Re: The correct derivation

[Mark Hadley]

Bryan,

I do not assume a single space. Quite the opposite. The derivation and illustration are all for two spaces. As is immediately obvious.

 

Bells makes no assumption about the spaces.

The maths derivation is high school algebra. It is valid from step to step, from start to finish. If you want to dismiss them show the fallacy.

Yesterday you claimed that I had not sent you a refutation, had not shown why you were wrong. That was false and insulting - as everyone can see. I sent these to you and you never responded. That's why you should apologise.

Your work has been refuted. Until you show a mistake in my maths, or a false step in Bells, or account for the absurdities revealed in the other email. 

Your ability to accept criticism matches Fred's You are more polite than Fred, but seek to profit from your false statements.

Mark

On Wed, 14 Jan 2026, 09:02 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

Dear Mark,

Thank you for going back to find your past argument. They all fall into the same category and I rejected them at that time,   You  assume a single space and therefore is fallacious. The correlations arise from two spaces. That point is now clearly articulated in the Two channels note, and it is there you must find errors. So I never accepted those responses, and I never agreed with them. You, Richards, Jan Ake  and all argued as you do, I did not agree, so it was a stalemate.   You must find errors in that note,

You ask for an apology  from me!!!!!!!!!  You repeatedly called me a liar, dismissed my arguments as a High School math error, and made personal comments, to which I did not reciprocate in kind. You are living in a fantasy world of Bell arrogance and a signal probability space.  I will never apologize when faced with such belligerent behavior. You arrogantly dismiss my responses, assuming you have answered it, when you had failed. 

Your only recourse is to find errors in my two channel paper, which you have not done. I want your capitulation to the two channels, which by your silence you have already tacitly given.

I hope this is clear: I respond only to technical arguments, like the abortive one you gave here.

Bryan

Best regards,
Bryan

On Tue, Jan 13, 2026 at 11:42 PM Mark Hadley <sunshine...@googlemail.com> wrote:

Dear Bryan,

I sent this as far back as 22feb 2023

It derives the correct formula for combining two populations.

You never responded.

I expect an apology: you have said dozens of times that nobody found fault in your work. Here it is in detail and unanswered.

Cheers

Mark 

[Mark Hadley]

Cos theta does not feature in the derivation of bells Inequalities

Nor my derivation of correlation coefficient 

Nor in any of the qualitative refutations.

This snippet of a paper does not relate to or affect any of the disproofs.

If you want to refute BI then use the CSHS version it's the simplest and clearest.

Let's be clear. I have disproven your work in four different ways. I claim they are all valid, but if only one of them is valid, then your work is disproven. 

To recover your work and credibility you need to deal with each disproof.

Mark

Mark

Mark 

On Wed, 14 Jan 2026, 12:09 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

Dear Mark

Bryan

On Wed, Jan 14, 2026 at 12:03 PM Mark Hadley <sunshine...@googlemail.com> wrote:

Bryan,

If you claim that Bell makes a hidden assumption. Show us where he uses that assumption in the derivation. Nobody else can see it. Until you show where that is used it remains a figment of your imagination.

My correlation derivation is a few simple maths steps. If you think that is wrong you should show us the step that is invalid in your physics

You have a very strange addition rules for correlations. It is remarkable, it has been challenged by people who have studied your paper. And it was never derived. It is to say the least fundamentally important to your result and quite remarkable. As a scientist you need to derive results that are important to you, disputed, and unique to your paper. The standard combination of distributions is well known and is derived, not assumed.

And you have not addressed the qualitative criticism of your addition formula - that it can give correlations larger than 1, and that the observed correlation is independent of the number of vector and covector events. That's how scientists and mathematicians do a sanity check on a strange equation.

So your work has been challenged in three independent ways and you have not been able to respond to them as a scientist. 

Your result has been disproven by Bell, it has been disproven by the derivation of correlation coefficients and shown to be nonsensical based on a qualitative analysis.

There is nobody, including you, who can defend your work against these refutations. And I think you know that, but cannot emotionally admit it.

Mark

On Wed, 14 Jan 2026, 10:23 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

Mark,

I only reply because you raised technical points.  Your analysis confirms Bell's statistics, and I refute that then and now. Your arguments in no way assail my two-channels.  Of course Bell assumes a single space, as I clearly show in my two-channel paper. You have only convinced yourself, or other gatekeepers of Bell, without moving me. 

You conflate the two sources into one exactly the same error as Christian and Diether made, Your arguments are no threat or answer me.  Also mine works and yours does not.

Bryan

[Mark Hadley]

[Bryan Sanctuary]

Mark,

I only reply because you raised technical points.  Your analysis confirms Bell's statistics, and I refute that then and now. Your arguments in no way assail my two-channels.  Of course Bell assumes a single space, as I clearly show in my two-channel paper. You have only convinced yourself, or other gatekeepers of Bell, without moving me. 

You conflate the two sources into one exactly the same error as Christian and Diether made, Your arguments are no threat or answer me.  Also mine works and yours does not.

Bryan

On Wed, Jan 14, 2026 at 10:30 AM Mark Hadley <sunshine...@googlemail.com> wrote:

[Bryan Sanctuary]

Dear Mark

Bryan

On Wed, Jan 14, 2026 at 12:03 PM Mark Hadley <sunshine...@googlemail.com> wrote:

Bryan,

If you claim that Bell makes a hidden assumption. Show us where he uses that assumption in the derivation. Nobody else can see it. Until you show where that is used it remains a figment of your imagination.

My correlation derivation is a few simple maths steps. If you think that is wrong you should show us the step that is invalid in your physics

You have a very strange addition rules for correlations. It is remarkable, it has been challenged by people who have studied your paper. And it was never derived. It is to say the least fundamentally important to your result and quite remarkable. As a scientist you need to derive results that are important to you, disputed, and unique to your paper. The standard combination of distributions is well known and is derived, not assumed.

And you have not addressed the qualitative criticism of your addition formula - that it can give correlations larger than 1, and that the observed correlation is independent of the number of vector and covector events. That's how scientists and mathematicians do a sanity check on a strange equation.

So your work has been challenged in three independent ways and you have not been able to respond to them as a scientist. 

Your result has been disproven by Bell, it has been disproven by the derivation of correlation coefficients and shown to be nonsensical based on a qualitative analysis.

There is nobody, including you, who can defend your work against these refutations. And I think you know that, but cannot emotionally admit it.

Mark

On Wed, 14 Jan 2026, 10:23 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

[Bryan Sanctuary]

Dear Mark,

Years ago I looked at Bell and when I understood it meant non-locality was the consequence, I said to myself, that is absurd.  That, however, is not enough.  I must prove it is absurd and so there must be an alternate explanation.  It is not enough to say my work is nonsense etc.  You must show it. I suggest that you look at the geometric product of Pauli matrices and see that it consists of a symmetric and antisymmetric part.  That leads to the notion of complementarity of spin, analogous to position-momentum.  You are thinking only of the symmetric contribution, like Bell did.  I have shown that the two distinct sectors of spin are complementary and dual.  In that sense, BI can be applied to each separately.  That is what the first section of the two channel note does, and it gives an algebraic CHSH of 4, which is 2+2.  I have never said that BI are wrong, only that it can be, and must be, applied to each complementary sector independently.

What is the complementary property of spin in your view?

At the fundamental level, you must show, somehow, that the two terms of the geometric product must be averaged, and not added.  I do not think that is possible.

I hope this helps.

Bryan

On Wed, Jan 14, 2026 at 6:11 PM Mark Hadley <sunshine...@googlemail.com> wrote:

Dear Bryan

1) your argument is nonsense. There is no concept of channel in the derivation of BI. All your results ( your clicks in the instrument) are included in BI. You have not found a step in BI that is invalid in your experiment. BI refutes your work until you show otherwise.

2) Even if you want to use BI, twice as you ludicrously suggest, then you are still wrong, your result has been refuted by the correlation calculation that I just resent. It's simple maths, every step is correct and easy to verify and it shows that your two channel result is wrong. You have not shown anything wrong with my derivation. And you have not derived your own result either - your astonishing, incredible , addition formula is presented without proof - unbelievable. My derivation had numbered equations. Which step do you think is wrong.

3) And simple commonsense arguments, the qualitative ones, show that the addition formula cannot be right. I presents two or three separate reasons why the addition formula does not make sense. You have not responded to either of them.

Mark

On Wed, 14 Jan 2026, 16:06 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

Mark,

In that two-channel, I show that spin has two channels.  

I then use Bell's own arguments and confirm that he says one can only agree with experiment by introducing non-locality,  At that point, I assert he missed the second channel, and including it means his non-local argument is replaced by that second channel. Therefore his non-local argument is not needed.  

However, Bell states his theorem, 

"If [a hidden-variable theory] is local, it will not agree with quantum mechanics; if it agrees with quantum mechanics, it will not be local.*

Since the coherence channel exists, non-locality is not needed, and his theorem fails.

I do nothing else.  That all you must refute.  Your arguments do other things, and are out of scope.  

Thanks for looking into it.

Bryan

*Bell, J.S. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy; Cambridge University Press:

Cambridge, UK, 2004; pp. 139/147.

On Wed, Jan 14, 2026 at 3:40 PM Mark Hadley <sunshine...@googlemail.com> wrote:

You are refuting Bell if you claim it does not apply to your physics.

It applies to any local hidden variable theorem. Applying it twice is absurd. I showed you before how your two distributions can be written mathematically as a single hidden variable distribution that satisfies BI.

You have claimed repeatedly, without proof, that Bell made a secret extra assumption, that invalidated his conclusion. This claim is refuted, his derivation and it's assumptions are well known and published, they do not require any assumptions about the distributions as you suggest.

Mark

On Wed, 14 Jan 2026, 13:10 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

Dear Mark,

I have not, and do expect, to ever refute BI.  

I am only saying Bell must be applied twice, once for each channel. 

Bryan

[Mark Hadley]

Dear Bryan,

Spin analysis and decomposition of the events into different streams have no relevance to the problem.

Look at my proof and derivation of combination of correlation coefficients. It has nothing to do with spin. You can label the populations with spin indices or call them apples and oranges. The maths is the same.  Your formula is wrong. I have disproved it in simple steps in the proof which are labelled - tell us if you don't understand or agree with any.

The BI or CHSH derivation introduce a function A( lambda) This function includes any and all variables in your problem. Whatever names you give them. However you analyse them. Both your streams/components/ indices, whatever you call them, all are included in A(lambda) CSHS proves that your result is wrong.

And my small examples show that your addition formula fails fundamental consistency tests.

Mark

On Wed, 14 Jan 2026, 19:27 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

Hi again,

Here is the geo product of Pauli matrices

\sigma_i \sigma_j = \delta_ij + \epsilon _ijk i\sigma_k

Since i cannot be equal and unequal to j simultaneously, the two terms are complementary.  That is the real power of the geo  product, it separates symmetric from antisymmetric. Physics defines spin by only the symmetric term, and I assert it should also include the antisymmetric term.

So what, in your view, are the complementary terms of spin?  I say they are \sigma and i\simga, and I think that is what you miss. You have not addressed that.  Bell used spin from only the symmetric part.  I assert that it is fundamentally incomplete. 

Everything I say and use has its origin in the geometric product above.

Bryan

On Wed, Jan 14, 2026 at 8:13 PM Mark Hadley <sunshine...@googlemail.com> wrote:

Bryan,

I don't just say your work is nonsense. I say it is wrong. Your equations are wrong. I prove that. Your understanding of Bell is simply wrong. Your combination of correlation coefficients is wrong. Mathematically wrong and I disprove it with high school maths. Your addition formula gives ludicrous consequences which I show with examples.

Neither I nor Bell are thinking of only one component. Bells result applies to the totality of components. Your idea that splitting the results into components can give new results or a way round BInis simply wrong.

If you find a step in CSHS that restricts it's application to one component then show us. It does not it involves a local hidden variable that includes ALL components that the experiment could conceivably have. It includes everything in your model and it predicts BI. Your decompositions adds nothing. It is simply irrelevant.

CSHS derivation disproves your result. The algebra of combined distributions disproves your result. They don't just call your ideas nonsense, hey show it is wrong. False mathematically incorrect.

[Bryan Sanctuary]

Dear Mark,

I have not, and do expect, to ever refute BI.  

I am only saying Bell must be applied twice, once for each channel. 

Bryan

[Bryan Sanctuary]

Dear Mark,

Thank you for going back to find your past argument. They all fall into the same category and I rejected them at that time,   You  assume a single space and therefore is fallacious. The correlations arise from two spaces. That point is now clearly articulated in the Two channels note, and it is there you must find errors. So I never accepted those responses, and I never agreed with them. You, Richards, Jan Ake  and all argued as you do, I did not agree, so it was a stalemate.   You must find errors in that note,

You ask for an apology  from me!!!!!!!!!  You repeatedly called me a liar, dismissed my arguments as a High School math error, and made personal comments, to which I did not reciprocate in kind. You are living in a fantasy world of Bell arrogance and a signal probability space.  I will never apologize when faced with such belligerent behavior. You arrogantly dismiss my responses, assuming you have answered it, when you had failed. 

Your only recourse is to find errors in my two channel paper, which you have not done. I want your capitulation to the two channels, which by your silence you have already tacitly given.

I hope this is clear: I respond only to technical arguments, like the abortive one you gave here.

Bryan

Best regards,
Bryan

On Tue, Jan 13, 2026 at 11:42 PM Mark Hadley <sunshine...@googlemail.com> wrote:

[Mark Hadley]

Dear Bryan

1) your argument is nonsense. There is no concept of channel in the derivation of BI. All your results ( your clicks in the instrument) are included in BI. You have not found a step in BI that is invalid in your experiment. BI refutes your work until you show otherwise.

2) Even if you want to use BI, twice as you ludicrously suggest, then you are still wrong, your result has been refuted by the correlation calculation that I just resent. It's simple maths, every step is correct and easy to verify and it shows that your two channel result is wrong. You have not shown anything wrong with my derivation. And you have not derived your own result either - your astonishing, incredible , addition formula is presented without proof - unbelievable. My derivation had numbered equations. Which step do you think is wrong.

3) And simple commonsense arguments, the qualitative ones, show that the addition formula cannot be right. I presents two or three separate reasons why the addition formula does not make sense. You have not responded to either of them.

Mark

On Wed, 14 Jan 2026, 16:06 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

Mark,

In that two-channel, I show that spin has two channels.  

I then use Bell's own arguments and confirm that he says one can only agree with experiment by introducing non-locality,  At that point, I assert he missed the second channel, and including it means his non-local argument is replaced by that second channel. Therefore his non-local argument is not needed.  

However, Bell states his theorem, 

"If [a hidden-variable theory] is local, it will not agree with quantum mechanics; if it agrees with quantum mechanics, it will not be local.*

Since the coherence channel exists, non-locality is not needed, and his theorem fails.

I do nothing else.  That all you must refute.  Your arguments do other things, and are out of scope.  

Thanks for looking into it.

Bryan

*Bell, J.S. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy; Cambridge University Press:

Cambridge, UK, 2004; pp. 139/147.

On Wed, Jan 14, 2026 at 3:40 PM Mark Hadley <sunshine...@googlemail.com> wrote:

You are refuting Bell if you claim it does not apply to your physics.

It applies to any local hidden variable theorem. Applying it twice is absurd. I showed you before how your two distributions can be written mathematically as a single hidden variable distribution that satisfies BI.

You have claimed repeatedly, without proof, that Bell made a secret extra assumption, that invalidated his conclusion. This claim is refuted, his derivation and it's assumptions are well known and published, they do not require any assumptions about the distributions as you suggest.

Mark

[Bryan Sanctuary]

Dear Mark,

I agree in the sense that Bell's work is classical, not quantum.  He and most of modern physics, however, agree Bell is about quantum, like the title of Bell's paper.  I simply say, they missed something, and what they miss in his classical theory is coherence (antisymmetry). Spin is more than up and down states in my view. Spin is a bivector, and is classical. It has a quantum limit. 

I think we are at a point where some cogitation is needed. My approach does not follow Bell because Bell's statistics are restricted to a single probability space, but I say it has two spaces, like pos-mom, energy-time and angle-ang mom.  I asked you what is the complementary variable of spin polarization? I would be interested in your reply.

Bryan

[Mark Hadley]

Bryan,

If you claim that Bell makes a hidden assumption. Show us where he uses that assumption in the derivation. Nobody else can see it. Until you show where that is used it remains a figment of your imagination.

My correlation derivation is a few simple maths steps. If you think that is wrong you should show us the step that is invalid in your physics

You have a very strange addition rules for correlations. It is remarkable, it has been challenged by people who have studied your paper. And it was never derived. It is to say the least fundamentally important to your result and quite remarkable. As a scientist you need to derive results that are important to you, disputed, and unique to your paper. The standard combination of distributions is well known and is derived, not assumed.

And you have not addressed the qualitative criticism of your addition formula - that it can give correlations larger than 1, and that the observed correlation is independent of the number of vector and covector events. That's how scientists and mathematicians do a sanity check on a strange equation.

So your work has been challenged in three independent ways and you have not been able to respond to them as a scientist. 

Your result has been disproven by Bell, it has been disproven by the derivation of correlation coefficients and shown to be nonsensical based on a qualitative analysis.

There is nobody, including you, who can defend your work against these refutations. And I think you know that, but cannot emotionally admit it.

Mark

On Wed, 14 Jan 2026, 10:23 Bryan Sanctuary, <bryancs...@gmail.com> wrote:

[Bryan Sanctuary]

Mark,

In that two-channel, I show that spin has two channels.  

I then use Bell's own arguments and confirm that he says one can only agree with experiment by introducing non-locality,  At that point, I assert he missed the second channel, and including it means his non-local argument is replaced by that second channel. Therefore his non-local argument is not needed.  

However, Bell states his theorem, 

"If [a hidden-variable theory] is local, it will not agree with quantum mechanics; if it agrees with quantum mechanics, it will not be local.*

Since the coherence channel exists, non-locality is not needed, and his theorem fails.

I do nothing else.  That all you must refute.  Your arguments do other things, and are out of scope.  

Thanks for looking into it.

Bryan

*Bell, J.S. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy; Cambridge University Press:

Cambridge, UK, 2004; pp. 139/147.

On Wed, Jan 14, 2026 at 3:40 PM Mark Hadley <sunshine...@googlemail.com> wrote:

You are refuting Bell if you claim it does not apply to your physics.

It applies to any local hidden variable theorem. Applying it twice is absurd. I showed you before how your two distributions can be written mathematically as a single hidden variable distribution that satisfies BI.

You have claimed repeatedly, without proof, that Bell made a secret extra assumption, that invalidated his conclusion. This claim is refuted, his derivation and it's assumptions are well known and published, they do not require any assumptions about the distributions as you suggest.

Mark

[Mark Hadley]

You are refuting Bell if you claim it does not apply to your physics.

It applies to any local hidden variable theorem. Applying it twice is absurd. I showed you before how your two distributions can be written mathematically as a single hidden variable distribution that satisfies BI.

You have claimed repeatedly, without proof, that Bell made a secret extra assumption, that invalidated his conclusion. This claim is refuted, his derivation and it's assumptions are well known and published, they do not require any assumptions about the distributions as you suggest.

Mark

[Mark Hadley]

Bryan,

I don't just say your work is nonsense. I say it is wrong. Your equations are wrong. I prove that. Your understanding of Bell is simply wrong. Your combination of correlation coefficients is wrong. Mathematically wrong and I disprove it with high school maths. Your addition formula gives ludicrous consequences which I show with examples.

Neither I nor Bell are thinking of only one component. Bells result applies to the totality of components. Your idea that splitting the results into components can give new results or a way round BInis simply wrong.

If you find a step in CSHS that restricts it's application to one component then show us. It does not it involves a local hidden variable that includes ALL components that the experiment could conceivably have. It includes everything in your model and it predicts BI. Your decompositions adds nothing. It is simply irrelevant.

CSHS derivation disproves your result. The algebra of combined distributions disproves your result. They don't just call your ideas nonsense, hey show it is wrong. False mathematically incorrect.

Mark

[Mark Hadley]

Bells is a classical result with relevance to quantum theory. We know that.

It says all classical physics makes this prediction.

That's the proof. That's why it disproves your work and your result. 

It did not miss anything. You have not added anything that can possibly change Bells results. That's why it is a disproof of your claim. That's what tells us your conclusion was wrong. And when we look we find your error. You combine distributions in a way that is wrong. And we have proven that you made a fatal error. Leading you to think you have found something new.

An understanding of spin is not required to derive or understand BI.

CSHS have a general probability distribution. It can be built from several other distributions. Your model is totally compatible with Bells assumptions. He uses A(lambda) to cover any probability function A of any coordinate space lambda. It is incredibly general and covers your set up with ease.

If you want to claim otherwise you would have to PROVE that your set up could not be described by any function A( lambda) for any A and any lambda. But I showed you exactly how you can create A(lambda) from your two distributions. It is simple to do. So CSHS applies and it disproves your work.

Bell disproves your result and the combination of correlation coefficients proves where your mistake is 

Mark

[anton vrba]

Dear Bryan, you say: Spin is a bivector, and is classical. It has a quantum limit.   

Please clarify these claims in precise mathematical and physical terms.

If spin is classical, then please describe explicitly how spin is represented as a bivector and how this bivector configures a photon without invoking configuration space (Hilbert space). A photon is classically defined by the Maxwell equations; where, exactly, does your bivector appear within Maxwell theory? What field, equation, or geometric object does it correspond to?

You also state that spin has a “quantum limit.” What do you mean by quantum limit? Is this a mathematically defined limiting procedure, a quantisation condition, a topological constraint, or something else? Please state it explicitly and show how it yields the observed quantum behaviour.

Unless these claims can be demonstrated mathematically and connected to empirical physics without handwaving or unproven heuristic assumptions, what you are proposing amounts to a new heuristic model of natural philosophy intended to replace the (also heuristic) Copenhagen interpretation of quantum mechanics.

Current physical models, while heuristic, are empirically corroborated and consistent with existing measurement frameworks. Replacing them with alternative heuristic models that simultaneously require changes to the measurement framework just does not play, in my opinion. As I have said before, Bell is a measuring tool, not physics. You are proposing changes both to the physics and to the measuring tools. The proper order, in my view, is first to convince the physics community of the physical model, and only then to revisit the measurement framework.

You have not yet convinced me of the physics.

BTW. I only understood Bell and CHSH after reading Freedmans Ph.D thesis {https://escholarship.org/uc/item/2f18n5nk},  a 100 page document.  The important part is understanding Bell and CGSH in conjunction with the measuring apparatus in all its fascists.  

Regards

Anton

------ Original Message ------

From "Bryan Sanctuary" <bryancs...@gmail.com>

To "Mark Hadley" <sunshine...@googlemail.com>

Cc "Bell inequalities and quantum foundations" <Bell_quantum...@googlegroups.com>

Date 1/14/2026 9:11:53 PM

Subject [Bell_quantum_foundations] Re: The correct derivation

[GeraldoAlexandreBarbosa]

Just to relieve tensions due to Bryan's classical double-"spin": see attachment with classical similar effects.

(From left to right: Geraldo Barbosa, Ricardo Horowicz, Luiz Davidovich)

Geraldo A. Barbosa, PhD

KeyBITS Encryption Technologies LLC

https://www.keybits.tech/

1540 Moorings Drive #2B, Reston VA 20190

E-Mail: GeraldoABarbosa@keybits.tech 

Cellphone: 1-443-891-7138 (US) - with WhatsApp

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[GeraldoAlexandreBarbosa]

Adding to my last email: Pay attention to the bottle and glass.

Geraldo A. Barbosa, PhD

KeyBITS Encryption Technologies LLC

https://www.keybits.tech/

1540 Moorings Drive #2B, Reston VA 20190

E-Mail: GeraldoABarbosa@keybits.tech 

Cellphone: 1-443-891-7138 (US) - with WhatsApp

On Thu, Jan 15, 2026 at 4:22 PM anton vrba <anto...@gmail.com> wrote:

[Richard Gill]

Bryan, Bell-CHSH allows as many channels as you like. 

It does assume binary measurement measurement outcomes and binary setting choices.

One calls this the 2 x 2 x 2 case: 2 parties, 2 settings per party, 2 outcomes per settin per party.

But it has been extended to more general experimental lay-outs. If Alice and Bob each have two measurement devices, and if they all have binary measurement outcomes, and binary setting choices, then we are in the 4 x 2 x 2 case. Have you looked up what generalised Bell inequalities are available for that case? If not, why not?

Sent from my iPad

On 15 Jan 2026, at 11:02, Bryan Sanctuary <bryancs...@gmail.com> wrote:



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[Richard Gill]

Bryan

Classical physical reasoning leads to the possibility to define actual and counterfactual outcomes as random variables defined on a single probability space. The “single probability space” does not need to be *assumed*. It can be mathematically *constructed*.

arXiv:2211.05569  [pdf, other]   quant-ph stat.AP

Bell's theorem is an exercise in the statistical theory of causality

Authors: Richard D. Gill

Abstract: In this short note, I derive the Bell-CHSH inequalities as an elementary result in the present-day theory of statistical causality based on graphical models or Bayes' nets, defined in terms of DAGs (Directed Acyclic Graphs) representing direct statistical causal influences between a number of observed and unobserved random variables. I show how spatio-temporal constraints in loophole-free Bell experiments, and natural classical statistical causality considerations, lead to Bell's notion of local hidden variables, and thence to the CHSH inequalities. 

[Richard Gill]

Bryan

Bell is about certain constraints which classical physical principles automatically imposes on observed correlations between macroscopic measurement outcomes in some simple experiments. 

Quantum concepts are not introduced, by design. Bell *deliberately* does not use words like “particle”, “wave”, “field”, “interference”, “spin”. He does not say anything whatsoever about what is going from a source to Alice and Bob’s apparatus.

If you think that *quantum theorists* have missed something then please get to work with some enterprising quantum experimentalists and get them to do the experiments which might confirm or disprove the existence of some new quantum properties of whatever “particles” might carry them. 

Bell’s work has long ago been generalised from the simple 2x2x2 case (2 parties, 2 settings, 2 outcomes). Tell us what case you will need, once the new kinds of detectors have been engineered.

Richard 

Sent from my iPad

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