Why is superdeterminism not the universally accepted explanation of nonlocality?

[ThomasT]

I'm including things like polarization vectors in the description of the particular function P for a particular entangled pair, rather than including them as an input of the function. This is just an arbitrary choice in how I'm defining things, so it shouldn't affect anything.

You've discretized the possible values of λ, a presumably continuous underlying parameter, in terms of dichotomized detector outputs. Is this the point where LR models of entanglement become incompatible with QM ... and the design of Bell tests?

 

[lugita15]

λ is meant to denote the polarization (angle) of the incident optical disturbance. a (or b) denotes the polarizer setting.

So, from standard optics, individual detection is the function, cos²(a - λ_a), or in the same way for the B side.

You still haven't told me how any of this determines whether a particular photon goes through the polarizer or not.

You've asked, quite rightly I think, which of your steps would a more comprehensive local deterministic view disagree with.

It's the step (in your steps) from which a linear correlation between θ and rate of coincidental detection is necessitated.

So, which step, in your opinion, is that?

I'll summarize the argument up to the point where I think logical necessity enters the picture: the two photons in a pair exhibit identical behavior at identical angle settings, therefore the particles have coordinated in advance which polarizer angles they will go through and which ones they won't. From just this much, I think linear correlation is necessitated. The rest is just spelling out the chain of logical deduction.

 

[lugita15]

You've discretized the possible values of λ, a presumably continuous underlying parameter, in terms of dichotomized detector outputs. Is this the point where LR models of entanglement become incompatible with QM ... and the design of Bell tests?

In step 3, all I'm saying is that the two photons, right when they are created, agree in advance what polarizer angles to go through and what angles not to go through. (I'm talking about individual polarizer settings, not angle difference.) How they choose which angles they want to go through and which ones not to is irrelevant. They could do it using some polarization vector or anything else. But the point is that they've made a definite decision about what angles are "good" and what angles are "bad". And it is just this information that I am calling P(θ).

I think if you do not believe that the particles have chosen the good and bad angles in advance, but you believe in identical behavior at identical polarizer settings, you cannot sensibly call yourself a local determinist.

 

[ThomasT]

You still haven't told me how any of this determines whether a particular photon goes through the polarizer or not.

Afaik, there's no way to determine that.

But, in one way of modelling it, the rate of individual detection (ie., the photon flux), per unit time, is,

cos²(a - λ_a)

I'll summarize the argument up to the point where I think logical necessity enters the picture: the two photons in a pair exhibit identical behavior at identical angle settings ...

Wrt your points this would be:

2. One of these [QM] experimental predictions is that entangled photons are perfectly correlated when sent through polarizers oriented at the same angle ...

From this I might infer that entangled photons are created via some common causal mechanism, and that their underlying properties are therefore related (which is in line with the QM treatment).

... therefore the particles have coordinated in advance which polarizer angles they will go through and which ones they won't.

Wrt, say, Aspect 1982, the QM treatment is that the polarizer-incident optical disturbances are related wrt the conservation of angular momentum. The net effect of this assumption is that wrt θ = 0° coincidental detection attributes will be (1,1) or (0,0).

This isn't in conflict with LR predictions, and doesn't necessitate a linear correlation between θ and rate of coincidental detection.

From just this much, I think linear correlation is necessitated.

Well, that can't be it. Because identical detection attributes at EPR settings don't necessitate a linear correlation between θ and rate of coincidental detection. So, it has to be some other step.

 

[ThomasT]

In step 3, all I'm saying is that the two photons, right when they are created, agree in advance what polarizer angles to go through and what angles not to go through. (I'm talking about individual polarizer settings, not angle difference.) How they choose which angles they want to go through and which ones not to is irrelevant. They could do it using some polarization vector or anything else. But the point is that they've made a definite decision about what angles are "good" and what angles are "bad". And it is just this information that I am calling P(θ).

Ok, and what I'm saying is that this "information" which determines the rate of individual detection is irrelevant wrt determining the rate of coincidental detection. Wrt the Aspect experiments the creation of an entangled pair imparts a relationship between them. They have a particular common or identical polarization which determines the rate of individual detection, and they have a relationship which, obviously, does not determine the rate of individual detection.

As I mentioned, the usual way of thinking about this is that, wrt say the Aspect experiments, λ refers to an underlying common polarization orientation ... which is, as far as I can tell, an acceptable inference given the experimental results.

From that inference one can construct a model of individual detection that's compatible with QM.

But if one tries to model coincidental detection in terms of that underlying parameter (the parameter that determines individual detection), then such a model will not be able to reproduce all the predictions of QM.

Now, go back to the visualization I suggested. You'll see that the parameter that determines individual detection, λ, the polarization of polarizer-incident photons, has nothing to do with, ie., is irrelevant wrt, coincidental detection.

What might we conclude from this? The assumption of identical underlying (and locally produced via emission process) polarization seems supported by experimental results. But, as we've seen, the polarization orientation has nothing to do with the rate of coincidental detection, and, additionally, the underlying parameter determining the rate of coincidental detection cannot be varying from pair to pair. Hence, the only logical conclusion is that the underlying parameters determining individual detection and coincidental detection are different underlying parameters.

I think if you do not believe that the particles have chosen the good and bad angles in advance, but you believe in identical behavior at identical polarizer settings, you cannot sensibly call yourself a local determinist.

I do believe that there is an underlying parameter that determines rate of individual detection. And it's an experimental fact that when θ = 0° then coincidental detection attributes will be either (0,0) or (1,1).

And, I also believe that rate of coincidental detection is not determined by λ. It can be anything. Doesn't matter. Coincidental detection is only determined by θ.

 

[lugita15]

Afaik, there's no way to determine that.

But, in one way of modelling it, the rate of individual detection (ie., the photon flux), per unit time, is,

cos²(a - λ_a)

It's not good enough to determine the rate of individual detection. The hidden variable must determine whether a given photon goes through a given polarizer at a given angle. Otherwise you don't have a deterministic theory.

Wrt, say, Aspect 1982, the QM treatment is that the polarizer-incident optical disturbances are related wrt the conservation of angular momentum. The net effect of this assumption is that wrt θ = 0° coincidental detection attributes will be (1,1) or (0,0).

This isn't in conflict with LR predictions, and doesn't necessitate a linear correlation between θ and rate of coincidental detection.

First of all, stick to the idealized setup please, because that's what my steps are designed for. Second of all, we've hit on a crucial point here: while it's true that both the quantum mechanics guy and the local determinist agree that at identical angles you only get (1,1) or (0,0), they disagree as to the interpretation of this fact. Quantum mechanics says that you have a wave function for the two particle system which gets collapsed, nonlocally of course, as soon as one of the particles is measured, and that is how the other particle knows to do the same thing as the first particle, even though they're separated by a great distance. In contrast, the local determinist would say that it's not some nonlocal collapse that is correlating their behaviors, but rather their past interaction in which they determined *in advance* what angles they would go through and what angles they would not go through. It is because of this difference that step 3 must hold for local deterministic theories but does not hold for quantum mechanics.

 

[lugita15]

I do believe that there is an underlying parameter that determines rate of individual detection.

That's not good enough. You have to believe that not just the rate of individual detection is predetermined, but also each and every individual detection result. You have to believe that for each individual entangled pair, the two particles in the pair decide in advance the pair's "good" angles and "bad" angles, meaning exactly which angles the photon will go through and which it won't. Without all that, how can you call yourself a local determinist?

 

[ThomasT]

It's not good enough to determine the rate of individual detection. The hidden variable must determine whether a given photon goes through a given polarizer at a given angle. Otherwise you don't have a deterministic theory.

The function, cos²(a - λ_a), does determine whether a given photon goes through a given polarizer at a given angle. At least that's the assumption. But λ_a can't be controlled.

 

[ThomasT]

... we've hit on a crucial point here: while it's true that both the quantum mechanics guy and the local determinist agree that at identical angles you only get (1,1) or (0,0), they disagree as to the interpretation of this fact.

I'm not sure that's the case.

Quantum mechanics says that you have a wave function for the two particle system which gets collapsed, nonlocally of course, as soon as one of the particles is measured, and that is how the other particle knows to do the same thing as the first particle, even though they're separated by a great distance.

How, exactly, does that work? What do you think is the conceptual basis for that assumption?

In contrast, the local determinist would say that it's not some nonlocal collapse that is correlating their behaviors, but rather their past interaction in which they determined *in advance* what angles they would go through and what angles they would not go through.

I'm curious. This is based on a knowledge of the historically documented behavior of light. What makes you think that the standard QM treatment isn't based on that very same knowledge, and associated inferences/assumptions?

It's already been demonstrated that the function correlating individual detection to λ and individual polarizer setting is compatible with QM.

It is because of this difference that step 3 must hold for local deterministic theories but does not hold for quantum mechanics.

3. From this you conclude that both photons are consulting the same function P(θ). If P(θ)=1, then the photon goes through the polarizer, and if it equals zero the photon does not go through.

This seems to me to be compatible with QM. Why do you think it isn't?

 

[ThomasT]

You have to believe that not just the rate of individual detection is predetermined, but also each and every individual detection result.

I believe that. And that belief is compatible with the QM formalism regarding individual results. It's just that λ can't be controlled. At least that's the assumption (based on extant experimental preparation).

You have to believe that for each individual entangled pair, the two particles in the pair decide in advance the pair's "good" angles and "bad" angles, meaning exactly which angles the photon will go through and which it won't. Without all that, how can you call yourself a local determinist?

I do believe something akin to that, just not in those terms. And so does QM. But QM recognizes that what's determining coincidental detection is the relationship between entangled photons. And that that's a parameter that individual measurements aren't measuring. Hence, the nonseparability of the parameters relevant to the coincidental measurement of entangled particles, and the nonseparability/nonlocality of the associated QM formalism.

 

[ThomasT]

I asked you: which of your steps would a more comprehensive local deterministic view disagree with?

Stating that, it's the step (in your steps) from which a linear correlation between θ and rate of coincidental detection is necessitated.

Then I asked: which step, in your opinion, is that?

And you answered that it's your Step 2.

Which says:

2. One of these [QM] experimental predictions is that entangled photons are perfectly correlated when sent through polarizers oriented at the same angle ...

And I pointed out that it's clearly evident (ie., obvious) that this observation, this step, doesn't imply a linear correlation between θ and rate of coincidental detection.

So, which of your steps does imply such a correlation?

 

[lugita15]

The function, cos²(a - λ_a), does determine whether a given photon goes through a given polarizer at a given angle. At least that's the assumption. But λ_a can't be controlled.

So how do you get from cos²(a - λ_a) to a 0 or a 1?

 

[lugita15]

How, exactly, does that work? What do you think is the conceptual basis for that assumption?

Wavefunction collapse has an illustrious history going back to Max Born and John von Neumann. They saw collapse as the most natural explanation for the fact the wavefunction could be calculated deterministically using the Schrodinger equation, but the results of quantum mechanical experiments could only be predicted probabilistically. And I think it was Schrodinger himself who came up with the idea that entangled particles are described a common wavefunction that stretches across space, and that any changes in the wavefunction propagate instantaneously.

It's already been demonstrated that the function correlating individual detection to λ and individual polarizer setting is compatible with QM.

Where has this been demonstrated?

This seems to me to be compatible with QM. Why do you think it isn't?

According to the conventional interpretation of quantum mechanics, you have a nonlocal wavefunction collapse that determines on the spot whether the particles should go through or not. Whereas a local determinist believes that the particles have agreed in advance what angles to go through or not go through.

 

[lugita15]

I believe that. And that belief is compatible with the QM formalism regarding individual results.

No, it's not. In the QM formalism, the question of what angles the photons goes through and what angles it doesn't go through is not predetermined in advanced, but is rather determined on the spot in a random manner when the wavefunction collapse occurs.

I do believe something akin to that, just not in those terms. And so does QM.

No, QM doesn't.

 

[lugita15]

I asked you: which of your steps would a more comprehensive local deterministic view disagree with?

Stating that, it's the step (in your steps) from which a linear correlation between θ and rate of coincidental detection is necessitated.

Then I asked: which step, in your opinion, is that?

And you answered that it's your Step 2.

No, I answered that it's my step 3, which says that the particles determine in advance what angles to go through and what angles not to go through. From there, it is my claim that logical deduction will get you to the conclusion that local determinism is incompatible with the notion that all the predictions of QM are correct.

 

[ThomasT]

So how do you get from cos²(a - λ_a) to a 0 or a 1?

Since λ_a is presumably varying randomly from photon to photon, then individual detection attributes (0 or 1) can't be predicted.

We're concerned with the rate of detection at A, which can be denoted as the function R(A).
Since λ_a is varying randomly, then the angular difference argument of R(A) is also varying randomly. So, averaging over that, you get R(A) = .5 (the rate of detection, or photon flux per unit time, without the polarizer, a, in place).

That is, R(A) predicts a random sequence of 0's and 1's for a run ... half 0's and half 1's.

Which is the same thing that QM predicts.

 

[lugita15]

Since λ_a is presumably varying randomly from photon to photon, then individual detection attributes (0 or 1) can't be predicted.

OK, but given λ_a for a particular photon pair, how do you get a 0 or a 1 out of that?

 

[ThomasT]

Wavefunction collapse has an illustrious history going back to Max Born and John von Neumann. They saw collapse as the most natural explanation for the fact the wavefunction could be calculated deterministically using the Schrodinger equation, but the results of quantum mechanical experiments could only be predicted probabilistically. And I think it was Schrodinger himself who came up with the idea that entangled particles are described by a common wavefunction that stretches across space, and that any changes in the wavefunction propagate instantaneously.

I don't think that's the most natural, or logical, way of looking at the experimental situation or interpreting the QM formalism. Instantaneous propagation seems to be a contradiction in terms. Reification of ψ carries some unnecessary baggage with it, and I doubt that most working physicists think in those terms.

Both LR and QM have coincidental detection determined by the incident photons consulting a common function. Ie., there's a common cause which produces the relationship between entangled photons that the polarizers are jointly measuring. The difference is that QM doesn't use λ (which refers to the polarization orientation of the polarizer-incident photons), presumably recognizing that the value of λ is irrelevant wrt determining rate of coincidental detection.

Where has this been demonstrated?

In my previous post. Or you can go back to Bell 1964.

So in this simple case there is no difficulty in the view that the result of every measurement is determined by the value of an extra variable, and that the statistical features of quantum mechanics arise because the value of this variable is unknown in individual instances.

Wrt,

3. From this you conclude that both photons are consulting the same function P(θ). If P(θ)=1, then the photon goes through the polarizer, and if it equals zero the photon does not go through.

I said,

This seems to me to be compatible with QM. Why do you think it isn't?

To which you replied,

According to the conventional interpretation of quantum mechanics, you have a nonlocal wavefunction collapse that determines on the spot whether the particles should go through or not. Whereas a local determinist believes that the particles have agreed in advance what angles to go through or not go through.

Both QM and LR have entangled photons consulting the same function. This is because they both assume a common cause. The stuff about nonlocal wavefunction collapse is just unwarranted and unnecessary, imo. The fact is that QM is acausal and (to paraphrase Bohm) nonmechanical wrt entanglement.

Why is the QM formalism the way it is? I'm not sure about that, but I think it does have to do with the assumption of a common cause. Also, as I think I've shown, the value of λ is irrelevant wrt determining rate of coincidental detection. And, anyway, QM doesn't have to be causal, since it's just calculating measurement probabilities.

Why can one still assume local determinism given the QM formalism? Because the QM formalism is acausal wrt entanglement. So, one might interpret entanglement as being due to nonlocal transmissions between entangled photons, or not. No way to know, afaik. Ultimately, the QM treatment wrt optical Bell tests is evaluated wrt the known behavior of light. And of course so should be any LR treatment of entanglement ... which is something that your line of reasoning seems to ignore.

 

[ThomasT]

OK, but given λ_a for a particular photon pair, how do you get a 0 or a 1 out of that?

The prediction, wrt LR and QM, is that the result at A, wrt any particular individual detection, will be either 0 (no detection registered) or 1 (detection registered).

 

[ThomasT]

No, I answered that it's my step 3, which says that the particles determine in advance what angles to go through and what angles not to go through. From there, it is my claim that logical deduction will get you to the conclusion that local determinism is incompatible with the notion that all the predictions of QM are correct.

It's been shown that the underlying parameter that determines coincidental detection (lets denote rate of coincidental detection as R(A, B)) is not varying from pair to pair. So, why would you think that an underlying parameter, eg. your P(θ), that is varying randomly from pair to pair is determining R(A,B)?

 

[ThomasT]

In the QM formalism, the question of what angles the photon goes through and what angles it doesn't go through is not predetermined in advance ...

But it can be interpreted that way. As noted, and demonstrated, an LR account of individual measurement is compatible with QM.

... but is rather determined on the spot in a random manner when the wavefunction collapse occurs.

Well, now you're talking about coincidental detection. Which is a different observational context. And, as I've mentioned, the QM treatment is nonmechanical wrt the projection along an axis associated with a detection. Afaik, this is based on the known behavior of light (eg., the law of Malus), and is retained because it works.

 

[lugita15]

ThomasT, I asked you, in what step of my argument do you believe that a local determinist can part ways? You say that it is whatever step that makes linear correlation an inevitability, and I claim that is at a very early stage in the argument that it becomes logically inevitable, but you disagree. Let me try once again to present my argument; I've refined the steps in an attempt to iron out any points of confusion or disagreement we've had so far. (And keep in mind that this is in the context of my idealized setup, not an actual practical Bell test like Aspect's.)

1. Suppose you are a local determinist who agrees with the experimental predicts of QM.
2. One of those experimental predictions is that the two photons in an entangled pair exhibit identical behavior when sent through identical polarizers at oriented at identical angles.
3. You conclude that the two photons have agreed in advance (in any way, whether by polarization vector or some other way) what polarizer angles they should go through and what polarizer angles they should not go through.
4. If they have decided to go through at particular angle θ, let us denote this by P(θ)=1, and if they have not decided to go through at the angle θ, let us denote this by P(θ)=0.
5. Let R(θ1,θ2) denote the percentage of mismatches (situations where one photon goes through and the other does not) if polarizer 1 is set to angle θ1 and polarizer 2 is set to angle θ2.
6. Using the definition of P in step 4, R(θ1,θ2) is the probability that P(θ1)≠P(θ2) for a randomly selected entangled pair.
7. It is an experimental prediction of quantum mechanics that R(θ1+C,θ2+C)=R(θ1,θ2) and R(θ1,θ2)=R(θ2,θ1), so we can just write R(θ1,θ2) as R(θ) where θ=|θ1-θ2|.
8. Using steps 6 and 7, the probability that P(θ1)≠P(θ2) for a randomly selected entangled pair is given by R(θ) where θ=|θ1-θ2|.
9. It is a mathematical fact that if you have two events A and B, then the probability that at least one of these events occurs (in other words the probability that A or B occurs) is less than or equal to the probability that A occurs plus the probability that B occurs.
10. You conclude that the probability that P(-30)≠P(30) is less than or equal to the probability that that P(-30)≠P(0) plus the probability that P(0)≠P(30), or in terms of R we can say R(60)≤R(30)+R(30)=2R(30)

So which step do you think there can be disagreeement on by a local determinist? Let me tell you that 1 is the assumption, 4 and 5 are definitions, 2 and 7 are experimental predictions of quantum mechanics, and 9 is a mathematical fact, so I think those are all beyond dispute. That leaves 3, 6, 8, and 10. But 6 follows from 4 and 5, 8 follows from 6 and 7, and 10 follows from 9.

That leaves 3, which I think is the point after which the argument becomes inevitable, but I am happy to hear if you think any of the other steps can be disagreed with.

 

[lugita15]

And let me also say that you're right in one respect about 3: it is not in and of itself an experimental difference between local determinism and quantum mechanics. Rather, it is a philosophical difference between local determinism and certain interpretations of QM which believe in nonlocal wavefunction collapse or nonlocal communication. Yet it is my claim that this particular philosophical, interpretational difference turns out to lead to differences in the actual empirical predictions of these two philosophical belief systems. And my purpose in this now 10-step argument is to demonstrate that there are such empirical differences: that a local deterministic universe cannot match all the experimental predictions of quantum mechanics.

 

[ThomasT]

That leaves 3, which I think is the point after which the argument becomes inevitable, but I am happy to hear if you think any of the other steps can be disagreed with.

Step 3. has to do with individual detection, and the randomly varying underlying parameter which determines that. And, as has been shown, that underlying parameter is irrelevant wrt coincidental detection.

So, ok, if you assume that that parameter is determining coincidental detection, then that might account for the incorrect conclusion that the correlation between θ and rate of coincidental detection is linear.

But it certainly doesn't inform wrt the locality or nonlocality of nature.

 

[ThomasT]

And let me also say that you're right in one respect about 3: it is not in and of itself an experimental difference between local determinism and quantum mechanics. Rather, it is a philosophical difference between local determinism and certain interpretations of QM which believe in nonlocal wavefunction collapse or nonlocal communication. Yet it is my claim that this particular philosophical, interpretational difference turns out to lead to differences in the actual empirical predictions of these two philosophical belief systems. And my purpose in this now 10-step argument is to demonstrate that there are such empirical differences: that a local deterministic universe cannot match all the experimental predictions of quantum mechanics.

I think that the differences have nothing to do with what's actually happening in the reality underlying instrumental behavior.

And of course I have no way of proving my contention any more than you do of proving yours.

I just think that mine is ... more reasonable, given what's known about the behavior of light.
That is, there's no compelling reason, imo, to adopt the assumption that entangled particles are communicating nonlocally, while there are, imo, some compelling reasons to suppose that they aren't. So, in the absence of empirical evidence to the contrary, we retain the assumptions of locality and determinism.

 

[lugita15]

So, ok, if you assume that that parameter is determining coincidental detection, then that might account for the incorrect conclusion that the correlation between θ and rate of coincidental detection is linear.

Where do I ever say in my now 10 steps that the parameter that determines the rate of individual detection must be the same as the parameter that determines the rate of coincidental detection? Remember, my 10-step argument is about the idealized setup, in which the data consists of individual detection results and the mismatches in this data.

 

[ThomasT]

Where do I ever say in my now 10 steps that the parameter that determines the rate of individual detection must be the same as the parameter that determines the rate of coincidental detection? Remember, my 10-step argument is about the idealized setup, in which the data consists of individual detection results and the mismatches in this data.

"individual detection results and the mismatches"

Mismatches are combined results. A different observational context than individual measurement.

It seems clear enough to me that what you're saying is that the underlying parameter that determines individual detection is the same underlying parameter that determines coincidental detection.

But how can this be if, wrt coincidental detection, the underlying parameter that determines individual detection can be anything and coincidental detection only varies as a function of θ?

 

[lugita15]

"individual detection results and the mismatches"

Mismatches are combined results. A different observational context than individual measurement.

Argh. I could try arguing with you yet again that a mismatch between two individual detection results is entirely determined BY those two individual detection results, but we've gone in circles around this numerous times.

So instead, let me ask you this: do you agree with the logic from step 5 to step 6 in my new 10 step proof? If you disagree with it, what's the problem you see with getting step 6? Because that is the closest thing I see to an assumption about coincidental detection and individual detection being connected. To me, step 6 seems obvious, but YMMV.

 

[ThomasT]

Argh. I could try arguing with you yet again that a mismatch between two individual detection results is entirely determined BY those two individual detection results ...

I'm not arguing that the combined results aren't composed of individual results. Obviously, mismatches or coincidental detections are composed of individual results. But you still don't get that the randomly varying underlying parameter that determines individual results can't be what's determining the combined results.

... do you agree with the logic from step 5 to step 6 in my new 10 step proof? If you disagree with it, what's the problem you see with getting step 6? Because that is the closest thing I see to an assumption about coincidental detection and individual detection being connected. To me, step 6 seems obvious, but YMMV.

5. Let R(θ1,θ2) denote the percentage of mismatches (situations where one photon goes through and the other does not) if polarizer 1 is set to angle θ1 and polarizer 2 is set to angle θ2.
6. Using the definition of P in step 4, R(θ1,θ2) is the probability that P(θ1)≠P(θ2) for a randomly selected entangled pair.

If the relevant underlying parameters, P(θ1) and P(θ2), determining whether the incident photons are transmitted or not by the polarizers is determined by a common function, then would the probability that P(θ1)≠P(θ2) be 0? Maybe not. But if we allow that that function isn't determining the rate of coincidental detection, then the probability of A≠B wouldn't be dependent on the values of P(θ1), P(θ2), or P(θ), would it?

What does YMMV mean?

 

[lugita15]

I'm not arguing that the combined results aren't composed of individual results. Obviously, mismatches or coincidental detections are composed of individual results. But you still don't get that the randomly varying underlying parameter that determines individual results can't be what's determining the combined results.

If computers are composed of circuits, then whatever determines the behavior of the circuits must determine the behavior of the computer. If molecules are composed of atoms, then whatever determines the behavior of atoms must determine the behavior of computers. Coincidental detection result data is (for my idealized setup) composed of individual detection result data, so whatever determines individual detection results must determine the coincidental detection results. It seems obvious to me, but in any case I don't think I used this fact in my proof.

If the relevant underlying parameters, P(θ1) and P(θ2), determining whether the incident photons are transmitted or not by the polarizers is determined by a common function, then would the probability that P(θ1)≠P(θ2) be 0?

We're talking about the probability that P(θ1)≠P(θ2) for a randomly selected entangled pair, so there's no reason why it should be zero.

And by the way, you may notice in my latest proof that I've abandoned the terminology of the two particles consulting the common function P(θ), to minimize confusion about whether I'm restricting the kind of hidden variable the particles can have. Instead, how I now express it is that the particles choose in advance what angles they should go through or not, and P(θ) is just our way of describing what angles they have selected to go through and what angles not to.

But if we allow that that function isn't determining the rate of coincidental detection, then the probability of A≠B wouldn't be dependent on the values of P(θ1), P(θ2), or P(θ), would it?

I'm not sure what you mean by this. What are A and B?

What does YMMV mean?

It's short for "your mileage may vary", meaning I experience this but you may experience something else.