## The CHSH inequality as a weakly objective result

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

### Re: The CHSH inequality as a weakly objective result

@FrediFizzx, is there a way to delete my account on this forum? I see no such option in the User Control Panel.
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Heinera

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### Re: The CHSH inequality as a weakly objective result

Heinera wrote:@FrediFizzx, is there a way to delete my account on this forum? I see no such option in the User Control Panel.

The roaches scatter when the light shines on them.
local

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### Re: The CHSH inequality as a weakly objective result

Heinera wrote:@FrediFizzx, is there a way to delete my account on this forum? I see no such option in the User Control Panel.

Nope. Just don't come on here. Don't post.
FrediFizzx
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### Re: The CHSH inequality as a weakly objective result

@minkew I really appreciate you tried to spot the mistakes in my arguments. Here are my response to your questionings

1) You talk about the domain of the functions A and B are different from the domain of their products.
The only way I can make sense of what you are saying is that since A and B are measured in
different laboratories when you take the product AB, A and B were generated in different events.
If you assume that then the Bell inequality cannot be derived. When we derive the Bell inequality we
must assume A and B in the product AB were generated in the same event therefore both have the same value of the hidden variable.
You must assume that in a theoretical derivation. Whether that is possible in a real experiment is another problem.
We must make sure that the theoretical derivation makes sense. Otherwise, it wouldn't make sense to test the theory with experiments.
Experimental loopholes are a different issue. I don't want to discuss experiments. Experiments are more complex than mere theory.

You also seem to contend that we only measure two discrete values -1 and +1. That is another theoretical assumption that is confirmed by experiment.
Of course, if do not assume that, I agree there is no Bell inequality.

2)I think that we agree on this point. Basically, you say that if we do not assume "Statistical independence"(SI) also know as "Measurement independence"
the inequality does not hold. I totally agree with this. Besides whether SI is necessary for "freedom" or whether freedom itself makes sense is contentious.
In other words, we agree here: if we do not assume SI, Bell's inequality cannot be derived.

3) You start with 8 spreadsheets. This unnecessarily complicates the analysis because according to point 1) if we assume that we already know coincident events
your 8 spreadsheets can be reduced to only 4 spreadsheets. As explained in 1) if we do not assume that we can "pair" the different measurements de inequality cannot be derived.
Thus to be able to derive the Bell inequality we must have only 4 spreadsheets, each for a possible product of AB as shown in the table of my other paper debunking "counterfactual definiteness"
https://arxiv.org/abs/1911.00343

What you call "fair sampling" is in fact "statistical independence" already touched upon in 2)
Justo

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### Re: The CHSH inequality as a weakly objective result

Justo wrote: The only way I can make sense of what you are saying is that since A and B are measured in
different laboratories when you take the product AB, A and B were generated in different events.

Yes, of course they were generated in two different measurement events, both acting on the single source pair emission event.

If you assume that then the Bell inequality cannot be derived.

It can be derived for separated measurements but its violation in that case implies violation of special relativity (see below).

When we derive the Bell inequality we must assume A and B in the product AB were generated in the same event therefore both have the same value of the hidden variable.

You are confusing the source event (emission) with the measurement events.

You must assume that in a theoretical derivation.

Not at all. In this paper, Graft demonstrates how to derive the quantum prediction in the cases of a) joint measurement (one event), and b) separated (marginal) measurements (two events).

https://arxiv.org/abs/1607.01808

Whether that is possible in a real experiment is another problem.

For separated measurements in EPRB, the joint prediction cannot apply unless Luders projection is invoked. But that violates special relativity for space-like separation. Gill covers his ears and babbles when he hears this. Passion at a distance is immune to relativity, don't you know?

We must make sure that the theoretical derivation makes sense.

For sure! Also that it is applied only to appropriate physical scenarios. The joint prediction cannot apply to EPRB absent projection that violates relativity.

What you call "fair sampling" is in fact "statistical independence" already touched upon in 2)

You are off-base here. Fair sampling is something that can apply to a single side's detector (e.g., at some angles the efficiency is lower), while statistical independence must refer to both sides. Fair sampling cannot be reduced to statistical independence.
local

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### Re: The CHSH inequality as a weakly objective result

local wrote:
Justo wrote: The only way I can make sense of what you are saying is that since A and B are measured in
different laboratories when you take the product AB, A and B were generated in different events.

Yes, of course they were generated in two different measurement events, both acting on the single source pair emission event.

If you assume that then the Bell inequality cannot be derived.

It can be derived for separated measurements but its violation in that case implies violation of special relativity (see below).

When we derive the Bell inequality we must assume A and B in the product AB were generated in the same event therefore both have the same value of the hidden variable.

You are confusing the source event (emission) with the measurement events.

You must assume that in a theoretical derivation.

Not at all. In this paper, Graft demonstrates how to derive the quantum prediction in the cases of a) joint measurement (one event), and b) separated (marginal) measurements (two events).

https://arxiv.org/abs/1607.01808

Whether that is possible in a real experiment is another problem.

For separated measurements in EPRB, the joint prediction cannot apply unless Luders projection is invoked. But that violates special relativity for space-like separation. Gill covers his ears and babbles when he hears this. Passion at a distance is immune to relativity, don't you know?

We must make sure that the theoretical derivation makes sense.

For sure! Also that it is applied only to appropriate physical scenarios. The joint prediction cannot apply to EPRB absent projection that violates relativity.

What you call "fair sampling" is in fact "statistical independence" already touched upon in 2)

You are off-base here. Fair sampling is something that can apply to a single side's detector (e.g., at some angles the efficiency is lower), while statistical independence must refer to both sides. Fair sampling cannot be reduced to statistical independence.

“Passion at a distance” is a poetic phrase introduced by Abner Shimony. If you believe the wave function is real then Luder’s projection violates relativity.

The problem with Graft’s ideas is that his way of thinking leads to predictions which are not observed in experiments.

“Fair sampling” is a statistical independence assumption concerning the mechanism whereby some particles are not detected. Whether or not they are detected should be statistically independent of the hidden variables they are thought to have taken with them from the source. Justo was talking, I believe about a different statistical independence assumption, namely that hidden variables are statistically independent of setting choices.

Failure of fair sampling leads to a subsample of completely observed photon pairs and in that subsample, it can cause statistical dependence between setting choices and hidden variables.

Violation of the one assumption can lead to violation of the other.
gill1109
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### Re: The CHSH inequality as a weakly objective result

local wrote:
Justo wrote: The only way I can make sense of what you are saying is that since A and B are measured in
different laboratories when you take the product AB, A and B were generated in different events.

Yes, of course they were generated in two different measurement events, both acting on the single source pair emission event.

And I said that I can make sense of what minkew said only if A and B are "NOT" from the same emission event.

local wrote:
Justo wrote: If you assume that then the Bell inequality cannot be derived.

It can be derived for separated measurements but its violation in that case implies violation of special relativity (see below).

No. If you derive the inequality assuming that data from A and B in AB did not originate in the same event you cannot derive the inequality (with bound 2)
This would be the case of having 8 unrelated spreadsheets instead of 4 spreadsheets containing correctly paired events AB

local wrote:
Justo wrote: When we derive the Bell inequality we must assume A and B in the product AB were generated in the same event therefore both have the same value of the hidden variable.

You are confusing the source event (emission) with the measurement events.

No, I am not. Measurement events in each laboratory must correspond to the same source event. If you can't do that, then you cannot test the Bell inequality.
Again, whether that can in fact be done in a real experiment is irrelevant. For the theoretical derivation, you must assume each factor in the product AB originated in the same event.

local wrote:
Justo wrote:You must assume that in a theoretical derivation.

Not at all. In this paper, Graft demonstrates how to derive the quantum prediction in the cases of a) joint measurement (one event), and b) separated (marginal) measurements (two events).

https://arxiv.org/abs/1607.01808

I am not talking about Graft paper and I am not talking about quantum predictions.

local wrote:
Justo wrote: Whether that is possible in a real experiment is another problem.

For separated measurements in EPRB, the joint prediction cannot apply unless Luders projection is invoked. But that violates special relativity for space-like separation. Gill covers his ears and babbles when he hears this. Passion at a distance is immune to relativity, don't you know?

Again, we are not talking about quantum predictions. The discussion I proposed in this thread and that minkwe and I are discussing is about Adenier's objection to the Bell inequality from the standpoint of classical physics and hidden variables.

local wrote:
We must make sure that the theoretical derivation makes sense.

For sure! Also that it is applied only to appropriate physical scenarios. The joint prediction cannot apply to EPRB absent projection that violates relativity.

Irrelevant for our discussion.

local wrote:
Justo wrote:What you call "fair sampling" is in fact "statistical independence" already touched upon in 2)

You are off-base here. Fair sampling is something that can apply to a single side's detector (e.g., at some angles the efficiency is lower), while statistical independence must refer to both sides. Fair sampling cannot be reduced to statistical independence.

I did not say otherwise. I said that minkwe does not distinguish fair sampling from statistical independence.

PS. I suggest you open a new thread if you want to discuss quantum predictions because that is a different topic.
Justo

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### Re: The CHSH inequality as a weakly objective result

gill1109 wrote: The problem with Graft’s ideas is that his way of thinking leads to predictions which are not observed in experiments.

There is no experiment without disqualifying problems that has shown -a.b. I have asked you at least three times to state which experiment you hang your hat on in making this claim. But you are too cowardly to give an answer. For one of the experiments that you often admiringly cite you admitted that it needs to be ten times bigger to be valid. This nonsense is the modern day equivalent of trying to make a perpetual motion machine. Where is the Nobel prize? (hat tip Joy)

Justo wrote: PS. I suggest you open a new thread if you want to discuss quantum predictions because that is a different topic.

Shove that where the sun doesn't shine. You don't tell me what to do, and you don't get private threads here. Use PM for private conversations. If you don't like what I post then either ignore it or go whine to the forum admin. Nothing is stopping minkwe from responding to you if he chooses to do so.

First you say it is important that the theoretical derivation makes sense and then you upbraid me for discussing that. You cannot ask classical theory to do something that even QM cannot. Fred makes the same point in a different way. Discussing the minute details of something that is moot is indeed dancing on a pinhead. Don't become another Gill!
Last edited by local on Tue Oct 19, 2021 5:45 am, edited 3 times in total.
local

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### Re: The CHSH inequality as a weakly objective result

Joy Christian wrote: So what is Gill's claim? It is that an event-by-event numerical simulation of the singlet correlations -cos(a, b) is impossible without exploiting loopholes or succumbing to nonlocality. He has gone to extreme lengths to undermine anything that threatens this claim. You can say this is Gill's theory or challenge.

OK, we agree it's not a theory, which is all that I said. And what you call Gill's challenge is not his. That is Bell's challenge! Gill can play at being Bell's bulldog if he wants. At least it is amusing.
local

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### Re: The CHSH inequality as a weakly objective result

local wrote:
Justo wrote: PS. I suggest you open a new thread if you want to discuss quantum predictions because that is a different topic.

Shove that where the sun doesn't shine. You don't tell me what to do, and you don't get private threads here. Use PM for private conversations. If you don't like what I post then either ignore it or go whine to the forum admin. Nothing is stopping minkwe from responding to you if he chooses to do so.

First you say it is important that the theoretical derivation makes sense and then you upbraid me for discussing that. You cannot ask classical theory to do something that even QM cannot. Fred makes the same point in a different way. Discussing the minute details of something that is moot is indeed dancing on a pinhead. Don't become another Gill!

I am not telling you what to do. I am only saying that the discussion of quantum predictions is another topic and that Graft's paper may well deserve its own thread. I don't see why you have to get offended by that.
What I said at the beginning is that Adenier's claim is widespread. Many Bell believers(for instance, Gill) understand the Bell linequality as Adenier did but conclude that the inequality is correct. I agree with Adenier and Joy Christian that the Bell inequality is meaningless understood in that way.
Justo

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### Re: The CHSH inequality as a weakly objective result

@Justo LOL! The Bell inequalities are meaningless no matter which way you try to understand them. It is because of ONE simple fact that you Bell fanatics can't seem to wrap your mind around. NOTHING can exceed the bound on the inequalities!!!!!!!!!!!!
.
FrediFizzx
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### Re: The CHSH inequality as a weakly objective result

Justo wrote: I am not telling you what to do. I am only saying that the discussion of quantum predictions is another topic and that Graft's paper may well deserve its own thread. I don't see why you have to get offended by that.

OK, thank you. However, review the thread. Am I the first or only person to go "off-topic"? Review the forum as a whole. Do threads typically rigorously adhere to a single topic? This forum is sort of free-wheeling, and that is a good thing. It's a unique resource in the QM foundations community.

minkwe is engaging with you in this thread. So it seems you are getting what you seek, despite the presence of things you consider to be off-topic. Looking forward to further discussion between you two!

What I said at the beginning is that Adenier's claim is widespread. Many Bell believers (for instance, Gill) understand the Bell linequality as Adenier did but conclude that the inequality is correct. I agree with Adenier and Joy Christian that the Bell inequality is meaningless understood in that way.

That's fine. Your posts are interesting and stimulating. They've inspired me to review your publications, which are also interesting and stimulating.
local

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### Re: The CHSH inequality as a weakly objective result

local wrote:That's fine. Your posts are interesting and stimulating. They've inspired me to review your publications, which are also interesting and stimulating.

Thank you local. I know threads tend to be chaotic with respect to staying on topic. I don't pretend people would engage only on what I meant at the beginning. I only pointed out that your point perhaps deserves it own thread. Although it may already had its own thread, I do not know that. I just can't check every other thread. Besides, there must be other people willing to discuss it. In the particular Adenier case, minkwe seems to be very involved in, just as I. Joy also used to, but now he changed to another argument.
Justo

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### Re: The CHSH inequality as a weakly objective result

Justo wrote:1) You talk about the domain of the functions A and B are different from the domain of their products.
The only way I can make sense of what you are saying is that since A and B are measured in
different laboratories when you take the product AB, A and B were generated in different events.
If you assume that then the Bell inequality cannot be derived.

That's not what I mean. I'm talking about the mathematics of functions. Take a very simple function, let us assume just for illustrative purposes that

$A(\overrightarrow{a}, \overrightarrow{\lambda}) = \frac{1}{sign(\overrightarrow{a} \cdot \overrightarrow{\lambda})$
This function has values $\pm 1$ but you can't derive Bell's inequality with this type of function precisely because the domain of the function which is the space of vector pairs $\{\overrightarrow{a}, \overrightarrow{\lambda}\}$ excludes where $\overrightarrow{a} \perp \overrightarrow{\lambda}$.

Now if you take a product of two such functions
$f(\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{\lambda}) = A(\overrightarrow{a}, \overrightarrow{\lambda}) A(\overrightarrow{b}, \overrightarrow{\lambda})$, The domain of the product is the space of vector triples $\{\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{\lambda}, \}$ which excludes the regions where

$\{\overrightarrow{a} \perp \overrightarrow{\lambda} \} \cup \{ \overrightarrow{b} \perp \overrightarrow{\lambda}\}$.

The first observation of the consequences of this is that any probability distribution on the domains of the individual functions won't apply to the products. I think this could be another way of stating what local has been saying about separated vs joint measurements.

Secondly, looking at your equation 29, each of the products will have a different domain and therefore the $C(\lambda_i)$ will only be a valid expression for the common part of the domains of those functions. In practice, what that implies is that some lambdas will be unmeasurable at certain angles and therefore $\sum p(\lambda_i) = 1$ is not correct in general.

Now I've just presented a very simple contrived example to illustrate the point but there is a multitude of much more interesting functions which can affect the domains to such an extent that Bell's inequalities appear to be "violated".

When we derive the Bell inequality we
must assume A and B in the product AB were generated in the same event therefore both have the same value of the hidden variable. You must assume that in a theoretical derivation. Whether that is possible in a real experiment is another problem.

Let us say you assume that you are only concerned with the common domain of all the functions in equation 29, then you have a big problem. There is no way to calculate those expectation values from the experiment because without knowing lambda, you have no way of subsetting the data from experiments to calculate proper expectation values marginalized to the common distribution. By using all the measured data, you are violating the very condition that permitted you to proceed with the derivation.

You also seem to contend that we only measure two discrete values -1 and +1. That is another theoretical assumption that is confirmed by experiment.

For the contrived example I provided above, you would also only measure two discrete values -1 and +1. The point is that there exists many scenarios where some lambdas will be unmeasurable at certain angles. The physical situation strongly suggests this to be the case in EPRB and it would make sense that the unmeasurable regions correspond to the transition between -1 and +1.

3) You start with 8 spreadsheets. This unnecessarily complicates the analysis because according to point 1) if we assume that we already know coincident events
your 8 spreadsheets can be reduced to only 4 spreadsheets. As explained in 1) if we do not assume that we can "pair" the different measurements de inequality cannot be derived.

It doesn't complicate it. It makes the required assumptions explicit. The weakly objective prescription requires that you start from 8 and then make the necessary assumptions to reduce it to 4. The assumptions required to do that turn out to be false. But even if you don't believe my proof that the assumption is false, you still have to admit that reducing the 8 spreadsheets to 4 is equivalent to the strongly objective prescription. That is why the derivation proceeds in such a case.

What you call "fair sampling" is in fact "statistical independence" already touched upon in 2)

I understand what you mean. But "Fair sampling" is just the claim that the sample distribution is similar to the population distribution. When you talk of statistical independence, you have to provide more information for it to make sense as you could be talking about a whole bunch of different things, some relevant and some not. Statistical dependence can result from unfair sampling.
minkwe

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### Re: The CHSH inequality as a weakly objective result

minkwe wrote:
Justo wrote:You also seem to contend that we only measure two discrete values -1 and +1. That is another theoretical assumption that is confirmed by experiment.

For the contrived example I provided above, you would also only measure two discrete values -1 and +1. The point is that there exists many scenarios where some lambdas will be unmeasurable at certain angles. The physical situation strongly suggests this to be the case in EPRB and it would make sense that the unmeasurable regions correspond to the transition between -1 and +1.

You guys are forgetting that since 2015, experimenters *impose* that two discrete values are generated in each wing of the experiment in each pre-determined time-slot. There is no *assumption* that we only get outcomes -1 and +1.

Justo wrote:Measurement events in each laboratory must correspond to the same source event. If you can't do that, then you cannot test the Bell inequality.

You can do that. Experimenters do do that.
gill1109
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### Re: The CHSH inequality as a weakly objective result

gill1109 wrote:
Justo wrote:Measurement events in each laboratory must correspond to the same source event. If you can't do that, then you cannot test the Bell inequality.

You can do that. Experimenters do do that.

Maybe I shouldn't have used the word "test". What I meant is that if you do not "theoretically" assume your results come from the same event you cannot "theoretically" derive the inequality. Whether you can do that or how to do that in an experiment is another issue, it is a problem relating to the experimental implementation.

Adenier objected to the theoretical conception of the inequality, in my opinion, for good reasons.
Justo

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### Re: The CHSH inequality as a weakly objective result

The Bell inequalities are meaningless no matter which way you try to understand them. It is because of ONE simple fact that you Bell fanatics can't seem to wrap your mind around. NOTHING can exceed the bound on the inequalities!!!!!!!!!!!!
.
FrediFizzx
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### Re: The CHSH inequality as a weakly objective result

Justo wrote:3) You start with 8 spreadsheets. This unnecessarily complicates the analysis because according to point 1) if we assume that we already know coincident events
your 8 spreadsheets can be reduced to only 4 spreadsheets. As explained in 1) if we do not assume that we can "pair" the different measurements de inequality cannot be derived.
Thus to be able to derive the Bell inequality we must have only 4 spreadsheets, each for a possible product of AB as shown in the table of my other paper debunking "counterfactual definiteness"
https://arxiv.org/abs/1911.00343

Maybe I wasn't clear, your 4 spreadsheets in that paper are essentially the same as my 8 spreadsheets. It doesn't change anything. You are still making the same assumptions implicitly. To see this, start from my 8 spreadsheets

$(A_1, \Lambda_w), (B_1, \Lambda_w), (A_1, \Lambda_x), (B_2, \Lambda_x), (A_2, \Lambda_y), (B_1, \Lambda_y), (A_2, \Lambda_z), (B_2, \Lambda_z)$

Then combine the outcomes from the same particle pairs (same lambdas) into 4 3xN spreadsheet
$(A_1, B_1, \Lambda_w), (A_1, B_2, \Lambda_x), (A_2, B_1, \Lambda_y), (A_2, B_2, \Lambda_z)$
It is even easier to illustrate my proof using this version now you have to rearrange a 3xN spreadsheet instead of a 2xN one. There is no way to make the $\Lambda_w$ match $\Lambda_x$ while simultaneously making $\Lambda_y$ match $\Lambda_z$.

Then your table 1 is just placing these side by side and adding gaps in-between. You also incorrectly state in the text below that
"Real experiments usually record the products Aik = A′(ai)B′(bk), not the individual “clicks”".

This is not true. They record individual clicks, not products. Products are only calculated during data analysis.
minkwe

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### Re: The CHSH inequality as a weakly objective result

minkwe wrote:
Justo wrote:Maybe I wasn't clear, your 4 spreadsheets in that paper are essentially the same as my 8 spreadsheets. It doesn't change anything. You are still making the same assumptions implicitly. To see this, start from my 8 spreadsheets

$(A_1, \Lambda_w), (B_1, \Lambda_w), (A_1, \Lambda_x), (B_2, \Lambda_x), (A_2, \Lambda_y), (B_1, \Lambda_y), (A_2, \Lambda_z), (B_2, \Lambda_z)$

Then combine the outcomes from the same particle pairs (same lambdas) into 4 3xN spreadsheet
$(A_1, B_1, \Lambda_w), (A_1, B_2, \Lambda_x), (A_2, B_1, \Lambda_y), (A_2, B_2, \Lambda_z)$
It is even easier to illustrate my proof using this version now you have to rearrange a 3xN spreadsheet instead of a 2xN one. There is no way to make the $\Lambda_w$ match $\Lambda_x$ while simultaneously making $\Lambda_y$ match $\Lambda_z$.

That is correct I am implicitly assuming that it is possible to accommodate data such that $\Lambda_w$ match $\Lambda_x$, etc.,in other words I assume that the four different experiments(spreadsheets) share the same domain of hidden variables.
That I recognize as a possible loophole in the derivation and include the subsection "4.2 A Possible Loophole" to justify it.
The argument is fairly simple. It is based on an observation made by Eugene Wignar in 1970. Wigner observed that only a small finite number of hidden variables can exist. For a CHSH experiment only 16 different hidden variables exist. That means when you perform more that 16 experiments, the hidden variable values must necessarily start repeating its values if they actually exists. Thefore, after a sufficiently large number of experiments, you will end up with the same values appearing in all four different experiments. Besides, assuming statistical independence the relative frequency will be the same for the possible values.
Justo

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### Re: The CHSH inequality as a weakly objective result

Justo wrote:That is correct I am implicitly assuming that it is possible to accommodate data such that $\Lambda_w$ match $\Lambda_x$, etc., in other words I assume that the four different experiments(spreadsheets) share the same domain of hidden variables.

That is not the only thing you assume. You don't just assume that they have the same domain of hidden variables, you also assume that you have the same order of values. The derivation usually proceeds in two steps:
$C(\lambda_j) = A(a_1, \lambda_j)B(b_1, \lambda_j) - A(a_1, \lambda_j)B(b_2, \lambda_j) + A(a_2, \lambda_j)B(b_1, \lambda_j) + A(a_2, \lambda_j)B(b_2, \lambda_j)$
1) By factoring out $A_1$ and $A_2$. This assumes that $A_1$( $A_2$ ) appearing in both terms is exactly the same thing.
$C(\lambda_j) = A(a_1, \lambda_j)[B(b_1, \lambda_j) - B(b_2, \lambda_j)] + A(a_2, \lambda_j)[B(b_1, \lambda_j) + B(b_2, \lambda_j)]$
and then
2) noting that one of the terms $B(b_1, \lambda_j) - B(b_2, \lambda_j)$ and $B(b_1, \lambda_j) + B(b_2, \lambda_j)$ will be zero. Again this assumes that $B_1$( $B_2$ ) appearing in both terms is exactly the same thing.

Again translating to the spreadsheets for illustration, this is equivalent to having ONLY 4 columns of outcomes in the spreadsheet $A_1, A_2, B_1, B_2$. But note that in the weakly objective scenario, there are 8. $A_{1w}, A_{1x}, A_{2y}, A_{2z}, B_{1w}, B_{1y}, B_{2x}, B_{2z}$, because even if you assume that the distributions of lambdas producing those outcomes are the same, you still have 8 independent random variables not four!

Therefore the two steps in the derivation above make the assumptions that the following equivalences must apply.
$A_{1w}, \equiv A_{1x}, A_{2y} \equiv A_{2z}, B_{1w} \equiv B_{1y}, B_{2x} \equiv B_{2z}$.

Note again that we are not only interested in the set of lambdas involved being the same, we are also interested in the order of lambdas being the same. You are assuming that the above reduction in degrees of freedom must apply. But that assumption is false because, the permutation, required to make $A_{1w} \equiv A_{1x}$ and $B_{1w} \equiv B_{1y}$ is incompatible with the permutation required to make $A_{2y} \equiv A_{2z}$ and $B_{2x} \equiv B_{2z}$. Due to the cyclic nature of the relationships, you will undo the order as you go around rearranging pairs. This is Vorob's theorem. You can't recover the joint 4xN distribution $A_1, A_2, B_1, B_2$ from separate independent pair-wise distributions $A_{1w}, A_{1x}, A_{2y}, A_{2z}, B_{1w}, B_{1y}, B_{2x}, B_{2z}$. You can't assume that a system with 4Nf degrees of freedom has Nf degrees of freedom.

Yet this is the hidden assumption involved in your derivation, and it is false.

That I recognize as a possible loophole in the derivation and include the subsection "4.2 A Possible Loophole" to justify it.

What you identify in 4.2 is not what I describe above. It is not a question of "exact reproduction". We don't expect the lambdas to repeat exactly for each of the 4 independent realizations. We only expect that "in-principle" we can rearrange the 4 spreadsheets such that the column labelled $A_{1w}$ almost match the column $A_{1x}$ in-principle. This assumption is necessary to justify the factorization identified as (1) above. Therefore this is not the same as De Baere's "reproducibility hypothesis". This assumption is distinct from and in addition to the assumption that the set of lambdas realized in each of the four independent weakly objective experiments be essentially the same.

In fact, we can ignore lambda completely and simply consider experimental outcomes and you will still have the problem of needing to reduce the 8 columns of outcomes $A_{1w}, A_{1x}, A_{2y}, A_{2z}, B_{1w}, B_{1y}, B_{2x}, B_{2z}$, into 4 columns of outcomes $A_1, A_2, B_1, B_2$, and you will still need to perform impossible permutations to accomplish that. No amount of manipulations of lambda or the length of the experimental runs can resolve this problem, which originates from incompatible degrees of freedom combined with cyclicity of pairs of columns. Vorob gives rigorous proof that this is the case.
minkwe

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