Given the SPF's extensive discussions re Bell's theorem, I would welcome critical comments here on this essay: Commonsense local realism refutes Bell's theorem - http://vixra.org/pdf/1403.0089v3.pdf

Based on elementary first-principles and undergrad maths and logic, I can find no errors there. So it might help to clarify the other critiques of Bell's theorem; especially those that take a less direct approach.

Thanks, Xray

Based on elementary first-principles and undergrad maths and logic, I can find no errors there. So it might help to clarify the other critiques of Bell's theorem; especially those that take a less direct approach.

Thanks, Xray

- Xray
**Posts:**44**Joined:**Mon Apr 21, 2014 2:23 pm

Splendid! So maybe you can create the two data files which Christian needs before June 11 and win 10 000 Euro? See the thread on the bet on Joy Christian's experiment. If you are right, it can be done with CLR.

http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=46

http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=46

- gill1109
- Mathematical Statistician
**Posts:**2812**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

Thanks Gill; I'll pass your challenge on to the author. I suspect he will want know how you, in charge of the source, will code the entangled particle pairs. In other words: What inputs do you want his theory to handle? I presume his outputs will be ±1.

So please give me the link to the precise specification of the source's outputs in the challenge? Via PM please.

Because -- just in case your challenge is judged to be unrealistic -- we should limit this thread to specific criticisms of the given essay.

Thus: Can you help me, please? Can you point me to any errors in the essay?

PS: I do believe that it's only undergrad maths that's employed.

So please give me the link to the precise specification of the source's outputs in the challenge? Via PM please.

Because -- just in case your challenge is judged to be unrealistic -- we should limit this thread to specific criticisms of the given essay.

Thus: Can you help me, please? Can you point me to any errors in the essay?

PS: I do believe that it's only undergrad maths that's employed.

- Xray
**Posts:**44**Joined:**Mon Apr 21, 2014 2:23 pm

Great article! Very fun to read. Got confused by some of the undefined acronyms but figured them out. Also, the function-set notation is completely foreign to me, I may not be alone.

Well worth the read though.

I hope we can discuss the article rather that peripheral bets. It can get tiring for every thread to be turned into one about bets.

Well worth the read though.

I hope we can discuss the article rather that peripheral bets. It can get tiring for every thread to be turned into one about bets.

- minkwe
**Posts:**1441**Joined:**Sat Feb 08, 2014 10:22 am

Xray wrote:Thanks Gill; I'll pass your challenge on to the author. I suspect he will want know how you, in charge of the source, will code the entangled particle pairs. In other words: What inputs do you want his theory to handle? I presume his outputs will be ±1.

So please give me the link to the precise specification of the source's outputs in the challenge? Via PM please.

Because -- just in case your challenge is judged to be unrealistic -- we should limit this thread to specific criticisms of the given essay.

Thus: Can you help me, please? Can you point me to any errors in the essay?

PS: I do believe that it's only undergrad maths that's employed.

Xray: the 10 000 Euro challenge (expires June 11, 2014 ... after that it's 5 000 Euro) is to create two computer files each containing N directions represented by spherical coordinates theta, phi. The files will be processed in the way described in the forum topic initial posting. Four correlations will be calculated, separately. The formulas, and the criterion for win/lose are all there. Joy Christian believes that such files will result from a certain experiment, and also by computer simulation of his model.

I will look at the CLR paper later.

- gill1109
- Mathematical Statistician
**Posts:**2812**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

Thanks Richard; I will really appreciate your critical review of CLR.

NB: Please be very critical, for, if there are no flaws there -- and I've yet to find one -- then Bell is the "silly" one; not von Neumann.

Note that CLR progressively identifies, and progressively corrects, Bell's errors -- beginning with his crucial first mistake. And CLR does this from first principles, using undergrad maths and logic. So no profound analysis should be required.

Thus, it seems to me, CLR opens the way for a topological refutation also. For higher-maths is generally capable of delivering results first found by lower-level considerations. (But I'm not qualified to critically comment on Joy's results; let alone CLR's.)

PS: I'm inclined to believe that your bet is safe. For reasons best given elsewhere.

NB: Please be very critical, for, if there are no flaws there -- and I've yet to find one -- then Bell is the "silly" one; not von Neumann.

Note that CLR progressively identifies, and progressively corrects, Bell's errors -- beginning with his crucial first mistake. And CLR does this from first principles, using undergrad maths and logic. So no profound analysis should be required.

Thus, it seems to me, CLR opens the way for a topological refutation also. For higher-maths is generally capable of delivering results first found by lower-level considerations. (But I'm not qualified to critically comment on Joy's results; let alone CLR's.)

PS: I'm inclined to believe that your bet is safe. For reasons best given elsewhere.

- Xray
**Posts:**44**Joined:**Mon Apr 21, 2014 2:23 pm

In the paper that this thread is about (stay on topic, Richard), I don't understand eq. 19 to 22. Maybe you can explain more about them?

- FrediFizzx
- Independent Physics Researcher
**Posts:**2905**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

FrediFizzx wrote:In the paper that this thread is about (stay on topic, Richard), I don't understand eq. 19 to 22. Maybe you can explain more about them?

I can make no sense whatsoever of equations (19) to (22). Nor for that matter, of many of the equations before those four. The author introduces an equivalence relation denoted "~" but actually he is talking about many different equivalence relations simultaneously. His notation is both too elaborate, and not elaborate enough, at the same time. It is so elaborate as to be almost unintelligible. Because it is almost unintelligible it is easy to hide simple conceptual errors. Those errors would have been obvious if his notation had made explicit what needs to be explicit.

The non-local hidden variables in this paper are simply hidden errors in the almost incomprehensible jungle of mainly superfluous symbols.

- gill1109
- Mathematical Statistician
**Posts:**2812**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

Well, on the more technical matters about (19)-(22), I'll see if I can get the author to reply. Personally, I found (26)-(37) easier to follow because there were more steps involved.

But some things that appear clear to me might help.

1. Re the claim that there are many different equivalence relations: There must be! In Appendix A.4, we read the definition of ~. Since there are four Q functions in EPRB -- in my terms: Q(+a), Q(-a),Q'(+b'), Q'(-b') --- there will be four equivalence relations on Λ under these four functions.

2. Re the claim that there are "mainly superfluous symbols": Why not be helpful and identify some of them? The essential dynamical symbols are given in Appendix [A.1], see (A.1). The equivalence relations are defined in [A.4]. Equivalent symbols are given in (A.5)-(A.6) by way of examples.

3. Richard, why not give an example of what needs to be explicit? Start with (A.1), (A.3), (A.5)-(A.6), (1)-(2) please; now what more would you like, explicitly please?

4. Richard, in that you've gotten to (19)-(22): that presumably means that you've seen Bell's first fundamental error explicitly identified and corrected. You've seen the nonsense in CHSH explicitly identified and corrected. So we need to be serious in defining issues that we want clarified -- because this looks like serious business to me.

5. Richard: "So elaborate to be ALMOST unintelligible?" Yet nowhere departing from undergrad maths and logic. Why not be explicit about your difficulties with equations before (19)?

6. As for this: "The non-local hidden variables in this paper are simply hidden errors in the almost incomprehensible jungle of mainly superfluous symbols." Surely your finding just one non-local hidden variable would sink a paper based on CLR? And surely superfluous symbols can be simply crossed out and reported; to thus help us all?

7. May I suggest to ALL (me included), that we be specific in identifying our concerns and difficulties. Even identifying what we accept? Like what I accept for certain: Bell's basic boo-boo in equating his 1964:(14a) to his (14b)? That CHSH is just plain silly; since the experiments do not match the maths; except if Bertlmann's socks need washing?

On these grounds ALONE I can see no rational basis, under Bell -- given the ground that has been cut from beneath him -- for expecting anything other that an equals-signs in (21)-(22).

PS: A final thought: Given the structure of the paper, we need only to find the first relevant error for it to collapse like a pack of cards. So are there any errors before (19)?

HTH, Xray

But some things that appear clear to me might help.

1. Re the claim that there are many different equivalence relations: There must be! In Appendix A.4, we read the definition of ~. Since there are four Q functions in EPRB -- in my terms: Q(+a), Q(-a),Q'(+b'), Q'(-b') --- there will be four equivalence relations on Λ under these four functions.

2. Re the claim that there are "mainly superfluous symbols": Why not be helpful and identify some of them? The essential dynamical symbols are given in Appendix [A.1], see (A.1). The equivalence relations are defined in [A.4]. Equivalent symbols are given in (A.5)-(A.6) by way of examples.

3. Richard, why not give an example of what needs to be explicit? Start with (A.1), (A.3), (A.5)-(A.6), (1)-(2) please; now what more would you like, explicitly please?

4. Richard, in that you've gotten to (19)-(22): that presumably means that you've seen Bell's first fundamental error explicitly identified and corrected. You've seen the nonsense in CHSH explicitly identified and corrected. So we need to be serious in defining issues that we want clarified -- because this looks like serious business to me.

5. Richard: "So elaborate to be ALMOST unintelligible?" Yet nowhere departing from undergrad maths and logic. Why not be explicit about your difficulties with equations before (19)?

6. As for this: "The non-local hidden variables in this paper are simply hidden errors in the almost incomprehensible jungle of mainly superfluous symbols." Surely your finding just one non-local hidden variable would sink a paper based on CLR? And surely superfluous symbols can be simply crossed out and reported; to thus help us all?

7. May I suggest to ALL (me included), that we be specific in identifying our concerns and difficulties. Even identifying what we accept? Like what I accept for certain: Bell's basic boo-boo in equating his 1964:(14a) to his (14b)? That CHSH is just plain silly; since the experiments do not match the maths; except if Bertlmann's socks need washing?

On these grounds ALONE I can see no rational basis, under Bell -- given the ground that has been cut from beneath him -- for expecting anything other that an equals-signs in (21)-(22).

PS: A final thought: Given the structure of the paper, we need only to find the first relevant error for it to collapse like a pack of cards. So are there any errors before (19)?

HTH, Xray

- Xray
**Posts:**44**Joined:**Mon Apr 21, 2014 2:23 pm

Xray wrote:4. Richard, in that you've gotten to (19)-(22): that presumably means that you've seen Bell's first fundamental error explicitly identified and corrected. You've seen the nonsense in CHSH explicitly identified and corrected. So we need to be serious in defining issues that we want clarified -- because this looks like serious business to me.

This does not look like serious business to me. This looks like a sadly typical viXra paper. I did not see Bell's first fundamental error explicitly identified and corrected. I do not need anything clarified.

Regarding notation, it would help if every different equivalence relation is identified as such, for instance by adorning the tilde symbol with a varying subscript. If the author has got something very simple and transparent to say let him say it in a simple and transparent manner. Drop all the fancy words and the fancy symbols which mainly go to show that the author is imitating inpenetrable mathematical texts, not trying to actually communicate mathematical ideas of substance. As far as I can see, by commonsense local realism the author means what most people mean by local realism, and hence any of the usual proofs of the CHSH inequality (for instance) can be run through, routinely.

If anyone thinks they understand this paper and think the author is right, then they should easily be able to win instant recognition and undying fame by winning the quantum Randi challenge or any similar challenge by simply converting the theory into a computer simulation script.

- gill1109
- Mathematical Statistician
**Posts:**2812**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

Richard, I have replied to your comment at viewtopic.php?f=6&t=50

I suggest your unfocussed content should continue there.

To be fair to all, and as I understand the reasonable protocol here: this Topic requires specific and focussed content that stays on topic.

I suggest your unfocussed content should continue there.

To be fair to all, and as I understand the reasonable protocol here: this Topic requires specific and focussed content that stays on topic.

- Xray
**Posts:**44**Joined:**Mon Apr 21, 2014 2:23 pm

Xray wrote:Richard, I have replied to your comment at viewtopic.php?f=6&t=50

I suggest your unfocussed content should continue there.

To be fair to all, and as I understand the reasonable protocol here: this Topic requires specific and focussed content that stays on topic.

Thanks Xray, you are right. I read your other post before this one, and already gave a focussed answer there. http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=50. I'll not repeat it here.

Here are some different focussed comments.

Section 4 of Watson's viXra paper entitled "Bell’s 1964 analysis refuted" shows that Watson does not understand the notion of an ensemble or population, does not understand the difference between an experimental average and a population mean value.

Watson has made the probability distribution of lambda discrete and uniform, and converted the integrals into sums. Fine. So in the population of hidden variables, there are exactly n different value of lambda, all equally likely.

In the part of Bell's paper we are talking about:

<AB> does not stand for the average of A times B over n runs

<AC> does not stand for the average of A times C over a different n runs.

<AB> stands for what the average of A times B would be, if infinitely often a lambda was drawn at random according to the probability distribution just mentioned, and A and B were both measured and A times B was averaged.

<AC> stands for what the average of A times C would be, if infinitely often a lambda was drawn at random and A and C were both measured and A times C was averaged.

Those two thought experiments are different experiments. Different sets of infinitely many runs. But the same values of lambda will turn up in both series, and they'll both turn up equally often.

- gill1109
- Mathematical Statistician
**Posts:**2812**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

gill1109 wrote:<AB> does not stand for the average of A times B over n runs

<AC> does not stand for the average of A times C over a different n runs.

<AB> stands for what the average of A times B would be, if infinitely often a lambda was drawn at random according to the probability distribution just mentioned, and A and B were both measured and A times B was averaged.

Richard, that is the point of Watson's argument. Bell's prescription assumes naively that particle pairs contributing to <AB> have the same probability distribution lambda as those contributing to <AC>. Watson is simply pointing out one counter-example in which Bell's assumption collapses fatally. As your friend Karl Hess has been trying to tell you for decades, simply add time as a variable, and you have a situation similar to what Watson is explaining.

So Richard, let us see if you now understand it. Say we add time as component of lambda. How on earth will the <AB> pairs have the same probability distribution for lambda as the <AC> pairs if no two particle pairs are identical? But you already know the answer because you write in your Larsson and Gill paper that:

Larsson & Gill wrote:http://arxiv.org/pdf/quant-ph/0312035v2.pdf

The problem here is that the ensemble on which the correlations are evaluated changes with the settings, while the original Bell inequality requires that they stay the same. In eﬀect, the Bell inequality only holds on the common part of the four diﬀerent ensembles ΛAC′ , ΛAD′ ,ΛBC′ , and ΛBD′

By simply adding time as a component, there is no common part of the four ensembles, a null set, and Bell inequality collapses. This is exactly the point being made by Watson. You do not deny that fact, you admit as much in your own paper. Talk of population mean and sample averages is just a diversionary tactic IMHO.

- minkwe
**Posts:**1441**Joined:**Sat Feb 08, 2014 10:22 am

minkwe wrote:By simply adding time as a component, there is no common part of the four ensembles, a null set, and Bell inequality collapses. This is exactly the point being made by Watson. You do not deny that fact, you admit as much in your own paper. Talk of population mean and sample averages is just a diversionary tactic IMHO.

This is not the point made by Watson. You are completely off topic.

Watson confuses sample and population, a quite different issue.

If you want to discuss Hess and Phillip contra Gill and Larsson start a new topic.

- gill1109
- Mathematical Statistician
**Posts:**2812**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

gill1109 wrote:This is not the point made by Watson. You are completely off topic.

Watson confuses sample and population, a quite different issue.

If you want to discuss Hess and Phillip contra Gill and Larsson start a new topic.

It is clearly on topic, you have clearly not understood Watson's point or are just trying to divert attention from it. The crucial point is that no two particle pairs are the same so Bell's derivation does not follow. How does population or sample averages avoid that problem? They don't. Read the paper again carefully. That is clearly the point Watson is making? The words "population" and "sample" are not mentioned in the paper so I don't know how did you came up with the suggestion that Watson is confusing the two? Just because you are a statistician does not mean everything is about populations and samples (carpenter and nail analogy appropriate here).

Hess and Phillip's point about time is clearly relevant, it is an example of the scenario Watson is talking about where every particle pair is unique. Larsson and Gill is clearly relevant. It is an admission by you in print that Watson is correct in his analysis. Bell's derivation does not follow unless the probability distribution of lambda is the same for all ensembles. This is Watson's point too. You will have to find something else in the paper to criticize because the above isn't a valid critique.

- minkwe
**Posts:**1441**Joined:**Sat Feb 08, 2014 10:22 am

minkwe wrote:Bell's derivation does not follow unless the probability distribution of lambda is the same for all ensembles. This is Watson's point too. You will have to find something else in the paper to criticize because the above isn't a valid critique.

That is not Watson's point. Watson, like you, does not distinguish between a population and a sample. Bell does, by the way. Very explicitly. In chapter 13 of "Speakable and unspeakable..."

But if you can't accept that, too bad - we agree to disagree.

BTW, a bit off topic, but did you do my silly R computer experiment yet? And have you read chapters 13 and 16 of "Speakable and unspeakable in quantum mechanics"? And have you advised Joy Christian to get out the bet which he has foolishly agreed to?

Just asking.

- gill1109
- Mathematical Statistician
**Posts:**2812**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

gill1109 wrote:Xray wrote:Richard, I have replied to your comment at viewtopic.php?f=6&t=50

I suggest your unfocussed content should continue there.

To be fair to all, and as I understand the reasonable protocol here: this Topic requires specific and focussed content that stays on topic.

Thanks Xray, you are right. I read your other post before this one, and already gave a focussed answer there. http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=50. I'll not repeat it here.

Here are some different focussed comments.

Section 4 of Watson's viXra paper entitled "Bell’s 1964 analysis refuted" shows that Watson does not understand the notion of an ensemble or population, does not understand the difference between an experimental average and a population mean value.

Watson has made the probability distribution of lambda discrete and uniform, and converted the integrals into sums. Fine. So in the population of hidden variables, there are exactly n different value of lambda, all equally likely.

In the part of Bell's paper we are talking about:

<AB> does not stand for the average of A times B over n runs

<AC> does not stand for the average of A times C over a different n runs.

<AB> stands for what the average of A times B would be, if infinitely often a lambda was drawn at random according to the probability distribution just mentioned, and A and B were both measured and A times B was averaged.

<AC> stands for what the average of A times C would be, if infinitely often a lambda was drawn at random and A and C were both measured and A times C was averaged.

Those two thought experiments are different experiments. Different sets of infinitely many runs. But the same values of lambda will turn up in both series, and they'll both turn up equally often.

gill1109 and Xray, thanks for the warm welcome, which I very much appreciate. I'm here to learn so the tougher your criticism the better for me. However, currently on assignment and away from my office, this quick response it to request that gill1109 specify any concerns as clearly as possible. Then, when I get to a decent larger-screen computer, I can give an accurate reply. I'm an engineer, not a statistician, so the quoted critique is not at all clear to me.

We are discussing Watson (2014, v.3). So, with that understanding, could you (gill1109) couch you criticism in more precise terms, please? For example, since every one of my equations is numbered, why not be specific and say things like this: In eqn. (99) Watson calculates the sample mean and wrongly equates it to Bell 1964:(101) ... because Bell is explicit in .... Or Watson's analysis of CHSH at eqn. (97) assumes ... contrary to their assumption ... .

PS: I have not yet analysed minkwe's replies, but my first reaction is that the introduction of "TIME" is unwarranted; certainly it is confusing to me if "time" is used as anything other than a "particle-pair" identifier. And in that regard, I suggest my use of an identifier from {wn + i|EPRB} is all that we need. minkwe, let me know if that is not the case, please; because there is no need for "time" in my theory.

Also: My priority will be to reply to Fredi and his questions re some equations; which gill1109 was also not clear about. So any clarifications in that regard will be helpful. As a guide, I expect to be back here in about 15 hours time. I apologise for that.

Gordon

- Gordon Watson
**Posts:**403**Joined:**Wed Apr 30, 2014 4:39 am

gill1109 wrote:That is not Watson's point. Watson, like you, does not distinguish between a population and a sample.

Well Watson's here now so he will tell us whether he even mentions populations or samples in his paper, let alone confuse the two.

BTW, a bit off topic, but did you do my silly R computer experiment yet? And have you read chapters 13 and 16 of "Speakable and unspeakable in quantum mechanics"?

Have you lost your mind already, this is quite silly and defintely off topic. Your R-experiment is silly and irrelevant as I've told you and demonstrated a million times, and you have no clue what I have read or haven't read so enough with the childishness already, it is tiring. Have some respect for other posters and keep discussion of your bets in the relevant threads. Is that too difficult for you?

- minkwe
**Posts:**1441**Joined:**Sat Feb 08, 2014 10:22 am

Gordon Watson wrote:We are discussing Watson (2014, v.3). So, with that understanding, could you (gill1109) couch you criticism in more precise terms, please? For example, since every one of my equations is numbered, why not be specific and say things like this: In eqn. (99) Watson calculates the sample mean and wrongly equates it to Bell 1964:(101) ... because Bell is explicit in .... Or Watson's analysis of CHSH at eqn. (97) assumes ... contrary to their assumption ... .

Hi Gordon, welcome to the forum!

I have many problems with your paper but the perhaps most important one is your transition around formula (6) from ensemble means, to averages in a particular sequence of runs. You fall into this trap through the device of trying to make things simple and discrete. So there are just n different possible values of lambda, each with the same probability. OK, no problem with that, as a pedagogical device.

But this does not mean that in n runs, one will get to see each of those n possible values exactly once, in turn. No: in each run, a new value of lambda is drawn completely at random (and "with replacement") from the list of n possibilities.

Do you know "Speakable and unspeakable"? I recommend very highly chapters 13 and 16. In chapter 13 there is even an explicit discussion of the transition from theory to experiment, namely from ensemble means to sample averages. The transition is of course only approximate (sample averages are not equal to population mean values) and it requires a random sampling assumption. This is where the random selection of settings comes in, which you do not mention anywhere, thereby missing one of the crucial components of the whole story.

- gill1109
- Mathematical Statistician
**Posts:**2812**Joined:**Tue Feb 04, 2014 10:39 pm**Location:**Leiden

FrediFizzx wrote:In the paper that this thread is about (stay on topic, Richard), I don't understand eq. 19 to 22. Maybe you can explain more about them?

Thanks for the question, Fredi. I should probably have prefaced CLR with this health warning: Despite the fact that CLR employs undergrad maths and logic, disentangling quantum entanglements may cause dizziness. See a doctor if the pain persists.

Let me add (with apologies) that I'll need a doctor myself after pecking out this interim reply on a small-screen computer!

As a prelim to explanation: From Section 2, we want the CLR/EPRB dynamics to do the talking: not words. In Appendix A.7, there's a brief comment on CLR dynamics: with particular reference to the IMPORTANT "If … Then …" component of its maths. Overarching the dynamics and the maths is this fact: each step in each equation is physically significant: so when you can't see a way through, ask a question re the physics applying at that specific point.

Further, compared to QM: CLR is based on functions (which are more general than operators) in R^3 (more real than Hilbert space). ... ... . So:

(19) is CLR's response to Bell (1964) as reflected in (1) and LHS (2). For (19) is built from CLR's version of (1) -- ie, (16)-(18) -- and then LHS (2).

(20) is self-explanatory -- as a function-set -- from (19), by observation. Thus (20) is a simple representation of EPRB as we seek to understand its correlated outputs. For each function in (20) -- like each function in CLR -- has dynamical consequences; ie, in (20), we see Alice's and Bob's SGDs at work via {SGD(a), SGD(b')| EPRB}; for (20) is just the set {Q, R, Q', R'| EPRB}.

Now, when we bring (16) and (17) together to determine <AB>, there is just one independent variable in the resulting combination: for λ + λ' = 0 in each EPRB spin-conserving decay. Which is very convenient because one independent variable allows us to apply the maths/implication of If … Then …. to record/employ a fact about the "dependent variable".

For what is revealed by a test on the independent variable has immediate factual consequences for what is then revealed about the twinned "dependent" variable. (Which is not some super-luminal/non-local effect BUT late/delayed FACTUAL news about a pristine property that was in existence immediately after the relevant spin-conserving decay; a property of one twin, NOW revealed by a test on the other twin. A property hidden until revealed by testing; such is the nature of many beables.)

So, in (21) we see the consequence of having one independent variable in our function-set: we can eliminate the Q that acts on the eliminated "dependent" variable.* So, the selected independent variable λ' (your choice; but the paper has pedagogically selected λ' ) in its response function R' is acted on by its Q', to reveal standard SGD(b') outcomes ±1; to THUS reveal the IF component of our maths: In this composite case, λ' = ±b'. The THEN then follows in the remaining response function: (-λ'.a) = (-(±b').a), respectively. So the reducing full-function-set consists of {(-(±b').a)(±1)} = {-b'.a} = -b'.a: QED. (Your understanding perhaps not helped here by my shortcut with ±b' combined. SO, as a good exercise, I suggest that you do them separately.)

Thus (22) follows: with this expanded note to the introductory phrase (for aid/clarity): "or, equivalently, completing (19) WITH THE SUPERFLUOUS* Q-FUNCTION DELETED, as at the start of (21): ...

Somewhat surprisingly, I believe that such analysis is clearer with increasing numbers of particles: so you may find that wresting with (26)-(37) is a better introduction-to/explanation-of (19)-(22). Particularly if you choose a different beable; contrary to the example in the paper. The guiding light being that here is only one correct answer; as per the paper.

NOW: What is the physical significance of CLR's reduction of (19) to (21) or (22)?

Under CLR's If/Then maths: IF pristine λ' is revealed by Bob's test to be equivalent to +b' -- THEN it is a fact (a pre-existing one) that pristine λ (= -λ') is equivalent to -b'. A result that can be experimentally confirmed by an Alice test employing SGD(-b) -- with no prime here because this is an Alice setting! But independent of Alice doing such a test: λs equivalent to -b' (= -b) will give a spread of results under an Alice test with her SGD set at anything other that ±b.

This spread is demonstrated in (23)-(24) where the familiar trig functions are seen. In other words; the set of λ ~ -b makes law-based transitions to ±a under Alice's test with SGD(a). This spread is understandable because the SGD squeeze-function maps an infinity of λs to just two DECs; so our factual knowledge is limited to those two facts -- plus the distributive LAW.

Concluding, for now, Fredi: I'm not sure how helpful the above will be. But -- if you have any interest in Bell's EPR work -- I encourage you to understand every move in every CLR equation. And I'm happy to help; at every step: together finding improvements and clarifications along the way.

PS: Moreover, by way of encouragement; apart from the fact that you'll be on Einstein's side: I doubt we'll find errors! For every CLR result accords with the experimental evidence: CLR at the same time revealing (from first principles) a chain of errors in Bell's work -- for which there is neither experimental confirmation nor logical justification.

* PPS: For the mathematically inclined, the CLR maths proceeds equally satisfactorily if the superfluous Q is NOT removed and the distributive is used. In that case, in EPRB, a selected Q-function is allowed to operate on the other Q-function by becoming a Q-functional. The selection of such a functional, it will be seen, is equivalent to the elimination of the redundant Q.

With best regards; and thanks again; Gordon

- Gordon Watson
**Posts:**403**Joined:**Wed Apr 30, 2014 4:39 am

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