A Simple Proof Of Bell's Theorem

alt

I've been following up a lot regarding the fascinating stuff I read here.

I recently came across this page;

http://quantumtantra.com/bell2.html

I know that the author and the other material on that site would be considered new age crackpottery by most here at PF, but anyway, that page for me is a simple and lucid explaination, just within my level of undesrtanding of these things. Had it been a little more complex, I would not have understood it. So I'm gald I found it, though I have a few questions;

- What do others think of the explanation ?

- Has it been proven by repeated experiment/s, or is it pure theory / maths ?

- If it HAS been proven by experiments, to what extent if any, has it been put to practical use ? I mean, instantanious communication between particles at any distance .. WOW !

Any feedback would be appreciated. Thanks.

DrChinese

Nick Herbert's explanation is fine. Bell tests are regularly performed (often in undergraduate labs) and support the predictions of quantum mechanics. Local realism has been experimentally rejected. Despite this conclusion, you cannot use the mechanism for FTL communication. The "signal" is random, not much you can get out of that.

The only thing I object to is Nick's claim to have the shortest proof. I think this is shorter, the actual proof being 8 paragraphs. It is certainly written by a better looking person (that being moi).

http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

ThomasT

Yes, Nick Herbert is something of a crackpot. Nevertheless, his assessment of the meaning of Bell's theorem (ie., the meaning of violations of Bell inequalities) might well be the mainstream view (although nobody really knows because no polls or surveys of physicists' opinions on this exist).

Some, maybe most, think it's incomplete and misleading. This isn't Herbert's field. In fact, this field, the meaning of Bell's theorem, is the concern of an extremely small minority of working physicists.

Has what been proven? That's the question. The opinion of many researchers in this field (of which Herbert is not one) is that violations of BIs do not imply nonlocality, or anything else about nature.

If WHAT has been proven by experiments? Nonlocality? No. It's unverifiable and unfalsifiable. It's just not physics.

Wow indeed. But there's no evidence for it. And there can never be any evidence for it. It's a completely nonphysical concept, if it can be considered a concept at all. Think about it. Communication implies some sort of propagation or transmission between spatially separated objects, and instantaneous says that two events are happening at exactly the same time. So, 'instantaneous communication' is something of a nonsensical oxymoron. Isn't it?

Ok. Having said the above, let me explain to you what Nick Herbert's simplification of Bell's theorem does and does not tell you.

What it does tell you is that if the light incident on the polarizers is modelled in terms of discrete and separable "instruction sets" or "coded messages", then it would be expected that the rate of coincidental detection would vary linearly as a function of the angular difference between the polarizer settings.

What it doesn't tell you is that it's been known for about 200 years that light doesn't behave in this way. The simplest way to illustrate this is to move one of the polarizers to the other side so that both polarizers are between the emitter and detector A or between the emitter and detector B. The results are exactly the same as when one polarizer is on side A and one polarizer is on side B. With both polarizers on one side, the rate of coincidental detection is the same as the rate of detection on the side that both polarizers are on.

Now, DrC might tell you that this is just a coincidence. DrC might even tell you that optical laws have nothing to do with optical Bell tests. But keep in mind that DrC is just a computer programmer of unknown competence. He's not a physicist. He's not an expert in optics, certainly not quantum optics. And, he's not an expert logician or mathematician. People who are experts in these fields have published papers showing that violations of BIs do not imply anything about nature. Who are you going to believe?

If you want to get at the truth of things, and be satisfied that you understand everything involved, then just ignore all the above and do your own research. Learn about classical and quantum optics. Read as much as you can of the relevant papers on Bell, GHZ, Hardy, etc. And come to your own conclusions. This should take you at least a couple of years unless you're ridiculously intelligent.

Then, hopefully, you can explain this stuff to me in a way that I can actually understand it.

JesseM

Nonsense. You have shown in the past that you have little detailed understanding of the subject, and that you have an axe to grind against Bell's theorem which seems to be based on mere intuition that there's something fishy about it rather than being able to point to any specific flaw, but that doesn't justify fantasizing that "most" physicists would take your side!
Again, nonsense. The opinion of virtually all mainstream researchers is that violations of Bell inequalities (which are predicted theoretically by QM) imply that all local realist models can be ruled out (you will find almost no mainstream researchers who dispute this on a theoretical level, though a few might say the imperfections of experiments to date give rise to serious doubt about whether violations of Bell inequalities would still be seen with an ideal experimental setup). Who are the "many researchers in this field" you refer to? Can you point to peer-reviewed papers which dispute the basic theoretical conclusion that violations of Bell inequalities (under the experimental conditions given by Bell) are sufficient to rule out local realism?
What has been proven theoretically is that local realism implies the Bell inequalities, and experiments show violations of Bell inequalities (although no experiment has been 100% ideal, the detection loophole and locality loophole have not both been simultaneously closed off by any experiment), so that's evidence that local realism is false. That's not the same as saying nonlocality is true, since you can dispense with "realism" (as in the Copenhagen interpretation) or you can dispense with other conditions assumed by local realism like the one that says each measurement has a single definite outcome (which isn't true in the many-worlds interpretation), but it's definitely "physics" to show theoretically that local realism --> Bell inequalities.
What are you talking about when you say "200 years"? None of the traditional classical optics experiments involve situations where each experimental setting (like a choice of polarizer angle) gives one of two possible measurement results, which is the type of experiment that Bell's theorem is dealing with. Of course you could artificially design a classical experiment where a green light would go off if the intensity of light passing through the polarizer was above a certain threshold and a red light would go off if the intensity was below that threshold, but the intensity could still vary continuously, and this type of classical experiment would not give any violations of Bell inequalities.
What is that supposed to "illustrate"? Are you considering a classical experiment where the measuring apparatus is designed in such a way that each measurement gives one of two possible results? If so, then again, putting the polarizers on the same side will not give any violation of Bell inequalities--do you disagree?
Why should he have to do that, given that there will be no classical violation of Bell inequalities here?
Peer-reviewed papers that have been accepted as valid by mainstream physicists? If so, where?

ThomasT

Nick Herbert's account is incomplete and misleading. Period.

I think that most physicists would say that nature is evolving according to local and discoverable underlying dynamics. Am I wrong?

Your reply doesn't address what I said. Are you deliberately trying to confuse people?

Experimental imperfections are not the problem. What's disputed is the logic involved in the development/derivation of Bell inequalities, and the subsequent interpretations thereof. I provided links to two papers that you can critique. So, critique them.

No. Not the totality of local realistic conceptions, but just the ridiculously simplistic ones associated with Herbert's "coded messages" or Mermin's (or whoever) "instruction sets". An alternative interpretation of the results is that what's been theoretically proven, and experimentally shown, is that light can't behave that way. But this comes as no surprise, because we already knew that it doesn't behave that way.

No, it isn't. You need to read the papers I recently linked to.

Just read the papers and let me know what you think.

Quantum polarimeters produce results according to the same optical law that classical polarimeters do. Don't obfuscate this. Herbert's account of Bell's theorem is quite on the mark, and it shows quite simply and easily a very profound problem with Bell's assumptions. We're expected to suppose that light is going to behave in Bell experiments in a way that it has never before been observed to behave. And, if it doesn't behave in this strange way, then we're supposed to conclude that nature is either nonlocal or that nature doesn't exist. It's absolutely absurd what you and DrC are promoting as 'science' to the public.

Read the papers by Hess, et al., then I'll have some more for you to look at.

Putting the polarizers on the same side (in an optical Bell test setup w/entangled photons) will give exactly the same results as having one on each side -- ie., a violation of BI. So, in the altered setup, you have a quantum polariscope on one side which determines the rate of coincidental detection. There's, presumably, nothing nonlocal going on in this setup. So, why, when we put one of the polarizers back on the other side, and get the same results, should we assume that there's something nonlocal going on in this, the original, setup? Or, consider an OPDC setup where a slight adjustment of a wave plate produces entanglement stats. Are we to suppose that this slight adjustment suddenly conjured up some new (and apparently arbitrarily varying) underlying ftl communicative medium or field (it can't be the EM field), or is there maybe a simpler explanation, maybe having something to do with, say, phases, or whatever?

Are you a peer of people like Hess, De Raedt, Sica, Dalton, Christian etc.? If so, then you can review and critique their papers for the benefit of the folks here at PF, and, who knows, maybe at least one of them will visit PF to discuss his/her paper with you.

Start with the papers that I linked to please. And please, no more nitpicky obfuscating responses on stuff that I've written. I'm just a curious amateur. I don't know what your qualifications are, but the people who wrote the papers I want you to critique are the real deal.

alt

Thanks, DrChinese. This is all very new and interesting to me. I have read your link and it is very informative.

ThomasT and JesseM, thank you also for the replies.

Even though you guys are at odds on this, I am really interested in what you're each saying, and avidly taking it all in - to the best of my QP abilities anyway (which are pretty much zilch).

DrChinese

I don't know about that one! Tens of thousands of copies of my commercial software have been sold over the years.

Also: Besides being charming and handsome, I am also modest. So don't dismiss me!!

DrChinese

You are welcome.

As a side note, I would recommend disregarding everything ThomasT says. Either that, or read it along with the morning comics.

nismaratwork

to this and your last post, you're vying with DaveC for the humor award!

Fun aside, thanks for the link to the other proof, I enjoyed that a great deal.

alt

Lol, thanks.

Antinomy itself is often instructive !

JesseM

If you define "local" in the sense of local realism (as opposed to some weaker sense like it being impossible for experimenters to exploit the laws of nature to send FTL signals), then yes, you are wrong.
It directly addresses what you said when you claimed "The opinion of many researchers in this field (of which Herbert is not one) is that violations of BIs do not imply nonlocality, or anything else about nature." Asking whether or not the universe's fundamental laws are local realist ones is certainly a question "about nature", and the opinion of virtually all mainstream researchers is that Bell's theorem proves on a theoretical level that violations of Bell inequalities imply local realism must be false.
Huh? You have provided no links!
You are simply wrong here (and given your own lack of knowledge of physics, it's ridiculous that you act so confident), the mainstream view is that the "totality of local realistic conceptions" have indeed been shown to be incompatible with violations of Bell inequalities. For example, local realist theories where the measurement outcome is determined by a combination of hidden variables associated with the particle (which may change as it travels) and hidden variables associated with the region of the experimental apparatus are ruled out too by BI violations. BI violations are incompatible with any theory that is "local realist" in the sense that A) all physical facts about a region of spacetime can be reduced to a collection of facts about local variables associated with individual points in spacetime, and B) the value of each local variable is only causally influenced by variables in its past light cone.
Don't rudely accuse me of obfuscation if you can't point out where I am wrong. Do you disagree that it would be impossible to replicate the violations of BI predicted by QM if we were doing purely classical optics experiments? (note that Maxwell's laws of classical electromagnetism, from which classical optics can be derived, are a fine example of local realist laws!) If quantum polarimeters produce BI violations and classical polarimeters never do, that would suggest they are not following the "same law".

Perhaps by "same law" you just mean that the classical Malus' law for polarized light and the law for entangled particles both involve a cos^2? The problem is that although the equation can be written in a similar form for both laws, the physical meaning of the symbols is completely different, so from a physical perspective they cannot be called the "same". If you write cos^2(a-b) in the classical context a would be the polarization angle of the light, b would be the angle of a single polarizer, and cos^2 would be giving the reduction in intensity of the light as it passes through the polarizer; but if you write cos^2(a-b) in the quantum context, a and b would both be polarizer angles, there would be no term for the polarization of the light, and cos^2 would be giving the probability that both photons give the same binary result (both passing through their polarizers, or neither).
What do you mean "never been observed to behave"? Again, in a classical optics experiment with a setup meeting Bell's conditions (spacelike separation between measurements, each measurement giving one of two possible binary results, etc.) you'd always observe light obeying the Bell inequalities, do you disagree?
It's absolutely absurd that you, obviously knowing very little about the literature and having only your own intuitions telling you there's something wrong with Bell's theorem (since you have never been able to point to a specific flaw or explain how a local realist model could lead to BI violations), confidently make the completely false claim that lots of physicists think Bell's theorem is flawed based on two papers you've found out of the vast literature on the subject (which you probably don't understand in any detail yourself). Confirmation bias is an amazing thing! But despite your wishful thinking, rest assured the notion that Bell's proof is invalid is a thoroughly fringe notion, and DrC and myself are just expressing the mainstream view which virtually all physicists who have studied the subject would agree with.
Are you referring to the paper that Gill, Weihs, Zeilinger, and Zukowski claim to refute here (full paper on arxiv.org here), as does Myrvold here and Mermin here and here? Hess' argument hasn't achieved any acceptance among quantum physicists, and in any case Hess' argument seems to be based on the notion that Bell neglected to consider time-dependent hidden variables, but the version of Bell's theorem which lets the hidden variables stand for all physical facts in the past light cones of the measurement at some specific time before the measurement (but after the last time the two light cones overlap), which is the version I have been discussing all along (for example, in posts 61 and 62 here...Bell also defined the hidden variables this way on p. 242 of Speakable and Unspeakable in Quantum Mechanics here), is perfectly consistent with the possibility of time-dependent hidden variables.
I was talking about a classical optical experiment, not a quantum experiment with entangled photons, as I made clear above. And if you were talking about a quantum experiment, then using that to "illustrate" your comment that "it's been known for about 200 years that light doesn't behave this way" is completely incoherent, since no one was performing experiments with entangled photons until the 20th century.
I'm not clear on what setup you're assuming in the quantum version. Does each photon actually travel through two polarizers in succession before reaching the polariscope, or do the two entangled photons go in the same general direction but have slightly different paths so one only passes through/is blocked by polarizer #1 while the other only passes through/is blocked by polarizer #2?
What "same results" are you talking about? To get a violation of BI you have to have multiple possible settings for the two polarizers, so you can sometimes observe the coincidence rate when both polarizers have the same angle but sometimes observe the coincidence rate when the polarizers are set to different angles. If we give the polarizers the possible angles 0, 60, and 120 (so that whenever the two polarizers have different angles a and b, cos^2(a-b) is always 0.25) when they are on the "same side" do we still have some trials where each member of the pair goes through a polarizer with a different angle? If so I have no idea why you'd say "There's, presumably, nothing nonlocal going on in this setup"--although it would be possible in this case for a local realist theory to replicate the statistics seen in QM, it would only be by exploiting the locality loophole, meaning there is some kind of "communication" going on between the particles where the polarizer angle encountered by one affects the behavior of the other (if on the other hand the particles have no causal influence on one another after leaving the source, then according to local realism the BI shouldn't be violated). If you then put the polarizers on opposite sides so that there's a spacelike separation between each photon encountering its polarizer, then either the communication has to break down (so the statistics change and you no longer see BI violations) or the "communication" will now have to work in an FTL manner.
Why "new"? I think in Bohmian mechanics the nonlocal quantum potential would always be guiding entangled particles, regardless of the statistics in the experiment, and doing so according to the same equation at all times. And as I've told you many times before, nonlocality is not the only alternative to local realism, one can also take options like the many-worlds interpretation.
The whole concept of "peer-reviewed" is that professional physicists do the reviewing, which gives the rest of us some sense of whether the paper is likely to have a coherent argument. If you want to continue to dispute my point that the mainstream view is that Bell's theorem is valid, you should be able to point to peer-reviewed papers, as opposed to any ol' paper someone uploaded to arxiv.org or whatever. You haven't actually linked to papers by these authors so I don't know if they're peer-reviewed (and I also don't know whether they are actually disputing Bell's theorem on a theoretical level or pointing to other issues like the possibility of a local realist theory that exploits experimental loopholes and would obey the BI in an ideal experiment, as in the Da Raedt model DrChinese was discussing here)
They may be "the real deal" in the sense of professionals, but if they question Bell's theorem on a theoretical level (and I don't know if the others besides Hess do), then they represent a decidedly fringe view which almost all mainstream physicists would disagree with. Please don't misrepresent a fringe view as being a commonplace opinion among physicists if you only have a few cherrypicked papers to back it up; these forums are designed to teach mainstream physics, so fringe views need to be flagged as such to avoid confusing readers.

DrChinese

I quack myself up.

ThomasT

You're most welcome alt. Did you actually agree with anything I said?

It seems to be pretty much how they roll here at PF.

I hereby retract my statement. Better yet just change "unknown" to "questionable". Just kidding. I'm sure you're quite good at what you do.

How could anyone possibly dismiss you? I was just trying to spice up the antinomy, whatever that means, in between my normal duties as intellectual swindler #2 (I'm going for the #1 spot).

Ok, it wasn't THAT funny.

ThomasT

So, you think that most physicists would say that nature is not evolving according to local and discoverable underlying dynamics?

I don't think that's the case. Anyway, here's those papers that I forgot to post here. The authors disagree with your assessment of Bell.

Possible Experience: from Boole to Bell
http://arxiv.org/PS_cache/arxiv/pdf/...907.0767v2.pdf
Published in: EPL, 87 (2009) 60007

Extended Boole-Bell inequalities applicable to quantum theory
http://arxiv.org/PS_cache/arxiv/pdf/...901.2546v2.pdf

The second paper, still in the works, on the extended Boole-Bell inequalities, provides a detailed account of why BIs are violated and why their violation doesn't imply nonlocality in nature.

First, just consider quantum and classical polarimeters without reference to BIs or optical Bell tests. They both produce results in accord with Malus Law.

Now consider, say, an Aspect or F and C optical Bell test setup where you have an emitter of entangled photons between two polarizers, a and b, between two detectors, A and B, respectively. Take the polarizer, a, and place it between the emitter and the polarizer, b. So, on the left side there is just a detector, A, and on the right side there are polarizers, a and b, between the emitter and detector B. The right side is now a polarimeter. It produces results in accord with Malus Law, as all polarimeters do. And the coincidence rate, ie. F(AB), is the same as with the original setup, .5 cos^2 |a-b|, in the ideal. But Herbert, vis Bell, requires that the polarimeter produce a linear relationship between |a-b| and rate of detection. This is why I said that light was being required by Herbert, vis Bell, to behave in a way contrary to what thousands of polarimetric experiments have shown.

Nobody's arguing about Bell's mathematical proof, or the validity of experiments in which BIs are violated. What's being contested is the interpretation of BI violation, leading to a more appropriate phrasing of Bell's theorem.


So, JesseM, thanks for bearing with me. I'm hoping that we might go through at least one of the papers I linked to. It will be very instructional for me, as well as probably for a lot of noncontributing readers of this thread. Maybe just take a page every few days or so and post your comments. Take it slow, because, you're right, at the moment I don't fully understand them.

alt

Hi ThomasT;

Yes, I read what you said. On another post, you also said "I'm just a curious amateur"

Well, if you are an amatuer, I'm a babe in the wilderness.

I'm gonna keep reading 'round here, and see if I can understand (let alone agree or disagree with) anything.

It's fun though :-)

JesseM

Yes, virtually all of them would say this. Please stop trying to use a few cherrypicked papers where the authors disagree with this conclusion to suggest otherwise. If you want to get a sense of what most physicists think rather than just confirm your own biases, try looking at textbooks which discuss Bell's theorem (you could go to google books and entering the keywords "bell's theorem" and "inequality"), you'll notice that all the textbooks written by physicists just present it as a straight fact that Bell's theorem shows QM is incompatible with local realism (or local causality or theories of local hidden variables, the exact wording may vary), they don't suggest any controversy about this conclusion.
Well it is, and as an amateur in the subject you can have little rational basis for believing otherwise--finding a few fringe authors who disagree doesn't qualify as a good basis. Again, try the experiment of looking in textbooks--would you disagree that if physics textbooks all present some theoretical conclusion as a fact, then it's a safe bet that this conclusion is one that the majority of physicists would agree with?
Your URLs are messed up, should be here and here. The first deals specifically with the Leggett-Garg inequality which is somewhat different from other Bell-type inequalites as it requires the additional assumption that in two successive measurements on the same system (with a timelike, not a spacelike separation), the first measurement will not influence the probability distribution for possible results on the second. And their example involving doctors and patients simply does not conform to the experimental conditions assumed by the inequality, see my discussion of this paper with billschnieder in post 941 and post 961 on this thread (I got sidetracked from continuing that discussion with bill but I intend to follow up on his last comment to me soon).

The first paper did appear in a peer-reviewed publication, the epl journal, perhaps because it didn't make the strong claim that the example they provided was actually a local realist violation of the Leggett-Garg inequality or any other Bellian inequality, you can interpret the paper as just being about an analogous inequality that fails to hold because the conditions under which the data is collected and indexed are different from those assumed in the derivation of Leggett-Garg (as well as the conditions assumed in the derivation of Boole's original inequality). The second paper, on the other hand, doesn't appear to have been published anywhere outside of arxiv.org (see google scholar search here), which doesn't require peer-review. I haven't read through it in detail, but skimming it the argument appears similar to the first paper, in that they are considering situations where the procedure by which data is collected and indexed does not conform to the conditions stipulated by Bell inequalities (or the conditions needed for a derivation of Boole's own inequality). In fact they mention the exact same example involving doctors and patients from the earlier paper on pp. 25-27 of this second paper. They do claim that it is possible to explicitly violate Bell's theorem with locally causal models on p. 28:
...but the details of these models aren't provided in this paper, so there's no way to verify if the models actually conform to all the necessary conditions for Bell's proof. Back in post 62 of the thread 'Bell Theorem' I once suggested a test of any claim that a local model can replicate BI violations, by simulating the model using a set of computers which are cut off from communicating with one another at some time prior to the experimenter's choice of what detector setting to use (analogous to the fact that in Bell's proof, the past light cones of the two measurements stop overlapping at some time prior to the choice of detector setting for each measurement--each computer's internal state at a given time can be simulating an arbitrary number of local variables in the past light cone of a measurement). I feel totally confident that the authors of the paper would not be able to come up with an algorithm that would allow for BI violations in this scenario (which can stand for any of the inequalities proposed by Bell for measurements on pairs of particles at a spacelike separation, although it does not stand for the Leggett-Garg inequality where the two measurements have a timelike separation):
(note that in some Bell inequalities it's assumed that there are 2 possible detector settings for each experimenter rather than 3, in which case each experimenter could only choose to enter 1 or 2 on the keyboard of their computer on each trial and the computer would have to return a measurement result).

If you have followed the authors' argument in this second paper and think they show it would be possible to get a BI violation in this type of simulation, please explain. Likewise if you think they show it would be possible to get a violation of a BI in a scenario that matches all the observable experimental conditions stipulated in the derivation of the inequality, please explain. But if you haven't really followed the argument of the paper, and are just citing a paper you don't understand to support your claim that there is widespread disagreement among physicists about whether Bell's theorem showed QM was incompatible with all local realist theories, then one can show the absurdity of this claim just by noting that you're pointing to a non-peer-reviewed paper uploaded to arxiv.org which has led to no real reactions from the physics community (in the form of other physicists citing the paper--only four papers have cited it so far, two by De Raedt), and that physics textbooks which mention Bell's theorem uniformly present it as a valid demonstration that QM is incompatible with local models.
Malus' law is a classical law dealing with classical electromagnetic waves of known polarization, and the reduction in intensity of the wave when it passes through a polarizer. Since the reduction in intensity of a light beam passing through a polarizer is in QM equal to the fraction of photons that make it through the polarizer, I think it would be reasonable to say there is a quantum version of Malus' law in a situation where a set of photons are all in the same known polarization eigenstate at some angle (say, because they already passed through a first polarizer at that angle) and we want to know the fraction of them that will make it through a polarizer at a different angle (so in this scenario each photon encounters two polarizers in succession). However, if you're talking about an experiment involving pairs of entangled photons which each pass/don't pass through a single polarizer, it is meaningless to say that "Malus' law" predicts anything about this situation, since Malus' law only applies when you know the polarization of the light, and we don't know anything about the polarization of each pair when they are generated (and after they pass through polarizers, the entanglement between their polarizations is broken). Do you disagree?
Why do you have an 0.5 in front of cos^2 (a-b)? I'm not sure of the exact "original setup" you're thinking of, but if the photons are entangled in such a way that identical measurements on each will yield the same results with probability 1, then if both polarizers are at the same angle so a-b=0, then the probability of getting the same result (either both passing through the polarizers, or both being reflected) should be cos^2(0) = 1, not 0.5 cos^2(0) = 0.5.

Assuming you should have written cos^2 (a-b) for the original setup, your altered setup above will not give the same results. After all, given perfectly efficient detection, on the left side every photon sent out by the emitter should be detected at A, since there are no polarizers to reflect photons sent to the left. On the right side, even if a and b are at the same angle so cos^2(a-b)=1, it is quite possible for some of the entangled photons to be reflected by polarizers at that angle rather than passing through, in which case the coincidence rate in this setup is not cos^2 (a-b).
Presumably Herbert/Bell would not require this in any arbitrary setup like the one you describe above, only in the particular setup assumed in the derivation of the Bell inequalities, where a represents a detector setting in the region of the measurement at A and b represents a detector setting in the region of the measurement of B, with a spacelike separation between the two random choices of detector settings.
Your statistics don't actually seem to work as noted earlier, and in any case this is just a strawman if "Herbert, vis Bell" (can you point to what particular statement of Herbert you are talking about?) was only talking about such a linear relationship in the type of experiment assumed by Bell where there was a spacelike separation between the random choice of setting a and the random choice of setting b.
Whe I said "the notion that Bell's proof is invalid", I was talking about the notion that Bell's proof does not correctly show (as it purports to) that all local realist models would produce results satisfying Bell inequalities in the experimental setup Bell was assuming. The idea that Bell's proof is invalid in this broad sense is indeed a "thoroughly fringe notion", and the type of textbook search I suggested (or a representative search of the literature, rather than a search for cherrypicked examples of papers by the tiny number of physicists who disagree) will show this. Again, if you want to have a discussion about some paper that questions Bell's proof I (or others like DrChinese) can do that, but please stop trying to present disagreement with Bell as something that is common among physicists when you have nothing but wishful thinking to back it up.
If there is some specific thing you have a question about we can discuss it, but I don't really feel like going through a detailed analysis of everything in those papers. You may find my comments about the first paper in the earlier discussion with billschnieder that I linked to helpful, and soon I plan to post a little more on the subject in response to bill's last post to me.

nismaratwork

Now this is interesting reading; like alt I feel I'm learning from reading this discussion. Back on track, life is good.