Another Response to Richard Gill's 10,000 Euros Challenge

[Joy Christian]

Hello Everyone,

The following is my response to the 10,000 Euros challenge announced by Richard Gill. You can read the terms of his challenge on these two posts here and here.

As required by the terms of the challenge, here are the two text files, AliceDirections.txt and BobDirections.txt, containing N spin directions I have postulated to occur within my proposed macroscopic experiment. Each of the lines in each text file contains two numbers, which represent the x and y coordinates of the spin directions for Alice and Bob, confined to the equatorial plane, in a right-handed coordinate system. The R script of the simulation which produces the spin directions, u_k(x,y) for Alice and -u_k(x,y) for Bob, can be found here. The script may also explain the notations and conventions used for the computations that follow.

For technical reasons of producing the text files from the R script, Fred and I have had to separate out the x and y components of the u matrix and bind them again in the matrix e = cbind(t,w), as defined in the R script. But I have also left the directions tables from the u matrix in the script, just in case someone has difficulty using the x and y components generated from the e matrix. Note also that Fred has checked the e-lists against the u output and the "good" truth table, and they match.

I have left the number of trials rather low, N = 10,000, but that should be fine for the present purposes. Needless to say, statistics can be improved with a larger N.

Since the challenge is about my proposed experiment (which is supposed to test my 3-sphere model for the EPR-Bohm correlation), it is very important that this response to the challenge is evaluated with a "good" deal of understanding of my model. Sadly, in my opinion, Richard Gill has not understood my model despite considerable efforts. I hope that he or anyone else who wishes to evaluate my response to his challenge, does so in the light of my latest paper on the subject.

In compliance with the terms of the challenge, correlations are computed for the four pairs of "Bell test angles", 0 and 90 degrees for Alice and 45 and 135 degrees for Bob, in the standard manner, using the standard formula for the correlation function,

E(a, b) = 1/N sum_k A_k B_k ,

with A_k = sign(a . u_k[good]) and B_k = -sign(b . u_k[good]) being the standard functions, appearing, for example, in the original model proposed by Bell in 1964.

The results of these computations turn out to be spectacular, with Bell-CHSH inequality violated, as can be appreciated from the correlation surface shown below:

[gill1109]

Beautiful picture!

I am a bit disturbed by "[good]" because I don't recall seeing it in the protocol of the challenge or in our discussions or in anyone's publications so far.

The two files only contain N = 6 912 directions, not 10 000

But that doesn't matter. The format is valid and you have clearly explained how you represent directions.

[Joy Christian]

I am a bit disturbed by "[good]" because I don't recall seeing it in the protocol of the challenge or in our discussions or in anyone's publications so far.

I think you have pretty good understanding of what "[good]" means, but let me spell it out for you:

(1) Your challenge is about my proposed experiment.

(2) My proposed experiment is about testing my local model for the EPRB correlation.

(3) 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 perceptions that we live in R^3 is just an illusion (a bit like looking around and thinking that the planet we live on is flat).

So, considering the above facts, the idea behind the above simulation of my experiment is to produce the N spin vectors that are consistent with the geometry and topology of the 3-sphere. The question then is, which set of N vectors is the correct set of N vectors consistent with the geometry and topology of the 3-sphere?

Call this correct set of N vectors the set of "[good]" N vectors. These are the N vectors that will be produced by Nature naturally, according to my hypothesis. The simulation thus produces them by choosing the "[good]" N vectors from R^3. You are infinitely more knowledgeable in the R script to know exactly what this means.

Nevertheless, let me quote myself from my initial post above to help you out:

Since the challenge is about my proposed experiment (which is supposed to test my 3-sphere model for the EPR-Bohm correlation), it is very important that this response to the challenge is evaluated with a "good" deal of understanding of my model. Sadly, in my opinion, Richard Gill has not understood my model despite considerable efforts. I hope that he or anyone else who wishes to evaluate my response to his challenge, does so in the light of my latest paper on the subject.

[gill1109]

Sadly, Richard Gill has not understood my model despite considerable efforts

That is true statement.

No problem. I see that "[good]" can safely be omitted. The files you sent me contain only "good" directions u and v.

I have downloaded the two files and then ran the following script

Code: Select all

ad <- read.csv("AliceDirections.txt")      # alice directions file
u <- rbind(ad$x, ad$y)           # just the x and y coordinates as a 2xN matrix
rownames(u) <- c("x", "y")
bd <- read.csv("BobDirections.txt")   # alice directions file
v <- rbind(bd$x, bd$y)           # just the x and y coordinates as a 2xN matrix
rownames(v) <- c("x", "y")

head(t(u))  # the first few directions in u
tail(t(u))   # the last few directions in u
head(t(v))  # the first few directions in v
tail(t(v))   # the last few directions in v

alpha <- 0 * pi / 180
beta <- 45 * pi / 180

a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
rho11 <- mean(sign(colSums(a*u))*sign(colSums(b*v)))   # E(0, 45)
rho11   # E(0, 45)

alpha <- 0 * pi / 180
beta <- 135 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
rho12 <- mean(sign(colSums(a*u))*sign(colSums(b*v)))

rho12    # E(0, 135)


alpha <- 90 * pi / 180
beta <- 45 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
rho21 <- mean(sign(colSums(a*u))*sign(colSums(b*v)))

rho21     # E(90, 45)


alpha <- 0 * pi / 180
beta <- 135 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
rho22 <- mean(sign(colSums(a*u))*sign(colSums(b*v)))
rho22     # E(90, 135)

rho11
rho12
rho21
rho22

I got the following results:

Code: Select all

> rho11
[1] -0.6261574
> rho12
[1] 0.6325231
> rho21
[1] -0.3541667
> rho22
[1] 0.6325231


This does not seem to be what you wanted.

So I have say:

The submission (received well before 09:30 a.m. June 11, 2014) was legal, but does not seem to win the challenge.

(A) Do you see problems with my R code?

(B) Or do you wish to appeal to adjudicators, right now?

In the latter case,

(C) Do you agree that we send the letter we composed earlier to the the three persons we had agreed on earlier?

PS To see why I think I got the data in the files into the script correctly, I show you this earlier part of the output:

Code: Select all

> head(t(u))
              x           y
[1,]  0.9609291  0.27679460
[2,]  0.6448492 -0.76430981
[3,]  0.9675821 -0.25255684
[4,] -0.8331492 -0.55304836
[5,]  0.9777434  0.20980434
[6,] -0.9987581 -0.04982248
> tail(t(u))
                 x           y
[6907,] -0.9992743  0.03809089
[6908,] -0.8874223 -0.46095741
[6909,] -0.9076572  0.41971220
[6910,] -0.9911818  0.13250887
[6911,]  0.9680468  0.25076962
[6912,] -0.6009117  0.79931538
> head(t(v))
              x           y
[1,] -0.9609291 -0.27679460
[2,] -0.6448492  0.76430981
[3,] -0.9675821  0.25255684
[4,]  0.8331492  0.55304836
[5,] -0.9777434 -0.20980434
[6,]  0.9987581  0.04982248
> tail(t(v))
                 x           y
[6907,]  0.9992743 -0.03809089
[6908,]  0.8874223  0.46095741
[6909,]  0.9076572 -0.41971220
[6910,]  0.9911818 -0.13250887
[6911,] -0.9680468 -0.25076962
[6912,]  0.6009117 -0.79931538


[Joy Christian]

Do you see problems with my R code?

What has my submission to do with your R code?

Anyone can run my R code to see that the correct numbers are

E_0_45 = -0.6946

E_0_135 = +0.7088

E_90_45 = -0.6999

and

E_90_135 = -0.7243,

giving

abs(E_0_45 - E_0_135 + E_90_45 + E_90_135) = 2.828.

[gill1109]

Do you see problems with my R code?

What has my submission to do with your R code?

You submitted two files.

I calculated four numbers according to the formulas which we agreed on.

If my four numbers are different from your four numbers, and if my code is a correct implementation of the formulas we agreed on, then your code must be wrong. It's as simple as that.

So you think my code is a wrong implementation of the formulas we agreed on?

Do you wish to proceed to arbitration?

[Joy Christian]

So you think my code is a wrong implementation of the formulas we agreed on?

I think that your code is irrelevant.

You can't say my code is wrong without saying what is wrong with it? It is a correct implementation of my model and my proposed experiment, as I explained above.

Do you wish to proceed to arbitration?

I will wait until Fred has a chance to look at your comments. He helped me to produce the text files, so his input in this is very important.

[gill1109]

So you think my code is a wrong implementation of the formulas we agreed on?

I think that your code is irrelevant.

You can't say my code is wrong without saying what is wrong with it? It is a correct implementation of my model and my proposed experiment, as I explained above.

Do you wish to proceed to arbitration?

I will wait until Fred has a chance to look at your comments. He helped me to produce the text files, so his input in this is very important.

Very good. Very reasonable.

I can say your code is wrong without saying what I think is wrong with it: the rules don't say that anyone is obliged to explain anything!

I do say that my code is a correct implementation of what we agreed ... and you appear to disagree. So we will have to agree that we disagree, and therefore we will call in outside help, precisely as the rules specify. But consult with Fred first. He can maybe advise you well, as to whether or not you have a chance that the adjudicators would take your side (note: we don't even know if the people we have in mind will be prepared to adjudicate. We may need to look for someone else. We had better stay on speaking terms, so that this can also be done in a civilized manner).

I can't predict which way it will go. Obviously, I know which way I want it to go, but I am not going to do anything to prejudice their independent decision.

[FrediFizzx]

(A) Do you see problems with my R code?

Yes, you have alpha on your rho22 set to 0 degrees. Should be 90. However, you are right those direction files from last night are not correct. New files will be forthcoming along with the correct script for reading and calculating them. Thanks for the basic read.csv formatting.

[FrediFizzx]

(A) Do you see problems with my R code?

Ok, yes there is another problem. You are doing the correlation calculations in flatland. As I have said a few times previously, the correlation calculations must respect 3-sphere geometry if we are correctly modeling Joy's experiment. Here are the links to the new direction files,

EPRsims/AliceDirectionsu.txt
EPRsims/BobDirectionsv.txt

There is nothing special about these direction files; they are just random directions through points on a circle like in Joy's latest simulation. Just as expected for the experiment.

And here is the R script code that reads those files and correctly calculates the correlations with the results copied and pasted in.

Code: Select all

ad <- read.csv("AliceDirectionsu.txt")      # alice directions file
u <- rbind(ad$x, ad$y)           # just the x and y coordinates as a 2xN matrix
rownames(u) <- c("x", "y")
bd <- read.csv("BobDirectionsv.txt")   # alice directions file
v <- rbind(bd$x, bd$y)           # just the x and y coordinates as a 2xN matrix
rownames(v) <- c("x", "y")

head(t(u))  # the first few directions in u
tail(t(u))   # the last few directions in u
head(t(v))  # the first few directions in v
tail(t(v))   # the last few directions in v

N <- 10^4
s <- runif(N, 0, pi)
p <- 1.21 * (-1 + (2/(sqrt(1 + (3 * s/pi)))))

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(v * b)  ## Inner products of cols of 'v' with 'b'
good1 <- abs(ca) > p & abs(cb) > p
N1 <- sum(good1)
(E_0_45 <- sum(sign(ca[good1]) * sign(-cb[good1]))/N1)
## [1] 0.7068713

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(v * b)  ## Inner products of cols of 'v' with 'b'
good2 <- abs(ca) > p & abs(cb) > p
N2 <- sum(good2)
(E_0_135 <- sum(sign(ca[good2]) * sign(-cb[good2]))/N2)
## [1] -0.6916364

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(v * b)  ## Inner products of cols of 'v' with 'b'
good3 <- abs(ca) > p & abs(cb) > p
N3 <- sum(good3)
(E_90_45 <- sum(sign(ca[good3]) * sign(-cb[good3]))/N3)
## [1] 0.7001815

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(v * b)  ## Inner products of cols of 'v' with 'b'
good4 <- abs(ca) > p & abs(cb) > p
N4 <- sum(good4)
(E_90_135 <- sum(sign(ca[good4]) * sign(-cb[good4]))/N4)
## [1] 0.7214182

## The Bell-CHSH inequality is violated:

abs(E_0_45 - E_0_135 + E_90_45 + E_90_135)
## [1] 2.820107


BAM!!! Can we move on now? Finally?

[gill1109]

Ok, yes there is another problem. You are doing the correlation calculations in flatland. As I have said a few times previously, the correlation calculations must respect 3-sphere geometry if we are correctly modeling Joy's experiment.

Dear Fred

That's what you have indeed several times; you said some product had to be a geometric product and so on .... I asked Christian for clarification, because that is not how I had read Christian's experimental paper (and I mean here: the experimental part of the experimental paper! It's just one page and it's self-contained). He told me that in the formula on page 4 of that paper, something like

E(a, b) = 1/N sum sign(a . u) sign(b . v),

the two dot-products are the ordinary scalar product of two real vectors, sign is the ordinary sign function, and the remaining multiplication, summation, division is ordinary arithmetic. The summation runs over the complete set of all N pairs of particles; the same set of N particle pairs for each correlation.

If you like we can no doubt find which post that was. I think in the topic devoted to Hugh Matlock's submission.

Richard

PS the script you post now is of course the script for simulating Pearle's (1970) model with a different subset of runs used for each pair of directions. We already know that Pearle's (1970) detection loophole model exactly reproduces the singlet correlations. Christian's experimental paper makes some quite different claims. No pairs of particles are lost. The same full set of N particle pairs is used to calculate each correlation according to the formula

E(a, b) = 1/N sum sign(a . u) sign(b . v),

there is no omitting of some pairs, and certainly not of a different collection of pairs per correlation, from some of the summations. Please read the text of the challenge, and the text of page 4 of Christian's first (short) experimental paper. That text of Christian's essentially contains instructions to the experimenter's IT assistants. It makes exactly clear what they have to program. I thought that Christian had confirmed my interpretation, many times. The text of the challenge tries to make it even more explicit. Sure, it could have been even more explicit ... Long ago I showed you the script I would use to process the experimental data and I asked "your side" for translations to other computer languages and confirmation that my script was OK. We got Perl, Excel, Mathematica and Python versions of my code from several different people, both people "on my side" and neutral bystanders, and we got no response whatsoever from "your side". You had your chance, months ago! Too late to suddenly say now that oh dear there is a misunderstanding it has to be done quite differently. Well - you can say that, and then we just call the whole thing off, and remain good friends. Little misunderstanding, little oversight. Everyone is co-responsible for breakdowns of communication.

[FrediFizzx]

Well, I know this is not what you and Joy exactly agreed upon but in my opinion it is the correct way to simulate the experiment. It works and you claim it works because of the detection loophole; we claim it works because of 3-sphere geometry. Only doing the real experiment will tell who is correct. Let's do it!

[gill1109]

Well, I know this is not what you and Joy exactly agreed upon but in my opinion it is the correct way to simulate the experiment. It works and you claim it works because of the detection loophole; we claim it works because of 3-sphere geometry. Only doing the real experiment will tell who is correct. Let's do it!

Sure, let's do the experiment! If the directions of the spinning hemispheres are equal and opposite and roughly uniformly distributed over S^2, then "your way" of processing the data will give the singlet correlations, "my way" won't. Note: each pair of particles will have to provide not only directions u and v but also a uniform random number between 0 and 1 or some appropriate transformation thereof, "p" or "s" or something. You can say that this is because of S^3 geometry by deriving Pearle's model from S^3 geometry. I was not happy with Christian's derivation in his new paper, but then I am not an expert in this kind of algebra. I thought there were a lot of ambiguities and anomalies in his analysis but heck you know his opinion of my math skills.

So it is not clear to me whether "p" and "s" are randomly generated during the data-processing or if they also come out of measurements of the particles. Are they part of the data-set or just part of a randomized algorithm?

Seems like we are close to closure. It would be nice if someone did the experiment.

[FrediFizzx]

Well, I know this is not what you and Joy exactly agreed upon but in my opinion it is the correct way to simulate the experiment. It works and you claim it works because of the detection loophole; we claim it works because of 3-sphere geometry. Only doing the real experiment will tell who is correct. Let's do it!

So it is not clear to me whether "p" and "s" are randomly generated during the data-processing or if they also come out of measurements of the particles. Are they part of the data-set or just part of a randomized algorithm?

It would just be "s" to figure out since "p" is dependent on "s". Ok Joy, the ball is in your court about "s" now.

[gill1109]

Well, I know this is not what you and Joy exactly agreed upon but in my opinion it is the correct way to simulate the experiment. It works and you claim it works because of the detection loophole; we claim it works because of 3-sphere geometry. Only doing the real experiment will tell who is correct. Let's do it!

So it is not clear to me whether "p" and "s" are randomly generated during the data-processing or if they also come out of measurements of the particles. Are they part of the data-set or just part of a randomized algorithm?

It would just be "s" to figure out since "p" is dependent on "s". Ok Joy, the ball is in your court about "s" now.

If there were not only a direction of spin but also a length of direction of spin, or an angle of rotation about that direction, that could give you the missing "degree of freedom".

[Joy Christian]

Richard,

In my opinion you are bobbing and weaving, skirting the main issues of physics, and grossly misinterpreting both my "experimental" papers and my model. There is nothing new in this, for I have been complaining about your misinterpretations of my work, and specifically about your misrepresentations of my "experimental" paper, for several years now, starting with our acrimonious discussion on FQXi a couple of years ago, where I repeatedly stressed that the four correlations in the context of my proposed experiment must be calculated separately. See, for example, this post of mine: viewtopic.php?f=6&t=31&p=1651&hilit=replacement#p1651.

But let us not worry about those contentious issues now. Instead, let us play your game your way, by your rules. Here is the cool gif you produced for displaying the four correlations in question: http://www.math.leidenuniv.nl/~gill/Images/movie2.gif. Do have a look at it again if you have forgotten about it? What do you see?

Now I have given you even better evidence for the four local-realistic correlations in my simulation: http://rpubs.com/jjc/19298. I have given you the entire surface of all possible correlations, for all choices of a and b, without data rejection, non-locality, or superdeterminism (cf. this paper http://arxiv.org/abs/1405.2355).

Now compare your gif with my correlation surface and tell me where do they differ. This is not a trick question. They do not differ at all; not even at a single point!

What is more, in the simulation I have also provided explicit calculations for the four specific correlations with the same set of directions u_k[good]. I have already explained what u_k[good] means in the context of my model and that of my proposed experiment to test it. Fred has now pointed out the errors in your calculations and provided the correct version of your calculations. Putting all this evidence together, it is fairly clear that I have successfully produced the N spin directions for Alice and Bob that you claimed were impossible to produce. In fact by now I have provided four different simulations exhibiting the same incontrovertible evidence.

In conclusion it is clear to me that you do not want to admit defeat. Instead, you want to keep calculating the correlations incorrectly and unphysically, in your own way, so that you do not have to admit defeat. Therefore, based on the evidence provided by both Fred and I, I formally request you to seek adjudication on my claim, which you have already accepted as a valid claim according to the rules you have set up (note that I deny your claim that we set up the rules together---we didn't).

[gill1109]

Fred has now pointed out the errors in your calculations and provided the correct version of your calculations

I think not. The way I see it, Fred has pointed out the errors (or at the very least: ambiguities) in Christian's instructions (page 4 of the experimental paper) of the last phase of his experiment. Christian insisted repeatedly that my reading of those instructions was correct. But at last, Fred has been able to convince Christian that he (Christian) was wrong.

This is precisely what I hoped to achieve. It has taken a long time, but at long last the coin has dropped. Congratulations, Fred.

Christian might like to explain to Fred how anyone is supposed to get "his" correlations, when "p" or "s" is not included in the two data-sets.

And: after doing an expensive experiment, should the experimenter then use a random number generator to "invent" the missing data? And use them to delete about 20% of the runs? (A different 20% for each correlation).

I shall send our hoped-for adjudicators the letter we had agreed on. Are the two data-sets the same as they were yesterday?

[Joy Christian]

Richard,

You missed my post above. Here is what I wrote. Please stop fantasizing in your flatland and read what I have actually written:

Richard,

In my opinion you are bobbing and weaving, skirting the main issues of physics, and grossly misinterpreting both my "experimental" papers and my model. There is nothing new in this, for I have been complaining about your misinterpretations of my work, and specifically about your misrepresentations of my "experimental" paper, for several years now, starting with our acrimonious discussion on FQXi a couple of years ago, where I repeatedly stressed that the four correlations in the context of my proposed experiment must be calculated separately. See, for example, this post of mine: viewtopic.php?f=6&t=31&p=1651&hilit=replacement#p1651.

But let us not worry about those contentious issues now. Instead, let us play your game your way, by your rules. Here is the cool gif you produced for displaying the four correlations in question: http://www.math.leidenuniv.nl/~gill/Images/movie2.gif. Do have a look at it again if you have forgotten about it? What do you see?

Now I have given you even better evidence for the four local-realistic correlations in my simulation: http://rpubs.com/jjc/19298. I have given you the entire surface of all possible correlations, for all choices of a and b, without data rejection, non-locality, or superdeterminism (cf. this paper http://arxiv.org/abs/1405.2355).

Now compare your gif with my correlation surface and tell me where do they differ. This is not a trick question. They do not differ at all; not even at a single point!

What is more, in the simulation I have also provided explicit calculations for the four specific correlations with the same set of directions u_k[good]. I have already explained what u_k[good] means in the context of my model and that of my proposed experiment to test it. Fred has now pointed out the errors in your calculations and provided the correct version of your calculations. Putting all this evidence together, it is fairly clear that I have successfully produced the N spin directions for Alice and Bob that you claimed were impossible to produce. In fact by now I have provided four different simulations exhibiting the same incontrovertible evidence.

In conclusion it is clear to me that you do not want to admit defeat. Instead, you want to keep calculating the correlations incorrectly and unphysically, in your own way, so that you do not have to admit defeat. Therefore, based on the evidence provided by both Fred and I, I formally request you to seek adjudication on my claim, which you have already accepted as a valid claim according to the rules you have set up (note that I deny your claim that we set up the rules together---we didn't).

[gill1109]

Joy, I understand perfectly well what you are requesting. I'm on a train to Sweden right now. When I have settled in at a decent computer with decent internet, I will email the letter we agreed on to the persons we had in mind. The first step is to find out if they agree to do this job. Be patient, please.

[Joy Christian]

Joy, I understand perfectly well what you are requesting. I'm on a train to Sweden right now. When I have settled in at a decent computer with decent internet, I will email the letter we agreed on to the persons we had in mind. The first step is to find out if they agree to do this job. Be patient, please.

Fair enough. My task is done. The ball is in your court now. I am moving on to the next stages of my proposed experiment, as Fred has been urging we should.

As for the “s” occurring in p = 1.21 * (-1 + (2/(sqrt(1 + (3 * s/pi))))), it is a hidden variable. An experimenter will not have access to it. Nature---i.e., the geometry and topology of S^3---will produce s and p naturally, and thereby constrain the spin directions u. All Alice and Bob will ever see are the "good" u directions, because the "bad" u directions simply do not exist in S^3. There is nothing mysterious about this if you understand the basic properties of S^3: http://arxiv.org/abs/1405.2355.