Inherited correlations: a new perspective on Bell’s Theorem

[Ghenadie Mardari]

Dear Friends,

I was invited by Richard Gill to join this group and to share my latest results about quantum entanglement. Hopefully, we can have an interesting discussion about them.

I believe that there is no fundamental gap between classical and quantum mechanics. Instead, there is a conceptual gap in the study of classical phenomena, where linear superposition is used to reduce irreducible processes. You can find more details in my latest two publications in Quantum Reports (one and two), where I did my best to explain why quantum entanglement is a classical phenomenon.

For this group, with its focus on Bell’s Theorem, the important conclusion is that quantum-like Bell violations are natural (ordinary) occurrences, because quantum mechanics is just a new branch of classical physics.

There is an obvious tension between this claim and the usual interpretation of Bell’s Theorem, so let us start by reviewing the basics of the argument for non-locality.

When remote observables are correlated, we have two explanations to consider:
1. There is a direct influence between them;
2. They have a common prior cause.

Only the second option allows for local causality between separated observables.

Yet, there are two possible ways for this to come about:

A. The “Realism” condition.
We are dealing with permanent properties that are real at the source and at the site of measurement. Their correlation at the point of measurement can be explained by a shared direct effect at the source. Ergo, a single profile of lambda determines the observable values (and their correlations). This is why Bell’s inequality applies (and is necessarily obeyed).

B. The (poorly named) “non-Realism” condition.
We are dealing with contextual properties that only become real at the site of measurement. They are not physically extant at the source. Their observable values (and correlations) are determined by indirect effects from dedicated hidden variables (different for each observable), coming from the prior common cause. Bell violations are naturally possible in this case.

To clarify, the concept of “Realism” is needed to disambiguate contextual values from contextual properties. In the latter case, the properties themselves are transient and conditional, not just their values. Yet, the underlying ontology is classical in both cases. Think of a piece of playdough that is reshaped in different ways at various points in time.

According to this explanation, Bell violations automatically rule out permanent properties that exist at the same time at the source. Yet mutually exclusive properties can still be correlated by indirect effects from a common source. So, how do we get from Bell violations to non-locality?

The answer, according to Bell, is that mutually exclusive properties – though local – require “new physics”. Contextual observables exist only at the point of measurement. Ergo, their “common cause” must be delivered from the past by dedicated hidden variables. This means that the correlation between hidden variables and corresponding measurement settings emerges from a direct effect of hidden variables on the experimental parameters. This is such an outlandish scenario that it must be rejected outright (according to Bell).

 
In conclusion, Bell violations are commonly interpreted with direct influences between observables (and non-local mechanisms) because the possibility of mutually exclusive properties was dismissed out of hand.

Accordingly, if we did not dismiss these mutually exclusive properties form consideration, we would have no reason to invoke nonlocality.

The question is: why should we consider this scenario? Isn’t super-determinism “crazy”?

The answer is that we do not actually need “new physics” in this case. We can also explain the correlations between hidden variables and measurement settings with the new concept of inherited correlations. In this new light, Bell violations with mutually exclusive properties can be perfectly local, in a classical sense.

The important nuance here is that a quantum measurement is not a “read only” observation. Instead, we begin with an invasive preparation that is preceded by a choice of experimental setting. Only at the final stage, downstream from the transformation, we have a “read only” event counter. In other words, the “measurement setting” is not a detector setting. It is a “transformation setting”. The effect of this clarification is to flip the causal arrow for the correlation between measurement settings and hidden variables.

Imagine two classical systems that are maximally correlated at a common source. A single parameter λ_0 determines their relationship. Bell violations are impossible, as long as the systems remain unperturbed. Yet, the two systems are going to be processed independently by Alice and Bob.

 
At one site, Alice makes a choice of setting, which leads to a transformation, followed by the act of detection. This means that the choice of experimental setting is upstream from the manifestation of dedicated hidden variables, associated with the induced transformation. (At the other site, the same procedure is followed by Bob.)

In short, Alice’s observables are determined by λ_0 plus a special new parameter λ_Alice. The final observations are explained by the sum of λ_0 + λ_Alice. Likewise, the observables of Bob are explained by λ_0 + λ_Bob. We have different hidden variables for each observable, and the hidden variables are correlated with the experimental settings. Yet, we do not have a causal effect of dedicated hidden variables on the measurement settings. Instead, we have a causal effect of the transformation settings on the dedicated portion of the hidden variable profile. The threat of “super-determinism” is eliminated.

In the final analysis, the strong correlations that lead to Bell violations are inherited from the shared profile λ_0. The “dedicated” portions of the hidden variables do not explain the strength of correlation and can even diminish it. Instead, their function is to introduce the incompatibility that (combined with strong correlations) produces Bell violations.

 
As a corollary of the above, there are three possible outcomes of a Bell experiment:
1. Bell violation due to non-locality.
2. No Bell violation, due to jointly distributed variables.
3. Local Bell violation due to incompatible variables.

 
If we do not dismiss the third option, then the natural explanation is that quantum correlations are local. Quantum mechanics predicts Bell violations for non-commuting variables (and only for non-commuting variables). Therefore, Bell experiments confirm the reality of mutually exclusive properties at the quantum level.

In a nutshell, this is the same old choice between “Locality” and “Realism”, given Bell violations, but without demonizing the concept of “Realism”.

If the "Realism condition" is falsified, it simply means that we are dealing with classical properties that are mutually exclusive.

[Mark Hadley]

I don't think this can violate BI it is already understood, assumed, and calculated that the hidden variables can include values from the source and each detector and anywhere on route. So long as the Alice variables do not depend on B settings.

look at Bohms theory where the measurement values are created at each arm.

Cheers

mark

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[Richard Gill]

I agree. I recommend Bell’s last paper “La Nouvelle Cuisine” where this is beautifully summarized.

Sent from my iPad

On 21 May 2026, at 12:41, 'Mark Hadley' via Bell inequalities and quantum foundations <Bell_quantum...@googlegroups.com> wrote:



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[Ghenadie Mardari]

You are correct: contextual values cannot violate the Bell inequalities. 

The "non-Realism" condition is specifically about contextual properties. 

Such violations are conclusively demonstrated in my latest paper with a classical fluid splitter.

They are made possible by irreducible system-level effects, just like in quantum mechanics.  

Think of it this way: if it was impossible to violate Bell's inequality with non-commuting variables, there would be no Bell's Theorem.

My contribution is merely to neutralize the "super-determinism" objection with a mechanism for inherited correlations.

Best wishes,

Ghenadie.

[Ghenadie Mardari]

If we assume that a "quantum observation" is a read-only measurement, then Bell's objection makes sense. 

It does seem like we need super-luminal causation to explain the correlations. 

In contrast, if we acknowledge that quantum operators describe invasive preparations (i.e., transformation settings, not detector settings), then we arrive at a purely classical explanation with local causality. 

The unsuspected mechanism involves inherited correlations for newly created properties, as explained in my post above.

Bell is often cited among the first people to acknowledge the local solution for mutually exclusive properties (with different hidden variables for different settings). Yet, he dismissed it as unreasonable, because of the implications for "observer free will" (or super-determinism). 

So, I am not contradicting known facts. I am only showing a new solution, given the two new concepts:
1. Irreducible macroscopic transformations.

2. Inherited correlations for contextual properties.

Best wishes,

Ghenadie.

[Richard Gill]

Sorry Ghenadie, in my opinion you are not contributing anything new. You cannot explain the violation of Bell inequalities in a classical way. If you want to prove me wrong, you can try to win my computer simulation challenge.

Sent from my iPad

On 21 May 2026, at 13:45, Ghenadie Mardari <gmar...@gmail.com> wrote:



[Alexandre de Castro]

Richard,
let us consider, hypothetically, that it were possible to obtain the predictions of quantum mechanics for the singlet state from a LHV.

Since the predictions of quantum mechanics for the singlet state violate Bell inequalities, could we say, in this hypothetical case, that the violation of Bell inequalities occurred in a classical way?

To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/6A7F8B4B-4BCF-4347-BCD1-28BB8B4F8A9E%40gmail.com.

[Richard Gill]

Since that hypothetical case would never occur you would never hear what I would say in that case, so I’m not going to commit now to any particular statement.

Sent from my iPad

On 21 May 2026, at 14:37, Alexandre de Castro <alx...@gmail.com> wrote:



[Ghenadie Mardari]

Dear Richard,

I think I have the goods.

My latest paper describes a classical toy model that naturally produces Tsirelson violations of the CHSH inequality.

A classical fluid-splitter produces the same alternative output distributions as a quantum electron beam in a Stern-Gerlach device. 

The correlations are rotationally invariant and satisfy the "freedom of choice" requirement.

How is this possible? There is a trick.

We need to consider irreducible system-level effects on individual behavior.

Similarly, Born's rule describes irreducible system-level effects on quantum behavior.

Accordingly, we can simulate two copies of my toy model with isolated computers.

The expected result is a quantum-like Bell violation.

I even have a Monte-Carlo simulation in the Appendix that supports this expectation.

So, respectfully, your challenge is not a concern for me.

The much bigger challenge is to convince my colleagues to look at my published paper.

(Alas, human nature...)

I think that the first step is to show that such a result makes sense, given the known facts. 

So, here are the facts that I consider relevant:

1. Bell's inequality is only valid for jointly distributed variables. 

Ergo, it only holds for properties that are real at the same time (the "realism condition").

2. Bell's inequality is naturally violated by mutually exclusive properties (which don't satisfy the "realism condition").

This is the whole basis for "super-determinism", "retro-causality" and other such "local" proposals.

3. The concept of "quantum measurement" is unfairly compared with classical read-only measurements.

Once this is corrected, the quantum singlet state can be interpreted as a classical process with local hidden variables.

You see, this puzzle had some missing pieces. 

Yet, now that I found them, things are different.

Sincerely,

Ghenadie.

[Richard Gill]

Sorry, I believe that you are completely deluded. Your latest paper does no such thing.

Irreducible system level effects (as predicted by quantum theory) on individual behaviour are non-local and non classical, in the following sense: you cannot simulate a successful loophole-free Bell-CHSH experiment on two separated computers, each receiving binary settings supplied by an external opponent.

Go ahead and try. You can win 5000 Euro from me, and get the Nobel prize for disproving Bell’s theorem, if you succeed. Just by publishing your computer programs so that anyone can test them will be enough. There is no need to check the maths.

If you are interested I can refer you to a peer reviewed, published, much cited paper by myself in which I figure out the number of trials so we each have less than 1 in a million chance of losing, according to our own claims/theories. The criterion is that you must get at least half way from 2 to 2 sqrt 2. I recall that 10 thousand trials with each of the four setting pairs is enough. I will deliver them in randomly mixed up order.

Sent from my iPad

On 21 May 2026, at 16:02, Ghenadie Mardari <gmar...@gmail.com> wrote:



[Parker Emmerson]

Ghenadie, the intuition about system-level transformations is interesting, but the Bell claim is still overstated. If the Monte Carlo simulation generates Bob from Alice with P(B=A∣a,b)=cos2(θ/2), then the quantum pairwise law has already been inserted at the conditional level. That is not a derivation from two isolated local response functions A(a,λ), B(b,λ). It is a context-by-context sampler of the desired pairwise correlations. So the paper may be useful as an analogy for mutually exclusive transformations and system-level effects, but it is not a counterexample to Bell in Richard’s two-program sense.

But Richard’s two-program sense is not neutral either. Richard is entitled to ask for a two-program scalar model. That is a legitimate challenge for one formal class. But if he also prohibits phenomenological-velocity methods, saddle maps, holomorphic functions, sweeping nets, noninjective fibers, and the resulting linear-algebraic/noncommutative event-formation route, then the challenge is no longer simply Bell’s theorem. It is a strengthened Bell–Richard theorem: Bell plus a prior ban on the non-Boolean architecture PV uses. Bell’s original argument did not analyze PV, did not know the push-out/exceptional-locus construction, and did not prove that every local event-formation account must be expressible as two scalar Boolean programs. Richard can test the scalar shadow; he cannot call the exclusion of the PV machinery a neutral consequence of Bell.

Phenomenological velocity is not introduced in Bell’s scalar form. PV is not another Bell-λ variable. It is a cancellation-sensitive algebraic/event-formation structure: a push-out unity, an embedded v-architecture, branch and fiber structure, noninjective fibers, holomorphic continuation, sweeping-net reconstruction, saddle-map geometry, and exceptional-locus semantics. In PV, the final +/−1 report is the Boolean projection of a richer pre-Boolean event architecture. If one forces that architecture into A(a,λ), B(b,λ), Bell applies, but that is exactly the flattening PV disputes.

This is where Kochen–Specker and Fine are not optional side remarks. Kochen–Specker warns that quantum-compatible event structures should not be assumed to admit global noncontextual Boolean valuations. Fine’s theorem says, in the CHSH setting, that the existence of a joint distribution for all four potential outcomes is equivalent to the Bell inequalities. Richard may demand such a table as a special challenge, but he may not also ban the PV machinery and pretend the result is still the full force of Bell’s original theorem. That is no longer Bell alone. It is Bell plus Richard’s extra restriction.

So Bell’s theorem is mathematically correct, but it is often philosophically over-advertised. It rules out the scalar Boolean bookkeeping representation. It does not, by itself, refute locality in the deeper physical sense, hidden structure in the PV sense, or noncommutative branch-sensitive event-formation. Bell kills the cave-wall shadow if one mistakes that shadow for a local hidden-variable model. It does not kill the higher-dimensional PV architecture casting the shadow.

That is why Richard’s challenge is Richard-compliant and Bell-functional, but not PV-neutral. It tests whether a theory consents to be flattened into Bell’s Boolean ledger. PV does not.

[Bryan Sanctuary]

Hello Grenadier,

Just a quick comment for now.  I agree that context is essential to EPR, and I would appreciate it if you can see some treads here from me on just that, and interested in your thoughts.

Richard has a standing bet for 5000 euros that only Fool's would take. I took it but said all would do is resolve EPR without nonlocaly.  (There is more than Boolean pairs). I did and it was published and Richard and other Bellists call me a quantum Crackpot, and quote '64 as definitive.

Richard and I decided to cancel the bet because I realized he would never give up Bell; would not pay. But I will bet him again with better rules if Richard believes he can still win.

Hey Richard wanna bet again?

Still, the bet is unimportant, but the physics is not.

I posted a calculation a few days ago showing the full smooth -cosine, and I will follow up soon. But that calculation used only a common phase between A and B. I use quaternions, not Boolean pairs. Quaternions obviate the non-locality assumption.

Very good to have your input

Bryan 

[Richard Gill]

Bryan

My bet is still open. Go ahead and see if you can win it. 

You made another bet with me. I won but you wouldn’t pay.

Richard

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On 21 May 2026, at 21:28, Bryan Sanctuary <bryancs...@gmail.com> wrote:



To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/CALLw9YxFfayszXRncXdg6oW8k_ij%3DNDNU0wNYvoPqXeaH_yQvQ%40mail.gmail.com.

[Richard Gill]

Dear Parker

I do not ban any PV architecture!

The two program challenge is a pedagogical tool to test the understanding of Bell-deniers. To help them understand Bell’s theorem by giving them an impossible task. It works. Bryan Sanctuary’s understanding has progressed so far that he no longer says Bell was wrong and he no longer works at creating a counterexample.

Bryan’s initial challenge to me was that he was going to convince the majority of physicists (within a year) that Bell was wrong. Zeilinger and similar would be forced to withdraw their papers. He lost, despite being granted a one year extension, but refused to pay up.

Read the section in this paper about quantum Randi challenges.

I asked Bryan for the program which he used to create the image which I include below because I want to see where he replaces the usual formula for a correlation of binary data with the theoretical answer he wants to obtain. It would be a shorter program if he merely used the data from a simulation of the EPR-b correlations (or from perfect quantum lab observations) to estimate a - b and then simply compute -cos(a - b). See his email and graphics below.

Bryan sees me as a gatekeeper for the quantum entanglement believing community, he sees them as a sort of maffia who are holding back progress in science and who suppress dissenters like himself. But I’m just a statistician who enjoys finding mistakes in long papers written by people who don’t understand Bell’s maths and logic. Which is very simple maths indeed. The mistakes have to be hidden deep in complexity. But they are always there.

I was able to refine Bell’s theorem using martingale methods to neutralise the memory loophole. My novel statistical analysis was used (and simplified and sharpened) by the four loophole-free experiments of 2000. All four experimental groups cited my contribution. I originally developed it as part of a two computer challenge to Luigi Accardi. He was exploiting the detection loophole and I insisted on no post-selection. His disproof of Bell’s theorem is long forgotten, now.

Richard

From: Bryan Sanctuary <bryancs...@gmail.com>
Date: 19 May 2026 at 14:49:22 CEST
To: Richard Gill <gill...@gmail.com>
Cc: Bell Inequalities and quantum foundations <bell_quantum...@googlegroups.com>
Subject: Re: Richard's programming challenge to Bryan



Dear Richard,

Richard,

You are impatient, it has only been a few days but you know how writing is. In the meantime, I attach the plot I got which compares the old add, on the left, with the latest simulation.  I get -cos using quaternions and I get the Boolean triangle using Bell. No nonlocality of course.  EPR data carries information in two ways: your Boolean pairs (Bell) and something I think you have not considered: the macroscopic build up of coherence within the collection bins.  The latter is only available after all the runs are completed.

So here is the plot which includes instantiated planes.  You will have to wait for a few days for the code.  I must write it up so it is clear, and that will take me time.

Bryan

Sent from my iPad

On 21 May 2026, at 18:28, Parker Emmerson <powerin...@gmail.com> wrote:



[Ghenadie Mardari]

Dear Richard,

Your analysis does not need to be false for my analysis to be correct, because mutually exclusive properties are a special kind of beast.

As I showed in my monogamy paper, the same flow of events (without any change whatsoever) can produce TWO kinds of patterns of correlation, simply depending on how many measurements are made at the same time. 

In a nutshell, two computers produce four spreadsheets (two for Alice and two for Bob) which demonstrably do not produce Bell violations when merged. Yet, when the same event lists are obtained by counting coincident events in pairs, the magic happens because of this special feature of mutual exclusivity.

This is a "loophole", if you will, that hasn't been considered before.

Note that my demonstration begins with alternative transformations over one and the same system. Mutually exclusive profiles of the same system are such that Bell violations are natural (because the rules of transformation are non-linear). So, it is not surprising for two copies of the same system to reproduce the same relationship, just like two socks (but with mutually exclusive qualities). 

I think that, when you read my papers, you will be intrigued and you will even want to help me put this simulation together.

Let's forget about prizes for a second. This is an exciting new idea.

Best wishes,

Ghenadie.

[Ghenadie Mardari]

Dear Parker,

You are correct that the quantum pairwise law is inserted in my simulation, but there is a nuance.

The point of this exercise is to reveal the equivalence between two scenarios:

A. The pairwise law emerges from non-local interactions between two systems.

B. The pairwise law emerges from the local relationships between two alternative irreducible transformations of the same system.

So, we have a formal equivalence between two scenarios: one in which Richard's challenge is not met, and one in which it is.

The details are in section 3 of my latest paper, but section 2 is also helpful.

Best wishes,

Ghenadie.

[Richard Gill]

My analysis says nothing whatever about yours (and vice versa)

What you suggest is a loophole which hasn’t been considered before, is in my opinion simply a feature of QM and probably of any theory which reproduces QM like correlations.

Sent from my iPhone

[Richard Gill]

Ghenadie, 

You had better respond to my challenge by writing the desired computer programs before claiming you have won it already.

Richard

Sent from my iPhone

On 22 May 2026, at 15:20, Ghenadie Mardari <gmar...@gmail.com> wrote:



[Richard Gill]

PS Ghenadie, your model is non local. I’ve told you the easy way for you to prove me and Bell wrong, and more importantly, to prove that to the whole world. At least … it would be easy if your claims were correct.

I certainly can’t help you achieve the impossible, and I don’t plan to waste time trying.

Sent from my iPhone

On 22 May 2026, at 15:45, Richard Gill <gill...@gmail.com> wrote:

My analysis says nothing whatever about yours (and vice versa)

[Richard Gill]

Dear Ghenadie

The point of my challenge is that it never can be met. Why? Because of Bell’s theorem. By which I mean a certain indisputably true mathematical theorem which Bell proved, to the mathematical standards of 1960’s to 1980’s theoretical physicists.

So your formal equivalence is only formal and depends on how we choose to define words like “local”.

I don’t wish to impose a definition on anyone. Bell himself chose to define “local causality” in a particular way. I already suggested that it could better be called “classical local causality”.

If you have a new definition, I’d like to hear it!

Yours

Richard 

Sent from my iPad

On 22 May 2026, at 15:20, Ghenadie Mardari <gmar...@gmail.com> wrote:



[Richard Gill]

Dear Parker

So you are the owner of the formulation of the deeper physical sense of locality?

Richard

[Ghenadie Mardari]

Dear Richard,

 

Please allow me to explain how I see Bell's Theorem in its historical context. 

Hopefully, this will help you (and other group members) to understand my claims better.

In the early days of quantum mechanics, people believed that non-commutativity was weird.

In particular, it seemed unacceptable that properties like quantum momentum and quantum position should be mutually exclusive.

Perhaps, this was just a limitation of our means of observation?

So, there is a well known suite of events, culminating with the EPR paradox, where people strived to find a "quantum theory" without properties that are mutually exclusive in a fundamental sense.

In response to this trend, Bell showed the following:

It is impossible to reproduce the predictions of quantum theory with properties that are simultaneously real prior to measurement.

Any such theory would be unavoidably weird, because it would require the same "spooky action" that EPR wanted to avoid.

In short, Bell's inequality is not about quantum theory. 

It is about any possible "local realist" alternative.

In particular, non-commuting quantum variables are immune to this argument, because the inequality is only valid for jointly distributed properties. 

You see, it is perfectly natural for mutually exclusive properties to violate Bell-type inequality with local causality. 

The problem, coincidentally, was that non-commuting properties were full of surprises.

So, it was very difficult to explain how they could achieve such local correlations.

After two decades of working on this problem, I formulated a macroscopic reinterpretation of quantum mechanics. 

I am attaching a short document with a few basic explanations, as a primer on this new approach. 

Have a wonderful week-end, everyone.

(It's unusually cold and rainy here in Maryland).

Ghenadie.

[Mark Hadley]

Dear Ghenadie,

You say "It is impossible to reproduce the predictions of quantum theory with properties that are simultaneously real prior to measurement." No the properties don't need to exist prior to measurement. Bell says little or nothing about properties. It is about the outcomes of measurements. +1 or -1 

Bells theory is indeed not about quantum theory, it does not use quantum theory (except to require that the measurements in the same direction are perfectly anti-correlated.)

You say... "In particular, non-commuting quantum variables are immune to this argument, because the inequality is only valid for jointly distributed properties. " It's not clear what you mean by non commuting variables. Mathematically calars always commute. It certainly applies to spin in different directions - that is how it is built. So it applies to measurements which IF WE USE QUANTUM theory need use non commuting operators.

You say  " You see, it is perfectly natural for mutually exclusive properties to violate Bell-type inequality with local causality. " Thats just false.

Mark

[Richard Gill]

Dear Ghenadie

That depends on how you define “local causality”. I don’t know what your definition is.

If by definition non commutativity while preventing signalling is “locally causal” then you have not explained anything. You have just “explained it away” by using your own private language.

Here in Holland we have a heat wave and it is Whitsun, so we get a day free.

Richard

Sent from my iPhone

[Richard Gill]

Bell did not use that fact in his more mature works. Hence his switch to CHSH. He was already aware in 1964 that he needed to avoid that assumption. In experiment, you never see perfect correlations.


Sent from my iPhone

[Mark Hadley]

I distinguish between theoretical ideal experiments and more realistic experiments. The former is suitable and preferable for exploring foundation issues. It gives maximum clarity. Particularly the relation between CSHS and quantum THEORY. ( that's what interests me )

Quite separately CSHS can be tested without invoking quantum theory.  That's very important if you want to test if the world is classical or not. I don't need convincing: QM is tested a million times every day and is always correct.

what is often overlooked is that there is only one successful quantum theory. If we had to discern  between competing theories then experimental detail would be so much more important.

Mark

[Richard Gill]

In that context, M. Pawlowski’s discovery of the concept of “Information Causality” was a huge breakthrough. He showed that Tsirelson’s inequality followed from an intuïtive information theoretic (hence statistical) concept. Giving the hope that QM itself could be derived from intuitive statistical concepts.

Sent from my iPhone

On 23 May 2026, at 17:59, Mark Hadley <sunshine...@googlemail.com> wrote:



[Ghenadie Mardari]

Dear Mark, 

You wrote:

On Sat, May 23, 2026 at 9:57 AM Mark Hadley <sunshine...@googlemail.com> wrote:

(...) 

You say... "In particular, non-commuting quantum variables are immune to this argument, because the inequality is only valid for jointly distributed properties. " It's not clear what you mean by non commuting variables. Mathematically calars always commute. It certainly applies to spin in different directions - that is how it is built. So it applies to measurements which IF WE USE QUANTUM theory need use non commuting operators.

(...) 

Mark

The topic of conditional commutativity is fascinating. 

The rule of thumb is that any combination of observables (regardless of their objective status) become jointly distributed when included in a joint measurement. At first sight, this is highly intuitive, because joint measurements automatically produce joint distributions. 

Yet, this entails some wild consequences in the case of mutually exclusive properties.

In this case, "mutually exclusive" means that our properties cannot be observed at the same time in the same context, even in principle. 

So, the joint measurement is performed across contexts (for example, with two entangled quanta at different stations).

Let us consider three observables A, B and C. We use them to make three pairwise measurements (A,B), (B,C) and (C,A).

This means that A is jointly distributed with B, B with C, and C with A.

The question is, do we have a global joint distribution (A,B,C), given the closed chain from above?

In other words, can we accommodate the three pairwise distributions in a single probability space?

Intuitively, it seems that the answer should be "yes", but the correct answer is "No, not necessarily".

It is precisely this incompatibility between pairwise distributions that leads to quantum-like Bell violations. 

(This has important consequences for the interpretation of Kochen-Specker contextuality, but that is a topic for another day.)

Here is the real surprise.

Suppose that we make a joint measurement with all three observables at the same time.

We have the same physical conditions and the same measurement devices as above. 

The only difference is that now we measure three properties at a time, instead of two.

Well, this time we get a global joint distribution (A,B,C), in apparent contradiction with the behavior from the previous example.

So, Bell violations are suddenly impossible.

This is so weird that it seems classically impossible.

Nonetheless, this is a classical phenomenon.

You can read more about this in my paper on quantum monogamy.

Best wishes,

Ghenadie.

[Ghenadie Mardari]

Dear Richard,

Thank you for your question about mutually exclusive properties.

On Mon, May 25, 2026 at 11:06 AM Richard Gill <gill...@gmail.com> wrote:

Good, we have the same definition of local causality.

How do you define “mutually exclusive properties”? Bell does not assume quantum mechanics, not do I.

Bell showed that certain QM predictions are incompatible with classical local causality.

In general, a family of mutually exclusive variables is one that does not have a global joint distribution.

The nuance, as suggested in my reply to Mark, is that we can have experimental (“subjective”) joint distributions that do not necessarily reflect an “objective” joint distribution. This fundamental discrepancy between objective and subjective distributions (in experiments without loopholes) was unfortunately overlooked before.

You see, joint sampling automatically produces joint distributions.
Yet, we have a choice here. 

We can sample a single context (with many variables), or we can sample several contexts at the same time (one variable per context).
So, the question is: how do we know that our observed (subjective) joint distribution corresponds to an objective joint distribution (from a single context)? For  example, how can we tell if quantum momentum and quantum position are real at the same time, given that Alice-Bob measurements commute?
The answer is that we can perform a Bell test.

Imagine that we make three overlapping joint measurements in a closed chain, producing three joint distributions: (A,B), (B,C) and (C,A).
We ensure that we have pairwise consistency, so Locality is not a concern.

To clarify, pairwise consistency means that (A,B) and (B,C) agree about the values of the shared observable B, and so on. This is true at the level of single events (not just on average). In plain language, if the same exact value is used in two overlapping joint measurements, then “Alice’s outcome is the same no matter what Bob is doing”.

So, here is the heart of the problem.
When we have three overlapping joint measurements with pairwise consistency, we have two possible scenarios:
1. (A,B), (B,C) and (C,A) are subsets of a global joint distribution (A,B,C).
2. (A,B), (B,C) and (C,A) are incompatible and do not have a global joint distribution.

In both cases, we have Locality, because of pairwise consistency. Yet, the “Realism” condition is only satisfied in the first case. 

Scenario 2 is where we have mutually exclusive variables and Bell violations are possible.

To sum up, Bell showed that QM predictions are incompatible with "Local Realism", not Locality in general.

Does this make sense?
I am ready to elaborate as much as necessary.

Best wishes,
Ghenadie.

[Richard Gill]

I disagree. Pairwise consistency does not imply locality. One cannot use “locality” as a property of an unspecified experiment.

One can discuss whether or not certain observed correlations have a locally causal explanation. That depends not only on what is measured, but also on when and where it is measured.

The mathematical issues of whether or not matching marginal distributions can be patched together to form a joint distribution are questions of convex analysis, linear programming, etc. 

I think you need to study Judea Pearl’s book and understand the modern statistical concepts of causality. Statistical science has come a long, long way in the last 60 years. Bell had a deep understanding of statistical issues but the language of physicists of his time was not adequate for grasping the complexities of conditional probability.

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On 26 May 2026, at 17:38, Ghenadie Mardari <gmar...@gmail.com> wrote:



[Ghenadie Mardari]

Dear Richard,

I think that we are close to a breakthrough in our conversation.

You see, we are trying to answer a simple question:
Given two independent computers, can we achieve a Bell violation?

The answer depends on mathematical facts.
In this case, we need to ask:
Given a closed chain of overlapping measurements (A,B), (B,C) and (C,A) with pairwise consistency, do we necessarily have a global joint distribution (A,B,C)?

This is the same thing as asking if Bell’s inequality always applies when direct influences between events are ruled out.

If the answer was “YES”, you would have a winning bet.
Yet now we know that the correct answer is “NO”.
Specifically, not in the case of joint measurements across contexts.
Therefore, you have a losing bet.

(This is what "Realism or Locality" means: we can find Local solutions by revising the Realism criterion).

To answer your comment, pairwise consistency (at the event level) has long been accepted as a direct expression of Locality, and for obvious reasons.

(It literally embodies the notion that Alice’s event is the same, no matter what Bob is doing).

 
The issue is that people took it for granted (ever since Specker’s analysis), that joint distributions follow with necessity from it. 

In other words, it was presumed (but never rigorously proven) that pairwise consistency was both necessary and sufficient for global joint distributions.

It is this partial understanding that motivated the quantum Randi challenge, as well as your bet. 

Nonetheless, we have a different game now, because we have a deeper understanding of both the mathematical facts and the corresponding physical mechanisms.

Progress happens.

Best wishes,
Ghenadie.

[Richard Gill]

Dear Ghenadie

You are repeating yourself. And I think you are talking nonsense, sorry about that!

Richard

[Richard Gill]

Dear Ghenadie, dear all

I think that the journal Quantum Reports (MDPI) is scamming you (and all those here using the same journal: Bryan Sanctuary, for instance).  Your papers are published within a month of submission. The review process essentially checks if you, the author, have done the difficult typesetting work professionally, done the literature references professionally. The referees probably merely remind you also to cite themselves and one of their friends, preferably for other MDPI publications. There is no way a paper full of complex and innovative maths and physics and computer science and statistics actually gets competent referee reports in that time window. Really, it is just a preprint archive. The impact factor is 1.3 which basically means the author and one of their friends are the only people who look at it. I don’t say “read it”. It’s a fantastic business model for MDPI.

The editorial board is paid by being allowed to publish their own papers in the journal, with minimal review. They also get to publish “special issues” whereby more of their friends get fast easy relaxed publication.

So: you paid the APC? (Article publishing charge). It’s a scam. Most of the APC is profit for MDPI. Most of the real work done is fully automated. MDPI invested in good computer programmers, IT and social media specialists, lawyers, AI for rapidly identifying referees. Their investment is paying off. You should sue MDPI from fraudulously taking your money.

I suggest we write a bogus paper by AI, using a second AI to disguise the writing of the first. Let’s see if it is published in a month. We can get the APC refunded afterwards by crowd funding.

I admit that I participated in this myself for a while, joining the editorial board of Entropy, and getting half a dozen papers published for free. I think the editorial board is better, and the impact factor is better, too.

Well: Elseviers, Springer etc are all just as bad.

Richard

Sent from my iPad

[Ghenadie Mardari]

Dear Richard,

It is normal for new facts to appear puzzling, but they begin to make more sense as the connections are revealed.

We are trying to define the scope of Bell’s inequality.
Does it apply to every possible local phenomenon, or does it only apply to the narrower subclass of “local realism”?

Historically, the weight of the evidence seemed to be in favor of the first option.
Once we have locality, joint distributions appear unavoidable.
Ergo, Bell’s inequality should apply.

Yet, these arguments had a perpetual asterisk attached to them, because of the obvious contradiction with the theorems of Vorob’ev from 1962.
This was a genuine puzzle. How can so many theorems be correct, if they contradict known mathematical facts?

The answer was only made possible by the recent discovery of quantum monogamy.

It turns out that Bell violations are possible for pairwise measurements, but not for global measurements (even in quantum theory). For example, using four entangled quanta to measure four non-commuting variables does not lead to violations of the CHSH inequality, either in theory or in practice.
Though, even this discovery was interpreted as an example of “quantum weirdness”, because it did not seem numerically possible to reproduce it consistently.

Nonetheless, I was able to construct a Vorob’ev cycle on a Mobius strip (after a discussion with you, by the way), leading to the missing piece of the puzzle.

It turns out that the same flow of events can lead to several patterns of correlation, in appropriate conditions. If we consider two observables at a time, we can get Bell violations (despite pairwise consistency), but the same result is impossible with three or more simultaneous observations.

You see, it did not make sense to expect Bell violations in classical systems, even across contexts.

Who would have guessed that the same objective reality leads to different patterns of correlation, just because we measure two properties instead of four?

Yet now we know that this is possible and the apparent conflict between classical and quantum probability is resolved.

Please take a closer look at my monogamy paper for more details.

Best wishes,
Ghenadie.

[Richard Gill]

Dear Ghenadie

In my opinion, there never was a conflict. The idea that there was a conflict was based on misunderstanding of basic mathematics.

Richard

Sent from my iPad

[Bryan Sanctuary]

Dear Richarad,

I have published 7 papers with mdpi since 2024 and I have never paid a penny to them.  Also, I get two or three reviews which the editors push so things speed up.

Bryan

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[Richard Gill]

Ah ha, they believe that your work, Bryan, is so controversial, that people like me will rush to publish refutations. And even if I don’t have to pay either, the controversy attracts readers and debate!

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On 28 May 2026, at 13:58, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Mark Hadley]

exactly

That's why I am not inclined to publish anything relating to Bryan's work. 

it does not deserve attention. 

it only got attention because if the bet, I would never have read it otherwise. That why it's annoying and fraudulent or entitled that Bryan has not paid up.

Cheers

Mark

[Richard Gill]

If Bryan honours his debt to me by paying up promptly, I’ll gladly write and submit a critique of his newest work.

Sent from my iPad

On 28 May 2026, at 14:55, Mark Hadley <sunshine...@googlemail.com> wrote:



[Bryan Sanctuary]

Hi Richard,

If you want me to pay, then I guess the bet is still on?  Is it?  If so, wait until my double slit paper comes out, another proof that I won. I will also soon send you the program that gives the Bell violation with NO add/average issue, another proof that you lost.  You did not win, you just declared you did. 

But I do not want your money, unless you insist the bet is still on. Rather,  I want to see you accept that Bell does not account for coherence, only polariztion.

You have already stated that my new result falls into one of your neat Bell cases. and dismissed it.  Please be patient.

Thanks

Bryan

[Mark Hadley]

Dear Bryan,

you have already lost years ago. The bet has a time limit that has passed.

you seem to think the bet stands or falls on your publications and that the failure can be redeemed. It cannot it is too late. Pay up.

your bet was doubly foolish. I don't think you will be able to make a scientific contribution. But your biggest mistake was not scientific. it was a complete misunderstanding of the speed if scientific development. The greatest minds of all times could not convince the majority of their peers in such a short timescale. 

regardless of your mathematical errors, you lost the bet because scientific revolutions are slow. A period of decades not a couple of years.

pay up. And console yourself that you failed on history. 

Cheers

Mark 

[Richard Gill]

No Bryan

I’m referring to the bet you made and lost by not becoming world famous for having disproved Bell’s theorem in two years from the time you made that bet with me.

Yesterday I wrote  “If Bryan honours his debt to me by paying up promptly, I’ll gladly write and submit a critique of his newest work.”

Right now you are several years in arrears.

Richard

Sent from my iPhone

On 28 May 2026, at 20:01, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Ghenadie Mardari]

Dear Richard,

When two theorems are correct but still make opposite predictions, that is the definition of a conflict in my book.

Moreover, this is a well-known conflict, going beyond the theorems of Vorob’ev.

 
For example, the original CHSH paper explicitly clarified that “rho-lambda” must be uniform and independent of measurement settings. Later, Shimony, Horne and Clauser showed that Bell violations are local when hidden variables are specialized (different HVs for different measurements). This is why Bell had to clarify the implications for “free will” in this case. It was not possible to cover the full spectrum of Bell violations with the umbrella of non-locality.
Also, this is why we have a zoo of “local” (but arguably no less strange) proposals, including the many-worlds interpretation.

In short, the problem is real, and this is why my solution deserves attention.

I think it is helpful to review how this debate started, with the EPR argument.

In a nutshell, QM describes non-commuting properties (such as momentum and position) as mutually exclusive. The question is: do we have properties that are objectively incompatible? What if these properties exist at the same time, and we are just not able to detect them at the same time?

To address this issue, EPR introduced their Reality Criterion:
If we can predict a measurement outcome with certainty, then we are dealing with an objective property (or, as they put it, there is an “element of reality” associated with it).

When we have entangled quanta, Alice can measure the momentum of her quantum and use that to predict the momentum of Bob’s quantum with certainty.
Alternatively, Alice could measure the position of her quantum and make a similarly confident prediction about Bob’s outcome.
Does it follow that quantum momentum and quantum position exist at the same time? 

Not necessarily, because Alice can only make one measurement at a time.

Nonetheless, ERP declared (in the closing paragraphs) that mutual exclusivity is unreasonable in this case. 

(This is how non-locality entered the stage).
So, their suggestion was that quantum momentum and quantum position are objectively compatible with each other. They must be real at the same time, prior to observation.

The corollary of this argument is that we should be able to formulate a theory without mutually exclusive properties as an alternative to quantum mechanics.
Yet, Bell showed that any such theory cannot reproduce the predictions of quantum mechanics.

To sum up, “simultaneous realism” requires non-locality to explain quantum correlations (according to Bell), but quantum theory itself is seen as “non-local” for interpretive reasons.
These are two different problems, and they can be solved independently from each other.

In short, the conflict between “no-go” theorems and Vorob’ev theorems is the same as the conflict between “local realism” and “quantum theory”, and the same as the conflict between Bell’s inequality and super-determinism. It all boils down to the statistical difference between objectively simultaneous and objectively exclusive properties.

Accordingly, if we wanted to concede that there is no conflict here, we would have to ignore a lot of well-established facts. 

More importantly, we don’t need to do that because there is a straightforward way to solve the conflict.

Best wishes,
Ghenadie.

[Mark Hadley]

Dear Ghenadie,

There is no conflict.

There is an explanatory gap.

People want to add reasonable interpretations to QM. But every plausible addition has problems elsewhere.

The original EPR argument was that it was implausible that incompatible properties were not elements of reality.

The new EPR argument and Bells, is a theoretical result that those elements of reality, in the normally understood sense, are incompatible with QM.

When anyone tries to talk about or explain or interpret QM I ask one simple question: what is their mechanism for contextuality? If it is serious work, that should be clear from the outset. If not then they dont understand the issues.

MArk

[Richard Gill]

I don’t see a conflict. You write a lot of nonsense here.

Sent from my iPad

[Ghenadie Mardari]

Dear Mark,

The mechanism of contextuality is the most interesting part of my work.

My papers address different aspects of irreducible system-level effects, but you can also find a plain language summary attached to this message.

You see, the EPR conclusion about “unreasonable” exclusivity was based on classical rules of thumb about linear superposition. So, we can find an interpretive solution to that aspect of non-locality.

Yet, we also have a separate conflict between different mathematical arguments.

If we define locality in plain language as “Alice’s outcome is the same, no matter what Bob is doing”, then it seems that Bell violations are never locally possible. We can always derive the CHSH inequality (for example), if we consider a system of four variables.

On the other hand, we have numerous theories (e.g., many-worlds, super-determinism, retro-causality), that explicitly satisfy the stated definition of locality and reproduce quantum-like Bell violations. Please note: they do it at the event level, so this is not just “average” non-signaling behavior.

You might think that “exotic” interpretations do fancy cheating, but it is not so simple.

In the case of cyclic measurement scenarios, pairwise measurements produce Bell violations without statistical anomalies, according to Vorob’ev’s Theorem.

So, the conflict is real. Though, it would have been very hard to solve if we didn’t discover quantum monogamy.

The big surprise is that mutually exclusive properties have two main patterns of correlation associated with the same measurement procedures. 

If we make global measurements, we get joint distributions. 

Yet, if we make pairwise measurements, we get incompatible coefficients of correlation.

In short, the “no-go” theorems are correct, but their limited domain of validity constrains their interpretive implications. 

Surprisingly, Bell-type inequalities apply to every system, but not all the time.

For the purpose of this discussion, the important conclusion is that we can get Bell violations with independent computers, once we understand how to do it.

Have a great weekend!
Ghenadie.

[Mark Hadley]

I don't believe many worlds offers anything to the Bell argument. They are just another sort of hidden variable. Though the many worlds interfer with each other in exactly the way to create the results of QM. Many worlds that split and never interacted would be local hidden variable theories. 

retro causality manifestly gives a non local mechanism. That's the approach I am working on.

Mark 

[Ghenadie Mardari]

Dear Mark,

The definition of Locality is that “Alice’s outcome is the same no matter what Bob is doing, and vice versa”.

 
The main selling point of super-determinism and retro-causality is that they can satisfy this definition of Locality.

So, if you have a non-local model with retro-causality, I can think of two explanations:
1. You changed the definition of Locality.
2. You formulated a model that is nonlocal by design.

More importantly, it has recently been shown that we can formulate local realist alternatives for any self-consistent non-local theory.

By the way, it is Richard who introduced me to the work of Paul Raymond-Robichaud, a few years ago. 

Back then, he seemed very comfortable with the notion that Realism (not Locality) is the concept to be questioned in quantum mechanics.

So, I am a little bit surprised by his current commitment to non-locality.

Best wishes,
Ghenadie.

[Richard Gill]

Ghenadie, that definition only makes sense if we can define “the outcome that Bob would have seen, had Alice’s setting been different” (and vice versa).

In a world where physics follows deterministic laws, this does make some sense. It must be accepted that thought experiments in which physical parameters or properties of physical states are treated as free parameters are informative, and allow us to study causation.

You are using a naive and not generally accepted notion of “local”.


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