Joint Definability and the SM vs. BiSM

[Bryan Sanctuary]

Hello All

A glitch at MDPI has put up an earlier version of my new paper, which will be corrected.  

However, the present discussion on the Forum might find some answers, and questions,  in the attached pdf of the paper. I put in a TOC so you can get a  quick overview.  

I hope this can generate some debate.

Bryan

[Richard Gill]

Bryan, in the abstract you write: “Because spin observables are not jointly definable on a single Kolmogorov probability space, the CHSH quadruple cannot be formed and Bell’s theorem is therefore not applicable”

1. Bell’s theorem (refined version with CHSH, see his later publications) is not about “observables”. It’s about measurement outcomes. About mathematical probability models for statistical data.

2. The single probability space you mention is derived by Bell from the physical concept of local causality.

So I take it you believe that local causality is not true.

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<quantumrepTOC-4537008.pdf>

[Richard Gill]

Dear all

Bryan writes in his

Assumption 1 (Bell–CHSH Structural Assumptions)  The derivation of the CHSH inequality rests on the following premises: 1) Locality, 2) Outcome definiteness, 3) Joint definability.

He omits the premise of statistical independence. And he omits the fact that Bell derived CFD from the physical assumption of local causality.

Premise 2, Outcome definiteness, is according to him "For each hidden parameter λ, the outcomes A(a, λ), B(b, λ) are well-defined elements of {±1}”

But that implies his third premise Joint Definability. "The four counterfactual variable A(a, λ), A(a′, λ), B(b, λ), B(b′, λ) are simultaneously defined measurable functions on a single Kolmogorov probability space (Λ, ρ)".

Premise 1 is Locality. "The outcome at one wing is independent of the distant setting”. 

Maybe he means statistically independent? He has mathematical (functional)  independence by Premise 2, existence of A(a, λ), B(b, λ) 

Richard

On 23 Sep 2026, at 19:57, Bryan Sanctuary <bryancs...@gmail.com> wrote:

[Richard Gill]

Unfortunately the supplementary material for Bryan’s new paper is also not yet online. Hence one cannot investigate Bryan’s computer simulation programs. 

Bryan claims to have a dynamical model for the measurement outcomes.

However, the theory he gives is a model for coincidences, ie the product of +/-1 outcomes.

The model does not generate the outcomes first, and only then, their product

The scalar projection of his "relative rotor" determines the coincidence probabilities and hence the observed correlation. The relevant quantity is

Q_{AB} = Q_A Q^{−1}_B     (18)

which gives

Q_{AB) = cos(a−b) + I e_2 sin(a−b) 

whose scalar part is cos(a−b).

The computations are correct. They are non-local.

On 23 Sep 2026, at 19:57, Bryan Sanctuary <bryancs...@gmail.com> wrote:

[Bryan Sanctuary]

Dear Richard,

The paper makes clear what the abstract says.  I do hope you can get into it.  It is really fascinating the way it all comes together.

Of course, I accept local causality. I dispute jointly defined outcomes for measurements made in different contexts. That is the step my model challenges .  I am doing what Nature does, and including what Bell nelects.  

This is expressed in section 5, the simulation.  Bell gives algebra, I include geometry on top of that algebra.  That is what I do.  

I think you might make the incorrect inference about local causality because I do have long range coherence, see, e.g.  Eqs (13)  (14)   (72), (73):  just as lasers have, and that might be misconstrued as violating local causality.  It does not. It is a product of two rotors. Spin in a rotor,

Thanks for the comment, happy to help

Bryan

[Richard Gill]

Thanks Bryan, I will be able to make my point clearer when I can access the supplementary materials. 

I have also published my derivation of CHSH from local causality and nobody has shown me any error in that derivation. 

It would be wonderful if you could put your finger on a serious mistake in there.

I truly believe it would get you a Nobel prize. Or the equivalent prize for mathematicians.

[Richard Gill]

Bell’s last paper (plus my retelling of it in a different mathematical language)

[Parker Emmerson]

Richard, Ghenadie,

The derivation is valid and the data exceed its bound; both are settled, and I hold both with you. What they settle together is that a record is not fixed by a state on the past slice alone. So the open question is physical: what fixes a record.

My construction answers it this way. The record at each wing is fixed by the completed event: the source record λ₀, the same for every setting pair and independent of both settings, together with both absorber boundaries. The model is deterministic given λ₀ and the two settings, and it exceeds the bound because the two boundaries both enter. That is a hypothesis about how records complete, and it stands or falls on measurement. At the singlet point it gives the quantum table, so nothing measured there separates it from any other surviving account. Away from that point it gives E = −tanh(κ·artanh cos θ), with κ = sech η and η the rapidity between the apparatus and the boundary structure, and a cubic bend in E against cos θ with f‴(0)/f′(0) = 2 tanh²η. A moving source is the dial; a cosmological anchoring predicts a sidereal modulation near 2 × 10⁻⁶; and a flat coincidence rate separates it from every selection account.

Richard, the mistake you ask for is not in your paper. What is open is which mechanism fills the gap the paper leaves, and the angular shape decides that.

Parker

[Richard Gill]

But I still don’t understand your work, Parker, and I have to get an innocent nurse out of jail.

There’s just been a new documentary on the case on British TV. Many new and exciting findings. I have a link to a recording if anyone is interested.

<The_Singlet_Correlation_as_a_Cross_Ratio_on_the_Absolute_2026_v3.pdf>

[Justo Pastor Lambare]

Can you please provide the link?

      Justo Pastor Lambaré

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[Bryan Sanctuary]

Hi Richard,

I attach the computer programs.  I will tell the editor the link does not work. Thanks.

 I looked at the papers that you uploaded to me, and I had already read them.  I think you simply recast Bell in a more modern way.  I see nothing that is  beyond Bell and nothing that obviates what I did.   I see you show local causes yield a joint model and the CHSH inequality, but no new inequalities and no geometry.  

But now you seem to claim that local causality plus independent settings gives a representation which obeys Bell.  So that seems to disagree with the experiment with the violation, I agree.  So can you please identify what is new in that paper and what I missed. 

Are you now changing your long held views and saying that locality gives joint definability and the CHSH comes from purely Boolean outcomes? 

Would like to get this straight, so thanks

Bryan

On Thu, Sep 24, 2026 at 8:53 AM Richard Gill <gill...@gmail.com> wrote:

Bell’s last paper (plus my retelling of it in a different mathematical language)

On 24 Sep 2026, at 14:31, Bryan Sanctuary <bryancs...@gmail.com> wrote:

[Bryan Sanctuary]

I cannot send you the zip file for the programs, it is blocked for security.  We must wait for MDPI to fix it.

Sorry for the delay

Bryan

[Bryan Sanctuary]

Changing the extension to .xxx does not remove the security block.  I attach the two FORTRAN programs that are enough to show, before I get MDPI to fix it

Bryan

[Richard Gill]

There is nothing new in my short paper. 

The experiments which violate CHSH in a loophole-free setting (time and distance constraints, random binary inputs, binary outputs, no post-selection etc etc) show that local causality is not true. It means that quantum correlations cannot be created in a classical way.

Bell’s local causality gives joint definability. Add in independent random settings and then CHSH must be hold.

Sent from my iPad

[Richard Gill]

The programs both compiled and ran correctly on my fairly recent MacBook Pro.

I was able to have an online AI tool convert the Fortran code to R and that ran correctly too.

So I am confident I will be able to understand the flow of the programs.

<simF_BiSM_v2.f90><simF_BiSM_v1.f90>

[Bryan Sanctuary]

Hi Richard,

Ok, so I did not miss anything in your paper.  Thanks.  You said "Bell’s local causality gives joint definability. "  I would say the opposite: local causality requires, not gives,  joint definability.  That is, Bell imposes a restriction on the outcomes, and that restriction is the need for joint definability, That was realized in 1982 by Fine.  But I thought your paper put Bell into a modern perspective. 

I have said, want to say it again, my model has correlation that exists between A and B. It stretches as far as Alice and Bob are apart.  It is a long range phase coherence  well known in bulk systems, like superconductivity, superfluidity, BE condensates and Lasers.  The EPR long range coherence is between only one pair: a common compensating phase is shared at the two locations.

Bryan

[Richard Gill]

Dear Bryan

Then you did not understand my paper. I assumed local causality (as Bell defined it) and derived joint definability.

Of course there is a restriction on the outcomes. They are binary. It is a model of a standard Bell-CHSH experiment: binary settings, binary outcomes. That is a fixed part of the experimental design.

This has got nothing at all to do with Fine.

Richard

[Bryan Sanctuary]

Hi Richard,

We are splitting hairs about Fine, but your paper made its point. 

So no worries,

Bryan

[Richard Gill]

Dear Bryan, dear all

I looked at Bryan's first Fortran program

He uses the formulas from his paper to compute the probabilities of equal vs unequal outcomes for a given value of lambda, and angles alpha and beta. This calculation is non local - it requires both settings to be known, together with information from the source, all in the same time and place

Then he simulates a random +/- 1 with those probabilties

This is a realisation of the product of two measurement outcomes, but the two outcomes themselves haven’t been created yet

Then you randomly create outcomes + + or - - with equal probability, if the product is +

You randomly create outcomes + - or - + with equal probability, if the product is -

Since Bryan is good at doing the algebra (or trigonometry, if you prefer) this is a legitimate and fairly simple way to simulate data from the EPR-B experiment on one computer.

Note the flow:

Pick angles alpha and beta; independently pick hidden variable lambda. Now simulate a pair of outcomes x, y = +/-1

The formulas used are the ones which Bryan derived. His model is a quite simple hidden variables model which reproduces the EPR-B statistics

Outcome x is a function of alpha and beta and a group of independent hidden variables lambda and a couple more more uniform [0, 1] variables

Outcome y is a function of alpha and beta and the same group of independent hidden variables lambda and a couple more uniform [0, 1] variables

We know that it is not possible to get these statistics by a simulation of the type:

Outcome x is a function of alpha and a group of independent hidden variables lambda; 

Outcome y is a function of betaa and those same independent hidden variables lambda.

The story that Bryan tells by doing these calculations by geometric algebra is not a story of a physical process. It is a story obtained by visualising algebraic operations in a Clifford algebra. The space in which Bryan draws those pictures is however not the physical space-time in which two photons or two electrons or whatever move toward two detectors.That is clear from the fact that Bryan doesn’t separate the simulation into two parts, keeping the process generating Alice’s output ignorant of Bob’s input, and vice versa.

Richard

On 24 Sep 2026, at 23:02, Bryan Sanctuary <bryancs...@gmail.com> wrote:

[Richard Gill]

Bell and Fine’s theorem can be expressed as one another’s converse

They make some common background assumptions. 

Bell then says    The data is generated by an LHV model => All CHSH inequalities are true. 

Fine says                    All CHSH inequalities are true => An LHV model could have generated the data.

[Bryan Sanctuary]

Dear All,

Richard is fundamentally WRONG, and WRONG again.  I predicted his comment.  He MUST SHOW IN THE PAPER WHERE I USE NONLOCALITY.  I do not use it.  Products of quaternions are NOT entangled, they multiply.  

Richard, you know this, so you have not read the paper, and you jumped to conclusions because of your extreme Bell Bias.  That is academic dishonesty and cannot make such pronouncements, especially when your goal is NOT knowledge, but to undermine.  On this issue, you must find exactly where there is nonlocality and prove it. 

N. B. , the coincidences are formed AFTER THE EXPERIMENT IS OVER.  

Did you hear that Gill OVER.   Please explain to us, how those bins of Boolean events, sent by email to a central location, and then analysed: HOW DO THE FILTER SETTINGS AT a AND b GET CHANGED???? Magic in Gill's book.  I can look at those bins anyway I want, but I just cannot change them.

Did you know there is a Malus Law for EPR pairs?  it is in the following.  Richard, read the last paragraph, and tell me where there is any nonlocality. 

Richard's categorization, without thought and without reading the paper, is WRONG.  If you still doubt it, then tell me WHERE IS THE NONLOCALITY in the following:

The dots have no math meaning, they only show that the product, which is carried by A and B, does not cancel because one is at station A and the other at station B. 

THIS IS TYPICAL GILL: OBFUSCATE, SHED DOUBT AND DECLARE A CATEGORY.  He does it for all work on Bell. 

I suggest Gill be very very careful with such comments.  He really is Blind and Blinkered by Bell.

QUATERNIONS MULTIPLY  THEY CORRELATE WITHOUT NONLOCALITY

Bryan

[Richard Gill]

Dear all.

Imagine that in a paper on BiSM, a respected author does a calculation involving both settings and a hidden variable. The question “where in the paper is nonlocality used” makes a category error. I think I might ask in reply: where do the quaternions multiply? Does that happen after the experiment when the data from both locations is processed? Personally, I imagine it happens outside of our day to day space-time in an abstract and eternal mathematical universe.

I’ll leave this to you all to argue about. My goal is knowledge, and my desire is to shed light. I’m sad Bryan sees this differently.

Yours

Richard

Sent from my iPad

On 25 Sep 2026, at 01:19, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Richard Gill]

When and where do they multiply?

[SMH Emamifar]

Parker,

Your question about what physically fixes each recorded outcome is exactly the point I find interesting.

Could we distinguish a pre-existing joint field structure from a newly transmitted disturbance? It seems worth keeping these separate when asking how the two analyzer boundaries enter the physical response.

Of course, that distinction alone does not derive the Bell correlations; the detector-response law still has to be specified.

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[Parker Emmerson]

Richard,

You asked Bryan when and where the multiplication happens. In my model I can tell you exactly: on the boundary of the source–absorber region. Take the source record λ₀, both absorber boundaries, both settings, and complete the event as one thing; the record at each wing is a piece of that completed event. That's where the multiplication is, and the program does it there because that's where I put it. So yes, it's one program with both settings in hand. It can't be split into your two computers, and any program that keeps every trial and gets past 2 fails that split the same way. The notebook's attached. It writes out timestamp, a, b, x, y for every trial, marginals ½, S = 2√2 at v = 0. Take any seed, change one dial, and watch the far record move.

You also wrote that Bell and Fine are converses. They are, and there's a version of that which says what the moving record actually is. Every no-signalling table, the quantum ones too, has a joint table on the sixteen cells (A₁, A₂, B₁, B₂) that gives back all four observed pair tables, provided the joint table is a signed measure (Abramsky–Brandenburger 2011, Al-Safi–Short 2013). A signed measure is just two positive ones, μ⁺ and μ⁻, with μ⁺ outweighing μ⁻ by exactly one. So think of it as weight on two sheets, a kept one and a reflected one; the minus sign only tells you which sheet a cell is on. Fine's theorem then says: CHSH holds exactly when all the weight sits on the kept sheet. A violation doesn't make the joint table disappear. It puts weight on the other sheet, and the data fix how much at minimum, which is what the CHSH number measures. The ≤ 2 proof is the one-sheet calculation: A₁(B₁+B₂) + A₂(B₁−B₂) is ±2 on every cell, so averaging over one sheet keeps you under 2. Average over both sheets and you get 2√2. And the "same B₁ in both contexts" step is just single-sheetedness; Fine proves they're the same assumption. So the reductio goes: assume one sheet, get 2, see 2√2, blame locality. What it actually found was the second sheet. Calling that an influence is a separate step, taken after the numbers.

So a mechanism has to say where the reflected sheet comes from. Mine: it's the trace of both absorber boundaries entering one selection. That's why the far record moves under replay while every marginal stays put.

Last thing, on the note I sent. The claim in it is smaller than it probably looked. What the data contain is E = −a·b at each labelled pair of settings, and the geometry whose invariants are exactly that is the Möbius geometry of the four null rays u ± a, u ± b on the absolute: the cross-ratio of those rays is D², and swapping the labels on one axis flips it to its reciprocal. On that object all five surviving completions look the same, because they all give the same invariant and there's no time order on it for them to disagree about. Their differences are real; they just live above the object, in mechanism, where the data don't reach. κ is the one deformation the data can see, and the minimal weight on the reflected sheet is what it reads.

Parker

[SMH Emamifar]

A small clarification regarding my previous email: the files I attached are a separate joint-flux reproducibility companion, not the Python premise audit mentioned in my message.

They demonstrate a declared nonfactorizable joint-response construction; they do not establish Bell-locality or a physical detector mechanism.

I am attaching the intended premise-audit report and source here. Apologies for the attachment mix-up.

Best regards,

SMH

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[SMH Emamifar]

Dear Bryan and colleagues,

I have been following the discussion of your geometric spin model, particularly the distinction between local causality and joint definability across measurement contexts.

I do not think an alternative account of Bell correlations must preserve every premise of a Bell-factorizable model. For binary outcomes, measurement independence (MI), parameter independence given a complete lambda (PI), and outcome independence given lambda (OI) jointly imply the CHSH bound |S| ≤ 2. A model reproducing a violation cannot retain all three in that form.

I am sharing a small Python premise audit, with its English report, source code, and editable cases. It checks the 16 local deterministic CHSH assignments and identifies minimal premise relaxations. This is an audit of known conditional mathematics, not a physical simulation of your model or evidence of faster-than-light communication.

My questions are:

What precise notion of local causality does your model retain when cross-context joint definability is rejected?

How do the shared phase and geometry produce the two separately recorded outcomes? In particular, where do the two analyzer settings enter the physical account: shared preparation, local detector responses, or the final comparison of records?

I would welcome your clarification and comments from the group.

Best regards,

SMH

[Bryan Sanctuary]

Dear Prof Emamifar,

Thank you for your comments and questions.  Here are my replies to your specific questions, and follow-ups are welcome.

Bryan

Question: What precise notion of local causality does your model retain when cross-context joint definability is rejected?

Answer: 

First, very generally, if you measure a spin at $a$, and then at $a'$, there is no certainty that if you return to measure $a$ again, you will get the same outcome. This is because the second measurement of $a'$ changes the phase \lambda, so the repeated measurement of $a$ can be different due to this phase change: 

This means, from the start, that different measurements needed for the CHSH quadruple can not be jointly definable.  This is analogous to say position and momentum.

Now for local causality from the bivector approach:

The starting point is to note that spin should include both symmetric and antisymmetric components,

I keep the bivector, i\sigma_2 = \sigma_3 \sigma_1, while universally spin is expressed as only \sigma. That is the operational difference.

Local causality is expressed by a common phase carried by an EPR pair after separation, section (2.1.3). and persist separately during propagation. Their  common \(\lambda\) cancels in the later algebraic comparison.  This provides long-range correlation between the two spins

That is, the isotropy of the singlet is maintained by these compensating phases that are carried by the spins of Alice and Bob. The phase remains as they move apart. 

Question: How do the shared phase and geometry produce the two separately recorded outcomes? In particular, where do the two analyzer settings enter the physical account: shared preparation, local detector responses, or the final comparison of records?

After the two separate (not entangled but phase correlated), the two filters instantiate a plane at each location.

Measurement is local at the remote locations of A and B, but their correlation leads to the cancellation of the phase, \lambda, when compared after the experiment is over.

The steps are those that the experiment and I believe Nature follow. I do something different (Equation (52)) from Bell, I construct the scalar from bins of Alice and Bob, Theorem 2.  This is best seen by looking at the description in Section 4.2.  There I explain exactly how I do the simulation and then program the EPR correlation.  It is notable that the clicks at A and B contain geometry, Equations (67-8).

Locality is ensured because the phases are carried by quaternions and their product gives the correlation with no entanglement. 

To get a general perspective of the difference between the SM (HEP) and the BiSM (LEP), please see section 5.4.

I hope this helps

Thanks again

Bryan

[SMH Emamifar]

Dear Bryan, Richard, Parker, and colleagues,

Thank you, Bryan, for your detailed response and your invitation to continue the discussion.

I would like to share the attached one-page logical flowchart. It emerged from our discussion, but it is independent of Zero Theory and does not propose a new interpretation or challenge the Bell–CHSH theorem.

Its purpose is to establish what a candidate physical explanation must account for before we begin constructing one.

Let us be generous to any proposed model. We may provisionally allow a common source, persistent phase coherence, or even a hypothetical physical connection that has not yet been discovered.

The central question remains: after separation, how do the newly selected analyzer settings enter the physical production of the two separately recorded outcomes?

The mathematical checkpoint is straightforward. With measurement independence and Bell factorizability,

[

\rho(\lambda\mid a,b)=\rho(\lambda),

]

[

P(A,B\mid a,b,\lambda)

P(A\mid a,\lambda)P(B\mid b,\lambda),

]

the CHSH bound follows:

[

|S|\leq 2.

]

The experimental violation means that these premises cannot all provide a complete account of the observed distributions. A model reproducing (S>2) must therefore identify precisely where it departs from the premises of that derivation.

This is not a demand that every proposed theory preserve Bell factorizability. Quite the opposite: a theory may depart from it, but the departure should be explicit rather than concealed by a different mathematical representation.

The attached algorithm asks four questions:

1. Does the model reproduce the four observed joint distributions, rather than only a final CHSH number?

2. How are the two individual detector outcomes physically generated after the settings are selected?

3. Which premise of the Bell–CHSH derivation is not retained, and at what step does this become apparent?

4. What physical law implements that departure, and what independent observation could distinguish the proposed mechanism from other explanations?

Bryan, your example of sequential measurements at (a), (a'), and (a) illustrates measurement disturbance. I would be interested in seeing how that argument applies to the separate detector records in a standard CHSH trial, and where the two analyzer settings enter the physical outcome-generation process.

Parker's boundary-based proposal provides a useful contrast because both absorber settings explicitly enter the proposed event-completion rule. That makes the relevant question visible: what physical process realizes this dependence?

I also want to preserve a distinction that is easily lost in this discussion: the absence of operational faster-than-light signalling does not, by itself, establish the absence of a fundamental nonlocal dependence. Conversely, observing a Bell violation does not directly measure the propagation speed of a hypothetical physical influence.

My purpose is not to choose between these explanations in advance. It is to ensure that we agree on the logical problem a physical theory is being asked to explain.

I would welcome your criticism of the attached flowchart, especially if you think any of its logical branches is incomplete or incorrectly stated.

Best regards,

S. M. H. Emamifar

[Richard Gill]

Dear SMH

I completely agree with you.

Richard

Sent from my iPad

On 28 Sep 2026, at 04:26, SMH Emamifar <smhema...@gmail.com> wrote:



S. M. H. Emamifar

On Wed, Sep 23, 2026, 9:27 PM Bryan Sanctuary <bryancs...@gmail.com> wrote:

Hello All

A glitch at MDPI has put up an earlier version of my new paper, which will be corrected.  

However, the present discussion on the Forum might find some answers, and questions,  in the attached pdf of the paper. I put in a TOC so you can get a  quick overview.  

I hope this can generate some debate.

Bryan

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[Richard Gill]

“It is notable that the clicks at A and B contain geometry, Equations (67-8).”

I couldn’t find a definition of bold X_{i,k} - “the retained local rotor geometry”

[anton vrba]

Hi SMH, I love your flow chart, it has a minor graphical-typo

The quantum prediction, the blue bar, is the Tsirelson's bound 2.828 and the experiments slightly below

regards

Anton

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

From "SMH Emamifar" <smhema...@gmail.com>

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

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

Date 9/28/2026 3:26:03 AM

Subject Re: [Bell_quantum_foundations] Joint Definability and the SM vs. BiSM

To view this discussion visit https://groups.google.com/d/msgid/Bell_quantum_foundations/CAAMGvzAfFnXBt9OnJgBLdMURRdbV5ZKvFvL%2BScV69DdU%3DG3jGA%40mail.gmail.com.

[Bryan Sanctuary]

Richard  cannot find the definitions of X_{i,k}, they are determined by the rules in section 4.3.1,   Those rules depend on the phase values  of \phi_{i,k} from which the symmetric POL and antisymmetric COH are constructed.  

So the Boolean values of A and B bins contain geometry, but that is only found POST-experiment when the two bins form coincidences.

Bryan

On Mon, Sep 28, 2026 at 5:04 AM Richard Gill <gill...@gmail.com> wrote:

“It is notable that the clicks at A and B contain geometry, Equations (67-8).”

I couldn’t find a definition of bold X_{i,k} - “the retained local rotor geometry”

Sent from my iPad

[Richard Gill]

So the the X_{i,k} are superfluous. You already have the \phi_{i,k}. 

You insist that those quaternions are only multiplied when the data from both sides are combined and the coincidences are evaluated? So the individual outcomes aren’t available yet? Does Nature do that? Or a quantum engineer?

Experimenters tell me that in actual experiments, they collect and process time-tagged +/-1 outcomes from each wing of the experiment, together with time-tagged binary settings. They don’t use equations (67)and (68). They don’t use any angles (phases) at all.

Sent from my iPad

On 28 Sep 2026, at 14:43, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Richard Gill]

How do bins vorm coincidences?


Sent from my iPhone

> On 28 Sep 2026, at 14:43, Bryan Sanctuary <bryancs...@gmail.com> wrote:
>

> two bins form coincidences

[Bryan Sanctuary]

Richard,

No, X_i are not superfluous, in fact essential to define, POL and COH  

Regarding the other point:  I follow exactly what the experiment does. But I use the rule Eq 52 two construct the scalar that has the long range phase correlation from each bin. That is new. Bell assumes Boolean, I assume a rotor up to detection. 

I know experimenters do not use Equations (67-68), because Nature has done it for them.  I am putting in the geometry that is contained in the bins (only evident post-analysis).  I very clearly say this is beyond Bell, and I can do it because I use quaternions.

This is explained throughout the paper, and specifically expressed in the simulation, section 4.1.  You now see that the Add/Average issue is completely answered. 

Have you now realized that there is no nonlocality in my work?  If not, we should discuss that point because it is clear evidence you have missed an important feature of the paper.

Bryan

[Parker Emmerson]

Richard, Bryan,

Richard, one thing ahead of Bryan's paper: my question to you is still open. Does any member of the package — settings independent of the source, a definite record at each wing, marginals fixed across the far dial — entail the factorising variable, and if so which member and by what argument? Put the other way: by what physical criterion does joint completion at the event count as nonlocality? I asked it on 20 September, and it's still waiting.

Your question to Bryan bears on it, because it splits the models cleanly. The dependence on both settings can enter in one of two places: in the analysis, when coincidences are counted, or in the event, when the records are made. If it enters in the analysis, the individual outcomes only come into existence when someone combines the files, and that's an engineer, not Nature. If it enters in the event, the record at each wing exists the moment the event completes, each one a time-tagged ±1, and the coincidence count just reads what's already there.

My model is the second kind. The sampler writes (timestamp, a, b, x, y) for every trial: a ±1 at each wing, the binary setting at each wing, and a time tag. The angles are the analysers' physical orientations; the record itself carries only settings and outcomes. The joint selection happens once, at completion, from the shared source record and both settings, and every marginal stays at ½ whatever the far dial does. That's the data format you describe, and it comes out of the model directly.

Bryan, a common geometry carried by the pair works in that second place. It fixes both records at the event, and each wing gets its ±1 there, before anyone compares files.

So the question stands where it was, Richard, and I'd ask you to answer it before classifying the next model.

Parker

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[Bryan Sanctuary]

Hi Parker,

Thank you.  

I comment on coincidence formation: 

"If it enters in the analysis, the individual outcomes only come into existence when someone combines the files, and that's an engineer, not Nature. "

I agree. I think the two are basically equivalent: we construct; while Nature does it naturally. Just take the raw experimental data and plot it.  You get the violation because that is what is observed.  Now you can analyse the data in two ways: Bell's just compares the random Boolean clicks (get the polytope); I first form the scalar from their geometry, and then digitize, (reproduces the violation), Equation 52, or summarizing:

Nature never combines the bins, but the long range correlation exists nonetheless.  As far as Nature is concerned, it just is.  However, we can construct observables from experimental data (that is what physics does), and so the engineer does it from the data. S/he does an experiment and reveals spin is a rotor.

Glad to discuss more.

Bryan

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[Richard Gill]

Dear Parker

I don’t have a model. I don’t have an explanation. I think it is beyond human understanding. I think that Aristotle was wrong: not everything has a cause. In experiments with “entangled” photons we actually are witnessing irreducible, non-local randomness.

Richard

[Richard Gill]

That is actually my own belief, Bryan! “It just is”.

Bell argued that there was no local realist explanation. Bryan agrees with Bell!

[Bryan Sanctuary]

To all:  Richard just stated he believes in quantum weirdness. In 2002, or 3, after giving a talk at U of Geneva and at lunch with Gisin, I asked him how nonlocality worked, that is, what did he tell his grad students?   Gisin replied:

"We are in a realm of physics which is beyond human comprehension"  

Richard is in the same club as Nicholas. Richard said " I think it is beyond human understanding."  

It is not, nothing is beyond our understanding , except revelation.

I have clearly answered this and revealed quantum weirdness is a result of long range phase coherence (like a laser).  I have replaced entanglement with a product of two quaternions.  (see the singlet sections 2.1.1 and 2.1.2}. So it is not weird, it is geometry.

Does that help you Richard? Soon I hope you see the light and understand Ballentine's statistical interpretation of the wave function. Then we can bury the hatchet. 

Bryan

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[Parker Emmerson]

Subject: Re: [Bell_quantum_foundations] Joint Definability and the SM vs. BiSM

Richard,

I'm not saying everything is within human understanding. But a Bell experiment is a material interaction, and I think the event and where it comes from can be understood, and the language forms we use to describe it are accurate.

Your answer also settles the question I asked. You didn't name a member of the package that entails the factorising variable, or a physical criterion that makes joint completion non-local. Your "non-local" rests on something else: that the correlation is uncaused. That's a position, and it's worth having on the record as one.

The experiments don't force it. They exclude one kind of cause: a state on the past slice fixing each record from its own setting. Bohm's theory, the retrocausal models and my construction all reproduce the singlet with a cause behind every record.

They also can't show the randomness is irreducible. Any probabilistic law is produced exactly by a deterministic rule acting on a hidden seed. That's the representation behind every probability model, including yours of Bell, where outcomes are functions of λ. My sampler does exactly that for the singlet, and its records (time tag, settings, ±1 at each wing) have the same statistics as pure chance. So "irreducible" isn't something an experiment witnesses. It's a reading of records that come out the same either way.

Device-independent randomness doesn't change this. A CHSH violation certifies unpredictability only for devices assumed to be no-signalling given the hidden state, which is parameter independence at the level of λ, the premise at issue. For my model, the seeds and both settings predict every outcome.

So your reading and mine are two readings of the same records at κ = 1. What separates them is measurable: below κ = 1 my law bends E against cos(a−b), and a dense angular scan can find the bend or rule it out.

The experiments show that one picture of causes was too narrow, not that causes run out. Aristotle survives Bell; the forward-only common cause doesn't.

Parker

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[Richard Gill]

I know the experiments don’t force the position which I presently prefer to take. I will very likely change my mind in the future.

[anton vrba]

Changing ones mind in the future on new evidence is the scientific process. 

The evidence has to be flawless, provable, integrate in physical laws like Maxwell's 1865 work, etc --- BiSM does none of that: it contains mathematical contradictions, it makes assertions without foundational backing, it does not attempt to integrate into known physics, it does not describe fields, it does not describe how measurements are made, etc.

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

From "Richard Gill" <gill...@gmail.com>

To "Parker Emmerson" <powerin...@gmail.com>

Cc "Bryan Sanctuary" <bryancs...@gmail.com>; "Emamifar SMH" <smhema...@gmail.com>; "Bell_quantum...@googlegroups.com" <Bell_quantum...@googlegroups.com>

Date 9/28/2026 5:37:01 PM

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[Parker Emmerson]

Richard, Anton,

Richard, then we're in the same place: the theorem sorts models into classes, and the data leave the reading open.

That bears on your paper with Anton on Bryan's model. Its checkable findings on the code and the local responses stand. What they show is that his model sits outside the factorising class, which his Reply already concedes. Outside that class is a different thing from non-local. Every model that reproduces the singlet computes the joint law from both settings, quantum mechanics' own algorithm included, and a joint completion can keep every mark of locality an experiment can check: every marginal fixed whatever the far setting, dependence fixed at completion rather than carried at a speed, the same law in every inertial frame. Mine keeps all three at κ = 1. So the finding is about the factorising class. If the conclusion said that, it would match what you wrote today, and the only reply left to it would be on substance.

What would move either of us is a measurement. Do you know a group with a high-visibility entangled source and time-tagged data taken at many analyser angles, not just the four CHSH settings? A fit of E against cos(a−b) for the cubic term would already put a bound on κ, and a run across a sidereal day is the test of the cosmological anchoring.

Parker

On Tuesday, September 29, 2026, anton vrba <anto...@gmail.com> wrote:

Changing ones mind in the future on new evidence is the scientific process. 

The evidence has to be flawless, provable, integrate in physical laws like Maxwell's 1865 work, etc --- BiSM does none of that: it contains mathematical contradictions, it makes assertions without foundational backing, it does not attempt to integrate into known physics, it does not describe fields, it does not describe how measurements are made, etc.

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

From "Richard Gill" <gill...@gmail.com>

To "Parker Emmerson" <powerin...@gmail.com>

Cc "Bryan Sanctuary" <bryancs...@gmail.com>; "Emamifar SMH" <smhema...@gmail.com>; "Bell_quantum_foundations@googlegroups.com" <Bell_quantum_foundations@googlegroups.com>

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[Bryan Sanctuary]

Anton said:

BiSM.......contains mathematical contradictions, it makes assertions without foundational backing, it does not attempt to integrate into known physics, it does not describe fields, it does not describe how measurements are made, etc.

I reply:

It is as clear as it can be: definitions are focussed. Based on Geometric Algebra. It shows the classical-quantum correspondence. Shows the clear relation between the SM and the BiSM. It does not describe fields like QFT. It clearly describes measurement by instantiation, etc. 

Historical note: Hamilton was stumped trying to rotate in 3D with operations in 3D (Euler type angles), He stopped by the bridge to carve because he needed one more dimension to do it:quaternions.  The situation is now similar: in order to understand spin, you need another dimension, and that makes spin a quaternion.  

I like that, it is consistent and makes sense.

Bryan

So Anton's comments seem a bit biased and not substantiated.

Bryan

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[anton vrba]

First: To understand Parker's commentary I attach the PDF  to explain "That bears on your paper with Anton on Bryan's model."

Dear Parker,  

You wrote "So the finding is about the factorising class. If the conclusion said that, it would match what you wrote today, and the only reply left to it would be on substance." the Sanctuary_Critique has to be agnostic to Richard's and my mindset.

Now to your question regarding time stamped data: To progress towards understanding QM we need to apply Occam's razor, forget time stamps, light cones etc., all the so called loopholes are equally spooky as Einstein's "spooky action at a distance".   We need to discover what entanglement is really all about, it does not begin and end with a singlet state, there are the three triplet states that BiSM does not care to mention, but need to be equally explained.

Regards 

Anton

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

From "Parker Emmerson" <powerin...@gmail.com>

To "anton vrba" <anto...@gmail.com>

Cc "Richard Gill" <gill...@gmail.com>; "Bryan Sanctuary" <bryancs...@gmail.com>; "Emamifar SMH" <smhema...@gmail.com>; "Bell_quantum...@googlegroups.com" <Bell_quantum...@googlegroups.com>

Date 9/28/2026 5:56:25 PM

Subject Re: [Bell_quantum_foundations] Joint Definability and the SM vs. BiSM

[Bryan Sanctuary]

Anton,

Entanglement is a property of the indistinguishability of identical particles.  End of story.  

However there are two views:

• the gnostic view:  entanglement is local and cannot persist after separation : quantum enlightenment, 
• the agnostic view:  entanglement is nonlocal and can persist after separation: quantum weirdness, 

Are you gnostic or agnostic wrt to QM?

Bryan

Bryan  

Cc "Bryan Sanctuary" <bryancs...@gmail.com>; "Emamifar SMH" <smhema...@gmail.com>; "Bell_quantum...@googlegroups.com" <Bell_quantum...@googlegroups.com>

Date 9/28/2026 5:37:01 PM

Subject Re: [Bell_quantum_foundations] Joint Definability and the SM vs. BiSM

I know the experiments don’t force the position which I presently prefer to take. I will very likely change my mind in the future.

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[Richard Gill]

The three other states can be created from the singlet by local unitary transformations

On 28 Sep 2026, at 20:23, anton vrba <anto...@gmail.com> wrote:

First: To understand Parker's commentary I attach the PDF  to explain "That bears on your paper with Anton on Bryan's model."

Dear Parker,  

You wrote "So the finding is about the factorising class. If the conclusion said that, it would match what you wrote today, and the only reply left to it would be on substance." the Sanctuary_Critique has to be agnostic to Richard's and my mindset.

Now to your question regarding time stamped data: To progress towards understanding QM we need to apply Occam's razor, forget time stamps, light cones etc., all the so called loopholes are equally spooky as Einstein's "spooky action at a distance".   We need to discover what entanglement is really all about, it does not begin and end with a singlet state, there are the three triplet states that BiSM does not care to mention, but need to be equally explained.

Regards 

Anton

<jbt3xw2q.png>

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

From "Parker Emmerson" <powerin...@gmail.com>

To "anton vrba" <anto...@gmail.com>

Cc "Richard Gill" <gill...@gmail.com>; "Bryan Sanctuary" <bryancs...@gmail.com>; "Emamifar SMH" <smhema...@gmail.com>; "Bell_quantum...@googlegroups.com" <Bell_quantum...@googlegroups.com>

Date 9/28/2026 5:56:25 PM

Subject Re: [Bell_quantum_foundations] Joint Definability and the SM vs. BiSM

Richard, Anton,

Richard, then we're in the same place: the theorem sorts models into classes, and the data leave the reading open.

That bears on your paper with Anton on Bryan's model. Its checkable findings on the code and the local responses stand. What they show is that his model sits outside the factorising class, which his Reply already concedes. Outside that class is a different thing from non-local. Every model that reproduces the singlet computes the joint law from both settings, quantum mechanics' own algorithm included, and a joint completion can keep every mark of locality an experiment can check: every marginal fixed whatever the far setting, dependence fixed at completion rather than carried at a speed, the same law in every inertial frame. Mine keeps all three at κ = 1. So the finding is about the factorising class. If the conclusion said that, it would match what you wrote today, and the only reply left to it would be on substance.

What would move either of us is a measurement. Do you know a group with a high-visibility entangled source and time-tagged data taken at many analyser angles, not just the four CHSH settings? A fit of E against cos(a−b) for the cubic term would already put a bound on κ, and a run across a sidereal day is the test of the cosmological anchoring.

Parker

On Tuesday, September 29, 2026, anton vrba <anto...@gmail.com> wrote:

Changing ones mind in the future on new evidence is the scientific process. 

The evidence has to be flawless, provable, integrate in physical laws like Maxwell's 1865 work, etc --- BiSM does none of that: it contains mathematical contradictions, it makes assertions without foundational backing, it does not attempt to integrate into known physics, it does not describe fields, it does not describe how measurements are made, etc.

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

From "Richard Gill" <gill...@gmail.com>

To "Parker Emmerson" <powerin...@gmail.com>

Cc "Bryan Sanctuary" <bryancs...@gmail.com>; "Emamifar SMH" <smhema...@gmail.com>; "Bell_quantum_foundations@googlegroups.com" <Bell_quantum_foundations@googlegroups.com>

Date 9/28/2026 5:37:01 PM

Subject Re: [Bell_quantum_foundations] Joint Definability and the SM vs. BiSM

I know the experiments don’t force the position which I presently prefer to take. I will very likely change my mind in the future.

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<Critique_Sanctuary_QR2026_4.pdf>

[anton vrba]

"The three other states can be created from the singlet by local unitary transformations"

Correct, but local unitary transformations are a form of mathematical post-processing. They describe how we can manipulate a state after the fact, not how Nature dynamically creates entanglement in the first place.

Take the foundational experiments of Freedman & Clauser, as well as Alain Aspect. They used a two-photon radiative cascade of the calcium atoms (J=0 -> J=1 -> J=0). In this system, the valence electrons remain passive spectators in an S=0 spin singlet state throughout the entire transition.

Instead, the entanglement is generated entirely by the conservation of orbital angular momentum, forcing the emitted photons directly into the state (|RR> + |LL>). When translated to linear polarization, this gives, depending on axis choice, the state (|HH> - |VV>) — which is a triplet-like Bell state, not a singlet.

Nature did not create a singlet and then apply a "local unitary transformation" to fit a geometric paradigm. It directly synthesised a correlated and entangled state. Any geometric theory of entanglement must fundamentally account for this orbital generation mechanism, rather than relying strictly on spin singlet geometry.

Regards
Anton

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

From "Richard Gill" <gill...@gmail.com>

To "anton vrba" <anto...@gmail.com>

Cc "Parker Emmerson" <powerin...@gmail.com>; "Bell_quantum...@googlegroups.com" <Bell_quantum...@googlegroups.com>

Date 9/28/2026 8:12:47 PM

[anton vrba]

P.S. In fact, if we look at state transformations in the lab, experimentalists like Paul Kwiat have demonstrated how to project and filter down states to isolate a singlet. There is no fundamental experiment that relies on starting with a raw photon singlet as the default parent state from which triplets must be derived. Nature is perfectly happy creating triplet-like entanglement right out of the gate.

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

From "anton vrba" <anto...@gmail.com>

To "Richard Gill" <gill...@gmail.com>; "Bryan Sanctuary" <bryancs...@gmail.com>

Cc "Parker Emmerson" <powerin...@gmail.com>; "Bell_quantum...@googlegroups.com" <Bell_quantum...@googlegroups.com>

Date 9/28/2026 9:44:59 PM

[Richard Gill]

Anton and my comments on SM vs BiSM are now posted on internet for the world to see.

Hopefully, Zenodo and arXiv will follow, plus letter to the editors of Quantum Reports.

Sent from my iPad

On 28 Sep 2026, at 19:30, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Richard Gill]

The claims which Anton wrote down, many of which I had previously made in numerous emails to you Bryan, are all substantiated.

Geometric algebra allows one to rewrite standard QM calculations using Clifford Algebra. That’s what you have done. Nothing of substance has been added, nothing changed. QM remains incomplete and the physical state of two distant parts of an entangled bipartite state can’t be separated. Experiment shows that Einstein’s dream can never be realised. If there is anything “behind” QM it is even weirder than QM itself, as Feynman even said. Those who believe they understand QM don’t.

Sent from my iPad

On 28 Sep 2026, at 19:30, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Parker Emmerson]

Dear all,

Attached is a notebook anyone can run; it needs numpy and matplotlib and takes about fifteen seconds. It puts two models through the same per-trial harness: random settings on every trial, a ±1 at each wing, every trial recorded as (timestamp, a, b, x, y).

Program A has Bell-local base responses plus a setting-dependent acceptance rule. On accepted trials it gives the singlet, |S| = 2.83; before acceptance |S| = 2, on the bound. Its price is an acceptance rate that dips to 0.8786 near 46° and 134°.

Program B is the PV-REC race: four clocks, one per candidate joint outcome, with rates from both settings, and the first to fire is the event. Every trial is kept, |S| = 2.83, the marginals are ½, and the coincidence rate is flat.

Richard, on your challenge. Your game has three computers, a source, Alice and Bob, and the source speaks once, before the settings exist, then stays silent. Under that rule my model gives |S| = 1.41. It also runs on three computers with the only links running source to Alice and source to Bob, the two legs of the transaction graph in my paper, if the source hears each wing's absorber boundary along its own leg. Then |S| = 2.828 ± 0.003, every marginal is at ½, and the log shows each wing receiving only its own outcome. The one message that breaks your time order is marked: the boundaries reaching the source after the settings exist. That's the time-symmetric step, stated rather than hidden.

That topology proves nothing by itself. The same three computers run a Popescu–Rohrlich box to |S| = 4, and the notebook does it. What the race adds is its law. Over every four-angle configuration its largest |S| is 2√2 exactly at κ = 1, lower for κ < 1, and 2 at κ = ½, so it sits inside the quantum set with one deformation below it, and a joint fit in the notebook recovers κ separately from visibility. My question is narrower than before: is the rule that the source falls silent at emission physics, or part of the game?

Bryan, the critique of your paper catalogues what's wrong, and the code findings stand: the BiSM_v2 populations sum to 2, the doubled harmonic comes from a sign convention, and the two statements about Eq. (71) contradict each other. What hasn't been said on the list is what the model gets right.

First, the two-level picture. Each wing records a local Boolean outcome and the phase relation shows up only in the statistics. Placed correctly, that's a theorem. Every positive 2×2 compatibility table factors uniquely as m_xy = C·A^x·B^y·D^{xy}; everything local to a wing lives in C, A and B, and the correlation is a function of D alone, E = (D² − 1)/(D² + 1) (The Bivector Grade of the Correlation Table, doi:10.5281/zenodo.21669459, Theorem 2.3). The one relocation: the bivector that carries the correlation is e_A ∧ e_B in the space of outcome labels, not a bivector of spacetime.

Second, the rotor sign. The ±R ambiguity acts on the table as a mirror pair: flipping one wing's labels inverts D and the sign of E, and flipping both leaves them fixed.

Third, the joint scalar. The critique's own B1 says "joint processing is not itself the issue … the issue is which function of the records is computed." Your cosine comes from a quantity formed from both settings at once, and that's the structure every model reproducing the singlet has, quantum mechanics' own algorithm included. The mistake was calling it local. Placed at the event, as a joint completion with both settings and every marginal fixed, it's a mechanism with a test, which is what the notebook's Program B runs. The critique's A7 gives the scope claim its correct form: a model outside the class is an instance of the theorem's conclusion, not an exception to it.

To test the notebook, change N or SEED in the first cell and rerun.

Parker

[Bryan Sanctuary]

Hi all

I had communicated with Richard and Anton saying I would respond, and we should coordinate our responses. They did not.  I could answer all their questions, but I will not.  First their critique is 9 pages long, and there are only a couple of points requiring clarification.  The rest is not really worth replying to, so I will defer to the paper. Who wants to read either, so my reply with be short.

They did raise a couple of points that need explaining more, and thanks to them for that, but there is nothing in their long paper that in anyway disproves what I am saying.  Their paper is not in the spirit of academic cooperation where we could have coordinated our positions.

So I will get to my reply soon.  Otherwise my paper stands alone except for those few points.

Bryan

[Richard Gill]

Dear Bryan, dear all,

That’s good, I’m looking forward to Bryan’s clarifications on what he finds important. I’m getting back to my real job (rescuing innocent nurses).

Do you know who is really the Bell gatekeeper? It’s arXiv. Our paper can’t be posted on arXiv till Bryan’s is. So I hope that will happen soon.

Richard

Sent from my iPad

On 30 Sep 2026, at 00:13, Bryan Sanctuary <bryancs...@gmail.com> wrote:



Hi all

I had communicated with Richard and Anton saying I would respond, and we should coordinate our responses. They did not.  I could answer all their questions, but I will not.  First their critique is 9 pages long, and there are only a couple of points requiring clarification.  The rest is not really worth replying to, so I will defer to the paper. Who wants to read either, so my reply with be short.

They did raise a couple of points that need explaining more, and thanks to them for that, but there is nothing in their long paper that in anyway disproves what I am saying.  Their paper is not in the spirit of academic cooperation where we could have coordinated our positions.

So I will get to my reply soon.  Otherwise my paper stands alone except for those few points.

Bryan

On Tue, Sep 29, 2026 at 12:14 PM Richard Gill <gill...@gmail.com> wrote:

Anton and my comments on SM vs BiSM are now posted on internet for the world to see.

Hopefully, Zenodo and arXiv will follow, plus letter to the editors of Quantum Reports.

[Richard Gill]

Very nice, Parker!

Sent from my iPad

On 29 Sep 2026, at 19:48, Parker Emmerson <powerin...@gmail.com> wrote:



<The_Singlet_Correlation_as_a_Cross_Ratio_on_the_Absolute_2026_v4.zip>

<pv_rec_vs_selection_chsh_v4.ipynb>

[SMH Emamifar]

Dear Richard, Parker, Bryan, Anton, and colleagues,

Parker's latest question made me reconsider something in the flowchart I previously shared.

My earlier question was:

After the settings a and b are freshly chosen, where do those settings enter the physical production of the two recorded outcomes?

I still think this is an important question. Parker's latest construction makes his own answer explicit: if the source is required to become causally silent before the settings exist, the Bell correlation is lost; in his proposed completion, later boundary conditions participate through a time-symmetric step.

Whether that particular mechanism is physically correct is a separate issue. But at least the dependency is now visible.

This made me wonder whether the usual physical formulation of the Bell puzzle may itself begin one step too late.

I do not mean that the Bell-CHSH theorem assumes that spacetime is fundamental. It does not. The CHSH bound follows from the relevant probabilistic and causal assumptions without introducing c.

With measurement independence and Bell-local screening,

rho(lambda | a,b) = rho(lambda),

and

P(A,B | a,b,lambda)

= P(A | a,lambda) P(B | b,lambda),

we obtain

|S| <= 2.

The experiments reject that screened model class.

My question is therefore not whether Bell's theorem is wrong.

It is this:

When we translate the theorem into the physical picture of two systems that have "separated", are we entitled to assume that their separation in laboratory spacetime is also a fundamental separation in the underlying physical ontology?

"Spacelike separated detector events" and "fundamentally separable physical subsystems" are not obviously the same statement.

Suppose, for example, that metric spacetime is not primitive, but is an emergent or observer-level description, while the more fundamental physical state is defined on a relational structure from which metric distance, duration, and propagation are themselves reconstructed.

Then Alice's and Bob's detector events may be kilometres apart and spacelike separated in the laboratory metric, without it automatically following that every underlying degree of freedom relevant to the pair is separated in that same sense.

This would not evade Bell's theorem.

A candidate theory would still have to explain:

- how a and b enter the underlying physical state,

- how the two local outcomes are generated,

- which Bell-local screening condition fails,

- why the observed marginals remain no-signalling,

- and how ordinary relativistic spacetime emerges at the observational level.

Nor would a relational ontology, by itself, imply that Bob can detect Alice's new setting before an ordinary causal signal arrives. That would be a further physical coupling and a separate empirical claim.

But it may change the mechanistic question.

Instead of beginning with:

"What travels from Alice to Bob fast enough to coordinate the outcomes?"

perhaps the prior question should be:

"What justifies treating Alice and Bob as fundamentally separated in the variables that physically generate the outcomes?"

In other words, before asking how a bridge crosses the distance, should we first establish that the relevant underlying ontology contains that distance as a fundamental separation at all?

I would be very interested in objections to this distinction, especially whether Bell locality itself rules it out independently of any assumption about the fundamentality of spacetime.

Best regards,

SMH

[Parker Emmerson]

Dear all,
A correction to two phrases in my last message.
I described Program A as having "Bell-local base responses." They're factorising: each outcome is a function of its own setting and the source. "Local" doesn't belong on that condition; it should be named by its content.
I also wrote that Bryan's "mistake was calling it local." That concedes that a joint quantity is non-local, which I don't hold. The error in the paper is attribution: it says the cosine comes from outcomes each station computes on its own, and the code shows it comes from a quantity formed from both settings. That joint quantity keeps every mark of locality an experiment can check: every marginal fixed whatever the far setting, and the dependence fixed at completion rather than carried at a speed.
Everything else in the message stands.
Parker

[Richard Gill]

I think the answers can be explored by considering the option of superdeterminism. 

See for instance Engel Wichmann’s recent paper https://doi.org/10.1016/j.nexres.2026.102343

[Bryan Sanctuary]

Hi Parker,

Please identify where my coding is nonlocal in BiSM_v1.   Also please look at equation 54 which shows the correlation is a product of two quaternions.  Therefore there is no nonlocality in my correlation. I just formed a scalar from the raw data.

You and Richard think nonlocality comes in when I choose two phases which are carried by A and B when I calculate the correlation.  You said  “Your cosine comes from a quantity formed from both settings at once … The mistake was calling it local.” I strongly disagree. You confirm:  “I agree with Bryan that local Boolean clicks do not themselves contain the cosine. The cosine belongs to a higher-level joint structure.” I strongly agree.

The cosine, however,  comes  after the experiment is over, (same as the experiment procedure).  For Alice  I sort out all the Boolean outcomes consistent with setting a,  Then I do the same for Bob, for setting b.  Then I form the difference in 5 degrees  |a-b| = 5 degrees, and calculate the correlation in steps across 2\pi. In each 5 degree segment, the relative population of the rotor changes geometrically for each segment and gives the -cosine, yes statistical, but not nonlocal. It is long-range phase coherence.

There is no nonlocality because the analysis is only post-experiment. After the bins are full, and the experiment is over, how can nonlocality sneak in? If people can accept that spooky entanglement exists over spacetime, surely you can accept that long-range phase coherence also exists, same as a laser, but without nonlocality.  Spooky is nonlocal, long-range phase coherence is not.

 Indeed then, you are right, the violation only occurs in the statistics, and not from the Boolean outcomes alone.  That is a big point of my paper and I am glad you see. It is the key and the difference between Bell and me.  

So the violation is due to the failure of Joint Definability.  I think we agree.

The code, BiSM_v1 simply shows the violation. BiSM_v2 analyses the raw data.  You have your view of BiSM_V2, and I suggest there are other ways to look at the raw data and make deductions.  You cannot say mine is wrong (if corrected to N --> 2N), because it follows from the data which I can manipulate in any way I want, and so can you (your points 1,2,3).  You can argue my Rules in section 4.3 if you want, or come up with your own, but all I say in BiSM_v2 is from those rules, and is an expression of their content and physical origin.  What is your physical origin, which I would like to know?

So two points, there is no nonlocality, and the curve resembling the Lemniscate is an expression of the raw data showing the double cover of SU(2) over SO(3). You can criticise it, you might not like it, you might think it irrelevant, but it is not wrong.

I did look at your notebook a bit and am not in a position to critique it much, Program A  reproduces the singlet only after conditioning the data.  You drop coincidences and I do not.  Program B:  did Richard or you say it is retrocausal?  Mine is not,  I do not see your underlying physical process. What can we deduce about the electron from your tables?   To me, your tables demonstrate that computationally reproducing 2\sqrt2 proves almost nothing about physical mechanisms. The question is then what physical structure produces your joint law?  Mine is that spin is a quaternion.

Interested in your comments of course.

Bryan

[Richard Gill]

Exactly!

You reproduced it computationally.

But when and where are two quaternions multiplied?

You say it only happens after someone looks at the combined data.

So you believe that the actual outcomes +/-1 don’t exist till long after all the records from both locations are brought together.

That’s an interesting point of view. You postpone the Heisenberg cut till the moment that someone casts their eyes on the complete data of the whole experiment. Till then, nothing had happened, everything was in quantum superposition. Isn’t there Oxbridge many worlds theory philosopher of science who believes that?

Of course it is a legitimate interpretation of quantum theory that wave function collapse occurs in the mind.



Sent from my iPad

[Richard Gill]

I would say that that statement is an oxymoron. Quaternion algebra is non commutative and quantum entanglement of two two-level systems is the reason that joint law is what it is. Problem is, it isn’t a *mechanism*. There is no mechanism.

Unless you believe in non-locality and superdeterminism. Then you can come up with other *reasons* for that joint law, eg certain symmetry assumptions. But again, nobody yet came up with a *mechanism*.

Sent from my iPad

> On 30 Sep 2026, at 14:06, Bryan Sanctuary <bryancs...@gmail.com> wrote:
>

[Richard Gill]

You calculate the probability of coincidence (equal vs unequal)
Then you simulate the event
Then you simulate the binary outcomes.

To get the outcomes for one trial, you need to know both the settings.



Sent from my iPad

> On 30 Sep 2026, at 14:06, Bryan Sanctuary <bryancs...@gmail.com> wrote:
>

[Bryan Sanctuary]

Richard,

I assume that reply was to me.

• I reproduce the violation the same way, I believe, that Nature follows. Bell and you do not.
• I NEVER said it happens when we look at the data, We have no stupid solipsism.
• I say the +/- outcomes occur at the detector. When do you think it happens? By fiat?
• I postpone nothing,  You are wrong about Heisenberg. You do not understand what I wrote.
• I have no wave function collapse, to suggest I do is a figment of your imagination.

I think by tomorrow I will give you a clarification of my joint definability proof.

Bryan

[Bryan Sanctuary]

Richard

Who are you writing to? If it is to me, it makes no sense.

Bryan

[Richard Gill]

Dear Bryan, cc. Bell group, 

What I write to you, Bryan, almost never makes any sense to you. Everybody here knows that already. Including you yourself! 🤪

Richard 

Sent from my iPad

On 30 Sep 2026, at 14:54, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Bryan Sanctuary]

Richard and anyone, show me in the paper, the exact place that I use nonlocality.  You cannot, and saying I do does not change it.  I use a product and that contains no nonlocality.  

Where is the nonlocality?  The L quaternion is at Alice and the R quaternion is at Bob.

I do not care who agrees with you.  You can all be wrong. For example, you all seem to think that Cl(1,3) is the  unique algebra, and you are all wrong.  For 100 years, you were all wrong.  It can also be Cl(2,2).  Also, most physics believes in nonlocality.  They are all wrong.  Once, a lot of people believed the Earth was flat.  They were all wrong.

So it matters to me not one iota that people agree with you. My paper shows they are all wrong.

Bryan

[Richard Gill]

We’ll soon find out what the editors of Quantum Reports think. We already know what Anton’s AI thinks. I hope Bryan you’ll soon post your paper on arXiv so that Anton and I can do the same.

A quaternion is not “at a place”. That’s category error. 

I don’t have any thoughts about Cl(2, 2) versus Cl(1,3). Are you talking about Clifford algebras over the reals?

Sent from my iPhone

On 30 Sep 2026, at 15:26, Bryan Sanctuary <bryancs...@gmail.com> wrote:



Richard and anyone, show me in the paper, the exact place that I use nonlocality.  You cannot, and saying I do does not change it.  I use a product and that contains no nonlocality.  

<image.png>

Where is the nonlocality?  The L quaternion is at Alice and the R quaternion is at Bob.

I do not care who agrees with you.  You can all be wrong. For example, you all seem to think that Cl(1,3) is the  unique algebra, and you are all wrong.  For 100 years, you were all wrong.  It can also be Cl(2,2).  Also, most physics believes in nonlocality.  They are all wrong.  Once, a lot of people believed the Earth was flat.  They were all wrong.

So it matters to me not one iota that people agree with you. My paper shows they are all wrong.

Bryan

[Richard Gill]

Dear Bryan

Cc Google group friends

Wikipedia tells me that when the vector space V is a finite-dimensional real vector space and the quadratic form Q is nondegenerate, Cl(V, Q) may be identified by the label Cl_p,q(R), indicating that V has an orthogonal basis with p elements with e_i² = +1, q with e_i² = −1, and where R indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. 

[Every nondegenerate quadratic form Q on a real vector space is equivalent to a diagonal form

where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form.^[44] The real vector space with this quadratic form is often denoted R^p,q. The Clifford algebra on R^p,q is denoted Cl_p,q(R).]

I read that Cl_1,3(R) is isomorphic to the multiplicative group of 2x2 matrices of quaternions, while Cl_2,2(R) is isomorphic to the multiplicative group of 4x4 real matrices. 

Each of those groups is a real vector space under addition and multiplication by real scalars. Do you agree or do you use a different definition from the one on Wikipedia?

Richard

On 30 Sep 2026, at 15:25, Bryan Sanctuary <bryancs...@gmail.com> wrote:

Richard and anyone, show me in the paper, the exact place that I use nonlocality.  You cannot, and saying I do does not change it.  I use a product and that contains no nonlocality.  

<image.png>

Where is the nonlocality?  The L quaternion is at Alice and the R quaternion is at Bob.

I do not care who agrees with you.  You can all be wrong. For example, you all seem to think that Cl(1,3) is the  unique algebra, and you are all wrong.  For 100 years, you were all wrong.  It can also be Cl(2,2).  Also, most physics believes in nonlocality.  They are all wrong.  Once, a lot of people believed the Earth was flat.  They were all wrong.

So it matters to me not one iota that people agree with you. My paper shows they are all wrong.

Bryan

[Parker Emmerson]

Dear all,

Attached is a short paper that puts the class-characterization reading of Bell's theorem into exact form. It grows out of the exchange on Bryan's model, and it cites that work throughout.

On each trial each of the four setting pairs either agrees or disagrees, so a trial carries four agreement signs. Split the sixteen possible patterns by parity. A record, the four outcomes x, x′, y, y′ fixed together, which is what every model in Bell's class has for each source state, always lands on the even half, where each CHSH form takes only the values ±2. So Bell's class is exactly the set of pattern distributions on the even half, and the CHSH minus sign is a reflection of one context.

Every quartet of correlations has a positive pattern distribution, and the least weight it must put on the odd half is (|S| − 2)/2, taken over the eight CHSH forms. That's proved by an exact enumeration of the vertices of the dual linear programme. For the singlet it is √2 − 1, attained by an explicit distribution; for a Popescu–Rohrlich box it is 1.

Bryan, this is where your joint-definability point lands. The joint law over the four outcomes forms exactly when the odd-half weight can be taken to vanish, so the failure of joint definability you identified is precisely a positive weight on the odd half: a property of the statistics that a single record cannot carry, as you've said of your model. Richard and Anton's critique places your cosine outside the record class. The paper supplies the measure of how far outside, √2 − 1 for the singlet.

Richard, on your two-computer challenge. Its rules make each computer a function of its own setting and the message, which is the record class, and its verdict |S| ≤ 2 is what those same rules entail. So it judges an entrant by the definition it requires the entrant to meet, and the verdict is fixed before any program runs. It is the reductio of Bell's theorem written as a game: the rules are the premise, an entrant's claim is the hypothesis, a run inside the bound is the contradiction, and the standing verdict is the conclusion of the rule that builds it. Each failed entrant is an instance of the class property, and a list of them adds nothing to the theorem that fixed them. What the challenge does well is catch a model that claims to be in the class while computing a joint quantity. A mechanism that declares itself outside the class isn't tested by it.

What carries content is a relation the rules don't already contain. For the Bell quartet that relation is the pair of absorber boundaries. Run my race with them excluded, the source silent after emission, and it returns your challenge's verdict, |S| ≈ 1.41. Run it with each boundary entering along its own leg, with the only links running from the source to each wing, and it returns 2.83, with every marginal at ½ and 70.7% of seeds on the odd half. The bound belongs to the rules; the violation belongs to the boundaries.

Richard, Bryan, on the Clifford algebras. The identifications are right: with Wikipedia's convention, p generators squaring to +1 and q to −1, Cl₁,₃(ℝ) ≅ M₂(ℍ) and Cl₂,₂(ℝ) ≅ M₄(ℝ), both of real dimension 16. Two refinements. They're isomorphisms of algebras rather than of multiplicative groups: GL₂(ℍ) and GL₄(ℝ) aren't vector spaces, since the sum of two invertible matrices needn't be invertible. And the isomorphism type forgets the quadratic form: Cl₃,₁(ℝ) is also M₄(ℝ), so which signature a model uses is fixed by which elements are declared to square to +1 and −1, and the matrix algebra alone doesn't record that.

On whether a quaternion is at a place: in its own space it is. Unit quaternions are the points of the three-sphere, the space of orientations, and a rotor carried by a particle sits where the particle is. What sits at neither wing is the relative rotor Q_A Q_B⁻¹, the relation between the two, and that's what carries the correlation. The singlet correlation is the same kind of object: a function of one cross-ratio of the four labelled analyser directions, with the distance between the wings absent from it.

The smallest Clifford case with a sign in it is Cl₀,₁(ℝ), one generator with i² = −1: the complex numbers. Squaring there is two-to-one, z and −z sharing a square, so a square root has to choose a sheet. That choice is where phenomenological velocity lives. Put the Lorentz coefficient κ into the two factors under a radical, multiplying one and dividing the other, and the product of the two principal roots equals the root of the product times a sign ε = ±1. Squaring erases ε. On the solutions, ε is sgn(lα) inside the light cone and +1 beyond it, so what is physical about v is the sign, which side of the cone, rather than the number. And ε is a nonseparable function of two signs: the ℤ/2 parity obstruction of a maximally CHSH-violating sign table, produced by a branch cut alone, before any probability enters. That parity is the odd half in the paper. The details are in What Squaring Erases.

S. M. H. Emamifar, Anton, a correction to what I wrote you: I described the arithmetic phenomenological velocity comes from as the split-complex numbers. It is the principal branch of the square root, as above, and the velocity's content is the sheet sign that squaring erases.

Two concrete asks, toward the test that would settle κ. Do you know of a group with time-tagged coincidence data from a high-visibility source, taken at many analyser angles rather than the four CHSH settings? And what test statistic would you use to bound the cubic term in E against cos(a−b), with visibility as a nuisance parameter? In the paper's terms, κ below 1 lowers the singlet's weight on the odd half in a way a visibility loss can't.

The verification script is in the appendix and in the zip. Please check the vertex enumeration, the mirror-weight formula and the two runs.

Parker

[Richard Gill]

Parker, that’s the whole point: “the verdict is fixed before any program runs”. The challenge is a pedagogical tool. The verdict is inevitable.

You’d be amazed how many quantum crackpots have tried to win it.

I think people who study Bell’s theorem and don’t realize they can’t win this challenge are weak in the head.

Bryan is smart enough to realize it can’t be done.

Gordon Watson should take up the challenge.

I correspond with several other people who are trying. I have warned them that they won’t succeed. Let them learn by trying.

Sent from my iPhone

[Richard Gill]

Good for you, Bryan! I admire your tenacity and your learning.


Sent from my iPad

> On 30 Sep 2026, at 15:26, Bryan Sanctuary <bryancs...@gmail.com> wrote:
>

[Bryan Sanctuary]

Parker,

Only a short comment, since your recent email is clear enough. I believe we agree on the essential mathematical point: the violation corresponds to a failure of joint definability. That is, the four pairwise correlations cannot arise as marginals of a single joint Boolean probability space of the Bell type.

Where we still differ is over whether the joint character of my analysis should be regarded as physical nonlocality. I maintain that post-analysis is not a physical nonlocal interaction. The local detector events have already occurred before the two records are brought together and paired. In the experiment, as you know, the experimenter does the same with time stamps, I just add geometry. 

Thus the scalar is joint in the straightforward mathematical sense that it is constructed from two records; that does not imply any physical influence or communication between the two wings, once you open the envelope, the correlation is observed, nothing is passed.

Bryan

[Parker Emmerson]

Richard, Bryan,

Richard, you've said the verdict of your challenge is fixed before any program runs. In the math: your rules define the class K of triples (f, g, ρ), with x = f(a, λ), y = g(b, λ) and λ ~ ρ drawn independently of (a, b). For every member and every λ, the agreement signs over the four setting pairs satisfy e₁e₂e₃e₄ = (f(a₀,λ) f(a₁,λ) g(b₀,λ) g(b₁,λ))² = +1, so the CHSH form e₁ + e₂ + e₃ − e₄ takes only the values ±2, and |S| ≤ 2 for every member. The game illustrates that theorem and demonstrates only that theorem: every failed entry is an instance of the identity. What it does well is detect a model claimed to be in K whose outputs in fact depend on both settings.

But "inevitable" belongs to the theorem, not to any run of the game, and the gap between the two is an infinite regress. Every run is finite, so a member of K can win by fluctuation. Bell's own model under your protocol, 20,000 runs of N = 100: 0.7% reach |Ŝ| ≥ 2√2, the first at run 36 with 2.883. The standard answer is that a lucky win fails replication, and it does: five fresh runs gave 1.47, 1.86, 2.34, 2.11 and 1.17, and one run of 10⁴ gave 2.005. But a replication is another finite run, open to the same fluctuation, and so is the replication of the replication. Every finite stage leaves the verdict uncertain. The regress is stopped by convention: a committed run length and a p-value threshold, chosen in advance. So what the game returns is a p-value, and "inevitable" is a statement about an infinite-trial limit that no play of the game reaches. The same regress, stopped the same way, stands behind every experimental Bell verdict, the Delft test's 245 trials at p = 0.039 included. So it doesn't decide between claims. What decides is which claims a committed run can move.
The scope of your universal. The identity above uses exactly one property: x is measurable with respect to (a, λ) and y with respect to (b, λ). That is your rule that the source falls silent at emission. It is silent on the class B in which the pair is a component of a completion map C(λ, a, b) ∈ {±1}² with P(x | a, b) = P(x | a) and P(y | a, b) = P(y | b). My race is a member of B: (x, y) is the argmin over the four candidates of T_xy = −ln U_xy / Γ_xy, with Γ_xy ∝ m_xy^κ normalised to a constant total, m_xy = ¼(1 − xy cos(a − b)), and U the source record. It gives |S| = 2.828 ± 0.003 at κ = 1, marginals ½, every trial kept, reproducibly across seeds. Realised on three machines, C is evaluated at the source vertex with a and b arriving along the two legs, and each wing machine receives only its own component. Each wing's output is the component C delivers, rather than a value computed on receipt of its own setting. The step outside K is that deferral: the evaluation of C after both boundaries exist. So your rules exclude exactly one operation, and with it allowed the singlet is an ordinary member of the class.

What B has that K lacks, measured with positive weights. Report the four agreement rates pᵢ rather than the netted S. Under K every trial's agreement pattern lies in the even coset P₊ = {e : e₁e₂e₃e₄ = +1}. The least weight any pattern distribution with the observed rates must place on the odd coset P₋ is w* = max(0, (S − 2)/2) over the eight CHSH forms, proved by exact enumeration of the vertices of the dual programme; for the singlet w* = √2 − 1. Replaying one U across the four setting pairs, the race puts 70.7% of U on P₋ at κ = 1. So K is exactly the class supported on P₊, and B is where the weight on P₋ lives.

The nature of the result, and its status. Bell's theorem is a class characterization: a quartet of correlations comes from a member of K exactly when a pattern distribution on P₊ reproduces it, and exactly when all eight CHSH forms satisfy |S| ≤ 2. That is Fine's theorem in pattern form, a positive statement exact on both sides of its hypothesis and proved directly. On that basis I deprecate it as a no-go theorem and retain it as the characterization, which carries every proved content of the theorem. The no-go reading adds two steps the theorem doesn't contain. The first is the description "a no-go proved by reductio," which is tautological: the rule (P ∧ Q ⊢ ⊥) → (P ⊢ ¬Q) is the canonical constructor of the no-go form P ⊢ ¬Q, so naming it says nothing about scope, which lies entirely in P. The second is the deletion of one premise after the collision with the data, named non-locality, which carries the class through the word "local" to every mechanism the word can be applied to. For measurement, the replacement is the mirror weight, a table's distance outside K. For physics, what lies outside K is a question of mechanism, and that's where B and κ come in. Version 7 of the paper, attached, sets this out as a deprecation notice.
Bryan, agreed on both counts: the violation is the failure of joint definability, and a joint quantity isn't by itself physical non-locality. In the sense that matters for signalling you're right: every marginal stays at ½ whatever the far setting. Where I'd place it differently is the pairing. The experimenter pairs records that contain a time stamp, a setting and a ±1, and the four tables built from those records already violate the bound. A pairing can only read what the records carry. If each wing's ±1 had been fixed by its own setting and a shared phase, the paired tables would be a mixture of fixed cards and would stay on the even coset; that's what the Bell control in your own program shows, with its triangle. So the failure of joint definability is a property of how the records were made, before anyone opens the envelopes, and the continuous phases your analysis adds aren't in the experimenter's records. That's why I put the joint structure at the event: the pair is completed together from the source record and both settings, with each marginal fixed. It's where your geometry can do the work you want it to do, and placed there it makes a prediction a scan can check.

Which claims a committed run can move. That nature lies outside K: the loophole-free tests moved it, to p-values small enough to close the fluctuation question, and we agree on it. That the randomness is irreducible and non-local: at κ = 1 every completion predicts the same table, so no run of any length raises this reading above its rivals, and the only result that could move it, κ < 1, would move it down. Its weight is set by declaration rather than by data. That κ < 1: a pre-registered dense angular scan moves it either way. With visibility profiled out and trials spread evenly over cos(a−b) in [−0.9, 0.9], the Fisher information for κ at κ = 1 is 0.0203 per trial, so a 3σ bound on 1 − κ at 10⁻³ takes about 4.4 × 10⁸ trials and at 10⁻⁴ about 4.4 × 10¹⁰: minutes and half a day with a bright source at 10⁶ pairs per second. The cosmological anchoring, 1 − κ ≈ 8 × 10⁻⁷, needs a run locked to the sidereal day.

Your challenge isn't on that list at all: it's the membership test for the class the theorem characterizes. Its outcomes carry evidence about one thing only, whether an entrant's program is really in K. Cited as evidence for anything beyond that, the theorem, a model outside K, or nature, it begs the question, because the conclusion is written into the rules every entrant must meet, and a list of failures is that conclusion repeated. Cited as inevitable, it overstates what any finite run returns, which is a p-value. The third claim is the only one in this thread that a measurement can raise or lower, and it's cheap to test. You're the person on this list best placed to write its test statistic and its finite-sample bound, and to name a group with time-tagged coincidence data at many analyser angles. Will you?

Parker

[Richard Gill]

Parker, I’ve worked on the probability theory. You need to read my survey paper in Statistical Science, and some other papers.

The challenge is a joke, a pedagogical tool. It has worked well.

If someone wants a bet, we will have to agree in advance on sample size, win/lose criterion, and wagers, time scale etc etc, and various technical “reproducibility” and “verifyability” criteria. I did the maths more than 20 years ago,

Sent from my iPad

On 30 Sep 2026, at 18:52, Parker Emmerson <powerin...@gmail.com> wrote:



Richard

[Richard Gill]

It depends on what the two records consist of, precisely.

In real Bell experiments Alice and Bob each have time-stamped lists of binary inputs and binary outputs. They don’t collect quaternions and multiply them after the experiment.

They sort the data into the four sub experiments according to the inputs, and average the products of outputs



Sent from my iPad

[Richard Gill]

Parker

I’ve done that more than 20 years ago in a number of publications. My martingale methodology was adopted and sharpened by the four loophole-free Bell experimenters of 2015-2017.

I’ve published papers with Gisin and with Zeilinger.

I’ve published statistical analyses of the data from those experiments using various different methods,

What do you mean by “name a group”?

Richard


Sent from my iPad

> On 30 Sep 2026, at 18:52, Parker Emmerson <powerin...@gmail.com> wrote:
>

[Bryan Sanctuary]

Parker,

Commenting on my part, I will state my mechanism that bridges the gap between the geometric correlation and the recorded Boolean outcomes. Let me know if you agree or have a different mechanism please.

I do not obtain that harmonic by averaging the original products (A_k, B_k), which gives the Bell triangle. I first reconstruct the scalar from which the coherence is expressed using the experimental coincidence populations. This gives R_k and I digitise that.

I welcome your insight into that extra step: which is instantiate first and digitize second, Eq. (52).

Thanks

Bryan

[Parker Emmerson]

Well the joke’s on me, I guess, because I’m the one who had to sort through it.

[Richard Gill]

Bryan, you are talking about computation, not about physics. 

Not about physical mechanisms.

I find it fascinating that you don’t see any distinction between those two worlds.

Sent from my iPhone

Richard

[Bryan Sanctuary]

Richard,

I have found a second way to linearize the KG equation, discovered the bivector, done classical mechanics on it, shown how it is consistent with the Dirac equation, shown the classical quantum correspondence, found the mechanism for quantum weirdness, related to the SM, clarified parity, and shown that the polytope becomes a harmonic, and that is just to begin with. 

For you to say I did no physics is absurd and shows your bias.  What more should I do? Please tell me. Make a list. Again, another unsubstantiated comment from you which is fundamentally wrong.  That is why I react negatively to you: your motives are misplaced.

Bryan

[Richard Gill]

Dear Bryan

My comment was thoroughly substantiated.

You ask what more should you do.

I’d like you to read Bell’s last paper (La Nouvelle Cuisine) and tell me what you think is the relation between his concept of local causality and your work.

I would like you to read my short arXiv preprint and tell me where you think I make mistakes. It’s https://arxiv.org/abs/2211.05569

I would appreciate it if you would answer my questions about those two Clifford Algebras.

Yours

Richard

Sent from my iPad

On 30 Sep 2026, at 21:33, Bryan Sanctuary <bryancs...@gmail.com> wrote:



[Bryan Sanctuary]

Richard,

I am not going back to read Bell or your papers again.  My paper discusses the experimental arrangements and derives the Malus coincidence probabilities within the rotor description.  

It is all there,

Bryan

[Richard Gill]

I didn’t think you would. I don’t think you ever did read those two papers.

And what about my question about Clifford algebras?

Sent from my iPhone

[Bryan Sanctuary]

Richard

What about Clifford algebra?  

Bryan

[SMH Emamifar]

Dear Bryan,

Your comment about the mechanism bridging the gap between geometric correlation and recorded Boolean outcomes made me look again at the comparison between your construction and mine.

I also read the Gill–Vrba critique more carefully. Their main objection, as I understand it, is not to the existence of the quaternion geometry itself, but to the physical and probabilistic bridge between the local detector-response construction and the jointly formed scalar that gives the cosine correlation.

That made me notice a possible overlap between our approaches which may be more precise than simply saying that the geometries look similar.

In your construction, the relative quaternion rotor gives the half-angle coincidence structure

P_neq = cos^2[(a-b)/2],

P_eq = sin^2[(a-b)/2].

In my construction, a half-angle structure appears earlier in the dependency chain, before the Bell pair is introduced.

The electron geometry gives an oriented 4π recurrence and a relative-frame half-angle law. Under a separate, explicitly conditional two-channel quadratic readout, this becomes

W_+(θ) = cos^2(θ/2),

W_-(θ) = sin^2(θ/2).

Only after that step do I introduce the antisymmetric pair preparation and the two-analyser joint construction, which then gives

E(a,b) = -cos(a-b).

I am not claiming that these are the same mechanism. In fact, what interests me is that the same half-angle / squared-response structure appears at apparently different levels of the two constructions.

So I think there is a fairly precise question we could ask:

Is there a genuine mapping between your quaternion half-angle geometry and the earlier electron/detector half-angle structure in my model, or does the similarity disappear once we specify exactly which physical object owns each angle and each probability?

This also seems to separate three questions which can otherwise become mixed together:

1. Where does the half-angle geometry originate?

2. How does that geometry become a binary detector response?

3. At what point does the explicitly joint/nonfactorizable Bell response enter?

In my own construction I am treating the third step explicitly as joint/nonfactorizable, rather than as a Bell-local counterexample, and I am leaving the remaining detector and source-microphysics bridges visible.

I would be very interested in your view on this specific comparison.

Best wishes,

S. M. H. Emamifar

[Richard Gill]

Dear Bryan

This is what I think you should do:

Consider equations (50), (51) and (52) in your paper, leading deterministically to Alice’s measurement outcome +/-1 as a function of the hidden variable lambda.

Do you agree that similar equations hold, involving the same lambda and Omega_free, but with something like B_b, Omega’_b and C’_b analogous to A_a, Omega_a and C_a?

[possibly “lambda” becomes “-lambda” on Bob’s side]

I write Omega prime and C prime for Bob-side objects to emphasize that these are not Alice’s set and function when she happens to use Bob’s setting b, but a different set and function.

I’d like to see (50’), (51’) and (52’) which do the same for Bob as (50), (51), (52) do for Alice.

Yours
Richard

[Richard Gill]

Dear SMH

Speaking for myself, I have indeed no objection at all to the use of Geometric Algebra to compute correlations in the the EPR-B situation. Bryan Sanctuary seems to admit the impossibility of a mechanistic, causal, description of local detector responses which respects real-world spatial-temporal constraints, since he agrees with Bell’s and Fine’s maths.

In the real world, according to the Bell picture, stuff goes from a source to two detectors. A measurement setting is introduced “from outside” into each detector. Each detector separately generates a response. Bell calls this “local causality”. Einstein had the same picture in his mind.

The story Bryan tells in his recently published magnum opus *appears* to respect that.

We know that experiment confirms QM predictions. So we are led to believe that Bell’s “naive” picture of how the world works is not correct. At least, not when applied to certain phenomena observed in quantum optics labs.

The picture Bryan paints is contradicted by his own computations!

I understand, SMH, that you too agree that “local causality” is not true and you are developing a mathematical approach which better fits to reality. Excellent!

Yours

Richard



Sent from my iPad

[Richard Gill]

PS. The story told by the simulations in Bryan’s paper is a different story. Quaternions depending on both settings are multiplied. A scalar part is extracted. This is used to simulate equality/not equality of measurement outcomes, but no measurement outcomes themselves, yet. After that, the outcomes are created by converting “equality” into “+ +” or “- -“ by the toss of a fair coin. Similarly for “inequality”.

Of course the simulation perfectly reproduces the EPR-B statistics. There are easier ways to do that. It proves nothing.


Sent from my iPad

[Bryan Sanctuary]

All

I must point it out.  

Richard's repeated comments are not designed to understand my work or yours, but to simply find anything negative, dicretitations are his aim, not science.  Nothing he says threatens my work but he rants on and spreads rumors.  This latest characterization shows he does not understand me  He still thinks my work is nonlocal, and from that error, he jumps to incorrect pronouncements.  The latest is a point: he gets it wrong, draws the conclusion and says:  "It proves nothing".

This disingenuous approach is what I object to.  It is basically only Gill who does it to many. Gill is simply out to destroy alternatives and defend Bell.  He can only interact with those who agree with him, and those who know little.  For any serious challenge, or from those who know a lot more than him,  he just tries to attack.  Since the police in the UK and Holland investigated him in his LucyL attitude, it seems he uses this belligerent approach  there too.  

 

I review a number of Post-Fine researchers: Richard, in the past, has attacked most of them with math obfuscation and then personally--those pople effected dispise Gill and will not communicate with him.  There is a long history, not just me. 

I cannot stop him, I can just point it out. He makes me angry of course. His 9 page AI rant against my paper is a huge waste of time and contains only two points that need answering.

He is in a club with Karl Mach and Gill is a Positivist by nature. 

Bryan

.

[anton vrba]

Bryan, your work is not science!, it is a series of Sanctuary impositions, with mathematical flaws, on Mother Nature.  

Your section 5.2. The SM vs. BiSM is simply bull shit, Your Definition 14  writes "A fermion is interpreted as a polarized state of this bivector. Mass and charge are proposed to arise as properties of the bound geometric product. Quantum behavior is proposed to emerge as a limiting description of the bivector dynamics rather than being postulated as its fundamental distinct ontology." formulate these proposal first rigorously only then can you state them!

Or: 5.2.1. Neutrinos  ...The cleaved object is neutral, chiral, massless, and without a mirror image. Energy resides only in propagation. A cleaved and free blade is indistinguishable from a neutrino. That is a imposition and not rigorously shown. Etc. 

Science is build from the ground up, the Standard Model is based on many years of work by many brought together into an apparent coherent frame, if we agree with the resulting theories produced is another matter.  Physicist are stuck in a 100 year old paradigm and are searching for a non existing graviton so that general relativity and quantum mechanics can be unified, in that paradigm they will search for the next 100 years without a result and they will all be awestruck how mysterious Nature is and will continue searching the subsequent 100 years.

Any meaningful contribution to physics would be working to resolve the above impasse, I see no evidence of that in your work.

Your rant against Richard is unjustified, he understands statistical mathematics better than a chemist, and he asks short pointed questions regarding your mathematics and if you cannot answer them it seems your only response is an attack on his personality.

And as a final point, we all on the forum observed your stance that you disproved Bell theorem, the Sanctuary--Gill bet is well documented. All challenges from  Richard ended in mudslinging like your email below.  It was a simple matter to write a comment to MDPI, based on the prior comments of Richard.  Since then you admit that Bell was not disproved but your work is out of scope of Bell's theorem and not applicable, I have not read an apology and concession from you to Richard on this forum in this regard.

regards

Anton

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

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

To "Richard Gill" <gill...@gmail.com>

Cc "Emamifar SMH" <smhema...@gmail.com>; "Parker Emmerson" <powerin...@gmail.com>; "anton vrba" <anto...@gmail.com>; Bell_quantum...@googlegroups.com

Date 10/1/2026 12:09:03 PM

Subject Re: [Bell_quantum_foundations] Joint Definability and the SM vs. BiSM

[Richard Gill]

Bryan, you asked me a question. You asked me what I would like you to do. I gave you a civil reply.

Please could you write out the analogues of equations (50), (51) and (52) for the source + Bob side of the model.

Richard


Sent from my iPhone

> On 1 Oct 2026, at 09:29, Richard Gill <gill...@gmail.com> wrote:
>
> Dear Bryan

> <image0.jpeg>
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> Please write them out.
>
> I think you’ll need to make your notation a bit more elaborate.
>
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> Sent from my iPad

[Parker Emmerson]

Richard, Bryan,

Here is the math behind my last messages, written out in full, and then what it means for locality.

1. Corner residues and the loop holonomy.
Settings a₀, a₁ for Alice and b₀, b₁ for Bob; the four pairs (aᵢ, bⱼ) are the seams of a loop a₀ – b₀ – a₁ – b₁ – a₀. A seeded completion gives, for each seed u and seam (i, j), outcomes (xᵢⱼ(u), yᵢⱼ(u)) ∈ {±1}², with agreement eᵢⱼ = xᵢⱼ·yᵢⱼ. A model run trial by trial reads one seam per seed; a program, or any model with a stated mechanism, can be read on all four seams at the same seed.
Corner residues: δᴬᵢ = xᵢ₀·xᵢ₁ and δᴮⱼ = y₀ⱼ·y₁ⱼ. A residue is −1 exactly when the outcome at that setting depends on which of its two seams completes it.
Holonomy: H = e₀₀·e₀₁·e₁₀·e₁₁.
Theorem: H = δᴬ₀·δᴬ₁·δᴮ₀·δᴮ₁ at every seed. Proof: group Alice's outcomes by corner and Bob's by corner.
All four residues are +1 exactly when the outcomes at that seed form one record (x, x′, y, y′), and then H = +1. So Bell's record class is the order-free class: the discrete form of Fubini's exchange of order.
Carried residue: any completion reproducing correlations E carries H = −1 on at least w*(E) = max(0, (S − 2)/2) of its seeds, taken over the eight CHSH forms, and carries at least that many residues per seed on average. The bound comes from the four agreement signs at each seed: they form a pattern in {±1}⁴, the seeds give a distribution over patterns with means E, and the least weight such a distribution places on patterns with product −1 is exactly w*(E). For the singlet, w* = √2 − 1.
Invisibility: with fixed marginals, P(x | aᵢ, b₀) = P(x | aᵢ, b₁) and likewise for Bob, so each corner's outcome has the same distribution along both seams. The residue is a seed-level relation, and every single-wing statistic stays as it is.

2. The exact residue of my race.
Uₓᵧ independent uniforms; c = cos(a − b); wₓᵧ = (¼(1 − xy·c))^κ; Γₓᵧ = wₓᵧ / Σw; Tₓᵧ = −ln Uₓᵧ / Γₓᵧ; the pair is the argmin. Its correlation is E = −tanh(κ·artanh c), the singlet at κ = 1.
Coupling law: at one seed, the agreements e₊ at c and e₋ at −c differ with probability tanh(κ·artanh c).
Proof: V = −ln U ~ Exp(1). Let α = min V over the agreeing pair and β = min V over the disagreeing pair, independent Exp(2), with the minimizing candidate in each pair the same at both parameters. With p ∝ (1 − c)^κ and q ∝ (1 + c)^κ the rates of the agreeing and disagreeing candidates at c, the parameter −c swaps them, and e₊ ≠ e₋ exactly when p/q < α/β < q/p. Since P(α/β ≤ s) = s/(1 + s), the probability is (q − p)/(q + p) = ((1 + c)^κ − (1 − c)^κ) / ((1 + c)^κ + (1 − c)^κ) = tanh(κ·artanh c).
Standard quartet a₀ = 0, a₁ = π/2, b₀ = π/4, b₁ = 3π/4: cos(a − b) = 1/√2 on three seams and −1/√2 on (a₀, b₁). Then δᴬ₁ = δᴮ₀ = +1 and H = e₊·e₋. On every seed with H = −1, exactly one of δᴬ₀, δᴮ₁ is −1, each with probability ½; on every other seed both are +1. And P(H = −1) = tanh(κ·artanh(1/√2)): 1/√2 at κ = 1 and √2 − 1 at κ = ½. Simulated at κ = 1, 0.9, 0.75, 0.6, 0.5: 0.7071, 0.6606, 0.5794, 0.4846, 0.4145, against 0.7071, 0.6602, 0.5790, 0.4845, 0.4142 from the formula. So the race's residue rate equals the magnitude of its correlation.

3. The same residue in a double series.
For l ≠ n, 1/(n² − l²) = (1/2n)·(1/(n − l) + 1/(n + l)), and summing over l with harmonic numbers gives Σ (l ≠ n) 1/(n² − l²) = −3/(4n²). Summing rows first gives −π²/8; columns first, +π²/8. The order residue is π²/4. For 1/|n² − l²|ˢ with s > 1 the sum converges absolutely and both orders agree. So in the family 1/|n² − l²|ˢ, the weight s = 1 is the one rung at which the value of the sum depends on the order of completion; I call that rung Ω_Λ. Calling a completion "at Ω_Λ" when its order residue is nontrivial: records are order-free on every seed; every completion of the singlet is at Ω_Λ on at least √2 − 1 of its seeds; my race is at Ω_Λ on exactly tanh(κ·artanh(1/√2)) of its seeds.

4. Clifford algebra: Spin and Pin lifts.
Richard, on your Clifford question. Write Clₙᵋ for the real algebra generated by ℝⁿ with v·v = ε|v|². In Wikipedia's notation, Cl₂⁺ = Cl₂,₀ ≅ M₂(ℝ) and Cl₂⁻ = Cl₀,₂ ≅ ℍ.
The even part of the plane's algebra: i = −ε·e₁e₂ has i² = −1, so Cl₂⁰ = span{1, i} ≅ ℂ, and Spin(2) = {cos φ + i·sin φ} is its unit circle. The covering ρ: Spin(2) → SO(2) is squaring: ρ(cos φ + i·sin φ) = rotation by 2φ, with kernel {±1}. A rotation path lifts to a closed loop in Spin(2) exactly when it turns an even number of times.
The principal square root picks one sheet of that cover, and √z·√w = ε(z, w)·√(zw) with ε ∈ {±1}. That sign is the cocycle of the double cover, and squaring erases it. It's also where phenomenological velocity lives. In that relation, κ = √(1 − v²/c²) multiplies one factor under a square root and divides the other, √(L₊κ)·√(L₋/κ) = ε·√(L₊L₋), with L₊ and L₋ the two factors of the radicand. On the solutions, ε = sgn(lα) inside the light cone and +1 beyond it. So what is physical about v is the sign, which side of the cone, and that sign is a choice of sheet of the cover above.
Pin: the reflection of the phase circle lifts into Pin⁺(2) ≅ O(2), where the lifted reflection is an involution, or into Pin⁻(2) = U(1) ∪ U(1)·j inside the unit quaternions Sp(1), where it has order four and squares to −1. For a quantum system with antiunitary time reversal T, T² = σ, the phases together with T realize Pin⁺(2) when σ = +1 and Pin⁻(2) when σ = −1, the spin-½ case. So quaternions enter spin-½ physics through time reversal, as Pin⁻(2) ⊂ Sp(1). Your Cl₁,₃(ℝ) ≅ M₂(ℍ) is quaternionic for the same family of reasons.

5. The Bell loop as a two-sheeted decoration, and its two classes.
The agreement signs eᵢⱼ are a ℤ/2 decoration of the loop a₀ – b₀ – a₁ – b₁ – a₀, and H is the sign the loop returns with. The decoration lifts, meaning eᵢⱼ = xᵢ·yⱼ for some values at the corners, exactly when H = +1. That's the same structure as the rotation path in section 4, which lifts to Spin(2) exactly when its sign is +1.
That gives two classes, each defined by what it carries. The untwisted class: H = +1 on every seed, every decoration a lift, every seed a record. This is Bell's record class. The twisted class: H = −1 on a positive share of seeds. The singlet belongs to the twisted class, with share at least √2 − 1, and so does the race, with share exactly tanh(κ·artanh(1/√2)).

6. What the two-computer challenge certifies.
Your rules make xᵢⱼ = f(aᵢ, λ) and yᵢⱼ = g(bⱼ, λ), so eᵢⱼ = f(aᵢ, λ)·g(bⱼ, λ): a lift, with H = +1 on every seed. The win condition asks for H = −1 on at least (S − 2)/2 of the seeds. So the challenge asks for a twisted band glued from untwisted strips. A band glued from untwisted strips is untwisted; that's what the strips are. What the challenge certifies is membership in the untwisted class, and it's good at catching a program that claims to be a lift while computing a joint quantity. Nature's tables belong to the twisted class, and so does the race, which carries the twist on tanh(κ·artanh(1/√2)) of its seeds with every marginal fixed. The challenge's rules set that class aside from the start.

7. Where the twist is carried, and its price.
The twist can be carried by the record instead of the completion. Let each outcome be a function of its own setting and a record, xᵢⱼ = f(aᵢ, λ) and yᵢⱼ = g(bⱼ, λ), so the completion is order-free, and let the record's distribution ρᵢⱼ depend on the setting pair, as it does when the record is fixed by constraints that include both settings. Richard, this is the route you pointed to with superdeterminism, read as a boundary constraint rather than a conspiracy.
Price: write each ρᵢⱼ as a common reference ρ plus a deviation, with δ the largest total-variation distance from ρ. The reference contributes at most 2 to S, and each deviation at most 2δ, since every correlation lies in [−1, 1]. So S ≤ 2 + 8δ, that is δ ≥ (S − 2)/8, and a linear programme over the sixteen records attains it.
At the standard quartet: κ = 1, |S| = 2.828, δ ≥ 0.104, which is (√2 − 1)/4; κ = 0.9, |S| = 2.641, δ ≥ 0.080; κ = 0.75, |S| = 2.316, δ ≥ 0.040; κ = 0.6, |S| = 1.938, where the record class suffices with δ = 0.
So the singlet's twist is always carried, in one of two places, each with a floor: in the completion, as an order residue on at least √2 − 1 of the seeds with a record independent of the settings; or in the record, as a setting dependence of at least (√2 − 1)/4 in total variation with an order-free completion.

8. Locality, Bell and Einstein.
Three things go by the name locality, and the math separates them.
(a) No signalling: each wing's statistics are independent of the far setting. This holds. The residue leaves every single-wing statistic as it is (section 1), and the race keeps every marginal at ½.
(b) Einstein's principle of local action: the real state of the far system is independent of what is done here. At the level of anything a wing carries, its state and its statistics, this holds too. The order residue belongs to the completed pair. It's a relation between two completions of one corner, the way a cross-ratio is a relation among four directions.
(c) Bell's local causality: each outcome is fixed by a complete past state and its own setting, with the source independent of the settings. That is exactly the untwisted class with a setting-independent record: every corner residue +1, every seed a record, H = +1. Nature's tables belong to the twisted class, carrying at least √2 − 1 of residue per seed for the singlet.
So Bell's theorem characterizes the untwisted class: order-free completions from a record independent of the settings, which carry H = +1 on every seed. The singlet belongs to the twisted class, and its completions carry the twist either in the completion, as an order residue, or in a record constrained by both settings, at the price in section 7. Bell named the untwisted class local causality, and the name carries the result further than the proof does: (a) and (b) hold.
Einstein argued that if (b) holds, quantum mechanics is incomplete. A completion exists: the race is one, with κ as the parameter quantum mechanics lacks. The form EPR assumed for it, separate values carried by each wing from the source, is a record. The elements of reality are completed pairs. At aligned settings the race fixes the pair with certainty, which is the EPR premise, and the certainty belongs to the completed pair.
And the completion runs on boundaries at both ends, so its order is the order in which a corner is completed across its two seams, with every marginal fixed.

Please check the coupling law in 2, the corner-residue identity in 1, and the price in 7; all three are short.

Parker

[Parker Emmerson]

Richard, Bryan,
Here is the form in which Bryan's position holds exactly, built from his own equations.
1. A Boolean outcome is a recovered sign. Alice's ±1 is A_a = sgn(cos(a − λ)), and sgn(z) = z/√(z²): squaring removes the sign, and dividing by the principal root of the square recovers it. At z = 0, here cos(a − λ) = 0, the recovery is 0/0. So each detector has a seam, the set of λ where its own data give its sign as 0/0.
2. Hold the seam rather than assigning it a value. Put a finite scale ν where classical arithmetic puts 0: sgn_ν(z) = z/√(z² + ν²). Each detector then has a seam band of finite width. Off the band, the outcome is Bryan's A_a(C_a(λ)), a function of the setting and λ. On the band, the wing's own data leave the sign held, with both values present. That is the form in which Bryan's Lemma 3 is true: the four Boolean outcomes are jointly defined on Ω_free off the seam bands, and held on them until instantiation.
3. Who resolves the held sign decides the class. If each wing resolves its own seam, the outcome is again a function of that wing's setting and λ, the model is a record, and the statistics are triangle-type, as Bryan's Bell control shows. If the pair resolves the seam together, against the relative phase, the resolution carries the twist. When the resolution keeps each wing's odds fixed, every single-wing statistic stays as it is: no signalling, and nothing acting on the far wing's state. That is Bryan's "no nonlocality," in the two senses that hold.
4. The sign a joint resolution fixes is non-separable: √z·√w = ε(z, w)·√(zw), with ε ∈ {±1} depending on both arguments. On a CHSH sign table that ε is exactly the ℤ/2 parity that the twisted class carries.
5. Bryan's Q_AB = exp[ie₂(a − λ)]·exp[−ie₂(b − λ)] = exp[ie₂(a − b)] is the right object for the resolution: λ cancels, so it is the pair's own invariant. The held signs are resolved against it, and the correlation is its scalar part. So his (54) belongs to the instantiation itself, where he puts it, rather than to an analysis afterwards.
6. The cost is measurable. A seed where neither wing is held at any of the quartet's settings has four fixed outcomes: a record, with the loop sign H = e₀₀·e₀₁·e₁₀·e₁₁ = +1. The singlet carries H = −1 on at least √2 − 1 of its seeds, so held seeds make up at least √2 − 1 of all seeds. A resolution law that meets this with fixed marginals is the exponential race: candidate rates ∝ (¼(1 − xy·cos(a − b)))^κ, the earliest clock fixing the pair, giving −tanh(κ·artanh cos(a − b)), which is −cos(a − b) at κ = 1, with every marginal at ½.
So the one change Bryan's equations need is at the seam. As written, (51) and (52) make C_a and A_a functions on all of Ω_free, which closes the seam, and that is why Bob's analogues of (50) to (52) give a record. With the seam held at a finite scale and resolved by the pair against Q_AB, his three claims hold together: joint definability fails on the seam bands, the correlation lives in the relative rotor, and every single-wing statistic stays fixed.
Parker

[Richard Gill]

Sorry Parker, your math is quite beyond me. I’m just an applied mathematician working in probability and statistics, with a hobby in quantum information theory.

Sent from my iPhone

On 1 Oct 2026, at 14:52, Parker Emmerson <powerin...@gmail.com> wrote:



[Parker Emmerson]

Richard,

Then here are the three parts that are plain probability.

1. For any program in your class, each computer's output is a function of its own setting and the message. At each seed, take the four agreement signs over the four setting pairs and multiply them: you get (x·x′·y·y′)² = +1, since each output appears twice. So every seed of every program in your class has that product +1. The singlet needs the product to be −1 on at least √2 − 1 of its seeds; that's a small linear programme.

2. My race is an exponential race. Four candidate pairs, independent exponential clocks, the first to fire wins. Take one seed and two settings with correlation parameters c and −c. The agreeing pair's minimum α and the disagreeing pair's minimum β are independent Exp(2), and P(α/β ≤ s) = s/(1 + s). From that, the probability that the agreement flips between the two settings is ((1 + c)^κ − (1 − c)^κ) / ((1 + c)^κ + (1 − c)^κ) = tanh(κ·artanh c). At the standard CHSH angles that puts the product of item 1 at −1 on exactly 1/√2 of seeds when κ = 1.

3. Superdeterminism, which you pointed to, has a price in total variation. Let the hidden variable's distribution depend on the setting pair, and let δ be the largest total-variation distance of those four distributions from a common reference. Then |S| ≤ 2 + 8δ, and a linear programme shows the bound is attained. For the singlet that's δ ≥ (√2 − 1)/4 ≈ 0.104.
What your rules prove, in the same terms: every program built to them carries the product +1 on every seed, so its CHSH value stays within 2. That's a true theorem about the class the rules define, and it's the one thing they prove. 

Three inferences drawn from the challenge go beyond it, and each has a name.

(a) Begging the question. The list of failed entries is cited as evidence about nature, or against a mechanism outside the class. But the conclusion is written into the rules every entrant has to meet, so the list only repeats it.

(b) Equivocating between the limit and the run. The outcome is called inevitable. That holds in the infinite-trial limit. Any finite run returns a p-value against a threshold fixed by convention, which is what your own terms for a bet require.

(c) Persuasive definition. The class is named "local," and the name is then carried to every mechanism the word fits. The rules test one class; the name extends the result to mechanisms the rules never tested.
Everything said on the strength of the challenge about locality, about nature, or about mechanisms outside the class rests on (a), (b) and (c), not on the rules.

(d) Infinite regress. A lucky win doesn't count because it fails replication, and it does fail. Run Bell's own model under your rules 20,000 times at 100 trials each and 0.7% of the runs reach |S| ≥ 2√2, the first at run 36 with 2.883; five fresh runs of the same program give 1.47, 1.86, 2.34, 2.11 and 1.17. But a replication is another finite run, open to the same luck, and so is the replication of the replication. Every finite stage leaves the outcome open. The regress is stopped by convention, a run length and a threshold fixed in advance, which is what your own terms for a bet require. So what the challenge returns is a p-value against a conventional threshold, and the same convention stands behind every experimental Bell result.

Stripped of the four fallacies, the challenge certifies membership in one class of models; every claim it has been used to support about locality, about nature, or about the people who enter it needs support it doesn't supply

If any of this is wrong, you're the right person to say where.

Parker

[Richard Gill]

Parker,  you can give fancy names to my computer challenges and wagers, but I am not particularly interested.  My challenge, and past wagers I have got involved in, were pedagogical tools. They were designed to help people learn by experience - by letting them hit up against a brick wall. They are a way to hold a conversation with a Bell-denialist. They don’t only help the person I am communicating with but also the onlookers. They provide entertainment for myself and others. It’s a trick. It’s a joke. It’s rather funny that you are taking it so seriously! But of course, that’s your prerogative to do so if you wish.

You call (a), (b), (c) and (d) fallacies. I call them features. You are the one calling the class “local”, but I don’t give it that name. The class is defined mathematically and you can call it whatever you like.

I have already done the probability calculations. I know the p-values.

[Richard Gill]

Dear Bryan

I’d still like to hear your yes/no answer to this question.

Consider equations (50), (51) and (52) in your paper, leading deterministically to Alice’s measurement outcome +/-1 as a function of the hidden variable lambda and her setting a.

Do you agree that similar equations hold, involving the same variable lambda and same set Omega_free, but with something like a function B_b, a set Omega’_b and a function C’_b analogous to A_a, Omega_a and C_a?

If the answer is no, please tell us what the analogous equations are for the other side of the experiment.

I’m guessing that Bob’s context-conditioned domain when using setting “b” is not the same set as Alice’s would have been had she used setting “b”.

It’s a simple notational question. It’s not a trick question.

Yours

Richard

Sent from my iPad

On 1 Oct 2026, at 13:09, Bryan Sanctuary <bryancs...@gmail.com> wrote:



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[Richard Gill]

Dear Bryan, dear friends

What a disgusting lie. Neither UK police nor NL police investigated me.

Corrupt UK police tried to prevent scientific information reaching the judge in the trial of Lucy Letby, which would have resulted in the immediate collapse of the trial had the defence been allowed to present it to the jury.

NL police merely acted as a courier service, getting my signature on a receipt to legally confirm that I had received the letter from UK police with its illegal allegations and threats. This was just intimidation, pure and simple.

Bryan, you sound quite deranged.

Richard



Sent from my iPad

> On 1 Oct 2026, at 13:09, Bryan Sanctuary <bryancs...@gmail.com> wrote:
>