Bell's theorem refuted via elementary probability theory

[gill1109]

GW, given your failure to provide a function set integrating to -a.b, shall we conclude that you now accept that there is no such set, and that your "proof" must therefore be flawed?

GW, “local” is right.

[Gordon Watson]

GW, given your failure to provide a function set integrating to -a.b, shall we conclude that you now accept that there is no such set, and that your "proof" must therefore be flawed?

GW, “local” is right.

https://vixra.org/abs/2011.0073

Title: Bell's Theorem Refuted via True Local Realism

Abstract Bell's theorem has been described as the most profound discovery of science; one of the few essential discoveries of 20th Century physics; indecipherable to non-mathematicians. However, taking elementary analysis to be an adequate logic here, we refute Bell's theorem, correct his inequality and identify his error. Further, we do this under the principle of true local realism, the union of true locality (or relativistic causality: no influence propagates superluminally) and true realism (or non-naive realism: some existents change interactively). We thus lay the foundation for a more complete physical theory: one in line with Einstein's ideas and Bell's hopes. Let's see.

Gordon
.

[gill1109]

GW, given your failure to provide a function set integrating to -a.b, shall we conclude that you now accept that there is no such set, and that your "proof" must therefore be flawed?

GW, “local” is right.

https://vixra.org/abs/2011.0073

Title: Bell's Theorem Refuted via True Local Realism

Abstract Bell's theorem has been described as the most profound discovery of science; one of the few essential discoveries of 20th Century physics; indecipherable to non-mathematicians. However, taking elementary analysis to be an adequate logic here, we refute Bell's theorem, correct his inequality and identify his error. Further, we do this under the principle of true local realism, the union of true locality (or relativistic causality: no influence propagates superluminally) and true realism (or non-naive realism: some existents change interactively). We thus lay the foundation for a more complete physical theory: one in line with Einstein's ideas and Bell's hopes. Let's see.

Gordon
.

Still no A, B and rho. The same misleading, ambiguous, unconventional notations. Gordon: you are not convincing anyone. You convinced yourself, and only yourself.

[local]

GW, "gill1109" is right.

[Gordon Watson]

GW, given your failure to provide a function set integrating to -a.b, shall we conclude that you now accept that there is no such set, and that your "proof" must therefore be flawed?

GW, “local” is right.

https://vixra.org/abs/2011.0073

Title: Bell's Theorem Refuted via True Local Realism

Abstract Bell's theorem has been described as the most profound discovery of science; one of the few essential discoveries of 20th Century physics; indecipherable to non-mathematicians. However, taking elementary analysis to be an adequate logic here, we refute Bell's theorem, correct his inequality and identify his error. Further, we do this under the principle of true local realism, the union of true locality (or relativistic causality: no influence propagates superluminally) and true realism (or non-naive realism: some existents change interactively). We thus lay the foundation for a more complete physical theory: one in line with Einstein's ideas and Bell's hopes. Let's see.

Gordon
.

Still no A, B and rho. The same misleading, ambiguous, unconventional notations. Gordon: you are not convincing anyone. You convinced yourself, and only yourself.

Richard,

1. Almost every sentence and every equation is precisely identified. After all, I am here to learn!

2. So please provide some clearly identified examples that support your claims re misleading and ambiguous notations. For, of course, I'll fix them.

3. Regarding my unconventional notation, please point me to their equivalents in the Bellian literature and I'll happily make changes.

4. As for the functions A and B : see Eqns (7)-(14) for their foundations.

5. As for ρ, surely it is clear from Eqn (19) that I use the standard ρ; the wholistic one that Bell uses: .

6. I hesitate to say more now (and prove you wrong) until I've had a chance to correct any obvious failings.

Thanks; Gordon.

[local]

Please just state your A and B functions in a form that we can integrate. All those arrows in your equations are totally opaque. Telling us the "foundations" is useless. Thank you.

Your rho function is acceptable in principle; we can fill in the details once we have your A and B.

[gill1109]

Richard,

1. Almost every sentence and every equation is precisely identified. After all, I am here to learn!

2. So please provide some clearly identified examples that support your claims re misleading and ambiguous notations. For, of course, I'll fix them.

3. Regarding my unconventional notation, please point me to their equivalents in the Bellian literature and I'll happily make changes.

4. As for the functions A and B : see Eqns (7)-(14) for their foundations.

5. As for ρ, surely it is clear from Eqn (19) that I use the standard ρ; the wholistic one that Bell uses: .

6. I hesitate to say more now (and prove you wrong) until I've had a chance to correct any obvious failings.

Thanks; Gordon.

You have still not learnt a thing.

I find equations (7) to (14) an incomprehensible mess of symbols.

Just tell me in simple words a recipe to compute an outcome on Alice's side given a value of lambda and a direction a.
Tell me in simple words a recipe to compute an outcome on Bob's side given a value of lambda and a direction b.
Tell us how you imagine Nature determining a value of lambda

Imagine you want someone to write computer programs for you, to simulate the two measurement stations, and the source. Three programs for three ordinary PC's.

You can't do it, right! You are living in a world of your own imagination but you can't make it accessible to anyone else.

Since I know perfectly valid arguments why it cannot be done, arguments which have stood the test of time and been vastly refined over more than 50 years, and which you do not even attempt to counter, I don't see why I should put any effort into your project.

Maybe you will have better luck trying to convince someone else who like you doesn't think that Bell was right, but unlike you who *is* able to write maths, or to give instructions to computer programmers to build something.

If the programs work, I will be convinced. If I can read the formulas so that I can quickly program them myself, and if they work, then I'll be convinced.

[Gordon Watson]

Please just state your A and B functions in a form that we can integrate. All those arrows in your equations are totally opaque. Telling us the "foundations" is useless. Thank you.

Your rho function is acceptable in principle; we can fill in the details once we have your A and B.

Thanks local, for your interest. Given your apparent agreement with Richard Gill, please let me know of any misleading and ambiguous notations, etc. I am here to learn and contribute.

Also, what does this mean and refer to: Telling us the "foundations" is useless. ??

1. The arrows are associated with the schematics mentioned in 2.1. They are intended to show the flight of each particle (pairwise correlated with its twin) through the system. See explanatory notes in 2.2, etc.

2. As I had earlier shown -- with any adequate functions A, B -- the integral delivers the weighted average of all possible results, weighted according to their probabilities.

3. Which is precisely the definition of an expectation: and precisely the result that you should obtain via the following functions:

, (GW-1)

, (GW-2)

where denotes the relevant equivalence relation -- see 3.1 -- with the qualifiers understood and dropped for simplicity.

Thanks again; Gordon
.

[local]

It's total gobbledegook to me. I can't make any sense out of it whatsoever.

How about make a simulation? Do you know how to program?

You're not going anywhere with that paper if no-one can make any sense out of it.

[local]

Later this evening or tomorrow I will give you an example calculation showing the functions and integration. Beware, it doesn't yield -a.b. Then hopefully you will be able to provide us with a similar derivation that yields -a.b.

[Gordon Watson]

It's total gobbledegook to me. I can't make any sense out of it whatsoever.

How about make a simulation? Do you know how to program?

You're not going anywhere with that paper if no-one can make any sense out of it.

So, in eqn (7), you cannot see the particles leaving the source and moving through the system? Etc?

I do not program

You wanted two functions that could be integrated to yield the expectation -a.b.

You now have them: me trusting that, as required, you see that each equals ±1 via the dot-product?

Gordon

[Gordon Watson]

Later this evening or tomorrow I will give you an example calculation showing the functions and integration. Beware, it doesn't yield -a.b. Then hopefully you will be able to provide us with a similar derivation that yields -a.b.

Good. Thanks. I am keen to see how close you can get to the QM result.

Incidentally: Do you have the QM calculation that provides the step-by-step result in Bell 1964:(3); ie, his eqn (3)?

Cheers; Gordon

[local]

Nah, I don't see it. All I see is some deltas, babbling about equivalence relations, unusual operators, equations inside integrals, etc. Neither Richard nor I can make any sense of it. Trust me, we are both Cantabrigians and smart guys. Oxonians are smart too.

Straight up Gordon, fill in the following equations:

A(a, lambda) = ???
B(b, lambda) = ???

Here is an example (not saying it is good for anything):

A(a, lambda) = abs(a - lambda) > PI/4 : 1; abs(a - lambda) <= PI/4 : 0

More typically, sin's and cos's are involved.

Don't talk to me about deltas, equivalence relations, equations within integrals, and all that stuff that both Richard and I find totally opaque. Just give the functions and I will write a simulation that shows it does not produce -a.b. OK?

Good. Thanks. I am keen to see how close you can get to the QM result.

Obviously, it will be something like -a.b / 2, i.e., half the required amplitude. I'm not trying to get close to 'the QM result', and anyway, I don't believe that -a.b is "the QM result". Aren't you following the threads?

Incidentally: Do you have the QM calculation that provides the step-by-step result in Bell 1964:(3); ie, his eqn (3)?

Don't change the subject. Give me your functions that I can use to write a simulation.

[FrediFizzx]

Nah, I don't see it. All I see is some deltas, babbling about equivalence relations, unusual operators, equations inside integrals, etc. Neither Richard nor I can make any sense of it. Trust me, we are both Cantabrians and smart guys!

Straight up Gordon, fill in the following equations:

A(a, lambda) = ???
B(b, lambda) = ???

Here is an example (not saying it is good for anything):

A(a, lambda) = abs(a - lambda) > PI/4 : 1; abs(a - lambda) <= PI/4 : 0

More typically, sin's and cos's are involved.

Don't talk to me about deltas, equivalence relations, equations within integrals, and all that stuff that both Richard and I find totally opaque. Just give the functions and I will write a simulation that shows it cannot produce -a.b. OK?

Good. Thanks. I am keen to see how close you can get to the QM result.

Obviously, it will be something like -a.b / 2, i.e., half the required amplitude. I'm not trying to get close to 'the QM result', and anyway, I don't believe that -a.b is "the QM result". Aren't you following the threads?

Incidentally: Do you have the QM calculation that provides the step-by-step result in Bell 1964:(3); ie, his eqn (3)?

Don't change the subject. Give me your functions that I can use to write a simulation.

1. As requested, and thus, as before:

A(a,λ) = (λ ~ a^±)•a. B(a,λ) = -(λ ~ b^±)•b. With ~ denoting an equivalence relation and • a dot-product.

If you cannot handle ~, let me know.

2. In the meantime, how about delivering on some of your commitments?

Will that simulation you promised be coming my way tonight or tomorrow?

And what is the point of a relation that is "not very good for anything"?

Please let me see some of the functions that involve sin and cos.

3. So you can get to one-half the QM result? Malus (C1810) could do that experiment and the calculation: if requested.

[He could also, in a related experiment, simulate the QM result: if he'd been asked!]

So if Malus can readily get one-half the QM result, how come you cannot see that a better source [beyond the then available technology] gives a better correlation today?

4. Further: you now tell me that you do not accept the QM results. I take it you thus deny the results for the EPRB, Aspect, GHZ, GHSZ, +++ experiments?

NB: My results encompass them all!

Best; Gordon

[local]

I suggest you try to get your work published. Good luck!

[Gordon Watson]

I suggest you try to get your work published. Good luck!

Thanks. I guess, despite my offer to help, you're not comfortable with equivalence relations. So, after a brief and possibly irrelevant interlude, I wonder if the functions at the end might help you, Gill, etc, understand my refutation of Bell's theorem etc more clearly?

..........

For, in short, via this

(GW-1, here)

and

(GW-2, here)

I presumed that you understood the physical significance of via §3.1 of my draft.

Then, of course, if denotes a short-form equivalence relation, you can see from the context 'which' equivalence relation is in play.

I also expected you to understand that was, via §2.3, a particle's post-interaction spin-axis; etc.

Thus, if then ; etc.

......................

Now the above notation was my way of "improving" on Bell's failed attempt -- see Bell 1964:(4) -- at using sign.

For I need no a-prime and NO result is left undetermined.

SO NOW: with a-prime abandoned, and accepting that the probability of such indetermination is zero, here are some A(a,λ) and B(b,λ) functions for you to integrate:

A(a,λ) = sgn. (GW-3)

B(b,λ) = -sgn. (GW-4)

QED via Bayesian inferencing; and, please, show each step in your integration.

Must run. Thanks again; Gordon

[local]

Thanks. I guess, despite my offer to help, you're not comfortable with equivalence relations.

Nah, I know what an equivalence relation is. It's just that your analysis is very obscure and there's no apparent reason for you to obfuscate it with equivalence relations and nonstandard notation.

I wonder if the functions at the end might help you.

I don't need your help, but yes, the functions will finally allow for us to integrate.

Now the above notation was my way of "improving" on Bell's failed attempt

LOL.

A(a,λ) = sgn
B(b,λ) = -sgn

Finally, something usable.

I wrote a short program to integrate the functions numerically, assuming that lamba is an angle distributed on [0..2*PI]. Here it is (not intended as production code but it works just fine):

Code: Select all

#include <stdio.h>
#include <math.h>
#include <memory.h>

constexpr auto PI = 3.1415926;

int results_A[1000];
int results_B[1000];

int main()
{
   double lambda;
   double a, b;
   int sum;
   double correlation;
   int trial, trials;
   FILE* f;

   fopen_s(&f, "d:\\tmp\\out.txt", "w");
   a = 0;
   for (b = 0.0; b < PI; b += 2 * PI / 360)
   {
      memset(results_A, 0, 1000 * sizeof(int));
      memset(results_B, 0, 1000 * sizeof(int));
      sum = 0;
      trials = 0;
      for (lambda = 0.0; lambda <= 2 * PI; lambda += 2 * PI / 360)
      {
         results_A[trials] = cos(a - lambda) >= 0.0 ? 1 : -1;
         results_B[trials] = cos(b - lambda) >= 0.0 ? -1 : 1;
         trials++;
      }
      // Correlate.
      for (trial = 0; trial < trials; trial++)
      {
         sum += results_A[trial] * results_B[trial];
      }
      correlation = (double)sum / trials;
      fprintf(f, "a=%.2f b=%.2f correlation=%.2f\n", a, b, correlation);
   }
   fclose(f);
   return 0;
}

Following is the output. It takes a = 0.0 and scans b over [0..PI]. You can also set a within [0..PI]. That just shifts the result. I could choose lambda randomly but it changes nothing. The output shows that only a linear ramp is produced, ala Bell. I could plot it for you but I have only limited time for nonsense. Your functions do not produce -a.b.

a=0.00000 b=0.00000 correlation=-1.00000
a=0.00000 b=0.01745 correlation=-0.98892
a=0.00000 b=0.03491 correlation=-0.97784
a=0.00000 b=0.05236 correlation=-0.96676
a=0.00000 b=0.06981 correlation=-0.95568
a=0.00000 b=0.08727 correlation=-0.94460
a=0.00000 b=0.10472 correlation=-0.93352
a=0.00000 b=0.12217 correlation=-0.92244
a=0.00000 b=0.13963 correlation=-0.91136
a=0.00000 b=0.15708 correlation=-0.90028
a=0.00000 b=0.17453 correlation=-0.88920
a=0.00000 b=0.19199 correlation=-0.87812
a=0.00000 b=0.20944 correlation=-0.86704
a=0.00000 b=0.22689 correlation=-0.85596
a=0.00000 b=0.24435 correlation=-0.84488
a=0.00000 b=0.26180 correlation=-0.83380
a=0.00000 b=0.27925 correlation=-0.82271
a=0.00000 b=0.29671 correlation=-0.81163
a=0.00000 b=0.31416 correlation=-0.80055
a=0.00000 b=0.33161 correlation=-0.78947
a=0.00000 b=0.34907 correlation=-0.77839
a=0.00000 b=0.36652 correlation=-0.76731
a=0.00000 b=0.38397 correlation=-0.75623
a=0.00000 b=0.40143 correlation=-0.74515
a=0.00000 b=0.41888 correlation=-0.73407
a=0.00000 b=0.43633 correlation=-0.72299
a=0.00000 b=0.45379 correlation=-0.71191
a=0.00000 b=0.47124 correlation=-0.70083
a=0.00000 b=0.48869 correlation=-0.68975
a=0.00000 b=0.50615 correlation=-0.67867
a=0.00000 b=0.52360 correlation=-0.66759
a=0.00000 b=0.54105 correlation=-0.65651
a=0.00000 b=0.55851 correlation=-0.64543
a=0.00000 b=0.57596 correlation=-0.63435
a=0.00000 b=0.59341 correlation=-0.62327
a=0.00000 b=0.61087 correlation=-0.61219
a=0.00000 b=0.62832 correlation=-0.60111
a=0.00000 b=0.64577 correlation=-0.59003
a=0.00000 b=0.66323 correlation=-0.57895
a=0.00000 b=0.68068 correlation=-0.56787
a=0.00000 b=0.69813 correlation=-0.55679
a=0.00000 b=0.71558 correlation=-0.54571
a=0.00000 b=0.73304 correlation=-0.53463
a=0.00000 b=0.75049 correlation=-0.52355
a=0.00000 b=0.76794 correlation=-0.51247
a=0.00000 b=0.78540 correlation=-0.50139
a=0.00000 b=0.80285 correlation=-0.49030
a=0.00000 b=0.82030 correlation=-0.47922
a=0.00000 b=0.83776 correlation=-0.46814
a=0.00000 b=0.85521 correlation=-0.45706
a=0.00000 b=0.87266 correlation=-0.44598
a=0.00000 b=0.89012 correlation=-0.43490
a=0.00000 b=0.90757 correlation=-0.42382
a=0.00000 b=0.92502 correlation=-0.41274
a=0.00000 b=0.94248 correlation=-0.40166
a=0.00000 b=0.95993 correlation=-0.39058
a=0.00000 b=0.97738 correlation=-0.37950
a=0.00000 b=0.99484 correlation=-0.36842
a=0.00000 b=1.01229 correlation=-0.35734
a=0.00000 b=1.02974 correlation=-0.34626
a=0.00000 b=1.04720 correlation=-0.33518
a=0.00000 b=1.06465 correlation=-0.32410
a=0.00000 b=1.08210 correlation=-0.31302
a=0.00000 b=1.09956 correlation=-0.30194
a=0.00000 b=1.11701 correlation=-0.29086
a=0.00000 b=1.13446 correlation=-0.27978
a=0.00000 b=1.15192 correlation=-0.26870
a=0.00000 b=1.16937 correlation=-0.25762
a=0.00000 b=1.18682 correlation=-0.24654
a=0.00000 b=1.20428 correlation=-0.23546
a=0.00000 b=1.22173 correlation=-0.22438
a=0.00000 b=1.23918 correlation=-0.21330
a=0.00000 b=1.25664 correlation=-0.20222
a=0.00000 b=1.27409 correlation=-0.19114
a=0.00000 b=1.29154 correlation=-0.18006
a=0.00000 b=1.30900 correlation=-0.16898
a=0.00000 b=1.32645 correlation=-0.15789
a=0.00000 b=1.34390 correlation=-0.14681
a=0.00000 b=1.36136 correlation=-0.13573
a=0.00000 b=1.37881 correlation=-0.12465
a=0.00000 b=1.39626 correlation=-0.11357
a=0.00000 b=1.41372 correlation=-0.10249
a=0.00000 b=1.43117 correlation=-0.09141
a=0.00000 b=1.44862 correlation=-0.08033
a=0.00000 b=1.46608 correlation=-0.06925
a=0.00000 b=1.48353 correlation=-0.05817
a=0.00000 b=1.50098 correlation=-0.04709
a=0.00000 b=1.51844 correlation=-0.03601
a=0.00000 b=1.53589 correlation=-0.02493
a=0.00000 b=1.55334 correlation=-0.01385
a=0.00000 b=1.57080 correlation=-0.00277
a=0.00000 b=1.58825 correlation=0.00831
a=0.00000 b=1.60570 correlation=0.01939
a=0.00000 b=1.62316 correlation=0.03047
a=0.00000 b=1.64061 correlation=0.04155
a=0.00000 b=1.65806 correlation=0.05263
a=0.00000 b=1.67552 correlation=0.06371
a=0.00000 b=1.69297 correlation=0.07479
a=0.00000 b=1.71042 correlation=0.08587
a=0.00000 b=1.72788 correlation=0.09695
a=0.00000 b=1.74533 correlation=0.10803
a=0.00000 b=1.76278 correlation=0.11911
a=0.00000 b=1.78024 correlation=0.13019
a=0.00000 b=1.79769 correlation=0.14127
a=0.00000 b=1.81514 correlation=0.15235
a=0.00000 b=1.83260 correlation=0.16343
a=0.00000 b=1.85005 correlation=0.17452
a=0.00000 b=1.86750 correlation=0.18560
a=0.00000 b=1.88496 correlation=0.19668
a=0.00000 b=1.90241 correlation=0.20776
a=0.00000 b=1.91986 correlation=0.21884
a=0.00000 b=1.93732 correlation=0.22992
a=0.00000 b=1.95477 correlation=0.24100
a=0.00000 b=1.97222 correlation=0.25208
a=0.00000 b=1.98968 correlation=0.26316
a=0.00000 b=2.00713 correlation=0.27424
a=0.00000 b=2.02458 correlation=0.28532
a=0.00000 b=2.04204 correlation=0.29640
a=0.00000 b=2.05949 correlation=0.30748
a=0.00000 b=2.07694 correlation=0.31856
a=0.00000 b=2.09440 correlation=0.32964
a=0.00000 b=2.11185 correlation=0.34072
a=0.00000 b=2.12930 correlation=0.35180
a=0.00000 b=2.14675 correlation=0.36288
a=0.00000 b=2.16421 correlation=0.37396
a=0.00000 b=2.18166 correlation=0.38504
a=0.00000 b=2.19911 correlation=0.39612
a=0.00000 b=2.21657 correlation=0.40720
a=0.00000 b=2.23402 correlation=0.41828
a=0.00000 b=2.25147 correlation=0.42936
a=0.00000 b=2.26893 correlation=0.44044
a=0.00000 b=2.28638 correlation=0.45152
a=0.00000 b=2.30383 correlation=0.46260
a=0.00000 b=2.32129 correlation=0.47368
a=0.00000 b=2.33874 correlation=0.48476
a=0.00000 b=2.35619 correlation=0.49584
a=0.00000 b=2.37365 correlation=0.50693
a=0.00000 b=2.39110 correlation=0.51801
a=0.00000 b=2.40855 correlation=0.52909
a=0.00000 b=2.42601 correlation=0.54017
a=0.00000 b=2.44346 correlation=0.55125
a=0.00000 b=2.46091 correlation=0.56233
a=0.00000 b=2.47837 correlation=0.57341
a=0.00000 b=2.49582 correlation=0.58449
a=0.00000 b=2.51327 correlation=0.59557
a=0.00000 b=2.53073 correlation=0.60665
a=0.00000 b=2.54818 correlation=0.61773
a=0.00000 b=2.56563 correlation=0.62881
a=0.00000 b=2.58309 correlation=0.63989
a=0.00000 b=2.60054 correlation=0.65097
a=0.00000 b=2.61799 correlation=0.66205
a=0.00000 b=2.63545 correlation=0.67313
a=0.00000 b=2.65290 correlation=0.68421
a=0.00000 b=2.67035 correlation=0.69529
a=0.00000 b=2.68781 correlation=0.70637
a=0.00000 b=2.70526 correlation=0.71745
a=0.00000 b=2.72271 correlation=0.72853
a=0.00000 b=2.74017 correlation=0.73961
a=0.00000 b=2.75762 correlation=0.75069
a=0.00000 b=2.77507 correlation=0.76177
a=0.00000 b=2.79253 correlation=0.77285
a=0.00000 b=2.80998 correlation=0.78393
a=0.00000 b=2.82743 correlation=0.79501
a=0.00000 b=2.84489 correlation=0.80609
a=0.00000 b=2.86234 correlation=0.81717
a=0.00000 b=2.87979 correlation=0.82825
a=0.00000 b=2.89725 correlation=0.83934
a=0.00000 b=2.91470 correlation=0.85042
a=0.00000 b=2.93215 correlation=0.86150
a=0.00000 b=2.94961 correlation=0.87258
a=0.00000 b=2.96706 correlation=0.88366
a=0.00000 b=2.98451 correlation=0.89474
a=0.00000 b=3.00197 correlation=0.90582
a=0.00000 b=3.01942 correlation=0.91690
a=0.00000 b=3.03687 correlation=0.92798
a=0.00000 b=3.05433 correlation=0.93906
a=0.00000 b=3.07178 correlation=0.95014
a=0.00000 b=3.08923 correlation=0.96122
a=0.00000 b=3.10669 correlation=0.97230
a=0.00000 b=3.12414 correlation=0.98338
a=0.00000 b=3.14159 correlation=0.99446

[local]

I just looked at Bell 1964. He explicitly gives the analytical solution for your functions. It is a linear function of theta (difference of the settings angles), just as my simulation shows.

Would you like to try again with a new set of functions?

[FrediFizzx]

I just looked at Bell 1964. He explicitly gives the analytical solution for your functions. It is a linear function of theta (difference of the settings angles), just as my simulation shows.

Would you like to try again with a new set of functions?

Yeah, I'm not sure why you even wasted your time with the simulation. ???
.

[local]

I didn't recall until I looked that Bell considered those exact functions. He gives the solution but not the method. But it is easy to integrate functions with sgn() by splitting things into multiple integrals. We can leave it as an exercise for GW.

BTW, the simulation took me only 15 minutes to write and test because I have similar simulations on hand. GW has spent years on this nonsense.

I'm beginning to think that GW can't be so absurdly stupid and must be trolling us. Statements like "QED via Bayesian inferencing" are pretty strong evidence of trolling. I will only re-engage if and when he presents us with another set of functions that he claims yields -a.b. I'll refute a maximum of three sets and then let it go. I suspect, though, that he now realizes that the jig is up.