On contextuality and the nature of settings

[Leo]

Greetings. New to this group. I asked to subscribe here as it is one of the very few active places where one can discuss Bell and stuff without risks of bans and the like for unorthodox views; I had very bad experiences on the matter in Physicsforums for example. 

I would like to start from a conceptual issue I have with the usual claim that Bell violations force us to abandon "counterfactual definiteness" or EPR realism (tied deeply with contextuality), which to my understanding has been promoted very recently by Richard Gill here. Please forgive me is this is a mischaracterization.
QM clearly assigns predictions to all possible measurement contexts simultaneously. Given an initial state, I can ask:
1)what correlations would I obtain if I measured AB?
2) what correlations would I obtain if I measured A'B'?
and QM gives definite answers to both before any measurement is made. These are precise predicted ensemble correlations.
Now here is the issue. A correlation is not an abstract object floating in space, it is constituted by individual trial outcomes. So if QM predicts with certainty the ensemble behavior associated with a context, then the constituent possible outcomes making up that ensemble must themselves possess some kind of existence condition in the EPR sense.
Importantly, this does not require a global joint assignment across incompatible contexts. I am not assuming a single hidden-variable table simultaneously containing values for A,A',B,B'. I am only saying that the context AB has a well-defined ensemble prediction and that therefore the possible constituent events of the AB ensemble must be well-defined in some sense. Likewise independently for A'B'. The usual orthodox reply is that QM only defines context dependent probabilities, not pre-existing single outcomes. The wavefunction encodes rules for generating outcomes under measurement, but not an underlying catalogue of already existing values.
The counterargument is that this seems operationally unstable. Historically and operationally, QM was constructed precisely from observed event structures and correlations (Heisenberg's matrix mechanics being a prime example). Quantum states themselves are reconstructed tomographically from the statistics of possible measurements. So the state appears to be nothing over and above the structured set of possible outcomes it encodes.
The orthodox position therefore becomes thar ensemble counterfactuals are meaningful and well defined, but single counterfactual outcomes are not.
My objection is that the first seems to presuppose the second. If the ensemble prediction for a counterfactual context is physically meaningful, then the constituent possible events of that context seem to require at least EPR-style reality conditions.
Bell’s theorem appears in this sense asymmetric: Bell starts from single run counterfactual assignments and derives constraints on ensembles, while QM starts from ensemble structure and refuses to extend it to single run counterfactual assignments.
The key question is whether QM's refusal is genuinely coherent, or whether the existence of well defined counterfactual ensemble structure already implicitly commits us to the existence of the corresponding counterfactual single event structure.

The next question then would be, how can multiple counterfactual event structures simultaneously exist without belonging to some unified global event space? Quantum mechanics encodes many incompatible contexts at once within a single quantum state, each possessing meaningful ensemble predictions and constituent possible events. If these contextual structures are all simultaneously meaningful, then it becomes natural to suspect that they are different aspects or projections of one underlying reality, of which the Hilbert Space is merely an abstract representation choice (for this, i refer to Koopman and von Neumanm: they showed that the Hilbert Space structure is entirely classical, with the main difference between QM and classical mechanics being non-commuting observables). Yet Bell shows that quantum contexts are locally classical but globally non-Boolean. Each context supports a consistent event algebra internally, yet these local algebras cannot be globally glued into one ordinary Boolean space of joint events. We can't assign +1/-1 to all contexts as long as those +1/-1 all have the same "scalar meaning" if you will.
This is analogous to what happens in geometry of charts for curved spaces, where locally Euclidean coordinate patches may fail to assemble into one global Cartesian structure because the underlying manifold possesses nontrivial topology. You can't flatten a sphere into a flat surface.
Honestly we didn't even need Bell to find this out. The crucial operation that is used in QM to obtain the entangled states is a tensor product, not merely a scalar product like what is required of Bell's factorization condition. QM recognizes the geometric relationships between the states, Bell doesn't: he merely reduces the geometry to a variable dependence (the "a" in A(a,lambda)). This is exactly equivalent to flattening the sphere.

Hence, my proposal: settings coordinates (the entries a,b in Bell's functions) are functions themselves, that when applied globally to all directions give non-linear maps. Take a very simple linear setup: organize in excel a column of, say, 15 cells with entry +1, and then 15 cells with entry -1. Repeat this twice just to have recursion. These would be the outcomes of functions A and B in Bell's sense. Now take the product between any two cells distant up to 30 cells (d=30), and do so multiple times shifting each time the line of outcomes relative to the two chosen cells.
You'll quickly find that the correlation follows the following law:

corr = 1 - 2*d/30 (up to a multiplicative - sign if you invert the order outcomes in the line itself).

In the limit 30->2PI, and identifying the linear distance with an angle Theta, this is exactly the triangle function in Bell's toy model:

corr = -1 + theta/pi

Now, lets associate each such correlation with what would be required to get the one predicted by QM (again, up to sign). We construct a map d->alpha, such that:

-arccos(corr) = alpha

where alpha is the angle between any two settings in real experiments. Now, this might seem ad-hoc. We just inserted the observed correlation by hand. But my point is subtler: this procedure does not change the distribution of lambda in Bell's function. Instead, it creates a non-linear map of setting directions, such that

A(a,lambda) -> A(f(a), lambda).

Under this map to the linear space, the factorization of Bell holds. Crucially, this identification requires the notion of a global coordinate chart in the linear space.
I want to point the attention to another subtle issue: this map works only for distances up to d=30. Once we get to 31, the arccos function acts on a negative number. In the angle parametrization, (the identification with alpha) this means alpha only goes up to 180°. For larger angles, we must use another correlation function,

corr = 3 - theta/pi for 2pi<theta<4pi.

This map essentially flips the labels +1/-1 on the line.

Some may recognize this structure as a folding/quotient map, typical of double covers.
I don't want to discuss Joy Christian's work as a whole, as that already has been done to death here and other places. I myself have no qualms about the supposed validity of his framework; I also recognize clear blunders in some of his papers, like the "new octonion--like divisio algebra" that he incorrectly identifies as a 7-sphere, while it's actually a cone. But anyway. I want to point out that this folding map is the one to one the map that brings us from S3 -> RP3, as can be seen in equation (71) in his paper "Macroscopic Observability of Spinorial Sign Changes under 2π Rotations". The derivation of the geodesic distance is correct.

So, in summary, my proposal is that Bell considers directions as belonging to an euclidian space, while in reality their relationship reflects projective geometry. Correlations are linear in the lifted space, while they appear distorted under projection (hence the -cos dependence). Maybe someone already considered this here and has illuminating comments.

[Bryan Sanctuary]

Hi Leo,

Welcome to the group.  I think you will find it interesting and the discussions frank.  You will find a knowledgeable group with varied views.  You will not be banned because of your views. 

 I am afraid your intro to your views, however, is a bit too long for me, at least, to read.  Can you condense your ideas  into a couple of focussed paragraphs?  That would help me. 

I must clarify one point. The idea that Bell violations may require abandoning naive counterfactual definiteness or simple EPR realism, through contextuality or incompatibility of measurement domains, is actually my position which I have argued for over many years, and Gill has rejected.  Until our debate, Richard  strongly defended the traditional Bell interpretation against that view. So while I am pleased to see the discussion moving in this direction, the underlying conceptual shift is not really new to my work, but it is to Richard's.

You will find discussion of this here on the forum, and surround my draft (attached again) and BTW, Richard says he agrees with my theorem. I also attached a plot of two of my simulations: on the left, the older work, I took the quaternion expression for the correlation and projected out the vector part and the bivector part, simulated each, added them and got a -cosine like curve.

This has been severely criticized by this group, led by Gill

The right hand plot is new, and it is part of the draft I have not finished. It shows a process I call "contextual instantiation" due to spin being a quaternion. There can be no issue since nothing is added. I impose geometry on Bell's  Booliean outcomes. I hope to soon post, as Richard has requested, the details of the calculation and program. 

Anyway, I look forward to your comments and views,

Bryan

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[Leo]

Thank you Bryan. Yes, I indeed took that counterfactal position from one of the threads in which you recently participated. Anyway, here's the summary without the details, courtesy of chatGPT:

Bell’s theorem is often taken to force us to abandon “counterfactual definiteness” or EPR-style realism. But quantum mechanics itself already assigns definite predictions to all possible measurement contexts simultaneously. Given a quantum state, QM specifies what correlations would occur for measurements like AB or A'B' before any measurement is performed. My concern is that ensemble correlations are constituted by possible individual outcomes, so if counterfactual ensemble structures are physically meaningful, then the constituent possible events seem to require at least some contextual form of reality as well. Orthodox QM avoids global hidden-variable assignments across incompatible contexts, but it still treats the counterfactual ensemble structure as well-defined. The question is whether this is genuinely coherent, or whether meaningful ensemble counterfactuals already implicitly commit us to meaningful counterfactual event structure.

This leads naturally to the geometric side of Bell's theorem. Bell effectively treats measurement settings as ordinary Euclidean coordinates, reducing everything to scalar variable dependence A(a,λ). But quantum mechanics organizes states through tensor-product geometry, where contexts are locally classical yet globally non-Boolean. This resembles curved geometry: local coordinate patches may each look Euclidean while failing to assemble into one global Cartesian structure. My proposal is that Bell correlations arise because measurement directions are related projectively rather than linearly. In a lifted space the correlations are linear, but under projection they appear as the familiar −cos(θ) dependence. Formally, this can be modeled through a nonlinear mapping A(a,λ) → A(f(a),λ), preserving Bell factorization in the lifted space while distorting correlations in the projected one. The resulting structure resembles the double-cover relation S³ → RP³, where spinorial sign changes and projective geometry naturally emerge.

[Richard Gill]

Dear Leo, dear all,

Your point of view is interesting and I haven’t started to digest it yet! But I do have an immediate reaction to one of your statements.

Bell, 1964, started from a quite different point from where he’d arrived by 1990, please take a look at “La Nouvelle Cuisine”, Chapter 24 in “Speakable and Unspeakable” 2nd edition. He does not start from counterfactual assignments and EPR arguments any more. He starts from his own concept of ”local causality”. From this he derives a mathematical form of counterfactual definiteness; much weaker than the form derived in 1964 from Einstein’s (more precisely EPR’s) concept of “elements of reality”. There is no longer any idea that measurement reveals pre-existing properties.

Apart from this, I would add that QM declines to explain single run factual outcomes. Since QM provides no single run mechanistic/causal model for what actually happens, there is no way to ask “what would have happened, if this or that were different, but everything else unchanged”.

Fine’s theorem tells us that if no-signalling is true and if all CHSH inequalities hold, one can construct a causal model for a Bell experiment, in which the LHV lambda is the vector of binary counterfactual outcomes of each of the binary settings at the two measurement locations. This is a pure mathematical fact. It is not a metaphysical statement that (under those conditions) measurement merely reveals pre-existing values of certain physical variables. A perfect fair coin-toss is adequately modelled by a deterministic physical process in which outcome depends deterministically on initial configuration. But the outcome is not pre-existing and merely revealed by experiment.

Richard




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[Bryan Sanctuary]

Hi Richard,

Well indeed I am interested in your reply.

Now that you have shifted position from the usual Bell non-locality argument, I think we are much closer than we have been, especially regarding the limitations of a single global Boolean structure for incompatible contexts. However, the correlations do not arise purely from conditions on counterfactual structures.  I use the real transported geometric phase coherence (I am sure it is an example of Berry phase, but have not shown it yet), that only becomes visible statistically through accumulated detector-bin populations. It is not a click-by-click process that Bell, and you, envision.

Bell treated that microscopic click structure as fundamental, whereas the physically important phase manifestation appears only macroscopically through the ensemble buildup, much as in the Double Slit experiment. That is new.

I want to ask you, since you agree my Theorem is mathematically correct, what does that mean to you? To me, it shows that Bell is not applicable to Bivector spin, and the violation is due to counterfactual non-definability, not non-locality.

Bryan

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[Leo]

Please Bryan stay on topic. 

Dear Richard,

Thank you for the response. I think, however, that my concern lies somewhat orthogonal to the usual realism-vs-operationalism discussion.

Regarding Bell’s later notion of local causality in "La Nouvelle Cuisine" : I agree that Bell moved away from the stronger EPR-style language of "measurement revealing pre-existing properties". My point was not specifically tied to that stronger notion. Rather, it concerns the structure already implicit in the response functions themselves, i.e. expressions of the form A(a,lambda) and B(b,lambda). Whether motivated through EPR or through Bell's later locality principle, one is still introducing a jointly parameterized counterfactual structure across settings.

My question is whether the operational meaning already granted by QM to incompatible measurement contexts can really remain confined purely to ensembles without implicitly committing us to some deeper event structure. So when I say QM seems asymmetrical relative to Bell, I mean precisely that Bell starts from single-run structure and derives ensemble constraints, whereas QM starts from ensemble structure while refusing extension to single-run counterfactual structure. Simply restating that QM declines to explain single outcomes does not yet address whether that refusal is conceptually stable once the ensemble counterfactual structure itself is treated as physically meaningful.

Likewise, regarding Fine's theorem: I fully agree with the mathematical result. But my concern is not whether one can construct a standard Boolean hidden-variable model whenever CHSH holds. Rather, I am questioning whether Bell's parametrization already assumes a globally linearizable structure for settings themselves.

My suggestion is that the variables entering Bell's functions are not operational settings directly, but nonlinear images of them:
A(a,lambda) -> A(f(a),lambda).

In the lifted space defined by f(a), Bell factorization and CHSH hold perfectly normally; correlations there are linear. The nonclassical cosine correlations arise only after projection back to the operational setting space. So the issue is not that Bell’s theorem "fails upstairs", but that the projection destroys global joint assignability downstairs because the relevant topology is nontrivial (the analogy I have in mind is precisely the relation between S^3 and RP^3, where locally consistent structures fail to descend globally without sign ambiguities).

I want to make clear that the nonlinear map is not merely local coordinate relabeling; it changes the global identifiability structure of settings and therefore the existence of globally consistent assignments after projection. 

In summary, I made this post to know other's opinion on the non-linear map for the settings idea. I don't think I've seen this considered in other literature, aside indirectly from Christian. It looks like a genuine new consideration in Bell discussions, as the settings variables are usually taken for granted.

[Mark Hadley]

Bryan,

You continue to talk nonsense and show a misunderstanding of Bell.

Your work on bivectors is ill conceived and contains high school errors. You have been shown the errors, given one page proofs of your mistakes and had it explained to you.

Do you want to explain to newcomers the vet that you entered into, list and won't pay up.

Mark 

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[Parker Emmerson]

Richard Gill’s challenge is valid only for the scalar local-program class it defines; indeed, PV’s unconditional Rung-0 theorem already agrees that no model in that class can reproduce S=2*Sqrt(2) . The logical defect arises when the challenge is presented as an answer to PV’s separate unconditional microcausal theorem: that theorem concerns a non-Boolean event algebra yielding the singlet law, not a Gill-admissible program of local scalar outputs. Unless Gill proves that every PV microcausal model has a faithful reduction to his program class, demanding that PV “win the challenge” is a type error: it tests a substituted Boolean/scalar object rather than the event-theoretic object actually claimed.

Richard may be treating one executable representation of Bell’s no-go argument as though it settled whether a differently formulated, unconditional local-event theorem answers Bell’s foundational concern.

This is why, although Bryan’s math is insufficient, his skepticism is not entirely unfounded, and more advanced mathematics may indeed demonstrate Richard’s challenge may be logically non-responsive to the unconditional PV theorem from the start.

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[Leo]

Can we please keep this on topic. Make your own threads if you want to discuss other things.

[Leo]

Sorry everyone, it seems as though some replies are not veing shown on the page of google threads to me. I just noticed Bryan sent a reply that did not appear here. Im new to groups so i assumed it worked like forum threads.

[Bryan Sanctuary]

Hi

I am not sure who wrote this: "This is why, although Bryan’s math is insufficient, his skepticism is not entirely unfounded, and more advanced mathematics may indeed demonstrate Richard’s challenge may be logically non-responsive to the unconditional PV theorem from the start."

but I would welcome to know why my math is insufficient? I would like to know your reason is all,

Thank

Bryan

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[Mark Hadley]

I didn't write it but we all know that your maths is inadequate. 

You can't combine correlations (averages) you can't do the maths, you can't understand the mistake, you can't understand the proofs shown to you. it's high school maths and beyond you.

And that's just the start 

[Bryan Sanctuary]

Dear Mark,

Thanks for the clarification.  You mix up my ignorance with academic disagreement.  You simply  have not convinced me. I have answered that you must include spin complementary, but it seems to end there. 

Bryan

[Mark Hadley]

Adding two correlation coefficients is just algebra. I've showed you the correct algebra. from the definition of a correlation coefficients, using high school maths, to get the result. it's the definitions and rules of arithmetic. saying "spin complementary" does not change the rules of addition and multiplication. it it does not make a wrong answer right.

[Bryan Sanctuary]

Mark,

I did not add two correlations.  I took the full correlation and projected out the vector and bivector components, simulated each, and then added back the two parts. You and others objected, so I did it your way, and did not add. As I said, that will soon come along with the program.

My response to your objection seems not to resonate with you.  I have said that two complementary components of  a variable do not form a convex set. You disagree. So leave it at that

Bryan

[Richard Gill]

Dear all

Parker wrote “more advanced mathematics may indeed demonstrate Richard’s challenge may be logically non-responsive to the unconditional PV theorem”. He appears not to realise that “Richard’s challenge” is a practical joke. It’s a pedagogical tool to help those challenged by high school algebra to discover the CHSH inequality for themselves. See the final section “Quantum Randi Challenges” of my well-cited 2014 paper “Statistics, Causality and Bell's Theorem”, Statistical Science, Vol. 29, No. 4, 512-528. https://arxiv.org/abs/1207.5103

I’m challenged by Parker’s algebraic geometry. I don’t understand a word he writes on “phenomenological velocity”. I suspect nobody on the forum does, and nobody dares to admit it. For all I know, it may be a brilliant practical joke too, cf. the Sokal hoax, https://en.wikipedia.org/wiki/Sokal_affair

Richard

Sent from my iPad

On 24 May 2026, at 19:14, Parker Emmerson <powerin...@gmail.com> wrote:



Richard Gill’s challenge is valid only for the scalar local-program class it defines; indeed, PV’s unconditional Rung-0 theorem already agrees that no model in that class can reproduce S=2*Sqrt(2) . The logical defect arises when the challenge is presented as an answer to PV’s separate unconditional microcausal theorem: that theorem concerns a non-Boolean event algebra yielding the singlet law, not a Gill-admissible program of local scalar outputs. Unless Gill proves that every PV microcausal model has a faithful reduction to his program class, demanding that PV “win the challenge” is a type error: it tests a substituted Boolean/scalar object rather than the event-theoretic object actually claimed.

Richard may be treating one executable representation of Bell’s no-go argument as though it settled whether a differently formulated, unconditional local-event theorem answers Bell’s foundational concern.

This is why, although Bryan’s math is insufficient, his skepticism is not entirely unfounded, and more advanced mathematics may indeed demonstrate Richard’s challenge may be logically non-responsive to the unconditional PV theorem from the start.

On Sunday, May 24, 2026, 'Mark Hadley' via Bell inequalities and quantum foundations <Bell_quantum...@googlegroups.com> wrote:

Bryan,

You continue to talk nonsense and show a misunderstanding of Bell.

Your work on bivectors is ill conceived and contains high school errors. You have been shown the errors, given one page proofs of your mistakes and had it explained to you.

Do you want to explain to newcomers the bet that you entered into, list and won't pay up.

Mark 

[Richard Gill]

Dear Bryan

Your theorem does not change anything for me. I’ve written out what I presently think about Bell’s work in Section 5.1 of a recent preprint.

https://arxiv.org/abs/2605.13154

Three ways to find comfort with the Bell proof and the results of the Bell experiments

Richard D Gill, Inge S. Helland, Bart Jongejan

Already, the abstract states “Gill accepts irreducible and non-local quantum randomness and finds the choice between locality and realism a false dichotomy. In his later works, Bell derives counterfactual definiteness from classical local causality, and that is what has to go. The metaphysical concepts "realism", "locality", "causality" need to be reconsidered”.

In Section 5.1, I write “ Bell’s theorem, together with its experimental confirmations, tells us that there are events in the real world which are uncertain and which have well-defined probabilities, but for which there does not exist a local, classical mechanism that “explains” why one outcome occurred rather than another. To my mind this suggests that Aristotle was wrong: some events do not have causes in the sense he had in mind. Irreducible randomness is part of the basement-level fabric of reality, and determinism has its limits. And moreover, irreducible randomness can be “non-local”.

Richard

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On 24 May 2026, at 17:31, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Richard Gill]

Dear Leo

The way I see it, the variables “a” and “b” entering Bell’s functions are labels. The actual physics inside the measurement devices are incorporated in “lambda”. In a CHSH experiment, Alice sets a toggle switch on the outside of her measurement apparatus to “setting 1” or “setting 2”. Inside the apparatus is where the setting is physically implemented.

Same thing with an experiment with settings, 0, 1, … 359 degrees. The numbers are marked on a dial. Alice turns the dial. Inside the machine, the setting is converted by electronics into an electromagnetic field operating for some time interval on a photon which one imagines arriving just then.

Please see this preprint:

https://arxiv.org/abs/2211.05569

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

Richard D. Gill

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. The word "local" applies to the way that the chosen settings influence the observed outcomes. The case of contextual setting-dependent hidden variables (thought of as being located in the measurement devices and dependent on the measurement settings) is automatically covered, despite recent claims that Bell's conclusions can be circumvented in this way.

Richard

Sent from my iPad

[Richard Gill]

Unfortunately, Google does not maintain Google groups in any serious way. It is slowly falling into disrepair. The correspondence between what appears in your email inbox and what Google’s web interface says is often out of date if not plain wrong.

Sent from my iPad

On 24 May 2026, at 19:25, Leo <leo_...@hotmail.it> wrote:

Sorry everyone, it seems as though some replies are not veing shown on the page of google threads to me. I just noticed Bryan sent a reply that did not appear here. Im new to groups so i assumed it worked like forum threads.

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[Leonardo Verzegnassi]

Dear Richard,

please read carefully my original post. I specifically mention that the idea is not merely a relabeling. The function f(a) in A(f(a), lambda) would take as input a direction, which can indeed be relabeled as you wish together with Alice's dial, and spits out a "different" direction in very simple terms, which then enters the measurement function to sput out an outcome. Here's a pedagogical example:

Imagine alice uses direction a, and bob direction b. For simplicity we assume f(b) = b. Alice generates her Bell function as A(a,lambda), calculates the resulting correlation, and finds that this is incompatible with say A(a', lambda) in Bell's sense (she can't find the joint), i.e. she gets a cosine correlation. But unbeknownst to her, the function she should have used has the form A(f(a), lambda), with say f(a)=a". Turns out, this modified function actually has a joint with b and a', because the relationship between the directions a'/b with a" is different than that with a. To simplify even more, the real theta entering correlations is not the one oberved by Alice and Bob as ab, but rather a"b.

Inviato da Outlook per Android

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From: Richard Gill <gill...@gmail.com>
Sent: Monday, May 25, 2026 7:45:33 AM
To: Leo <leo_...@hotmail.it>
Cc: bell_quantum...@googlegroups.com <bell_quantum...@googlegroups.com>
Subject: Re: [Bell_quantum_foundations] On contextuality and the nature of settings

 

[Richard Gill]

Dear Leo,

Please read my preprint carefully. I do not talk about a “mere” relabelling.

The physics is all inside “lambda”. And it comes from inside the measurement devices, not just the source.

Richard 

Sent from my iPad

On 25 May 2026, at 08:00, Leonardo Verzegnassi <leo_...@hotmail.it> wrote:



[Leonardo Verzegnassi]

Dear Richard,

I've read your preprint. It does not address my point. 

Lambda is supposed to be a "common cause" variable shared in the past light cone common to Alice and Bob. My function of settings is nothing of the sort. It's a nonlinear remapping of what the settings actually mean. 

In your paper you use binary settings with a lever that can go only up or down. Aside from the fact that such a setup would not violate Bell's inequalities unless you introduce some sort of directional relationship (as it is, it is equivalent to a pair of parallel polarizers), my point is that what you call setting a maps to a different setting a' in another space, in a non linear way.

Im not sure how to explain this in a way that is readily comprehensible, but I'll try.

Take a simple linear path of a person walking. 

We call setting "a" his initial position, setting "b" his position at d=10 m, and setting "c" his position at d=20 m. This is a linear map from setting label to distance d. 

Now, lets apply the non linear map to all three settings. This map (chart b) does:

a = 0 -> a' =0 (leaves a untouched in the new chart)

b = 10 -> b' = 20

c = 20 -> c' = 80

Now the new distance d between settings is not the same as that in the linear map. 

My suggestion is that in experiments we are using chart b as it were chart a. Chart a obeys bell  inqualities. Chart b doesn't. The joint distribution belongs only to chart a, it is distorted in chart b so as to appear contextual. Of cpurse this simple example does not capture the lack of a global section typical for projections of non trivial bundles. 

Inviato da Outlook per Android

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From: Richard Gill <gill...@gmail.com>
Sent: Monday, May 25, 2026 8:18:14 AM
To: Leonardo Verzegnassi <leo_...@hotmail.it>
Cc: Bell_quantum...@googlegroups.com <Bell_quantum...@googlegroups.com>

[Mark Hadley]

utter nonsense Leo.

lambda are any parameters used to determine the +/- outcome. any at all so long as Alice one dies not depend on Bobs settings. That's any parameters at all.

a is the setting at Alice has. This is unambiguous for an ideal experiment and you can assume it is calibrated for an actual experiments.

The whole calculation of +1 or -1 is in A. Bell does not care what the function is. You can operate in a before you use it, but conceptually A is still a function a and some variables lambda.

Bell allows any functions or projections to be part of the calculation. that can't change BI.

one small point where I disagree with Richard is the I'd say the physics is in the function A.

Mark

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[Mark Hadley]

I'm talking about your seminal work. The paper that YOU said should be used to evaluate your work. The infamous equation 7 I think it was. which several of us independently spotted the mistake in.

You definitely added two correlation instead of taking the weighted average.

Mark

[Richard Gill]

Lambda describes the state of all the relevant physical quantities in the long rectangular box extending from Alice to Bob. It includes all the stuff which the physicists have built which lie between the inputs and outputs. 

Sent from my iPad

[Mark Hadley]

I don't see it that way. For example DeBroglie Bohm pilot wave theory, lambda is the particle position. The function A uses the wave to describe which outcome results from any particular location. lambda =x is a simple variable which determines the relevant trajectory that is followed. The twisting and turning if the trajectory is in the function A.

I don't think it's a big point. 

Cheers

Mark

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[Richard Gill]

Mark is right. The physical mechanisms would indeed be expressed in the function A.

The physical variables (properties of physical stuff, from everywhere in the experiment) are in lambda.

“lambda” is not local. The functions A and B are local. Your encodings can be part of A and B.

Bell explains it best, I think, in “La Nouvelle Cuisine”.

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On 25 May 2026, at 09:29, Mark Hadley <sunshine...@googlemail.com> wrote:



[Richard Gill]

De Broglie Bohm for EPR-B, two particles, cannot be written within Bell’s framework. I think Bell would agree. 

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On 25 May 2026, at 09:52, Mark Hadley <sunshine...@googlemail.com> wrote:



[Leonardo Verzegnassi]

Mark, my point is not a nonlinear tweak inside A(a,lambda). It is more like a projective distortion of coordinates: the Bell-classical structure lives in a lifted space, while the experimental settings a,b are only the projected coordinates. Just as distances can look different after projection a relation that is linear upstairs can appear nonlinear downstairs. So Bell's inequality can hold in the lifted space even though the projected operational variables show the usual quantum correlations. The issue is not the shape of A but that a and b are not globally faithful coordinates of the space where the joint assignment lives. Locally there is not way to tell, because each setting is a local fixed point in the projection. A bit like a raisin in an expanding loaf.

Also, historically Bell distinguishes sharply between a,b and lambda: lambda represents variables associated with the shared past light cone/common cause structure, whereas a,b are the freely chosen local settings. My proposal is not about smuggling extra structure into lambda, but about reconsidering what the setting variables themselves actually parametrize.

Inviato da Outlook per Android

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From: Richard Gill <gill...@gmail.com>
Sent: Monday, May 25, 2026 9:59:22 AM
To: Mark Hadley <sunshine...@googlemail.com>
Cc: Leonardo Verzegnassi <leo_...@hotmail.it>; bell_quantum...@googlegroups.com <bell_quantum...@googlegroups.com>

[Mark Hadley]

it's a non local hidden variable theory. The wave and hence A depends on the whole apparatus including a and b settings.

[Richard Gill]

Dear Leo

Unfortunately, Bell’s theorem shows you are wrong. How Bell’s theorem has historically been interpreted is irrelevant. 

Maybe you would like to show that you are right with a computer simulation?

Richard 

Sent from my iPad

On 25 May 2026, at 10:03, Leonardo Verzegnassi <leo_...@hotmail.it> wrote:



[Leo]

Here's a more relevant example, directly tied to the EPR setup. 

In the lab frame, say Alice has angle 0° with direction a and Bob has angle 45° with direction b.
The quantum correlation would be 

corr: -cos(ab) = -0.707

Bell's linear model predicts:

corr: -1 + ab/pi = -0.5

But now suppose the operational settings are projected coordinates of another space. In the lifted space we choose settings a',b' such that their linear distance is not 45°, but instead satisfies:

- 1 + a'b'/pi = -0.707

which gives approximately a'b' = 52.7°.
So the map becomes
a=0° -> a'=0°
b=45° -> b'=52.7°

So downstairs the experiment sees 45°, but upstairs the Bell-classical structure lives at an effective separation of about 52.7°. The point is that the operational angle and the distance governing the hidden-variable correlations need not be the same object if the operational settings are projections of a different global geometry.  

[Leo]

Dear Richard, 
I don't intend to be adversarial. My historical comment was uniquely a reply to your and others statements about lambda. 
Saying "Bell's theorem shows that I'm wrong" is neither an honest reply from engagement with my argument nor an actual argument.

[Richard Gill]

Well then please try to rise to my challenge and implement your ideas in a computer program. I think you don’t realise the enormous flexibility which Bell’s formulation allows the “user”.

If you think it can be done, then do it. 

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On 25 May 2026, at 10:28, Leo <leo_...@hotmail.it> wrote:

Dear Richard, 

To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/fd00de59-78e0-4eae-994c-1fce8efc72a1n%40googlegroups.com.

[Richard Gill]

Dear Leo

Sorry, I should have written “Bell’s theorem tells *me* you are wrong”. My actual argument was that your mappings from labels to physical angles can be put into the functions A and or B.

I wonder how you are going to do that when the labels “a” and “b” which will be used in any one run are not known in advance.

The simple maths of CHSH allows enormous flexibility in the interpretation and specification of the domain of the functions A_1, A_2, B_1, B_2

I’m writing A_a for the function A( \cdot, “a”) etc.

Richard

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On 25 May 2026, at 10:28, Leo <leo_...@hotmail.it> wrote:

Dear Richard, 

To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/fd00de59-78e0-4eae-994c-1fce8efc72a1n%40googlegroups.com.

[Richard Gill]

Dear Leo

This looks like Bryan’s approach: pairs of settings close to equal or close to opposite need to be stretched apart, pairs of settings close to orthogonal need to be pushed closer. We want to get rid of the pointy-ness of the triangle wave (zigzag, sawtooth) so we have to make it cross the midline (correlation zero) faster.

You can’t achieve that by operating on the mappings from setting labels to setting angles separately.

As Bryan says you need to collect all the data from both wings of the experiment together first.

This is a lesser-known loophole: the loophole of not analysing the data in the normal way. Redefine “correlation”.  The “move the goalposts” loophole.

Richard

Sent from my iPad

On 25 May 2026, at 10:24, Leo <leo_...@hotmail.it> wrote:

Here's a more relevant example, directly tied to the EPR setup. 

To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/5fcbbde3-d509-4010-a2fc-440e9fe7aee4n%40googlegroups.com.

[Mark Hadley]

Leo,

You are wrong. Bell does not predict the outcome. 

Bell requires results of +1 or-1

Bell requires them to be anticorelated if a and b are equal.

Those two conditions are the set up. Bell does not give a formula for the correlation only an inequality for a combination of correlations.

To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/5fcbbde3-d509-4010-a2fc-440e9fe7aee4n%40googlegroups.com.

[Richard Gill]

Dear Leo, Mark, Bryan

Here’s a simulation of Bell’s simple LHV and an example of post-processing the observed correlation so as to get the negative cosine. My guess is that Bryan is also doing something like this.

I suggest calling it the  “moving the goalposts loophole” by using a private definition of “correlation”.

By Bell’s theorem, there’s no way I could redefine the functions A and B or the hidden variable lambda to get the negative cosine.

[another way is of course using the detection loophole - discard suitably post-selected trials so as to massage the correlations in a desired direction. We statisticians know all about how to lie with statistics]

Yours

Richard

[Leonardo Verzegnassi]

Mark,

I don't understand your reply. It seems completelt disconnected from what i stated.

Richard,

I am mostly unaware of what Bryan is trying to do. I recognize the issue that the simple remapping i presented requires a fixed notion of one of the two detectors as the "0". Bob requires alice's direction to map his own direction consistently. I am unaware though of whether all mappings of similar conception are ruled out in a Bell-theoretic sense. Usual derivations on upper bounds like CHSH don't delve into such procedures. 

Anyway, I also want to remind everyone of the first part of my post, concerning the conceptual position on counterfactuals.

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From: Richard Gill <gill...@gmail.com>
Sent: Monday, May 25, 2026 11:51:20 AM
To: Mark Hadley <sunshine...@googlemail.com>; Bryan C. Sanctuary <Bryan.s...@mcgill.ca>; Leo <leo_...@hotmail.it>; Bell inequalities and quantum foundations <bell_quantum...@googlegroups.com>

Subject: Re: [Bell_quantum_foundations] On contextuality and the nature of settings

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[Leo]

Mark, 
I understand now your reply, for some reason it appeared to me you replied to a different post. 
The function  corr: -1 + ab/pi appears in one of Bell's toy models, the one with the sign function. It is the triangle shaped curve. I used it as an example.
Also, you probably know that is not all that's required by Bell. Bell requires a specific product form as a causality condition, an independence from settings of lambda and the notion that counterfactuals are meaningful. The triangle shape come out as a specific, intuitive choice of the measurement functions.

[Richard Gill]

Bell does not exclude any remapping of Alice’s inputs, nor does it exclude any remapping of Bob’s inputs.

But a remapping of Alice’s which depends on Bob’s actual input is obviously excluded, and vice versa.

If you move close inputs even closer to one another, and nearly opposite inputs to be even more opposite, you’ll be forced to move close to perpendicular inputs away from perpendicular. You can thereby fake the quantum correlations. You need to make the pointy parts of the saw-tooth more rounded, so you have to move stuff away from the part of the saw-tooth where the correlation is close to zero, 

There are theorems which give the minimal amount of data points which need to be moved to hit the target correlation.

Can you read my the two documents I just posted? A little situation experiment and the formulas written out in LaTeX instead of R code.

[Richard Gill]

Dear Leo

Bell (in his mature works) assumes what he calls local causality together with “no-conspiracy”. I think you need to study modern approaches to causality such as the famous book by Judea Pearl. You could start with my preprint

https://arxiv.org/abs/2211.05569

Feel free to ask me any questions you like.

Bell did not assume that counterfactuals are meaningful, whatever that means.

Richard

To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/ed3ba3ae-1df8-4c0c-be3e-ed32ab07123cn%40googlegroups.com.

[Mark Hadley]

I suggest you look at modern derivations of CSHS for ideal experiments. Those treatments have the greatest clarity and show clearly how little is assumed.

The example you quote us not of any fundamental significance. it's not part of CSHS derivation.

It requires the A functions to exist. which is just saying the individual outcomes could be predicted.

Mark

To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/ed3ba3ae-1df8-4c0c-be3e-ed32ab07123cn%40googlegroups.com.

[Leo]

Richard, 

by counterfactual are meaningful I mean that there exists a joint for all possible settings, P(A,A',B,B'). This is a core assumption of Bell, and it is essentially a modern version of EPR realism.
No conspiracy is a separate assumption, which I stated as " an independence from settings of lambda". 
I will read up on Pearl. I must mention that I have Bell's book "Speakable and Unspeakable" and that I have read it.
I still remind everyone about the first part of my post, where I asked essentially about the philosophy of counterfactuals. If anyone has anything to say about that, please do.

[Mark Hadley]

Dear Leo,

I have never seen  P(A,A',B,B'). defined or used. It is not part of the CSHS derivation.

MArk

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[Austin Fearnley]

Hi Leo

" ..... I asked essentially about the philosophy of counterfactuals. If anyone has anything to say about that, please do."

I am an amateur physicist, with a mathematics/statistics background, but have simple thoughts about counterfactuals.
Many years ago Richard explained them to me online: aka "dragged me through a hedge backwards"!  Not sure I understood it all.

I have followed Bell and CHSH derivations and agree with them but have no inclination towards really understanding them in depth.
I am much more interested in entanglement, which is not the only feature of Bell, and have worked by making computer simulations of entanglements using Excel VB.

Counterfactuals clearly do not work with QM. If they did then the quantum Randi challenge would not be effective.
In my first simulations I had an entangled particle pair with polarisation vectors p and p'.
Alice measures a particular p and gets a particular measurement A.  If Alice were subsequently to measure a second particle, also with polarisation vector p, then she would be very likely not to get the same outcome measurement.  Only if Alice's measuring device was by chance also set to be aligned with p would she get the same measurement outcome.

IMO a polarisation vector is like a quantum state, say |up>.  It is not a vector pointing up.  It is certainly not pointing down.  And 'on average' it does point up. So you might expect some positive correlation in a long run of pairs.

[Bart Jongejan]

Dear Leo, dear all,

I am also new to this group.

I want to react to Leo's contribution by cherry-picking three passages.

(1)
> [...] abandon "counterfactual definiteness" or EPR realism (tied deeply with contextuality), which to my understanding has been promoted very recently by Richard Gill here

In https://arxiv.org/abs/2605.13154 Richard, Inge S. Helland and I agree on the mathematical correctness of Bell's theorem and we agree on "reality is what is real". From that point our views diverge. Of us three, I am the one who maintains that Nature might evolve deterministically.

(2)
> -arccos(corr) = alpha
> where alpha is the angle between any two settings in real experiments. Now, this might seem ad-hoc.

In section 5.2 in the aforementioned preprint, I propose a simple HV theory  that arrives at that equation in a principled way. The theory can be used to explain the results of a Bell trial in terms of a HV, but cannot predict them. The software at https://github.com/BartJongejan/simulated-Bell-data  demonstrates the theory. The program spits out simulated Bell trials, two settings and two values per trial. You can ask it to create a simulated Bell experiment consisting of tens of millions of such trials. One can then investigate whether the CHSH inequality is violated and by how much. And of course one can look for statistics that raise suspicions about the locality of Alice's and Bob's measurements.

One can arrive at the transformation -arccos(corr) = alpha by reasoning about large numbers of isotropically distributed settings for Alice's and Bob's apparatuses. This transformation is only valid if there are three spatial dimensions. The degree to which the CHSH inequality is violated is a function of the number of spatial dimensions, according to the proposed theory.

Richard maintains that my theory is not a physical theory, and I think he is right. That brings me to another passage in your post.

(3) > This is analogous to what happens in geometry of charts for curved spaces, where locally Euclidean coordinate patches may fail to assemble into one global Cartesian structure because the underlying manifold possesses nontrivial topology. You can't flatten a sphere into a flat surface.

I agree.
I think that a physical theory of hidden variables has to state that particle spin is curved space-time and that our classical Euclidean perception of space-time is emergent. About 25 years ago I have posted two speculative papers (not peer reviewed) about such a theory. They are on arXiv. That theory proposes an extremely simple curved space-time as a model for a spinning particle. The metric attributes some nice properties to geodesics that are in line with the theory I mentioned above, but the metric has also very serious problems.

[Leo]

Dear Richard,

Unfortunately I deleted your reply by accident, so this is going to be a reply to your last email.

I disagree with your claim that counterfactual definiteness is not doing additional work beyond local causality.

Local causality, as used in Bell’s derivation, is a condition on the probabilities of actually realized outcomes for a given pair of setting.

This alone only constrains correlations for the chosen settings. It does not, by itself, define or require outcomes for alternative, unperformed settings. I do recognize that often in Bell discussions the two things are conflated.

The joint object P(A,A',B,B'), or equivalently simultaneous values for all settings, is not contained in local causality as such. It arises only when one additionally assumes that the same lambda determines responses for all possible settings within a single probability space, i.e. that counterfactual outcomes are jointly well-defined given lambda.

So I do not agree that local causality already contains counterfactual definiteness. To me the latter enters when one extends the model from actual settings to a global assignment over all possible settings.

[Leo]

Dear Bart,

I read your paper with much interest. I think your construction puts on concrete ground what the idea of lacking well defined counterfactuals means. But, and on this I have to agree with Richard's comment in the same paper, the distribution of outcomes seems dependent on both alice's and bob's settings. I didn't quite follow entirely the part where you talk about graphs and weights, perhaps because it's now late in the day for me, so maybe there's a nuance here I haven't understood.

[Richard Gill]

Dear Leo

I disagree.

“For a given pair” means “for any (or all) given pairs”. To do it formally one would scatter the text with quantifiers “for all” and “there exists”.

The pair a, b (and the further background variable c) are mathematical variables. The model is supposed to hold for any a, b, c. For all a, b, c, the probability distribution of lambda given a, b and c is supposed not to depend on “a” and “b”.

The model is a statistical model. A model of the joint probability distribution of A and B given a, b, and c. Where a, b, c are parameters, variables. The assumed statistical conditional independencies induce constraints on the *family* of “observed” statistical distributions of A and B given “a” and “b” and further background variables  “c”.

Richard

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On 25 May 2026, at 19:03, Leo <leo_...@hotmail.it> wrote:

Dear Richard,

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[Richard Gill]

The joint distribution of the pair of outcomes does depend on the pair of settings. Their correlation depends on the pair.

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On 25 May 2026, at 21:26, Leo <leo_...@hotmail.it> wrote:

Dear Bart,

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[Bart Jongejan]

Dear Leo

Thank you for having read my section 5.3 in Richard's, Inge's and my paper. Comments are so welcome! 

The graphs and weights part is central to my model. That, and the observation that Bell's thought experiment can be improved (in my opinion) by considering not just four settings A, A', B, and B', but large numbers of randomly created settings that Alice and Bob can choose between. The improvement is in not having to be worried about whether Alice (or rather Alice's copy of the hidden variable) already knows too much locality-wise about Bob's setting if she knows for sure that Bob's setting is parallel to the plane in to which her own settings A and A' are constrained. And vice versa from Bob's point of view. By isotropically distributing large numbers of potential settings over all available spatial directions, Alice's and Bob's copies of the HV have zero information about the other party's actual setting. One could say that my model rests on the removal of a loophole.