Bell's theorem refuted via elementary probability theory

[Gordon Watson]

In anycase, don't you see that your superscripts for are superfluous and confusing?? Otherwise please explain what the following means:

2. Your example has no rational meaning for me: except that it must be false. For, with your single-channel detectors passing only post-interaction spin-up particles, that first minus-sign is problematic.

Exactly! That equation is implied by your notation which you used to express perfect anti-correlation in section 2.2 of your paper. If it doesn't make sense even to your then you shouldn't be using that notation.

Let me cut to the chase here and see if I can quickly sort things out:

1. The detectors are 100% efficient. We need no tricks to beat Bell.

2. §2.2.(ii) introduces and specifies the output channels.

3. It seems things would have been clearer had I written (more clumsily): for the device and for the function.

4. Thus: In my terms, vide §2.2.(ii), as soon as you write , you have specified a single-channel detector that records only post-interaction particles with spin UP in the direction.

5. Do you now see why your use of my notation makes no sense to me?

6. With further, and similar, clarifications to come.

All the best; Gordon

[minkwe]

1. The detectors are 100% efficient. We need no tricks to beat Bell.

Thus no need to talk about absorbing particles and letting others go through then?

2. §2.2.(ii) introduces and specifies the output channels.

3. It seems things would have been clearer had I written (more clumsily): for the device and for the function.

That's actually worse. Maybe you are not getting the point. If you want to show that there are two channels, draw a diagram. Or describe it like this:

Each station is equipped with dual-channel detectors with channels relative a setting vector , with corresponding outcomes
These outcomes can be generated from pristine particles carrying properties using representative detector functions of the form:



Where performs the transformation by projecting particle spin to a direction either parallel or anti-parallel to the detector orientation.

Due to momentum conservation, the particle pair going to opposite stations carry opposing properties such that the following relation holds;

Don't you think this is clearer? What is missing from this description?

4. Thus: In my terms, vide §2.2.(ii), as soon as you write , you have specified a single-channel detector that records only post-interaction particles with spin UP in the direction.

Yes, which reveals the problem with your notation. You are introducing things in your analysis, that have no content. It is clouding the point you want to make.

5. Do you now see why your use of my notation makes no sense to me?

No. I'm using your notation exactly as it is implied. The point is to show you that your choice of notation implies nonsensical things like what I wrote above.

[minkwe]

So how about going with ; etc?

Your notation looks too much like a probability for me.

Why is not enough? Why must you prefix it with ? What additional information does it convey?

[gill1109]

In anycase, don't you see that your superscripts for are superfluous and confusing?? Otherwise please explain what the following means:

2. Your example has no rational meaning for me: except that it must be false. For, with your single-channel detectors passing only post-interaction spin-up particles, that first minus-sign is problematic.

Exactly! That equation is implied by your notation which you used to express perfect anti-correlation in section 2.2 of your paper. If it doesn't make sense even to your then you shouldn't be using that notation.

Let me cut to the chase here and see if I can quickly sort things out:

1. The detectors are 100% efficient. We need no tricks to beat Bell.

2. §2.2.(ii) introduces and specifies the output channels.

3. It seems things would have been clearer had I written (more clumsily): for the device and for the function.

4. Thus: In my terms, vide §2.2.(ii), as soon as you write , you have specified a single-channel detector that records only post-interaction particles with spin UP in the direction.

5. Do you now see why your use of my notation makes no sense to me?

6. With further, and similar, clarifications to come.

All the best; Gordon

Gordon

"minkwe" is completely right.

Have you noticed that no-one who takes the trouble to try to read your paper in detail can work out what you are doing?

If *you* would remove superfluous labels off your notation, but be so good as to put variables or indices which are actually needed back on again, you would discover that you are mainly writing complete nonsense.

By needed, I mean that when you define functions or variables or probability measures or sets in such a way that they actually depend on the values of things introduced earlier, it would be wise not to suppress that dependence from your notation.

For instance, "a" could be the name of a variable which is used as a container to hold possible directions (angles) chosen by Alice. You need to distinguish the concept, the name you give it, and the value (the instantiation) which might be in that "container" at a particular moment.

[Gordon Watson]

1. The detectors are 100% efficient. We need no tricks to beat Bell.

Thus no need to talk about absorbing particles and letting others go through then?

2. §2.2.(ii) introduces and specifies the output channels.

3. It seems things would have been clearer had I written (more clumsily): for the device and for the function.

That's actually worse. Maybe you are not getting the point. If you want to show that there are two channels, draw a diagram. Or describe it like this:

Each station is equipped with dual-channel detectors with channels relative a setting vector , with corresponding outcomes
These outcomes can be generated from pristine particles carrying properties using representative detector functions of the form:



Where performs the transformation by projecting particle spin to a direction either parallel or anti-parallel to the detector orientation.

Due to momentum conservation, the particle pair going to opposite stations carry opposing properties such that the following relation holds;

Don't you think this is clearer? What is missing from this description?

4. Thus: In my terms, vide §2.2.(ii), as soon as you write , you have specified a single-channel detector that records only post-interaction particles with spin UP in the direction.

Yes, which reveals the problem with your notation. You are introducing things in your analysis, that have no content. It is clouding the point you want to make.

5. Do you now see why your use of my notation makes no sense to me?

No. I'm using your notation exactly as it is implied. The point is to show you that your choice of notation implies nonsensical things like what I wrote above.

All that absorbing, clumsy, nonsensical worse-stuff, etc., is the result of me showing how I interpreted your use of my notation.

I use none of it, and none such was implied.

Nevertheless, I see that improvements are required. And I will make them. Your example is clearer, and where I see us headed.

Please keep such coming. They are very helpful.

PS: I expect to deliver the remaining answers to your earlier technical questions tomorrow.

[Gordon Watson]

So how about going with ; etc?

Your notation looks too much like a probability for me.

Why is not enough? Why must you prefix it with ? What additional information does it convey?

I use to tell me it's a particle; and not a probability under your .

{...} lists the particle's relevant properties, given the setting.

From your point-of-view: What am I missing here?

[Gordon Watson]

Then, within the detector there are 2 functions:

(i). Via the polariser we have ; the output being determined by the equivalence class or to which belongs. See §3.1; me now thinking that it might be clearer to replace with ??

(1). Please no. (2). Polarizers should work the same way irrespective of which arm of the experiment they are located at. (3). A polarizer Transforms a particle so the details should show how the relevant particle properties change. (4). I would change your notation to . (5). But you still haven't shown exactly how the transformation happens. (6). Anyway, you write instead of. Please explain why you think the former is more correct, otherwise, the latter is a much better notation. (7). Again, please stay clear of using functions to denote particles.

(1). This pleading is not clear to me. Maybe it will not be necessary after the following comments.

(2). Of course; and mine do. Do you somehow believe they do not?

(3). Agreed. Further explanation to come in the context of your later, more specific, questions in this set. A particle-polariser interaction reveals an equivalence class to which the particle belonged. We can thus infer the equivalence class to which its pristine twin belongs. In this way, local properties deliver local results.

(4). OK.

(5). See (3) above.

(6). I write that way to capture the symmetries in relevant items. Example, pending your response to the matter of particle-specification in a prior post:

I write more symmetrically and for the equivalence classes*** whereas your suggestion may lead to and .

(7). I remain amazed. I write, clearly, that is NOT a function. However, if that troubles you it will most certainly trouble others. Hence the proposal in (6) above.

*** More on my use of them tomorrow, DV.

Gordon

[Gordon Watson]

"minkwe" is completely right.

Have you noticed that no-one who takes the trouble to try to read your paper in detail can work out what you are doing?

If *you* would remove superfluous labels off your notation, but be so good as to put variables or indices which are actually needed back on again, you would discover that you are mainly writing complete nonsense.

By needed, I mean that when you define functions or variables or probability measures or sets in such a way that they actually depend on the values of things introduced earlier, it would be wise not to suppress that dependence from your notation.

For instance, "a" could be the name of a variable which is used as a container to hold possible directions (angles) chosen by Alice. You need to distinguish the concept, the name you give it, and the value (the instantiation) which might be in that "container" at a particular moment.

Thanks Richard,

I have noticed the troubles with understanding my draft. And I'm working to fix that.

Apart from that, what (alas) with being a fairly concrete thinker, I have little idea what else you mean. Could you follow minkwe's example, and give me some examples of superfluous labels, suppressions, etc?

Also, since I attempt to use in the same way that Bell does, could you explain as a container?

Are you referring to Alice choosing two directions, say and ?

Thanks.

[minkwe]

I use none of it, and none such was implied.

Well, I'm not making stuff up. At the end of 2.2 you have:



What I wrote follows directly from it. I'm thinking you have a completely different understanding of what you think your notation means compared to everyone else reading it.

Nevertheless, I see that improvements are required. And I will make them. Your example is clearer, and where I see us headed.

Do you? I'm not so sure about that.

[minkwe]

(1). This pleading is not clear to me. Maybe it will not be necessary after the following comments.

The pleading is required because I don't think you get your problem with notation. Even the new ones you introduce are bad and sometimes worse. I say "please no", because you keep writing particles as functions.

(2). Of course; and mine do. Do you somehow believe they do not?

If you give them different symbols then it appears they are not the same. An outcome of +1 at Alice is exactly the same value as an outcome of +1 at Bob. Just because they were obtained at different stations, from different particles does not mean I must now write outcomes as and . I exaggerate so that perhaps you can see the absurdity.

(3). Agreed. Further explanation to come in the context of your later, more specific, questions in this set. A particle-polariser interaction reveals an equivalence class to which the particle belonged. We can thus infer the equivalence class to which its pristine twin belongs. In this way, local properties deliver local results.

That's important for your diary but completely irrelevant to the point you are trying to make. It's just fluff.

(6). I write that way to capture the symmetries in relevant items. Example, pending your response to the matter of particle-specification in a prior post:

Again it is irrelevant since you haven't actually shown how such symmetry is important for the analysis. Just more fluff.

I write more symmetrically and for the equivalence classes*** whereas your suggestion may lead to and .

That's just a re-statement of your preference, not an explanation of the reason behind your preference.

(7). I remain amazed. I write, clearly, that is NOT a function. However, if that troubles you it will most certainly trouble others. Hence the proposal in (6) above.

I'm also amazed that you still do not get it. Would you be happy to see the following text in a paper:

"Consider the particle "

It clearly says P(...) is not a probability. It's not just about what bothers me. Now you've attached your wagon to and won't let go. I gave you the standard notation for representing a collection of properties and you want to prefix it with a , which is never done, again creating a monstrosity that anyone reading your paper would scratch their heads trying to figure out, for absolutely no reason.

[Gordon Watson]

I use none of it, and none such was implied.

Well, I'm not making stuff up. At the end of 2.2 you have:

. (1)

What I wrote follows directly from it. I'm thinking you have a completely different understanding of what you think your notation means compared to everyone else reading it.

As I wrote earlier http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=451&start=140#p12209,

2. So my detector-function anticipates/foreshadows floccinauci's idea



that, as floccinauci writes,

fixes two notational problems in Bell's original paper: (1) The symbols and are labels, not variables since their values do not change during the experiment. (2) I've used instead of to denote expectation, which is more standard.

Thus, in my terms and confident that I know what my notation means here, (1) is simply

. (2)

And so, here, in my terms: since one cannot get from (2) to your claims, you are clearly making stuff up.

As opposed to me, who (still amazed), here vows to work harder at reducing my stuff-ups!

HTH; Gordon
.

[Gordon Watson]

The way ahead; with special thanks to minkwe, floccinauci and Fred.

Believing that I now have sufficient insight to arrive at my eqn (26) more clearly, I plan to stick more closely to Bell's notation.

Still finding and helpful to the way I see/understand EPRB, the "unifying claim" in (6) remains



Which is nothing new. But it will at least be clearer that, under TLR, can no longer be viewed as a non-local latent variable.***

An appropriately modified (19)-(20) still leads to (26).

HOWEVER, (7)-(14), with issues to be further discussed here, will be different: making it clearer that Bell's "impossible functions" [(1)-(4)] are not impossible.

*** Bellian source to be found.

HTH; yours, more selectively; Gordon
.

[minkwe]

The way ahead; with special thanks to minkwe, floccinauci and Fred.

Believing that I now have sufficient insight to arrive at my eqn (26) more clearly, I plan to stick more closely to Bell's notation.

Still finding and helpful to the way I see/understand EPRB, the "unifying claim" in (6) remains



Which is nothing new. But it will at least be clearer that, under TLR, can no longer be viewed as a non-local latent variable.***

An appropriately modified (19)-(20) still leads to (26).

HOWEVER, (7)-(14), with issues to be further discussed here, will be different: making it clearer that Bell's "impossible functions" [(1)-(4)] are not impossible.

*** Bellian source to be found.

HTH; yours, more selectively; Gordon
.

A bit of progress. But what is the integration variable? As written the expression is incorrect. I understand why you are introducing superscripts but you are trying to fix a problem with that does not exist. The particles leave the source with the fact that they end up at two stations with the same property does not make non-local. is general enough to capture the scenario in which there is a conservation of momentum with opposite properties going to different stations. Bell's are better in that sense.

[Gordon Watson]

The way ahead; with special thanks to minkwe, floccinauci and Fred.

Believing that I now have sufficient insight to arrive at my eqn (26) more clearly, I plan to stick more closely to Bell's notation.

Still finding and helpful to the way I see/understand EPRB, the "unifying claim" in (6) remains



Which is nothing new. But it will at least be clearer that, under TLR, can no longer be viewed as a non-local latent variable.***

An appropriately modified (19)-(20) still leads to (26).

HOWEVER, (7)-(14), with issues to be further discussed here, will be different: making it clearer that Bell's "impossible functions" [(1)-(4)] are not impossible.

*** Bellian source to be found.

HTH; yours, more selectively; Gordon
.

A bit of progress. But what is the integration variable? As written the expression is incorrect. I understand why you are introducing superscripts but you are trying to fix a problem with that does not exist. The particles leave the source with the fact that they end up at two stations with the same property does not make non-local. is general enough to capture the scenario in which there is a conservation of momentum with opposite properties going to different stations. Bell's are better in that sense.

With less need for explanation, how about this:



where is the space of twins, , pairwise correlated via the same-instance conservation of total angular momentum.

The above reduces to the Bell formulation. But makes it clear that our work begins with TLR [distinct local arguments] and the common detector-function .
.

[gill1109]

With less need for explanation, how about this:



where is the space of twins, , pairwise correlated via the same-instance conservation of total angular momentum.
The above reduces to the Bell formulation. But makes it clear that our work begins with TLR [distinct local arguments] and the common detector-function .

No, it doesn’t reduce to the Bell formulation. And you are wasting space defining +beta = beta, and -beta = -(beta). Try:



Bell starts right there, and shows that the second equality leads to a contradiction (the first is a definition).

[Gordon Watson]

With less need for explanation, how about this:



where is the space of twins, , pairwise correlated via the same-instance conservation of total angular momentum.
The above reduces to the Bell formulation. But makes it clear that our work begins with TLR [distinct local arguments] and the common detector-function .

No, it doesn’t reduce to the Bell formulation. And you are wasting space defining +beta = beta, and -beta = -(beta). Try:



Bell starts right there, and shows that the second equality leads to a contradiction (the first is a definition).

TO GILL, FROM WATSON, RE NO.
HAVE MONEY. WILL BET.
AGAINST YOU. ON ME.
HOW MUCH? WHAT ODDS?
A.S.A.P.

[gill1109]

With less need for explanation, how about this:



where is the space of twins, , pairwise correlated via the same-instance conservation of total angular momentum.
The above reduces to the Bell formulation. But makes it clear that our work begins with TLR [distinct local arguments] and the common detector-function .

No, it doesn’t reduce to the Bell formulation. And you are wasting space defining +beta = beta, and -beta = -(beta). Try:



Bell starts right there, and shows that the second equality leads to a contradiction (the first is a definition).

TO GILL, FROM WATSON, RE NO.
HAVE MONEY. WILL BET.
AGAINST YOU. ON ME.
HOW MUCH? WHAT ODDS?
A.S.A.P.

What exactly do you want to bet on?

It’s important to fix how and when the winner will be determined.

For instance, I have 64 000 Euro says that you can’t reproduce the singlet correlations in a networked computer simulation. As I see it, Bell’s theorem says I’ll win. There are however stringent rules on the protocol of the simulation experiment. No cheating, no experimental loopholes.

According to my calculations the odds in that bet are overwhelmingly in my favour. But anyway, if I should happen to *lose*, the impact of the experiment on science will be so huge that I’ll quickly get the money back by appearing on talk shows. Moreover, the winner will get the Nobel prize, and won’t be interested in the cash he or she gets from the bet. A graceful winner will be happy that I just give 5000 Euro to “Medecins sans Frontieres”.

If you can come up with A, B and rho which do the job, then program them, and win the bet.

I thought you claim that Bell’s theorem is wrong. Come up with proof! I bet you have none!

[Gordon Watson]

(ii). Via the analyser, say , we have .

In other words, you are saying you don't know how the transformation happens. Does the polarizer analyzer combination need any additional information except the setting and particle property to produce an outcome ? If not you need to explain why the result is probabilistic and not deterministic.

I take probability theory (PT) to be the logic of science. (And I like the related math to do much of my talking.) PT allows me to encode incomplete information, en route to establishing more complete information.

So I am not saying that I don't know how the transformation happens. In my view, under EPRB, particle-polariser interactions expose the equivalence class (EC) to which each incoming (pristine) particle belonged.

An EC is an element of physical reality. So if I knew it, which is possible (maybe with the help of others and their tests on a twin particle), I could predict results with certainty.

However, in the absence of such knowledge -- ie, in the presence of uncertainty -- I can only predict that half the ECs will be of one type; half the other type. Nevertheless, as I try to show, PT allows me to move from such uncertainty to an increased (and more helpful) certainty.

Note that each polariser is matched to the spin of the related particles. So each result is determined by the relevant EC. Then (as I say), in the absence of such, I work probabilistically. And, in my view, satisfactorily.

So the detector function is the composition of and .

Yes, I figured. This is a longwinded way of going about it. Traditionally, Bell's function is understood to include everything required to generate an outcome . Thus, while everything you have described polarizers and analyzers and detectors may be interesting to you, they are superfluous to anyone else familiar with these types of analyses. You will do your audience a lot of good by going straight to . I've deliberately omitted because it is problematic. I think you are focusing too much on equivalence classes because it allows you to segway to the probability analysis. However, combining it together with in that way is not proper. At best it is an abuse of notation because the common understanding of the meaning of in Bell literature is not restricted to equivalence classes of lambda that produce specific outcomes. If you must do that, then you must introduce the concept and new notation to boot. I think I see now clearly why you are struggling with superscripts. But there is an easier way. You don't have to do that at all.

I don't see that my focus on ECs is excessive. As for the Bell literature, please show me a tested that did not reveal an EC. It seems to me that it would not be EPRB if a produced other than a specific outcome?

Yes, I struggle with , superscripted or otherwise! Reason: I believe, (i) we advance teaching and understanding when we expose symmetries; (ii) we clarify Bell's ideas when we use the licence given by Bell at the end of II. Formulation; (iii) it seems that my shorthand needs to be clearer.

PS: I sent you Season's Greetings re using different subscripts AND (hopefully properly) taking advantage of Bell's allowing that stands for any number of variables!

See this post from "floccinauci" viewtopic.php?f=6&t=451&start=120#p12137.

Especially since all the polarizer/analyzer/detector stuff is superfluous anyway, you can go directly to the probability treatment. No need to mention anything about the details of the functions if you don't know them.

However, if you are going to demonstrate that a function is able to generate the probability result, which is an interesting exercise in its own right (probably more so), then if and when you provide the functions, they better contain details and not rely on the same probabilities to generate outcomes. You need to show the mechanics.

Then, when it seemed some could not complete the related integral, I wrote (for comparison with Bell's use of the sgn function):***

Please no. If I give you a specifc vector at Alice's station, and a specific setting vector . Please can you calculate the outcome for that specific particle, step by step using . If you can't, please explain why you can't.

I'm inclined to return as close as is sensible to Bell's notation. The ECs are not probabilistic; my knowledge of them is.

Thanks for the offer of specific vectors. Maybe I need to be clearer that I use that term in the context of GA.

But see here how I use PT: The probability of you knowing such a variable is zero. However, if you are a lab-mate of mine that tests twins, the probability that you know the EC of the particle that is on its way to me is non-zero. So best offer that very helpful EC.

*** In my view, the sgn function is too simplistic for the dynamics involved. My function allows for the dynamics associated with spin, torque, precession.

You haven't shown us anything about this "dynamics". This is what I've been asking for.

Anyone can say "the sign function is too simplistic, the function allows for the dynamics associated with spin, torque, precession". How is this different from what you are saying above, until you provide such a function? It is not sufficient to just replace Bell's notation with yours and claim dynamics. The sign function so far has the advantage that it is transparent about what it thinks is going on, even if it is wrong.

For me, I feel that I took the better view: that it was better to seek functions that were informative and hopeful. Not one that was false and inappropriate.

It seems to me, the ECs are all about the dynamics.

PS: It would help me greatly if (where sensible) you limited each post to a single issue. That would help me answer earlier and more expansively when I'm pressed for time.

Thanks, for that in advance and the above; Gordon
.
.

[minkwe]

I take probability theory (PT) to be the logic of science. (And I like the related math to do much of my talking.) PT allows me to encode incomplete information, en route to establishing more complete information.

That's not what the phrase "Logic of Science" is supposed to mean when referring to PT. It doesn't mean non-probabilistic answers are illogical.

So I am not saying that I don't know how the transformation happens. In my view, under EPRB, particle-polariser interactions expose the equivalence class (EC) to which each incoming (pristine) particle belonged.

How? Either you know how and don't want to tell us, or you don't know how and don't want to tell us.

If I say , I've provided exactly how is transformed together with the setting to generate the outcomes . Then I've also shown that there are just two equivalence classes of lambda, as there must be because we already know that the outcomes can only be one of by definition. The important point about specifying the mechanism of transformation is to show how the universe of vectors are divided into the two equivalence classes.

However, if I refuse to provide the mechanism (aka details of the function) and just insist that there are equivalence classes, I would have added zero content to the Bell discussion.

I don't see that my focus on ECs is excessive. As for the Bell literature, please show me a tested that did not reveal an EC. It seems to me that it would not be EPRB if a produced other than a specific outcome?

I think you miss the point. It is not that ECs are lacking, but that they are so obvious as to be uninteresting for the analysis. Bell introduces equivalence classes by definition. The interesting part is not the presence of ECs but in specifying which belong to the same equivalence class in the context of a setting . And you show that by presenting the functions, otherwise talking about ECs is uninteresting.

Yes, I struggle with , superscripted or otherwise! Reason: I believe, (i) we advance teaching and understanding when we expose symmetries; (ii) we clarify Bell's ideas when we use the licence given by Bell at the end of II. Formulation; (iii) it seems that my shorthand needs to be clearer.

Who is it are you trying to teach? Have you figured that these purported students need to learn about symmetry at the same time as trying to understand a topic for which symmetry so obvious as to not be interesting? A good teacher shows focus and restraint when conveying information. Just because you are thinking about it doesn't mean it needs to be written down. What exactly is the main point you want to make. Focus on that. If your brain is overflowing with great ideas, write many different papers, each one focused on one key idea.

I'm inclined to return as close as is sensible to Bell's notation. The ECs are not probabilistic; my knowledge of them is.

So you don't know what the functions are, contrary to what you wrote above? I'm confused.

Thanks for the offer of specific vectors. Maybe I need to be clearer that I use that term in the context of GA.

Geometric Algebra allows you to do analysis in a coordinate-free manner. It doesn't mean you wouldn't know what to do with coordinates when given.

But see here how I use PT: The probability of you knowing such a variable is zero. However, if you are a lab-mate of mine that tests twins, the probability that you know the EC of the particle that is on its way to me is non-zero. So best offer that very helpful EC.

That's actually very funny, or sad depending on your perspective. By providing them to you, you have to assume that I know them so saying I can't possibly know them, is not a reason for not answering my question. What if I knew them, how would you proceed? If you know the mechanics of your functions this question should be very easy.

For me, I feel that I took the better view: that it was better to seek functions that were informative and hopeful. Not one that was false and inappropriate.

Are you still seeking these functions or have you found them already?

It seems to me, the ECs are all about the dynamics.

And the details of the functions show how the ECs are generated. So the dynamics are in the functions, not the ECs.

PS: It would help me greatly if (where sensible) you limited each post to a single issue. That would help me answer earlier and more expansively when I'm pressed for time.

Feel free to respond to just one issue at a time. I just prefer to respond in this way especially since I want to respond in one seating without having to deal with the rate-limit.

[gill1109]

But see here how I use PT: The probability of you knowing such a variable is zero. However, if you are a lab-mate of mine that tests twins, the probability that you know the EC of the particle that is on its way to me is non-zero. So best offer that very helpful EC.

Of course, we can in principle do a standard Bayesian calculation of what Alice should believe about her measurement outcome, given her setting, and given her knowledge (expressed as a prior probability distribution) of Bob's setting and Bob's outcome. This does not help us understand the physics, nor help us understand Bell's theorem. Long ago, Ed Jaynes said that he thought that Bell was doing his probability theory wrong. But then Steve Gull pointed out that, this time, Jaynes was mistaken. You are barking up the wrong tree, Gordon! (And Minkwe's criticism is spot on).