Sci.Physics.Foundations

Dear friends

Has anyone seen Sabine Hossenfelders’ latest YouTube production, https://youtu.be/ytyjgIyegDI ? “Does superdeterminism save quantum mechanics? Or does it kill free will and destroy science?” A glance at her Google Scholar page will show that she has a recent huge production of papers on this theme many with co-authors Jonte Hance (Bristol) and Tim Palmer (Oxford).
https://scholar.google.com/citations?hl ... by=pubdate
The papers with Tim Palmer are often long and mathematically very complex. Clance is more a ‘foundations of QM’ person. I recommend their joint paper https://arxiv.org/abs/2109.02676 The wave-function as a true ensemble J.R. Hance, S. Hossenfelder

“Recommend” is not quite the right word. I think it is confusing and confused. I think its arguments are circular (if not completely illogical) and really that both this paper and Sabine’s video shows that she and her coauthors just don’t understand the arguments. Instead she’s on a crusade especially directed at influential older male physicists.

Her fans love this work: Quantum mysteries explained and resolved! It’s all a big hoax!

My opinion on the science: I think she is simply promoting non local deterministic models. The free will business is a red herring. Hance has some weird ontology to do this, Palmer has chaos theory and weird topologies. Both are smoke screens to hide from themselves as well as from us that the emperor (their actual argument) actually has no clothes.

Wishing everyone a happy mid-winter break and all the best for the New Year!

Yours
Richard

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https://arxiv.org/abs/2109.02676
The wave-function as a true ensemble
J.R. Hance, S. Hossenfelder

Abstract: In quantum mechanics, the wave-function only predicts probabilities of measurement outcomes, not individual outcomes. This suggests that it describes an ensemble of states with different values of a hidden variable. Here, we analyse this idea with reference to currently known theorems and experiments. We argue that the ψ-ontic/epistemic distinction fails to properly identify ensemble interpretations and propose a more useful definition. We then show that all ψ-ensemble interpretations which reproduce quantum mechanics violate Statistical Independence. Finally, we explain how this interpretation helps make sense of some otherwise puzzling phenomena in quantum mechanics, such as the delayed choice experiment, the Elitzur-Vaidman bomb detector, and the Extended Wigner's Friends Scenario.

A bunch of hogwash nonsense. All physical action in Nature is always local. Easy to prove.
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FrediFizzx wrote: ↑Wed Dec 22, 2021 1:08 am A bunch of hogwash nonsense. All physical action in Nature is always local. Easy to prove.

OK, so prove it and publish your proof, and tell Sabine Hossenfelder, Tim Palmer, and Jonte Hance, that their work is superfluous. Sabine and her co-authors are also arguing that it's local, but they need an awful lot of very heavy artillery to get this point across. Apparently, they think it is not obvious and not easy to prove.

gill1109 wrote: ↑Wed Dec 22, 2021 8:13 am

FrediFizzx wrote: ↑Wed Dec 22, 2021 1:08 am A bunch of hogwash nonsense. All physical action in Nature is always local. Easy to prove.

OK, so prove it and publish your proof, and tell Sabine Hossenfelder, Tim Palmer, and Jonte Hance, that their work is superfluous. Sabine and her co-authors are also arguing that it's local, but they need an awful lot of very heavy artillery to get this point across. Apparently, they think it is not obvious and not easy to prove.

Ok, more nonsense on top of nonsense. Superdeterminism is stupid when we already know it is not necessary to be deterministic. We have two simulations that are fully deterministic by just knowing the values of 3 vectors, a, b and s. So, already proven!!! Don't really care about publishing since I'm already publishing here on the forum. And don't really care about telling anyone past this forum since I already know the truth about Nature.
That is good enough for me.
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FrediFizzx wrote: ↑Wed Dec 22, 2021 1:08 am
A bunch of hogwash nonsense. All physical action in Nature is always local. Easy to prove.

You are right, Fred. I watched Sabine's video, which has some 200,000 views by now. But what she is proposing is just wrong.

She is claiming that we should relax one of the assumptions Bell made, which is that the distribution of hidden variables should not depend on the settings a and b. More precisely, Bell made the following assumption (call it "free will" or "statistical independence" or whatever):

p(a, b; h) = p(a, b'; h) = p(a', b; h) = p(a', b'; h) = p(h),

where "h" stands for hidden variables and p(a, b; h), etc., are probability densities that appear in the four correlation functions. In our work, we always set p = 1/n uniformly for n trials, which is even stronger than the above assumption by Bell because for us p does not even depend on h.

Now Sabine says that we should not assume all four probability densities to be the same. They should depend on the settings in each case. But that is "hogwash", as you put it. Why? Because that would mean that, in general, the probability distribution of the hidden variables would change when the settings are changed. That makes her model retrocausal at best and nonlocal at worst. It is not a solution to anything. It is just one big confusion.

I am saddened by this because I like her and I like some of her other works. She has been a good friend to me in the worst of times. But in this work, she has ventured outside her field of expertise and blundered. At least that is what I think after watching her video. I have no time to read her papers.
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Joy Christian wrote: ↑Wed Dec 22, 2021 4:40 pm Now Sabine says that we should not assume all four probability densities to be the same. They should depend on the settings in each case. But that is "hogwash", as you put it. Why? Because that would mean that, in general, the probability distribution of the hidden variables would change when the settings are changed. That makes her model retrocausal at best and nonlocal at worst. It is not a solution to anything. It is just one big confusion.

I think there is more to it than the way you guys put it. While I dislike the characterization of "superdeterminism", I believe there is a subtler point that applies. The important question is why the distribution changes not the fact that it changes. Bell's assumption that the distribution must be the same is what is wrong. Why must it be the same?

"When the settings change" is not a simple innocent thing. Changing the settings amounts to a different configuration of the equipment essentially performing a different experiment, with different conditions being applied to determine what counts as being "entangled". Naturally, the distributions should be expected to be different. Why should they be the same? Note that we are not merely talking about the source particles here, we are talking about distributions that apply to measurement results obtained under those treatments. Take a look at any Bell-test experiment and see how the data are treated. It's not a simple thing like just turning on the source. There are elaborate procedures for heralding the entanglement, and these procedures necessitate that the distributions are different.

It does not have to be due to retrocausality, it could simply be due to the fact that different settings combinations sample different subsets of the pre-experiment distribution, just like in the delayed-choice experiments.

minkwe wrote: ↑Wed Dec 22, 2021 4:46 pm
Note that we are not merely talking about the source particles here, we are talking about distributions that apply to measurement results obtained under those treatments.

I am talking about the source that is set in the overlap of the backward lightcones of Alice and Bob. The hidden variables "h" is then a set of initial states of the system, with their distribution given by the function p(h). This function must be independent of any settings that can be chosen at later times, beyond the overlap, even seconds before the particles arrive at the detectors. If it is allowed to be a function of the settings as well, as in p(a, b; h), then that would clearly lead to retrocausality as well as nonlocality. Because Alice can then change her setting and thus the function p(a, b; h) so that Bob would obtain a different result than what he would have obtained with the previous setting. Moreover, by changing their settings Alice and Bob can change the entire set of past initial states defined by p(a, b; h), thus making the whole business retrocausal. So I think what Sabine is suggesting is a total mess.
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Joy Christian wrote: ↑Wed Dec 22, 2021 7:57 pm I am talking about the source that is set in the overlap of the backward lightcones of Alice and Bob. The hidden variables "h" is then a set of initial states of the system, with their distribution given by the function p(h).

Yes but that distribution is not relevant to Bell. The only thing relevant is the distribution of lambda corresponding to the outcomes actually obtained. This distribution can be different from the one you are talking about without any retrocausality or "superdeterminism", or nonlocality, or conspiracy. My point is that the two should not be conflated, as is commonly done.

This function must be independent of any settings that can be chosen at later times, beyond the overlap, even seconds before the particles arrive at the detectors.
If it is allowed to be a function of the settings as well, as in p(a, b; h), then that would clearly lead to retro-causality as well as nonlocality.

Yes, but this function is irrelevant for Bell. The equivalent function which describes the distribution of lambdas corresponding to the outcomes obtained at settings (a,b) does depend on those settings because the experimental apparatus is setup precisely to filter the original distribution for a joint measurement. P(a,b|h) = P(a|h)(b|a,h). Note the significance of P(b|a,h) and the irrelevance of P(h).

Because Alice can then change her setting and thus the function p(a, b; h) so that Bob would obtain a different result than what he would have obtained with the previous setting. Moreover, by changing their settings Alice and Bob can change the entire set of past initial states defined by p(a, b; h), thus making the whole business retrocausal. So I think what Sabine is suggesting is a total mess.

That is not correct. Alice cannot change Bob's result by changing her setting. Rather. She can select a different subset of Bob's already obtained outcomes ( cf. P(b|a,h) ). This is just the misnamed delayed-choice all over. In fact it's not even Alice and Bob who are changing anything. It's just the data analyst playing games.

minkwe wrote: ↑Wed Dec 22, 2021 11:06 pm

Joy Christian wrote: ↑Wed Dec 22, 2021 7:57 pm I am talking about the source that is set in the overlap of the backward lightcones of Alice and Bob. The hidden variables "h" is then a set of initial states of the system, with their distribution given by the function p(h).

Yes but that distribution is not relevant to Bell. The only thing relevant is the distribution of lambda corresponding to the outcomes actually obtained. This distribution can be different from the one you are talking about without any retrocausality or "superdeterminism", or nonlocality, or conspiracy. My point is that the two should not be conflated, as is commonly done.

Yes. We are talking about two different scenarios. What you have in mind is what the experimenters do. How they analyze data and what they evaluate as correlations. All that is important, but that is not my concern, and I don't think Bell was concerned about that either. What I am concerned about is what we expect from a local hidden variable theory in which things happen as a part of the structure of the world. As such, the distribution of the initial states or hidden variables in the overlap of the backward lightcones of Alice and Bob are of central concern, regardless of any measurements made by anyone. This distribution, p(h), has thus nothing to do with any experimental settings. And yet, if and when measurements are indeed made by Alice and Bob, then the very same hidden variables must play a role in determining their observed results A(a, h) and B(b, h). This is what is demanded by theoretical concerns of local hidden variable theories, going back to the argument by EPR. So while what you and Sabine are saying may be relevant for actual experimental practice, in theory p(h) must only depend on h, since it -- p(h) -- is structurally (or ontologically) predetermined in the overlap of the backward lightcones of Alice and Bob regardless of their existence.
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Joy Christian wrote: ↑Thu Dec 23, 2021 2:02 am Yes but that distribution is not relevant to Bell. The only thing relevant is the distribution of lambda corresponding to the outcomes actually obtained. This distribution can be different from the one you are talking about without any retrocausality or "superdeterminism", or nonlocality, or conspiracy. My point is that the two should not be conflated, as is commonly done.

Yes. We are talking about two different scenarios. What you have in mind is what the experimenters do. How they analyze data and what they evaluate as correlations. All that is important, but that is not my concern, and I don't think Bell was concerned about that either. What I am concerned about is what we expect from a local hidden variable theory in which things happen as a part of the structure of the world. As such, the distribution of the initial states or hidden variables in the overlap of the backward lightcones of Alice and Bob are of central concern, regardless of any measurements made by anyone. This distribution, p(h), has thus nothing to do with any experimental settings.[/quote]
I agree with this part.

And yet, if and when measurements are indeed made by Alice and Bob, then the very same hidden variables must play a role in determining their observed results A(a, h) and B(b, h).

I don't agree with this part. Not unless you make additional assumptions about the measurement functions A(a, h) and B(b, h). This is not necessarily the case, given the way Bell defines A(a, h) and B(b, h). If these functions each have domains that have in any way smaller subsets of the source hidden variable distribution, then the fact that the source distribution is not affected by settings is irrelevant, to the fact that the effective hidden variable distribution varies with the settings. I believe this is the scenario that ever comes into the discussion concerning Sabine's work.

This is what is demanded by theoretical concerns of local hidden variable theories, going back to the argument by EPR.

The only thing that is demanded of any local hidden variable theory is that all effects travel at the speed light at most. There is nothing that requires the functions A(a, h) and B(b, h) to have a domain that is identical to the source distribution.

So while what you and Sabine are saying may be relevant for actual experimental practice, in theory p(h) must only depend on h, since it -- p(h) -- is structurally (or ontologically) predetermined in the overlap of the backward lightcones of Alice and Bob regardless of their existence.

This is not just a matter of experimental practice. It goes to the heart of the theory also. While it would be absurd to suggest that Alice and Bob's settings change the source distribution p(h), this is not what is being claim. Yet, the effective distribution is naturally different for different setting pairs even while the source distribution is unchanged.

minkwe wrote: ↑Thu Dec 23, 2021 2:38 pm

Joy Christian wrote: This function must be independent of any settings that can be chosen at later times, beyond the overlap, even seconds before the particles arrive at the detectors. If it is allowed to be a function of the settings as well, as in p(a, b; h), then that would clearly lead to retrocausality as well as nonlocality.

Yes, but this function is irrelevant for Bell. The equivalent function which describes the distribution of lambdas corresponding to the outcomes obtained at settings (a,b) does depend on those settings because the experimental apparatus is setup precisely to filter the original distribution for a joint measurement. P(a,b|h) = P(a|h)P(b|a,h). Note the significance of P(b|a,h) and the irrelevance of P(h).

Joy Christian wrote: And yet, if and when measurements are indeed made by Alice and Bob, then the very same hidden variables must play a role in determining their observed results A(a, h) and B(b, h).

I don't agree with this part. Not unless you make additional assumptions about the measurement functions A(a, h) and B(b, h). This is not necessarily the case, given the way Bell defines A(a, h) and B(b, h). If these functions each have domains that have in any way smaller subsets of the source hidden variable distribution, then the fact that the source distribution is not affected by settings is irrelevant, to the fact that the effective hidden variable distribution varies with the settings.

So at the heart of your argument is the following (I don't know whether Sabine has the same argument or not because I haven't read her paper):

We have a source probability distribution p(h) that does not depend on the settings a and b. I presume all parties agree with this much.

But, you are then saying that the probability distribution corresponding to the outcomes A(a, h) and B(b, h) obtained for the settings (a,b) depends on those settings, inducing the transition p(h) --> p(a, b|h), as a result of the act of selecting the settings (a,b).

The above, in red, are not the words you have used, but that is what is happening physically in your scenario. But that is clearly a retrocausal act. The original probability distribution p(h), predetermined in the overlap of the backward lightcones of Alice and Bob, changes into an entirely new probability distribution p(a, b|h). This change in the distribution of hidden variables is not as innocent as simply "filtering" the original distribution when making joint measurements. That act changes the past distribution p(h) into the present distribution p(a, b|h) retrocausally. That is even worse than I thought.
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minkwe wrote: ↑Thu Dec 23, 2021 2:38 pm The only thing that is demanded of any local hidden variable theory is that all effects travel at the speed light at most. There is nothing that requires the functions A(a, h) and B(b, h) to have a domain that is identical to the source distribution.

Sorry Michel, but yes there is something, and it is moreover related to how data is analysed. And John Bell was concerned by that, too.

If the domain of the functions is not a fixed product set, then physically this means the outcomes belong to the set {-1, +1, “no outcome”} and physically there is now an issue for the post-experiment data analyst. Do they post-select? Or do they use a Bell-type inequality for a three-outcome experiment?

Let me also remark that Sabine’s video shows she’s making a very big mistake. Suppose I define lambda = (x,y,a,b) then it is trivially true that p(a,b,x,y) = integral_lambda p(x | a, lambda) p(y | b, lambda) p(a, b | lambda) p(lambda) d lambda. Therefore, saying that “The assumption ‘p(a, b | lambda) = p(a, b)’ is a weird technical assumption” is plain wrong. It is not weird, technical, or meaningless. It is an equally powerful physical assumption as realism and locality.

I don’t like models with zero predictive or explanatory power which just allow one to say that black is white in a very sophisticated and complicated way. And then go on to use the model to say that Bell, Zeilinger and Gisin are fools, is going a bit too far. And to suggest that this is because they are older white males, and that a new generation of women scientists will soon arise and prove it to the whole world, is not smart. With all respect to Tim Palmer, Jonte Hance, and Sabine Hossenfelder!

Wishing you all a wonderful and peaceful mid-winter feast!

Joy Christian wrote: ↑Thu Dec 23, 2021 10:04 pm We have a source probability distribution p(h) that does not depend on the settings a and b. I presume all parties agree with this much.

But, you are then saying that the probability distribution corresponding to the outcomes A(a, h) and B(b, h) obtained for the settings (a,b) depends on those settings, inducing the transition p(h) --> p(a, b|h), as a result of the act of selecting the settings (a,b).
The above, in red, are not the words you have used, but that is what is happening physically in your scenario.

Exactly, that is what I'm saying. There is nothing strange about this.

But that is clearly a retrocausal act.

Absolutely not. To see this let me give you just one simple example of such a function. There are many others but this is just a simple one to illustrate.

A(a, h) = -B(b, h) = 1/sign(a.h)

The probability distribution of h which applies to the outcomes observed for A(a,h) is not the same as the probability distribution observed for the functions B(b, h) and yet again both distributions are different from the one which applies to the joint outcome A(a,h)*B(b, h). All this without any retrocausality.

The original probability distribution p(h), predetermined in the overlap of the backward lightcones of Alice and Bob, changes into an entirely new probability distribution p(a, b|h).

Absolutely not. The original distribution does not change at all. This is not the claim.

This change in the distribution of hidden variables is not as innocent as simply "filtering" the original distribution when making joint measurements. That act changes the past distribution p(h) into the present distribution p(a, b|h) retrocausally. That is even worse than I thought.

This is not what is being claimed at all. What is claimed is simply that the domain of the function A(a,h) depends on both "a" and "h", and the domain of the product A(a, h)*B(b, h) depends on ("a", "b", and "h").

minkwe wrote: ↑Sat Dec 25, 2021 12:00 pm
What is claimed is simply that the domain of the function A(a,h) depends on both "a" and "h", and the domain of the product A(a, h)*B(b, h) depends on ("a", "b", and "h").

That would be true only if the functional arguments for the settings "a" and "b" have the same physical and mathematical status as that for the argument for "h" in the distribution function p(a, b|h). But they do not if the settings (a, b) are the result of some spontaneous events. In practice, they are indeed results of spontaneous events, unlike h, which is a variable in the space H of all hidden variables. That is what Bell assumed. So you would have to argue that no such spontaneous events --- like acts of free will --- are possible. There is always some unknown mechanism that brings about the settings (a, b).
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Joy Christian wrote: ↑Sat Dec 25, 2021 12:43 pm

minkwe wrote: ↑Sat Dec 25, 2021 12:00 pm
What is claimed is simply that the domain of the function A(a,h) depends on both "a" and "h", and the domain of the product A(a, h)*B(b, h) depends on ("a", "b", and "h").

That would be true only if the functional arguments for the settings "a" and "b" have the same physical and mathematical status as that for the argument for "h" in the distribution function p(a, b|h). But they do not if the settings (a, b) are the result of some spontaneous events. In practice, they are indeed results of spontaneous events, unlike h, which is a variable in the space H of all hidden variables. That is what Bell assumed. So you would have to argue that no such spontaneous events --- like acts of free will --- are possible. There is always some unknown mechanism that brings about the settings (a, b).
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It doesn't matter what mechanisms bring about (a, b). For the simple function A(a,h) = 1/sign(a.h), by freely choosing the vector "a", Alice is eliminating all vectors "h" which are perpendicular to "a". This does not mean, Alice has retrocausally changed the distribution of "h" produced by the source. It simply means that when you integrate the function A(a,h), any p(h) which appears in that integral cannot possibly be the same one as the source distribution. It must be one that depends on "a" also. Worse still, if you integrate the product, A(a,h)*B(b,h), any p(h) appearing under that integral must be dependent on both ("a", and "b") because by choosing the setting "b", Bob, has eliminated from this p(h) any vectors "h" perpendicular to "b". Therefore the p(h) which appears in Bell's equation (2), is not the same as the source distribution for functions of the type A(a,h) = -B(a,h) = 1/sign(a.h) -- and I didn't even think hard to find this one. There are many others.

But the point is that it is possible for the distribution p(h) which applies to the product A(a,h)*B(b,h) to depend on both ("a" and "b") without retrocausality or any other absurdity.

ρ(λ|a,b) ≠ ρ(λ) for the above function.

minkwe wrote: ↑Sat Dec 25, 2021 6:15 pm ...
But the point is that it is possible for the distribution p(h) which applies to the product A(a,h)*B(b,h) to depend on both ("a" and "b") without retrocausality or any other absurdity. ...

I'm sorry, but that statement sound a bit "fishy" to me. If p(h) originates with the source, then I really don't see how it could depend on both "a" and "b". Those vectors have not even happened yet!
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FrediFizzx wrote: ↑Sat Dec 25, 2021 6:59 pm

minkwe wrote: ↑Sat Dec 25, 2021 6:15 pm ...
But the point is that it is possible for the distribution p(h) which applies to the product A(a,h)*B(b,h) to depend on both ("a" and "b") without retrocausality or any other absurdity. ...

I'm sorry, but that statement sounds a bit "fishy" to me. If p(h) originates with the source, then I really don't see how it could depend on both "a" and "b". Those vectors have not even happened yet!

Indeed. One can write down many things mathematically, but do they mean anything physically? After all, we are trying to model physical experiments, or better still, physical phenomena. The question then is, while one can indeed write down p(a, b|h) mathematically, is it meaningful physically?

The claim of "superdeterminism" seems to be that the distribution function p(a, b|h) is a result of the acts of measurements and that is ok. But it is not ok, because a and b are not arguments like h. While h could be a genuine variable and thus have a distribution p(h), in practice a and b are not variables at all. They are parameters that can remain fixed throughout all trials and could be results of spontaneous events, such as acts of free will.

Moreover, we have an explicit example in the 3-sphere model where p(a, b|h) = p(h) = 1/n = constant for all trials n. We don't need "superdeterminism."
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Joy Christian wrote: ↑Sat Dec 25, 2021 12:43 pm

minkwe wrote: ↑Sat Dec 25, 2021 12:00 pm
What is claimed is simply that the domain of the function A(a,h) depends on both "a" and "h", and the domain of the product A(a, h)*B(b, h) depends on ("a", "b", and "h").

That would be true only if the functional arguments for the settings "a" and "b" have the same physical and mathematical status as that for the argument for "h" in the distribution function p(a, b|h). But they do not if the settings (a, b) are the result of some spontaneous events. In practice, they are indeed results of spontaneous events, unlike h, which is a variable in the space H of all hidden variables. That is what Bell assumed. So you would have to argue that no such spontaneous events --- like acts of free will --- are possible. There is always some unknown mechanism that brings about the settings (a, b).

Joy, you are absolutely right here!

Michel: having the domain of A depend of a and lambda in an inseparable way can only be interpreted physically as some kind of post-selection. We throw away some trials depending on some relation between a and lambda.

FrediFizzx wrote: ↑Sat Dec 25, 2021 6:59 pm

minkwe wrote: ↑Sat Dec 25, 2021 6:15 pm ...
But the point is that it is possible for the distribution p(h) which applies to the product A(a,h)*B(b,h) to depend on both ("a" and "b") without retrocausality or any other absurdity. ...

I'm sorry, but that statement sound a bit "fishy" to me. If p(h) originates with the source, then I really don't see how it could depend on both "a" and "b". Those vectors have not even happened yet!

Fred is absolute right too. Very fishy!

FrediFizzx wrote: ↑Sat Dec 25, 2021 6:59 pm

minkwe wrote: ↑Sat Dec 25, 2021 6:15 pm ...
But the point is that it is possible for the distribution p(h) which applies to the product A(a,h)*B(b,h) to depend on both ("a" and "b") without retrocausality or any other absurdity. ...

I'm sorry, but that statement sound a bit "fishy" to me. If p(h) originates with the source, then I really don't see how it could depend on both "a" and "b". Those vectors have not even happened yet!
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I don't think you guys are understanding me. I suspect that Richard understands but as usual he wants to play games because I've explained this to him before and he started scrambling to find a disingenuous way out.
Okay Fred, let us do a little exercise. Let , and be vectors. Forget for a moment what those vectors mean physically. Focus on the maths.






What is the domain of , what is the domain of , what is the domain of

minkwe wrote: ↑Sun Dec 26, 2021 8:13 am

FrediFizzx wrote: ↑Sat Dec 25, 2021 6:59 pm

minkwe wrote: ↑Sat Dec 25, 2021 6:15 pm ...
But the point is that it is possible for the distribution p(h) which applies to the product A(a,h)*B(b,h) to depend on both ("a" and "b") without retrocausality or any other absurdity. ...

I'm sorry, but that statement sound a bit "fishy" to me. If p(h) originates with the source, then I really don't see how it could depend on both "a" and "b". Those vectors have not even happened yet!
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I don't think you guys are understanding me. I suspect that Richard understands but as usual he wants to play games because I've explained this to him before and he started scrambling to find a disingenuous way out.
Okay Fred, let us do a little exercise. Let , and be vectors. Forget for a moment what those vectors mean physically. Focus on the maths.





What is the domain of , what is the domain of , what is the domain of

I understand what you are saying. No technical discussion is necessary to understand the issue. It has been known and discussed extensively for the past fifty years. There are two options for the derivations of Bell inequalities:

(a) the probability distribution p(h) is independent of the settings a and b. This has been assumed by Bell to derive the bound of 2 on CHSH.

(b) the probability distribution depends on the settings a and b so that p(a, b|h) =/= p(h). This is the assumption of "superdeterminism."

So the options are clear cut. With option (b) the bound of 2 on CHSH cannot be derived. Therefore one can construct a locally causal theory.

The only remaining questions are: (i) What does option (b) mean physically and conceptually? And (ii): What price one must pay for the option (b).
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