## Pedagogical proofs of Bell's theorem

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

### Re: Pedagogical proofs of Bell's theorem

My favourite proof of the CHSH inequality is given in my short paper https://arxiv.org/pdf/2103.00225.pdf
Comment on “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space” (Now at version 3). No spreadsheets. Just some basic probability theory. Some simple mathematical assumptions involving eight binary random variables A, B, X, Y, X_1, X_2, Y_1, Y_2

A and B take values in {1, 2}. The other variables take values in {-1,+1}

Independence: A, B is statistically independent of X_1,X_2,Y_1,Y_2

LHV: X = X_A, Y = Y_B

In an experiment, one observes (many times) realisations of the quadruple (X, Y, A, B) (outcomes and settings).

X_1, X_2, Y_1, Y_2 are hidden variables (or if you prefer, functions of hidden variables).

One estimates four conditional correlations rho(i,j) := E(XY | A = i, B = j)

I’m interested in criticisms of the mathematics.
gill1109
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### Re: Pedagogical proofs of Bell's theorem

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Joy Christian
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### Re: Pedagogical proofs of Bell's theorem

Joy Christian wrote:We are all born ignorant, but one must work hard to remain stupid

Joy has clearly been working very hard. But not hard enough.
https://checkyourfact.com/2019/06/19/fact-check-benjamin-franklin-born-ignorant-remain-stupid/

I quote from “check your fact.com”:
The alleged quote has been attributed to Franklin, a signer of the Declaration of Independence and one of the nation’s most prominent founders, countless times online. However, there is no evidence that he ever made this statement. A search of Franklin’s collected works turned up nothing similar.

“I’ve never run across it. Doesn’t sound like Franklin to me,” H.W. Brands, historian and author of “The First American: The Life and Times of Benjamin Franklin,” told The Daily Caller News Foundation in an email. In its current form, the expression has only been circulating in print since 1999, according to etymologist Barry Popik. Variations of the quote, however, date back much further. The saying resembles the words of 18th century French philosopher Claude Adrien Helvétius, for example. In his “A Treatise on Man,” Helvetius writes, “Man is born ignorant; he is not born a fool; and it is not even without labour that he is made one.”

FrediFizzx wrote:@Joy LOL!! Ignorance is curable but stupid is forever.

What an incredibly high level of intellectual debate. Very helpful, Fred. Thanks again.
Last edited by gill1109 on Wed Sep 01, 2021 12:14 am, edited 2 times in total.
gill1109
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### Re: Pedagogical proofs of Bell's theorem

@Joy LOL!! Ignorance is curable but stupid is forever.
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FrediFizzx
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### Re: Pedagogical proofs of Bell's theorem

gill1109 wrote:
FrediFizzx wrote:@Joy LOL!! Ignorance is curable but stupid is forever.

What an incredibly high level of intellectual debate. Very helpful, Fred. Thanks again.

You're Welcome. One of my favorite quotes. Uncle Al used to say it all the time on the old SciPhysics* Usenet groups.
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FrediFizzx
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### Re: Pedagogical proofs of Bell's theorem

gill1109 wrote:My favourite proof of the CHSH inequality is given in my short paper https://arxiv.org/pdf/2103.00225.pdf
Comment on “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space” (Now at version 3). No spreadsheets. Just some basic probability theory. Some simple mathematical assumptions involving eight binary random variables A, B, X, Y, X_1, X_2, Y_1, Y_2

A and B take values in {1, 2}. The other variables take values in {-1,+1}

Independence: A, B is statistically independent of X_1,X_2,Y_1,Y_2

LHV: X = X_A, Y = Y_B

In an experiment, one observes (many times) realisations of the quadruple (X, Y, A, B) (outcomes and settings).

X_1, X_2, Y_1, Y_2 are hidden variables (or if you prefer, functions of hidden variables).

One estimates four conditional correlations rho(i,j) := E(XY | A = i, B = j)

I’m interested in criticisms of the mathematics.

Richard, come-on, I'm surprised after so many years on this forum you still present a proof like that in a paper.

You are talking about "In an experiment" so please explain the experiment that this expression represents:
$Z = X_1Y_1 - X_1Y_2 - X_2Y_1 - X_2Y_2$
Each term is equivalent to a pair of coin tosses essentially. How many pairs of coin tosses are involved in the above expression?

Do you really not understand what it means to factor out random variables like
$Z = X_1(Y_1 -Y_2) - X_2(Y_1 + Y_2)$

How many pairs of coin tosses are involved in the above expression?
minkwe

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### Re: Pedagogical proofs of Bell's theorem

minkwe wrote:
gill1109 wrote:My favourite proof of the CHSH inequality is given in my short paper https://arxiv.org/pdf/2103.00225.pdf
Comment on “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space” (Now at version 3). No spreadsheets. Just some basic probability theory. Some simple mathematical assumptions involving eight binary random variables A, B, X, Y, X_1, X_2, Y_1, Y_2

A and B take values in {1, 2}. The other variables take values in {-1,+1}

Independence: A, B is statistically independent of X_1,X_2,Y_1,Y_2

LHV: X = X_A, Y = Y_B

In an experiment, one observes (many times) realisations of the quadruple (X, Y, A, B) (outcomes and settings).

X_1, X_2, Y_1, Y_2 are hidden variables (or if you prefer, functions of hidden variables).

One estimates four conditional correlations rho(i,j) := E(XY | A = i, B = j)

I’m interested in criticisms of the mathematics.

Richard, come-on, I'm surprised after so many years on this forum you still present a proof like that in a paper.

You are talking about "In an experiment" so please explain the experiment that this expression represents:
$Z = X_1Y_1 - X_1Y_2 - X_2Y_1 - X_2Y_2$
Each term is equivalent to a pair of coin tosses essentially. How many pairs of coin tosses are involved in the above expression?

Do you really not understand what it means to factor out random variables like
$Z = X_1(Y_1 -Y_2) - X_2(Y_1 + Y_2)$

How many pairs of coin tosses are involved in the above expression?

Dear Michel, I am really glad that you took a look at my proof. I am surprised that you were not able to follow it. It's a small piece of absolutely elementary probability theory. It can be treated in a first year university STEM course in probability and statistics.

Joy Christian has agreed that the mathematical core of Bell's work is correct. My model and the calculations I do with it is my understanding of the mathematical core of Bell's proofs of the Bell-CHSH inequality. Joy has a quite different aim, namely to question physical conceptions of space and time (locality, realism, causality).

Fred Diether doesn't understand my proof but I do believe you, Michel, have more aptitude for abstract mathematics, and much more interest in abstract mathematics, than he seems to have.

The expression $Z = X_1Y_1 - X_1Y_2 - X_2Y_1 - X_2Y_2$ does not "represent" an experiment. "Z" is a random variable. It is a function from a probability space (Omega, F, P) to the real line. No "coin tosses" are represented or used in that expression.

I am considering a certain possible real world experiment. A thought experiment, easy to implement in a school classroom. I explain it in the paper. It results in the observation of independent realisations of a quadruple (X, Y, A, B). Those four random variables have a joint probability distribution. That probability distribution is described in terms of some further random variables (X1, X2, Y1, Y2). The eight altogether have a joint probability distribution which satisfies the conditions which I have written down.

In fact, this is simply the urn model with slips of papers described by Heinera.

The urn experiment runs as follows. A probability distribution is fixed for (X1, X2, Y1, Y2). Thus: 16 probabilities adding up to one. A probability distribution is fixed for (A, B). Four probabilities adding up to one. We now, independently, create realisations (x1, x2, y1 y2) and (a, b). We then define x = xa and y = yb. We report the quadruple (a, b, x, y).

You could say that (x1, x2, y1 y2) are numbers written on a slip of paper drawn from an urn and sent to Alice and Bob. In the mean time, Alice and Bob chose "indices", or labels, a, b = 1 or 2. Alice then reports either x1 or x2 depending on whether a was equal to 1 or 2. Bob behaves similarly. The trial results in two numbers +/-1 and two labels = 1 or 2. After each trial the slip of paper is returned to the urn, the urn is mixed again, and we repeat.

I hope you understand that for four numbers x1, x2, y1, y2 it is a fact that x1 y1 - x1 y2 - x2 y1 - x2 y2 = x1 * (y1 - y2) - x2 * (y1 + y2). If we moreover evaluate the expression when the xi and yj take values +/-1, then the result is always +2 or -2.

I have given talks to high school children explaining the proof, they loved it. I have given talks to Buddhist scholars and neuroscientists explaining the proof, they loved it. It should not be difficult. The question is finding the right language to use when explaining it to different kinds of people.

My paper is under review with IEEE Access. There are 13 referees and an editor. All of the referees but one, and the editor-in-chief, have no issues with my proof of the CHSH inequality. A big majority of the reviewers think the paper should be published. A few have other opinions. For instance, one referee thinks that Bell's inequality is both true and not true. The standard axioms of mathematics are wrong. Several referees think that quantum mechanics is fine and we should not be wasting time talking about local realism. All referees but one agree that Christian's maths is wrong. If anyone is interested I am happy to confidentially show you the reports. If my paper is published then in fact all referee reports will be published too.
gill1109
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### Re: Pedagogical proofs of Bell's theorem

gill1109 wrote:
A big majority of the reviewers think the paper should be published. ... All referees but one agree that Christian's maths is wrong. If anyone is interested I am happy to confidentially show you the reports. ... If my paper is published then in fact all referee reports will be published too.

To begin with, you are known to lie effortlessly, so there is no reason to believe anything you are saying. Secondly, IEEE Access does not publish referee reports. So you have just provided proof that you have no scruples about lying, lying publicly, and lying repeatedly.

But despite this, suppose that "A big majority of the reviewers think [your] paper should be published" and "All referees but one agree that Christian's maths is wrong", then these reviewers are incompetent idiots. If anyone of them is reading this, then note well: I think you are an incompetent idiot. For Gill's paper, just as all of his other papers about my work, is junk paper. It contains numerous lies, distortions, and misrepresentations. It has nothing to do with my work at all. It should never be published anywhere. It is not worth the paper it can be printed on.
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### Re: Pedagogical proofs of Bell's theorem

gill1109 wrote:The expression $Z = X_1Y_1 - X_1Y_2 - X_2Y_1 - X_2Y_2$ does not "represent" an experiment. "Z" is a random variable. It is a function from a probability space (Omega, F, P) to the real line. No "coin tosses" are represented or used in that expression.

I am considering a certain possible real world experiment. A thought experiment, easy to implement in a school classroom.

It's not an experiment, yet it's a possible real-world experiment. In short, you understand the question and just like playing games.

I have given talks to high school children explaining the proof, they loved it.

This is not high school. We aren't easily bamboozled here.

I have given talks to Buddhist scholars and neuroscientists explaining the proof, they loved it. It should not be difficult. The question is finding the right language to use when explaining it to different kinds of people.

I see you like talking to people who can't challenge you. If you really want to know how good your proof is, then stop playing games and answer the question.

Does $Z = X_1Y_1 - X_1Y_2 - X_2Y_1 - X_2Y_2$ represent an experiment or not. Whether it is a thought experiment or a "possible experiment" is irrelevant. You are envisioning an experiment. Why else would you mention the word "experiment"? Why else would you think your expression has anything to do with Bell's theorem. Please stop playing games. Despite me asking you multiple times, you have never described this "thought experiment". You always dodge the question by claiming you are not talking about an experiment, only to turn around and say it is a thought experiment.
minkwe

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### Re: Pedagogical proofs of Bell's theorem

gill1109 wrote:The expression $Z = X_1Y_1 - X_1Y_2 - X_2Y_1 - X_2Y_2$ does not "represent" an experiment. "Z" is a random variable.

How many independent random variables make up Z?

gill1109 wrote:That probability distribution is described in terms of some further random variables (X1, X2, Y1, Y2). The eight altogether have a joint probability distribution which satisfies the conditions which I have written down.

Okay so at least you admit that there are 4 independent random variables ($X_1, X_2, Y_1, Y_2$) in Z. Now please answer how many independent random variables are there in the outcomes of a Bell test experiment? Of course you won't answer. You'll instead start talking about high-school students and Buddhists.
minkwe

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### Re: Pedagogical proofs of Bell's theorem

minkwe wrote:
gill1109 wrote:The expression $Z = X_1Y_1 - X_1Y_2 - X_2Y_1 - X_2Y_2$ does not "represent" an experiment. "Z" is a random variable.

How many independent random variables make up Z?

gill1109 wrote:That probability distribution is described in terms of some further random variables (X1, X2, Y1, Y2). The eight altogether have a joint probability distribution which satisfies the conditions which I have written down.

Okay so at least you admit that there are 4 independent random variables ($X_1, X_2, Y_1, Y_2$) in Z. Now please answer how many independent random variables are there in the outcomes of a Bell test experiment? Of course you won't answer. You'll instead start talking about high-school students and Buddhists.

1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.

2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).

My model is a model of one trial of a Bell experiment. The four “hidden variables” X1, X2, X3, X4 are not statistically independent of one another. They may have any joint probability distribution you like. The pair of settings (A,B) is assumed independent of the quadruple of hidden variables (X1,X2,X3,X4). The pair (X,Y) is a deterministic function of the settings and of the hidden variables.
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### Re: Pedagogical proofs of Bell's theorem

Perhaps to make this clearer to @minkwe: This is basically the model of the CHSH urn experiment.
Heinera

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### Re: Pedagogical proofs of Bell's theorem

Heinera wrote:Perhaps to make this clearer to @minkwe: This is basically the model of the CHSH urn experiment.

Yes!
gill1109
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### Re: Pedagogical proofs of Bell's theorem

gill1109 wrote:1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.

Simple answer: There are 4 random variables ($X_1,X_2,X_3,X_4$), no need for weaselly words.

2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).

Simple answer: There are 8 random variables in a Bell test Experiment $X^1_1, Y^1_1, X^2_1, Y^2_2, X^3_2, Y^3_1, X^4_2, Y^4_2$ no need to play word games.

There is no such thing as a CHSH urn experiment.
minkwe

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### Re: Pedagogical proofs of Bell's theorem

minkwe wrote:
gill1109 wrote:1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.

Simple answer: There are 4 random variables ($X_1,X_2,X_3,X_4$), no need for weaselly words.

2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).

Simple answer: There are 8 random variables in a Bell test Experiment $X^1_1, Y^1_1, X^2_1, Y^2_2, X^3_2, Y^3_1, X^4_2, Y^4_2$ no need to play word games.

There is no such thing as a CHSH urn experiment.

Yes there is. I defined it mathematically. This mathematical model is a mathematical model of Heinera’s urn experiment, it’s a mathematical model of a Bell simulation model using LHV and satisfying freedom (no conspiracy), it’s a mathematical model of one trial of a real loophole-free Bell experiment assuming local realism.

In fact it is just Bell’s model of local hidden variables, augmented with explicit random selection of settings.

Start with a probability space which is a product space of two independent components. (Omega, F, P) with Omega = Omega1 x Omega2, F = F1 x F2, P = P1 x P2 (I won’t bore you with the definitions of product of two sigma-algebras and product of two probability measures - these are not Cartesian products of sets).

Omega1 is Lambda, the space of the hidden variable lambda, P1 is Bell’s probability measure rho.

On it are defined X1, X2, Y1, Y2; these are Bell’s functions A with settings fixed at a1, a2, and B with settings b1, b2. Four functions of lambda.

Omega2 is the space on which are defined A, B

I don’t think “random variable” is a weasely word. It’s a mathematical term. https://en.m.wikipedia.org/wiki/Random_variable
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### Re: Pedagogical proofs of Bell's theorem

gill1109 wrote:
minkwe wrote:
gill1109 wrote:1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.

Simple answer: There are 4 random variables ($X_1,X_2,X_3,X_4$), no need for weaselly words.

2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).

Simple answer: There are 8 random variables in a Bell test Experiment $X^1_1, Y^1_1, X^2_1, Y^2_2, X^3_2, Y^3_1, X^4_2, Y^4_2$ no need to play word games.

There is no such thing as a CHSH urn experiment.

Yes there is. I defined it mathematically. This mathematical model is a mathematical model of Heinera’s urn experiment, it’s a mathematical model of a Bell simulation model using LHV and satisfying freedom (no conspiracy), it’s a mathematical model of one trial of a real loophole-free Bell experiment assuming local realism.

In fact it is just Bell’s model of local hidden variables, augmented with explicit random selection of settings.

Start with a probability space which is a product space of two independent components. (Omega, F, P) with Omega = Omega1 x Omega2, F = F1 x F2, P = P1 x P2 (I won’t bore you with the definitions of product of two sigma-algebras and product of two probability measures - these are not Cartesian products of sets).

Omega1 is Lambda, the space of the hidden variable lambda, P1 is Bell’s probability measure rho.

On it are defined X1, X2, Y1, Y2; these are Bell’s functions A with settings fixed at a1, a2, and B with settings b1, b2. Four functions of lambda.

Omega2 is the space on which are defined A, B

I don’t think “random variable” is a weasely word. It’s a mathematical term. https://en.m.wikipedia.org/wiki/Random_variable

Admit it, there are 4 independent random variables in your proof and 8 in an actual Bell test experiment. Will you admit this simple obvious fact? Or will you try to obscure it with irrelevant nonsense.

I ask a question about random variables and you talk about "hidden variables" and every type of variable except the one I asked about. Then when I call you out on it, you imply I'm saying "random variable" is a weasely word. Yet you can't accept the simple fact that there are 4 independent random variables in your proof and 8 in a Bell test experiment.
minkwe

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### Re: Pedagogical proofs of Bell's theorem

minkwe wrote:
gill1109 wrote:
minkwe wrote:
gill1109 wrote:1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.

Simple answer: There are 4 random variables ($X_1,X_2,X_3,X_4$), no need for weaselly words.

2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).

Simple answer: There are 8 random variables in a Bell test Experiment $X^1_1, Y^1_1, X^2_1, Y^2_2, X^3_2, Y^3_1, X^4_2, Y^4_2$ no need to play word games.

There is no such thing as a CHSH urn experiment.

Yes there is. I defined it mathematically. This mathematical model is a mathematical model of Heinera’s urn experiment, it’s a mathematical model of a Bell simulation model using LHV and satisfying freedom (no conspiracy), it’s a mathematical model of one trial of a real loophole-free Bell experiment assuming local realism.

In fact it is just Bell’s model of local hidden variables, augmented with explicit random selection of settings.

Start with a probability space which is a product space of two independent components. (Omega, F, P) with Omega = Omega1 x Omega2, F = F1 x F2, P = P1 x P2 (I won’t bore you with the definitions of product of two sigma-algebras and product of two probability measures - these are not Cartesian products of sets).

Omega1 is Lambda, the space of the hidden variable lambda, P1 is Bell’s probability measure rho.

On it are defined X1, X2, Y1, Y2; these are Bell’s functions A with settings fixed at a1, a2, and B with settings b1, b2. Four functions of lambda.

Omega2 is the space on which are defined A, B

I don’t think “random variable” is a weasely word. It’s a mathematical term. https://en.m.wikipedia.org/wiki/Random_variable

Admit it, there are 4 independent random variables in your proof and 8 in an actual Bell test experiment. Will you admit this simple obvious fact? Or will you try to obscure it with irrelevant nonsense.

I ask a question about random variables and you talk about "hidden variables" and every type of variable except the one I asked about. Then when I call you out on it, you imply I'm saying "random variable" is a weasely word. Yet you can't accept the simple fact that there are 4 independent random variables in your proof and 8 in a Bell test experiment.

There are not four statistically independent random variables in my proof.
In a Bell test experiment with randomly chosen settings one observes four random variables.

Michel, what do you mean by “independent”?
Are you using the word in its sense in elementary calculus, or in its sense in elementary statistics and probability?
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### Re: Pedagogical proofs of Bell's theorem

minkwe wrote:There is no such thing as a CHSH urn experiment.

The delusions of the Bell deniers are now reaching comical proportions. It's like flat earthers claiming "satelites do not exist."
Heinera

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### Re: Pedagogical proofs of Bell's theorem

Heinera wrote:
minkwe wrote:There is no such thing as a CHSH urn experiment.

The delusions of the Bell deniers are now reaching comical proportions. It's like flat earthers claiming "satelites do not exist."

Just for the record, I fully agree with minkwe that "There is no such thing as a CHSH urn experiment."
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Joy Christian
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### Re: Pedagogical proofs of Bell's theorem

Heinera wrote:
minkwe wrote:There is no such thing as a CHSH urn experiment.

The delusions of the Bell deniers are now reaching comical proportions. It's like flat earthers claiming "satelites do not exist."

Talk about delusions. Bell's theory has been dead for 14 years now but the Bell fanatics just plug along with their lies and deceptions like nothing ever happened. But now we have the last nail in the coffin of Bell's dead junk physics theory.

viewtopic.php?f=6&t=481&p=14359#p14346
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