My claim to collect the 10,000 Euros offered by Richard Gill

[gill1109]

My goodness, someone is getting upset. First he switches to a very large font, now he is using large red letters. Next will be capitals.

Unfortunately, writing over and over again "I am the king of France" does not make it true, whatever size letters you write them in.

If u and v are just different names for the same spin direction, then please delete the superfluous items from the files, and claim your 10 000 Euro. You have about two weeks to go. I'm sure someone can help you fix the problem.

[Joy Christian]

If u and v are just different names for the same spin direction, then please delete the superfluous items from the files, and claim your 10 000 Euro.

There is nothing to delete, amend, revise, or correct in my submission.

There is only one set of directions for Alice, the negative of which is for Bob. I have submitted only 1 file for Alice, not 2, and likewise only 1 file for Bob, not 2.

u and v specify exactly the same set of N directions in the physical space, meeting the terms of the challenge documented by Richard Gill himself.

I have won the challenge. Richard Gill owes me 10,000 Euros.

I repeat: Richard Gill owes me 10,000 Euros.

In addition to these simulations, I have recently won the 10,000 Euros offered by Richard Gill for theoretically producing the 2n angular momentum vectors, and , appearing in the equation (16) of my proposed experiment (see also this page). He had claimed that it was mathematically impossible to produce such 2n vectors and had challenged me to produce them as a "proof of concept" for my proposed experiment. I defeated his challenge on the 3rd of May 2014 by explicitly producing the 2n vectors in these two simulations.

It is very important to note that in the second simulation above I have used vectors defined by the ordered set



for calculating the first two of the four correlations in the simulation,

and vectors defined by the ordered set



for calculating the last two of the four correlations in the simulation.

Richard Gill claims that these sets of vectors specify different sets of directions in the physical space. But evidently they specify exactly the same set of directions in the physical space. They both define a unique distribution of points on a circle of radius . This should be quite easy for any mathematician to see, by

simply noting that .

and are thus different names of one and the same spin direction (say ) in the physical space.

[gill1109]

JJC, I don't listen to someone who shouts at me. You are presently on my "foe list" so I don't have to be revolted by your posts. Go cry to Mama. Maybe Lucien Hardy would like to intercede on your behalf?

I don't know who you are trying to fool, but giving Bob two possible counterfactual outcomes, and selecting which one is "the" outcome according to which of two settings is chosen by Alice, is the most blatantly non-local "LHV for the singlet correlations" I have ever seen. It certainly deserves some kind of prize but I am not going to say what kind. You made my vacation very enjoyable, very memorable.

I am not the UK welfare state, so, much as I am amused by your antics, I am not going to put money in your busker's cap.

[Joy Christian]

I don't listen to someone who shouts at me.

Fine. Others are listening. For those who are, here are the facts:

Page 4 of my experimental paper contains only two unambiguous equations:



Here it is important to recall that

The experimental paper, page 4, states that we then calculate 1/N sum_j sign(a . lambda_j ) sign(b . -lambda_j).

...the vectors a, b, lambda_j, -lambda_j [are] unit vectors in R^3, the dot signified the usual scalar product, and sign [means] sign.

Note: There is only one correlation function, E(a, b), in equation (16), not four.

An introductory paragraph of the paper exposes the ambiguity in Bell's observables:



The following statement on my blog spells out the resolution of the Gill challenge:

In addition to these simulations, I have recently won the 10,000 Euros offered by Richard Gill for theoretically producing the 2n angular momentum vectors, and , appearing in the equation (16) of my proposed experiment (see also this page). He had claimed that it was mathematically impossible to produce such 2n vectors and had challenged me to produce them as a "proof of concept" for my proposed experiment. I defeated his challenge on the 3rd of May 2014 by explicitly producing the 2n vectors in these two simulations.

It is very important to note that in the second simulation above I have used vectors defined by the ordered set



for calculating the first two of the four correlations in the simulation,

and vectors defined by the ordered set



for calculating the last two of the four correlations in the simulation.

Richard Gill claims that these sets of vectors specify different sets of directions in the physical space. But evidently they specify exactly the same set of directions in the physical space. They both define a unique distribution of points on a circle of radius . This should be quite easy for any mathematician to see, by

simply noting that .

and are thus different names of one and the same spin direction (say ) in the physical space.

Conclusion: I have won the challenge. Richard Gill owes me 10,000 Euros.

I repeat: Richard Gill owes me 10,000 Euros.

[Joy Christian]

Hi Everyone,

I have written another version of the second of the above two simulations: http://rpubs.com/jjc/18915.

Both versions of the simulation can be understood as respecting the parity change intrinsic to my 3-sphere model for the EPR-Bohm correlation.

The vectors for Alice and for Bob are now defined by the ordered sets



and

.

Note the differences in the definitions. The last two of the four correlations in the simulation are now calculated using left-handed basis for the spins instead of right-handed basis . Not surprisingly, all four correlations once again match exactly with the corresponding quantum mechanical predictions.

[gill1109]

Looks to me that your 90 degrees rotation of u and v in order to create the second two pairs of correlations http://rpubs.com/jjc/18915 might somehow be connected to Alice's second angle 90 degrees being 90 degrees on from her first angle 0 degrees. We see that *how* Bob computes his outcomes depends on *which* setting is being used by Alice. I would call this action at a distance, or perhaps, conspiracy (super-determinism).

Please show us how you would now compute, say, E(23, 40). ie pick any two other alpha, beta, and show us the quantum correlation E(alpha, beta).

If you are *really* serious that this is a claim to the 10 000, do you want me to send our letter to the adjudicators to ask them to evaluate this submission?

[gill1109]

Please show us how you would now compute, say, E(23, 40). ie pick any two other alpha, beta, and show us the quantum correlation E(alpha, beta).

Any ideas? How can I compute a whole correlation *surface* from your two files with two different representations of both Alice's and Bob's directions? Why two? Because Alice had only two different measurement directions? Suppose Alice may choose anything from 0 to 360 in steps of 5 degrees, and Bob may choose anything form 0 to 360 in steps of 5 degrees?

Please extend your script with a calculation of all of these (1 + 360 / 5)^2 pairs of measurement directions.

[Joy Christian]

Please extend your script with a calculation of all of these (1 + 360 / 5)^2 pairs of measurement directions.

Is this a new rule added to the challenge? The N^th rule? Does N have an upper bound?

You were obviously never serious about the challenge. To realize that, go back to the very first time you set out the rules of the challenge. Then follow through all of your hundreds of posts about the challenge to see how many times you have added a new rule, or twisted the existing ones. It will be quite an eye opening exercise.

[gill1109]

Please extend your script with a calculation of all of these (1 + 360 / 5)^2 pairs of measurement directions.

Is this a new rule added to the challenge? The N^th rule? Does N have an upper bound?

You were obviously never serious about the challenge. To realize that, go back to the very first time you set out the rules of the challenge. Then follow through all of your hundreds of posts about the challenge to see how many times you have added a new rule, or twisted the existing ones. It will be quite an eye opening exercise.

I was always serious. I always knew that the challenge could not be won and at last you seem to have realised that I was right. The challenge was to create two data-files which would "win" the experimental bet. For this purpose, it turned out that your instructions to the experimenter's IT team were ambiguous, to put it kindly. It seems now that you repudiate the earlier "naive" interpretation (real vectors, ordinary dot product ...), which you earlier seemed to agree with; but we have made a lot of progress: page four of the experimental paper needs to be expanded.

I am not adding any new rules. I am just inquisitive.

But now to the future. New topic. New question.

How do you compute E(a, b) for an arbitrary a, b different from the four pairs considered in your R script?

I think it's a sensible, natural, scientific, question.

No hurry...

PS here is how you can "fix" the experimental paper, following the Pearle model. Consider a spinning sphere. It spins around a direction U; and at a random moment, it is in a position of being rotated through an angle Theta, around U, from some reference position, as viewed in some reference coordinate system.

Now let lots of pairs of balls spin in equal and opposite directions U, V, and suppose their rotation angles at a fixed moment Theta, Phi are also equal and opposite. Create two large files of pairs (U_k, Theta_k); (V_k, Phi_k); k = 1, ..., N; N = (say) 10 000. Arrange that V_k = - U_k for all k, and Theta_k = - Phi_k (mod 2 pi) for all k. Arrange that the U_k are a random sample from the uniform distribution on S^2 and the Theta_k are an independent random sample from the uniform distribution on [0, 2 pi].

Now for given measurement directions a, b define

c_k = a . U_k
d_k = b . V_k
s_k = (2/sqrt(3*(Theta_k/2 pi)+1)) - 1
t_k = (2/sqrt(3*((2 pi - Phi_k)/2 pi)+1)) - 1
A_k = sign(c_k) if abs(c_k) > s_k otherwise 0
B_k = sign(d_k) if abs(d_k) > t_k otherwise 0

Notice that the product A_k B_k = -1, 0, or +1.
Define E(a, b) = sum(A_k B_k) / sum(abs(A_k B_k)) ie the average of the nonzero products.

So your task is merely to explain to the experimenters why this non-conventional definition of correlation is actually the right one to use. Well, of course it is the right one in the sense that in the large N limit it'll give you E(a, b) = - a . b. But what is the meaning of the "rejection" of the pairs of spheres with A_k B_k = 0? What is the relation of the pair (U_k, Theta_k) to usual notions of angular momentum? How is this all related to the geometric algebra and the innovative definition of correlation (using bivector standard deviations) of the one page paper?

PPS it might be useful to study Caroline Thompson's chaotic spinning ball model, which is a nicely physically motivated way leading to correlations like this. Somehow, when you observe the balls, you interact with them, and sometimes they even disappear altogether.

[Joy Christian]

I was always serious. I always knew that the challenge could not be won and at last you seem to have realised that I was right. The challenge was to create two data-files which would "win" the experimental bet. For this purpose, it turned out that your instructions to the experimenter's IT team were ambiguous, to put it kindly. It seems now that you repudiate the earlier "naive" interpretation (real vectors, ordinary dot product ...), which you earlier seemed to agree with; but we have made a lot of progress: page four of the experimental paper needs to be expanded.

My paper is just fine as it is. Do you see anyone who has actually read my papers---like Fred, Michel, Hugh, or Ben---complaining about ambiguities in them? No. Because there aren't any. The ambiguities are all in your mind because you are stuck in R^3. Unless you liberate yourself from it, you will never understand my papers.

But now to the future. New topic. New question.

How do you compute E(a, b) for an arbitrary a, b different from the four pairs considered in your R script?

I think it's a sensible, natural, scientific, question.

You compute E(a, b) from the observed experimental data exactly in the manner I have specified in equation (16) of my "experimental" paper.

On the other hand, theoretically you may compute E(a, b) in accordance with the topology of the 3-sphere, as it is done here, http://rpubs.com/jjc/16567, and explained here, http://arxiv.org/abs/1405.2355. Or you can compute them using Geometric Algebra, as it is done in this paper: http://arxiv.org/abs/1301.1653.

These computations have nothing to do with the "challenge" per se, which is concerned about theoretically producing the N vectors observed in the experiment.

What I am saying is that Nature will produce the desired N vectros in my proposed experiment, regardless of the "challenge" of producing them theoretically.

[gill1109]

Equation (16) is E(a, b) = 1/N sum_{j = 1}^N {sign (λj · a)} {sign(−λj · b)}

and this is indeed what everything turns around.

Problem 1. In equation (16),

(i) Are a, b and λj unit vectors in R^3 ?

(ii) Is "." the scalar dot product?

(iii) Is "sign" the usual sign function?

(iv) Is everything else ordinary arithmetic?

If not, please explain what they are, instead.

Problem 2. Please explain how to compute (16) when the input data are the two files produced in your latest simulation, which each contain pairs of directions called u and v. Your R script http://rpubs.com/jjc/18915 does not use formula (16). Instead, it uses Alice's setting a in order to determine whether to use Bob's directions u or v, and it only allow two choices for a, and two choices for b. What is the formula for arbitrary a, b?

[Joy Christian]

Equation (16) is E(a, b) = 1/N sum_{j = 1}^N {sign (λj · a)} {sign(−λj · b)}

So: are a, b and λj unit vectors in R^3 ?
Is "." the scalar dot product?
Is "sign" the usual sign function?
Is everything else ordinary arithmetic?

If not, please explain what they are, instead.

Secondly, please explain how to compute (16) when the input data are the two files produced in your latest simulation, which each contain pairs of directions. Your R script http://rpubs.com/jjc/18915 does not use formula (16). Instead, it uses Alice's setting a in order to determine whether to use Bob's directions u or v.

Did you not just read my previous reply? Apparently not.

Very well:

Are a, b and λj unit vectors in R^3 ?

Yes.

Is "." the scalar dot product?

Yes.

Is "sign" the usual sign function?

Yes.

Is everything else ordinary arithmetic?

Yes.

Secondly, please explain how to compute (16) when the input data are the two files produced in your latest simulation, which each contain pairs of directions. Your R script http://rpubs.com/jjc/18915 does not use formula (16). Instead, it uses Alice's setting a in order to determine whether to use Bob's directions u or v.

The calculations in the above simulation are quite transparent. As I said in my previous reply, you have to distinguish between the experimental procedure from the theoretical generation of the N vectors being used in this particular simulation. Here we are trying to imitate Nature. This simulation is not Nature herself. It is a numerical model of an experiment that is supposed to test a theoretical model of Nature. Therefore, for this particular set of theoretically generated N vectors, the first two correlations have to be calculated using the right-handed basis, and the last two correlations have to be calculated using the left-handed basis. If you see this as a deficiency, then it is a deficiency of this particular simulation, and not a deficiency of Nature herself. Read my previous reply again to get my point.

[gill1109]

Equation (16) is E(a, b) = 1/N sum_{j = 1}^N {sign (λj · a)} {sign(−λj · b)}

So: are a, b and λj unit vectors in R^3 ?
Is "." the scalar dot product?
Is "sign" the usual sign function?
Is everything else ordinary arithmetic?

If not, please explain what they are, instead.

Secondly, please explain how to compute (16) when the input data are the two files produced in your latest simulation, which each contain pairs of directions. Your R script http://rpubs.com/jjc/18915 does not use formula (16). Instead, it uses Alice's setting a in order to determine whether to use Bob's directions u or v.

...

Are a, b and λj unit vectors in R^3 ?

Yes.

Is "." the scalar dot product?

Yes.

Is "sign" the usual sign function?

Yes.

Is everything else ordinary arithmetic?

Yes.

...

The calculations in the above simulation are quite transparent. As I said in my previous reply, you have to distinguish between the experimental procedure from the theoretical generation of the N vectors being used in this particular simulation. Here we are trying to imitate Nature. This simulation is not Nature herself. It is a numerical model of an experiment that is supposed to test a theoretical model of Nature. Therefore, for this particular set of theoretically generated N vectors, the first two correlations have to be calculated using the right-handed basis, and the last two correlations have to be calculated using the left-handed basis. If you see this as a deficiency, then it is a deficiency of this particular simulation, and not a deficiency of Nature herself. Read my previous reply again to get my point.

Splendid.

Indeed, in order to get the "right answer", you have to use the right hand basis for two of the four correlations (the ones with a = 0 degrees), and the left hand basis for the other two (the ones with a = 90 degrees). I do see this as a deficiency of the simulation. I object to you switching left hand to right hand basis for Bob's observed spin directions, depending on which direction Alice chooses to measure. If you want the adjudicators to adjudicate on this, I will bring it to their attention when I see them (all three) next week.

I am very glad that you have completely clarified the meaning of (16). The "naive" interpretation was the intended interpretation. Formula (16) was not ambiguous after all. This means that Hugh's submission definitely does not win the challenge. But if he wants the adjudicators to adjudicate on that as well, I will ask them if they are prepared to do so, as well.

You did not answer my second question:

Secondly, please explain how to compute (16) when the input data are the two files produced in your latest simulation, which each contain pairs of directions. Your R script http://rpubs.com/jjc/18915 does not use formula (16). Instead, it uses Alice's setting a in order to determine whether to use Bob's directions u or v.

In other words: take the two data sets coming from your simulation. Can you use the same two data sets to calculate other correlations E(a, b)? If so, how?

[Joy Christian]

[You did not answer my second question:

Secondly, please explain how to compute (16) when the input data are the two files produced in your latest simulation, which each contain pairs of directions. Your R script http://rpubs.com/jjc/18915 does not use formula (16). Instead, it uses Alice's setting a in order to determine whether to use Bob's directions u or v.

In other words: take the two data sets coming from your simulation. Can you use the same two data sets to calculate other correlations E(a, b)? If so, how?

No, I cannot. That was not the purpose of this particular simulation. The purpose was simply to meet the conditions of the challenge (which, in my opinion, I have).

[gill1109]

You did not answer my second question:

Secondly, please explain how to compute (16) when the input data are the two files produced in your latest simulation, which each contain pairs of directions. Your R script http://rpubs.com/jjc/18915 does not use formula (16). Instead, it uses Alice's setting a in order to determine whether to use Bob's directions u or v.

In other words: take the two data sets coming from your simulation. Can you use the same two data sets to calculate other correlations E(a, b)? If so, how?

No, I cannot. That was not the purpose of this particular simulation. The purpose was simply to meet the conditions of the challenge (which, in my opinion, I have).

Splendid. Now all my questions are answered.

It seems that you claim that Nature might generate two data-sets which can be succesfully processed using formula (16) while a computer simulation cannot do that; at least, so far, you did not succeed in finding one which works.

In other words, you cannot win the challenge about the data set simulation, according to my rigid interpretation of (16), but you could win the bet about the results of a real experiment, data analysed according to the rigid interpretation of (16).

No need to revise the experimental paper.

As far as I am concerned, we can re-instate the bet now (if you are also prepared to possibly *lose* 5 000 Euro) since all ambiguities about the post-processing of the data have been resolved.

[Joy Christian]

It seems that you claim that Nature might generate two data-sets which can be succesfully processed using formula (16) while a computer simulation cannot do that; at least, so far, you did not succeed in finding one which works.

Yes, I do believe Nature will naturally generate the two data sets in my proposed experiment, just as she does in the "quantum" experiments.

Theoretically it is a question of finding the correct distributions of the directions u(r,s,t,...) and v(r,s,t,...), where u and v are dual to the bivectors I.u and I.v (with I being a volume form). I remain unimpressed by Bell-type arguments of impossibility in this regard, which Nature is well known to disrespect.

I am not saying that a computer simulation cannot produce the data sets. But having seen what codes can and cannot do, I remain more confident in my analytical results than in their numerical implementations.

In other words, you cannot win the challenge about the data set simulation, according to my rigid interpretation of (16), but you could win the bet about the results of a real experiment, data analysed according to the rigid interpretation of (16).

I will win the bet about the experiment for sure, and I may also win the challenge (if I haven't won already). If I don't, then "I will feel sorry for the Dear Lord."

As far as I am concerned, we can re-instate the bet now (if you are also prepared to possibly *lose* 5 000 Euro) since all ambiguities about the post-processing of the data have been resolved.

I have no problem with that. There were no ambiguities in my mind in any case.

[Heinera]

I still feel there is a question left to be resolved here: From the two lists of vectors generated by the eperiment, are we or are we not allowed to compute several correlations corresponding to different detector settings on the same lists? And if not, who shall decide which detector settings can be used?

[gill1109]

No ambiguities.

Suppose the experiment has delivered us the data sets u_1, ... u_N and v_1, ..., v_N.

Define

A = A(j) = sign(a . u_j),
A' = A'(j) = sign(a' . u_j),
B = B(j) = sign(b . v_j),
B' = B'(j) = sign(b'. v_j)

For each j, A B + A B' + A'B - A'B' = +/- 2.

Therefore averaging over j = 1 ... N

E(a, b) + E(a, b') + E(a', b) - E(a', b') lies between -2 and +2

[gill1109]

I still feel there is a question left to be resolved here: From the two lists of vectors generated by the eperiment, are we or are we not allowed to compute several correlations corresponding to different detector settings on the same lists? And if not, who shall decide which detector settings can be used?

Seems to me there is no ambiguity. We can compute as many correlations we like according to whatever detector settings we like from the two lists. For the challenge and the bet, we focussed on two particular settings for Alice and two for Bob. But the experimental paper (page 4) makes clear that the same collection of videos of N exploding balls is used for calculating all possible correlations.

[Heinera]

I still feel there is a question left to be resolved here: From the two lists of vectors generated by the eperiment, are we or are we not allowed to compute several correlations corresponding to different detector settings on the same lists? And if not, who shall decide which detector settings can be used?

Seems to me there is no ambiguity. We can compute as many correlations we like according to whatever detector settings we like from the two lists. For the challenge and the bet, we focussed on two particular settings for Alice and two for Bob. But the experimental paper (page 4) makes clear that the same collection of videos of N exploding balls is used for calculating all possible correlations.

OK, so long as this is clear. Any kind of cherrypicking subsets of vectors to suit the detector settings must be ruled out.