The Mistakes by Bell and von Neumann are Identical

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Wed Jul 15, 2020 12:38 am

Joy Christian wrote:Image
You can find the above comments from Einstein on page 89 of the collection of papers by Shimony, cited as Ref. [6] in my paper.

Please give me the reference to the original paper by Shimony. The Cambridge University Press book you refer to is only available in hard-cover, costs maybe 100 Euro and will take time to get hold of.

In the meantime, I can only quote Bohr: "How wonderful that we have met with a paradox. Now we have some hope of making progress.".

I am certain, for many reasons, that Bell's theorem is not homologous to von Neumann's failed proof. We must be careful to distinguish quantum dogma (the pronouncement "thou shalt not..." uttered by the high priests of quantum physics) from the bare facts predicted by core quantum theory, and moreover confirmed experimentally. Secondly, when discussing hidden variables models, we must remember that a hidden variables theory as envisaged by Einstein would be a classical-like theory which underlay quantum theory, thus making the same (or at least, experimentally indistinguishable) predictions as quantum theory. So if quantum theory predicts that the sum of the expectation values of two observables is equal to the expectation value of a third, then any hidden variable theory which actually does reproduce (accurately enough) quantum theory, must satisfy the same rule (or at least, satisfy it, to a close approximation).

It seems to me that Einstein's argument is wrong. The linearity does have to hold. It is something else which goes wrong. We have made an interesting discovery. Our idol was not always right...

So far you are just quoting authority to support your point of view. You are not responding to my argument. A true scientist does not accept things on the authority of great scientists (or great religious authorities) of the past. A true scientist accepts things on the basis of logical argument and empirical observation.

Added a short time later:

Einstein says that the linearity of expectation values supposed to hold in quantum theory need only hold under states which exist in quantum theory. That is true, but that is not the issue here. Moreover, he is careless in his words. The linearity of what? We must distinguish carefully between objects existing in certain mathematical theories. The theories are different. They correspond in some way to things observed in nature but they are not the same as those things. This is a typical mistake of physicists, to get reality mixed up with the abstract (formal, mathematical) objects in the language (mathematics) they use to describe reality.

In a hidden variable theory, the true state of nature is described by further, presently un-observable, variables. Einstein says that when you further condition on those variables, the additivity between the things which underly those observables need no longer hold. The point is, the quantum observable is not the same thing as the values observed when the observable is measured. In my spin example, sigma_x + sigma_z = sqrt 2 sigma_u is an identity relating three self-adjoint observables. Suppose we came up with a hidden variables theory for the three spins, which stated that on measuring any of those observables, we merely observe the value of a pre-existing random variable (ie a function of the true underlying state of the system). Then there would exist three random variables on a single probability space such that the probability distribution of any one of the three was the probability distribution of the outcomes of measuring the corresponding observable. Call these three random variables X, Z, and U. The three random variables would have to take the values +/-1 only with probabilities which can be deduced from their three expectation values trace rho sigma_x etc.

It would have to be the case that E(X) + E(Z) = E(sqrt 2 . U). As I already said, please check yourself that trace(rho sigma_x) + trace(rho sigma_z) = sqrt 2 trace(rho sigma_u). The linearity does hold, both in the hidden variables theory and in quantum theory! The problem is that the sum of two random variables taking the values +/-1 is a random variable taking the values -2, 0, and +2; while sqrt 2 U is a random variable taking the values +/- sqrt 2.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Wed Jul 15, 2020 1:31 am

gill1109 wrote:
Please give me the reference to the original paper by Shimony. The Cambridge University Press book you refer to is only available in hard-cover, costs maybe 100 Euro and will take time to get hold of.

Shimony's original paper is in Foundations of Quantum Mechanics, Proceedings of the International School of Physics `Enrico Fermi' (1971), pp 182--94, published by Societa Italiana di Fisica.

But Shimony's original paper does not contain the comments I have quoted. They were added by Shimony on the last page of the original paper when the collection of his papers was published by Cambridge University Press. Here is a scan of the last page of Shimony's paper as it appears in the CUP collection: http://einstein-physics.org/wp-content/ ... nstein.pdf.

gill1109 wrote:
So far you are just quoting authority to support your point of view. You are not responding to my argument. A true scientist does not accept things on the authority of great scientists (or great religious authorities) of the past. A true scientist accepts things on the basis of logical argument and empirical observation.

Nonsense. I have provided a detailed original argument in my paper. Note that I have actually provided an explicit proof of the handwaving argument put forward by Bell in section 3 of chapter 1 of his book. Bell's argument against von Neumann's assumption in question is less sketchy than Einstein's comments I have quoted above but more sketchy than my argument.

***
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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Wed Jul 15, 2020 3:34 am

Joy Christian wrote:
gill1109 wrote:
Please give me the reference to the original paper by Shimony. The Cambridge University Press book you refer to is only available in hard-cover, costs maybe 100 Euro and will take time to get hold of.

Shimony's original paper is in Foundations of Quantum Mechanics, Proceedings of the International School of Physics `Enrico Fermi' (1971), pp 182--94, published by Societa Italiana di Fisica.

But Shimony's original paper does not contain the comments I have quoted. They were added by Shimony on the last page of the original paper when the collection of his papers was published by Cambridge University Press. Here is a scan of the last page of Shimony's paper as it appears in the CUP collection: http://einstein-physics.org/wp-content/ ... nstein.pdf.

gill1109 wrote:
So far you are just quoting authority to support your point of view. You are not responding to my argument. A true scientist does not accept things on the authority of great scientists (or great religious authorities) of the past. A true scientist accepts things on the basis of logical argument and empirical observation.

Nonsense. I have provided a detailed original argument in my paper. Note that I have actually provided an explicit proof of the handwaving argument put forward by Bell in section 3 of chapter 1 of his book. Bell's argument against von Neumann's assumption in question is less sketchy than Einstein's comments I have quoted above but more sketchy than my argument.

***

Thanks for the further details on Shimony’s paper and the book of his collected works.

You still haven’t pointed out what is wrong with my argument. Nor has anybody else.

I have provided detailed criticism of your arguments in https://arxiv.org/abs/1203.1504, Does Geometric Algebra provide a loophole to Bell's Theorem? (with correction note), Entropy 2020, 22(1), 61 (21 pp.); https://doi.org/10.3390/e22010061 Did you submit a rebuttal of that paper to Entropy or to another journal?

Please explain where my argument (for the linearity of expectation in hidden variables models) goes wrong, and please tell us your hidden variables model for sigma_x, sigma_z, and their sum sqrt 2 sigma_u for some u.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Wed Jul 15, 2020 4:49 am

***
I don't care about your arguments or what you have published in Entropy. What Einstein, Bell-1966, and others point out about the assumption in von Neumann's theorem of 1933 and Bell's theorem of 1964 is extremely clear, simple, and devastating, for both theorems. They point out that sum of expectation values is not equal to the expectation value of the sum for hidden variables theories when the corresponding quantum mechanical operators do not commute. That means that no matter what the other assumptions, logic, or conclusions of these theorems are, we can always construct a hidden variable theory that does not respect the faulty assumption. It is that simple. In other words, Bell's non-constructive theorem is an invalid argument.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby Heinera » Wed Jul 15, 2020 6:09 am

Joy Christian wrote:***
What Einstein, Bell-1966, and others point out about the assumption in von Neumann's theorem of 1933 and Bell's theorem of 1964 is extremely clear, simple, and devastating, for both theorems. They point out that sum of expectation values is not equal to the expectation value of the sum for hidden variables theories when the corresponding quantum mechanical operators do not commute.


Since Bell was not an idiot, he solved this by adding an extra assumption to his hidden variable theorem. Can you guess what it is?
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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Wed Jul 15, 2020 6:10 am

Joy Christian wrote:***
I don't care about your arguments or what you have published in Entropy. What Einstein, Bell-1966, and others point out about the assumption in von Neumann's theorem of 1933 and Bell's theorem of 1964 is extremely clear, simple, and devastating, for both theorems. They point out that sum of expectation values is not equal to the expectation value of the sum for hidden variables theories when the corresponding quantum mechanical operators do not commute.

You misread all those folk. Their mathematical statements are entirely correct and you are completely misinterpreting them. Already, the statement of Einstein/Shimony refers to a computation of expectation values *under a hidden state*, and not to such a computation under states which can be prepared in the laboratory. Mathematical identities between self-adjoint operators on a Hilbert space do not have to automatically translate into the same identities between the fundamental dynamical variables in some classical-like statistical-physical theory of what is going on "behind the scenes". This was von Neumann's mistake and John Bell saw through it, easily.

You might use the same names, "state"; "position", "momentum", "spin" and so on, in describing both theories, but you are then using natural language to somehow enliven a mathematical discussion of two mathematical theories of completely different nature, one of which (QM) is supposed to be somehow emergent from the other (a not yet invented classical-like statistical mechanics model). You did not notice the "small print". The devil is in the details. The same words - state, spin - are just guides to intuition and to suggest a connection with reality; they don't define the connections with reality.

My example proves that you are indeed mistaken. We have met with a paradox, and hence we do have some hope of making progress
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Wed Jul 15, 2020 6:19 am

Heinera wrote:
Joy Christian wrote:***
What Einstein, Bell-1966, and others point out about the assumption in von Neumann's theorem of 1933 and Bell's theorem of 1964 is extremely clear, simple, and devastating, for both theorems. They point out that sum of expectation values is not equal to the expectation value of the sum for hidden variables theories when the corresponding quantum mechanical operators do not commute. That means that no matter what the other assumptions, logic, or conclusions of these theorems are, we can always construct a hidden variable theory that does not respect the faulty assumption. It is that simple. In other words, Bell's non-constructive theorem is an invalid argument.

Since Bell was not an idiot, he solved this by adding an extra assumption to his hidden variable theorem. Can you guess what it is?


Assuming locally does not magically make the sum of expectation values equal to the expectation value of the sum for hidden variable theories relevant for non-commuting observables.

***
Last edited by Joy Christian on Wed Jul 15, 2020 6:41 am, edited 2 times in total.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Wed Jul 15, 2020 6:27 am

gill1109 wrote:
Joy Christian wrote:
I don't care about your arguments or what you have published in Entropy. What Einstein, Bell-1966, and others point out about the assumption in von Neumann's theorem of 1933 and Bell's theorem of 1964 is extremely clear, simple, and devastating, for both theorems. They point out that sum of expectation values is not equal to the expectation value of the sum for hidden variables theories when the corresponding quantum mechanical operators do not commute.

You misread all those folk. Their mathematical statements are entirely correct and you are completely misinterpreting them.

The comments by Einstein via Shimony's paper I have quoted cannot be more clear:

Image

"Einstein then said that there is no reason why this premise should hold in a state not acknowledged by quantum mechanics if R, S, etc. are not simultaneously measurable."

The "extra assumption" added by Bell that Heinera refers to (namely locality) does not make the above devastating objection by Einstein go away.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby Heinera » Wed Jul 15, 2020 6:41 am

Joy Christian wrote:[...]
The "extra assumption" added by Bell that Heinera refers to (namely locality) does not make the above devastating objection by Einstein go away.

***

Locality and no-conspiracy, which together implies statistical independence, which make "the above devastating objection by Einstein go away."
Last edited by Heinera on Wed Jul 15, 2020 6:42 am, edited 1 time in total.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Wed Jul 15, 2020 6:42 am

Heinera wrote:
Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:[...]
The "extra assumption" added by Bell that Heinera refers to (namely locality) does not make the above devastating objection by Einstein go away.

***

Locality and no-conspiracy, which together implies statistical independence, which make "the above devastating objection by Einstein go away."

No, it does not.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Thu Jul 16, 2020 1:50 am

Joy Christian wrote:The comments by Einstein via Shimony's paper I have quoted cannot be more clear

They could be made more clear. Apparently, they need to be made more clear. Einstein makes a proviso and you ignore it.

You also ignore my example.

I understand your reluctance to get into mathematical details, no-one wants to put in the necessary hard work and attention to detail. Of course you prefer to argue by authority. But it is easy to select from the words of authorities in order to put statements into their mouths which they would never endorse! That’s a game which is as old as the hills.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Thu Jul 16, 2020 2:37 am

gill1109 wrote:
Joy Christian wrote:The comments by Einstein via Shimony's paper I have quoted cannot be more clear

They could be made more clear. Apparently, they need to be made more clear. Einstein makes a proviso and you ignore it.

You also ignore my example.

I understand your reluctance to get into mathematical details, no-one wants to put in the necessary hard work and attention to detail. Of course you prefer to argue by authority. But it is easy to select from the words of authorities in order to put statements into their mouths which they would never endorse! That’s a game which is as old as the hills.

This is the same nonsense you have been repeating. Far from arguing by authority, I have put forward a detailed original argument, albeit while standing on the shoulders of authorities, that supersedes even Bell's argument against von Neumann, which you have been ignoring. The incontrovertible point of my argument is this: Bell's theorem is based on the assumption that the sum of expectation values is equal to the expectation value of the sum. This assumption is not valid for hidden variable theories. Moreover, it cannot be derived from "statistical independence" as Heinera has suggested. Therefore Bell's theorem against local hidden variable theories is as invalid as von Neumann's theorem against general hidden variable theories.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby Heinera » Thu Jul 16, 2020 2:59 am

Joy Christian wrote:The incontrovertible point of my argument is this: Bell's theorem is based on the assumption that the sum of expectation values is equal to the expectation value of the sum. This assumption is not valid for hidden variable theories. Moreover, it cannot be derived from "statistical independence" as Heinera has suggested.


That is just you not understanding what statistical independence means.

It means that regardless of the particular detector settings, the limiting distribution of the lambdas will be the same. Then the sum of expectation values is equal to the expectation value of the sum, trivially.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Thu Jul 16, 2020 3:31 am

Heinera wrote:
Joy Christian wrote:
The incontrovertible point of my argument is this: Bell's theorem is based on the assumption that the sum of expectation values is equal to the expectation value of the sum. This assumption is not valid for hidden variable theories. Moreover, it cannot be derived from "statistical independence" as Heinera has suggested.

That is just you not understanding what statistical independence means.

It means that regardless of the particular detector settings, the limiting distribution of the lambdas will be the same. Then the sum of expectation values is equal to the expectation value of the sum, trivially.

Every schoolchild knows that. But as I said before, statistical independence does not help one overcome the fact that the sum of expectation values is not equal to the expectation value of the sum for hidden variable theories when the corresponding quantum mechanical operators do not commute. A hidden variable theory is a theory based on dispersion-free states. Unlike quantum mechanical states, dispersion-free states have no statistical character. Moreover, unlike a quantum mechanical operator, every observable in a hidden variable theory must have a definite value, and that definite value must be one of the eigenvalues of the corresponding quantum mechanical operator. It is for these reasons that Einstein, Bell, and many others have pointed out that for hidden variable theories the sum of expectation values cannot be equal to the expectation value of the sum when the corresponding quantum mechanical operators do not commute. Statistical independence cannot help to overcome this fundamental objection. Read section 3 of the very first chapter of Bell's book to understand this argument.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby Heinera » Thu Jul 16, 2020 4:37 am

Joy Christian wrote: But as I said before, statistical independence does not help one overcome the fact that the sum of expectation values is not equal to the expectation value of the sum for hidden variable theories when the corresponding quantum mechanical operators do not commute.
***

Of course it does help. But all your failed arguments aside, the paper or yours that Richard published a rebuttal to describes a purely classical experiment where non-commutation of quantum mechanical operators simply does not apply. It is you that suggest we should photograph the ball fragments so that they can be measured for all angle settings simultaneously. There is nothing in that experiment that doesn't commute.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Thu Jul 16, 2020 5:05 am

Heinera wrote:
Joy Christian wrote:
But as I said before, statistical independence does not help one overcome the fact that the sum of expectation values is not equal to the expectation value of the sum for hidden variable theories when the corresponding quantum mechanical operators do not commute.

Of course it does help. But all your failed arguments aside, the paper or yours that Richard published a rebuttal to describes a purely classical experiment where non-commutation of quantum mechanical operators simply does not apply. It is you that suggest we should photograph the ball fragments so that they can be measured for all angle settings simultaneously. There is nothing in that experiment that doesn't commute.

Statistical independence does not help in overcoming Einstein's objection, as I have explained in much more detail in my paper: https://arxiv.org/abs/1704.02876.

The corresponding quantum mechanical operators do not commute even in that classical experiment. That is the whole point. But anyone who has never read my papers will never see that:

(1) https://link.springer.com/article/10.10 ... 014-2412-2,

(2) https://royalsocietypublishing.org/doi/ ... sos.180526,

and

(3) https://ieeexplore.ieee.org/document/8836453.

The argument that has failed is that of Bell and his followers, as these published papers demonstrate. At least two more of my papers on the subject are under consideration by journals.

The critiques of my work by Gill you mention are irrelevant. They contain extremely elementary mistakes, which I have repeatedly exposed on this forum and elsewhere. See, for example,

(1) https://arxiv.org/abs/1203.2529,

(2) https://arxiv.org/abs/1501.03393,

and

(3) https://www.academia.edu/38423874/Refut ... ls_Theorem.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Mon Jul 20, 2020 6:13 am

Joy Christian wrote:Einstein, Bell, and many others have pointed out that for hidden variable theories the sum of expectation values cannot be equal to the expectation value of the sum when the corresponding quantum mechanical operators do not commute.

None of those authorities has ever made that statement. You still have not given us an exact reference. The closest you came up with was words of Shimony quoting from memory spoken words of Einstein in which moreover there was a proviso which you neglected to consider. That is hearsay, not authority. A garbled reminiscence of an after dinner conversation between elderly gentlemen.

I gave you mathematical proof that it must be true exactly if the HV theory exactly reproduces QM predictions, and approximately otherwise. You did not deign to point out an error in my mathematical argument!

No problem. You are welcome to your opinion and I wish you luck in getting it published.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby Joy Christian » Mon Jul 20, 2020 7:01 am

gill1109 wrote:
Joy Christian wrote:
Einstein, Bell, and many others have pointed out that for hidden variable theories the sum of expectation values cannot be equal to the expectation value of the sum when the corresponding quantum mechanical operators do not commute.

None of those authorities has ever made that statement. You still have not given us an exact reference.

I have already given several exact references in my paper and in my posts above. The most popular such reference is section 3 of chapter 1 of Bell's book. That is pretty exact.

BTW, in 1980 when the conversation took place, Abner Shimony was not an "elderly gentleman." He was barely 50 years old and in his prime. He published many landmark papers after 1980.

gill1109 wrote:
You are welcome to your opinion and I wish you luck in getting it published.

Luck is not required. Sooner or later the truth will prevail. I have already published three papers (linked above) in respectable journals. Two more on the subject are under consideration.

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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Wed Jul 29, 2020 9:41 pm

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:
Einstein, Bell, and many others have pointed out that for hidden variable theories the sum of expectation values cannot be equal to the expectation value of the sum when the corresponding quantum mechanical operators do not commute.

None of those authorities has ever made that statement. You still have not given us an exact reference.

I have already given several exact references in my paper and in my posts above. The most popular such reference is section 3 of chapter 1 of Bell's book. That is pretty exact.

BTW, in 1980 when the conversation took place, Abner Shimony was not an "elderly gentleman." He was barely 50 years old and in his prime. He published many landmark papers after 1980.

gill1109 wrote:
You are welcome to your opinion and I wish you luck in getting it published.

Luck is not required. Sooner or later the truth will prevail. I have already published three papers (linked above) in respectable journals. Two more on the subject are under consideration.

***

Still no response to my mathematical argument. Yes, you will get all your papers published sooner or later, but as long as they contradict well-understood mathematical facts, they are not going to have much impact. They do present a useful challenge to students and a nice illustration of common misconceptions of Bell’s work. I think your work has had a great deal of positive influence on quantum science.
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Re: The Mistakes by Bell and von Neumann are Identical

Postby gill1109 » Wed Jul 29, 2020 9:52 pm

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:
Einstein, Bell, and many others have pointed out that for hidden variable theories the sum of expectation values cannot be equal to the expectation value of the sum when the corresponding quantum mechanical operators do not commute.

None of those authorities has ever made that statement. You still have not given us an exact reference.

I have already given several exact references in my paper and in my posts above. The most popular such reference is section 3 of chapter 1 of Bell's book. That is pretty exact.

BTW, in 1980 when the conversation took place, Abner Shimony was not an "elderly gentleman." He was barely 50 years old and in his prime. He published many landmark papers after 1980.

gill1109 wrote:
You are welcome to your opinion and I wish you luck in getting it published.

Luck is not required. Sooner or later the truth will prevail. I have already published three papers (linked above) in respectable journals. Two more on the subject are under consideration.

***

Here is Section 3 of Chapter 1 of Bell’s book. He does not say what you claim he says.

“Consider now the proof of von Neumann that dispersion free states, and so hidden variables, are impossible. His essential assumption is: Any real linear combination of any two Hermitian operators represents an observable, and the same linear combination of expectation values is the expectation value of the combination. This is true for quantum mechanical states; it is required by von Neumann of the hypothetical dispersion free states also. In the two-dimensional example of Section 2, the expectation value must then be a linear function of α and β. But for a dispersion free state (which has no statistical character) the expectation value of an observable must equal one of its eigenvalues. The eigenvalues (2) are certainly not linear in β. Therefore, dispersion free states are impossible. If the state space has more dimensions, we can always consider a two-dimensional subspace; therefore, the demonstration is quite general. The essential assumption can be criticized as follows. At first sight the required additivity of expectation values seems very reasonable, and it is rather the non-additivity of allowed values (eigenvalues) which requires explanation. Of course the explanation is well known: A measurement of a sum of noncommuting observables cannot be made by combining trivially the results of separate observations on the two terms–it requires a quite distinct experiment. For example the measurement of σx for a magnetic particle might be made with a suitably oriented Stern–Gerlach magnet. The measurement of σy would require a different orientation, and of (σx + σy) a third and different orientation. But this explanation of the nonadditivity of allowed values also established the nontriviality of the additivity of expectation values. The latter is a quite peculiar property of quantum mechanical states, not to be expected a priori. There is no reason to demand it individually of the hypothetical dispersion free states, whose function it is to reproduce the measurable peculiarities of quantum mechanics when averaged over. In the trivial example of Section 2 the dispersion free states (specified λ) have additive expectation values only for commuting operators. Nevertheless, they give logically consistent and precise predictions for the results of all possible measurements, which when averaged over λ are fully equivalent to the quantum mechanical predictions. In fact, for this trivial example, the hidden variable question as posed informally by von Neumann in his book is answered in the affirmative. Thus the formal proof of von Neumann does not justify his informal conclusion: It is therefore not, as is often assumed, a question of reinterpretation of quantum mechanics–the present system of quantum mechanics would have to be objectively false in order that another description of the elementary process than the statistical one be possible.’ It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.”

— Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy by J. S. Bell
https://amzn.eu/9Iy7S6c

Notice, Bell says explicitly that the problem is not the linearity of expectation values. This is true for QM and true for HV theories. The problem, as I told you, is that the sum of eigenvalues is not necessarily an eigenvalue of the sum.

I suggest you read your own references more carefully.
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