <- colSums(u * a) ## Inner products of cols of 'u' with 'a'
cb <- colSums(u * b) ## Inner products of cols of 'u' with 'b'
(E_90_45 <- sum(sign(ca) * sign(-cb))/N)
## [1] -0.699
(N)
## [1] 7070
alpha <- 90 * pi/180
beta <- 135 * pi/180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
ca <- colSums(u * a) ## Inner products of cols of 'u' with 'a'
cb <- colSums(u * b) ## Inner products of cols of 'u' with 'b'
(E_90_135 <- sum(sign(ca) * sign(-cb))/N)
## [1] -0.7276
(N)
## [1] 7070
## The Bell-CHSH inequality is violated:
abs(E_0_45 - E_0_135 + E_90_45 + E_90_135)
## [1] 2.829
Please note that the terms of the Gill challenge are duly satisfied.
Only a single set of 7,070 u-vectors is used, in the standard calculations of the correlations.
The reason why Gill is getting wrong results is because he has departed from my model and constructed his own unphysical model , which gives him the wrong results.