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[FrediFizzx]

I am not sure what this R code is showing? Is it agreeing or disagreeing with Fred's result?
.

It is disagreeing because of not matching trial numbers.

Code: Select all

 A <- function(a, th) {
+   ifelse(abs(cos(a - th)) > 0.25 * cos(th / 2)^2,
+          -sign(cos(a - th)),
+          -sign(sin(a - th)))
+ }
>
> B <- function(b, th) -A(b, th)
>
> N <- 10^6
> th <- runif(N, 0, 2*pi)
>
> a <- 0
> b <- pi/4
>
> mean(A(a, th) * B(b, th))
[1] -0.493414
> -cos(a - b)
[1] -0.7071068

.

[Joy Christian]

I am not sure what this R code is showing? Is it agreeing or disagreeing with Fred's result?
.

It is disagreeing because of not matching trial numbers.

Code: Select all

 A <- function(a, th) {
+   ifelse(abs(cos(a - th)) > 0.25 * cos(th / 2)^2,
+          -sign(cos(a - th)),
+          -sign(sin(a - th)))
+ }
>
> B <- function(b, th) -A(b, th)
>
> N <- 10^6
> th <- runif(N, 0, 2*pi)
>
> a <- 0
> b <- pi/4
>
> mean(A(a, th) * B(b, th))
[1] -0.493414
> -cos(a - b)
[1] -0.7071068

Ok. But that is the problem with the R code, not with the analytical formulae I have written down. In the analytical formulae, the question of matching trial numbers does not arise. Given a value of theta, the results A and B are determined and then the mean E(a, b) is calculated by summing over theta. Thus the value of theta itself specifies the trial Intex for both A and B.
.

[FrediFizzx]

Ok. But that is the problem with the R code, not with the analytical formulae I have written down. In the analytical formulae, the question of matching trial numbers does not arise. Given a value of theta, the results A and B are determined and then the mean E(a, b) is calculated by summing over theta. Thus the value of theta itself specifies the trial Intex for both A and B.
.

Well, it is actually a problem with the R code programmer. You might want to add a note about the trial numbers getting messed up because of the orphans left behind by the analytical functions.
.

[Austin Fearnley]

Hi Fred

Here are my results for A and B for 360 values of theta. Not sure how I can send an excel spreadsheet? I aimed for a positive correlation rather than a negative one. Correlation is 0.495650509

Austin

Code: Select all

θ   A   B
1   1   1
2   1   1
3   1   1
4   1   1
5   1   1
6   1   1
7   1   1
8   1   1
9   1   1
10   1   1
11   1   1
12   1   1
13   1   1
14   1   1
15   1   1
16   1   1
17   1   1
18   1   1
19   1   1
20   1   1
21   1   1
22   1   1
23   1   1
24   1   1
25   1   1
26   1   1
27   1   1
28   1   1
29   1   1
30   1   1
31   1   1
32   1   1
33   1   1
34   1   1
35   1   1
36   1   1
37   1   1
38   1   1
39   1   1
40   1   1
41   1   1
42   1   1
43   1   1
44   1   1
45   1   1
46   1   1
47   1   1
48   1   1
49   1   1
50   1   1
51   1   1
52   1   1
53   1   1
54   1   1
55   1   1
56   1   1
57   1   1
58   1   1
59   1   1
60   1   1
61   1   1
62   1   1
63   1   1
64   1   1
65   1   1
66   1   1
67   1   1
68   1   1
69   1   1
70   1   1
71   1   1
72   1   1
73   1   1
74   1   1
75   1   1
76   1   1
77   1   1
78   1   1
79   1   1
80   1   1
81   1   1
82   -1   1
83   -1   1
84   -1   1
85   -1   1
86   -1   1
87   -1   1
88   -1   1
89   -1   1
90   -1   1
91   -1   1
92   -1   1
93   -1   1
94   -1   1
95   -1   1
96   -1   1
97   -1   1
98   -1   1
99   -1   1
100   -1   1
101   -1   1
102   -1   1
103   -1   1
104   -1   1
105   -1   1
106   -1   1
107   -1   1
108   -1   1
109   -1   1
110   -1   1
111   -1   1
112   -1   1
113   -1   1
114   -1   1
115   -1   1
116   -1   1
117   -1   1
118   -1   1
119   -1   1
120   -1   1
121   -1   1
122   -1   1
123   -1   1
124   -1   1
125   -1   1
126   -1   1
127   -1   1
128   -1   1
129   -1   1
130   -1   1
131   -1   1
132   -1   1
133   -1   -1
134   -1   -1
135   -1   -1
136   -1   -1
137   -1   -1
138   -1   -1
139   -1   -1
140   -1   -1
141   -1   -1
142   -1   -1
143   -1   -1
144   -1   -1
145   -1   -1
146   -1   -1
147   -1   -1
148   -1   -1
149   -1   -1
150   -1   -1
151   -1   -1
152   -1   -1
153   -1   -1
154   -1   -1
155   -1   -1
156   -1   -1
157   -1   -1
158   -1   -1
159   -1   -1
160   -1   -1
161   -1   -1
162   -1   -1
163   -1   -1
164   -1   -1
165   -1   -1
166   -1   -1
167   -1   -1
168   -1   -1
169   -1   -1
170   -1   -1
171   -1   -1
172   -1   -1
173   -1   -1
174   -1   -1
175   -1   -1
176   -1   -1
177   -1   -1
178   -1   -1
179   -1   -1
180   -1   -1
181   -1   -1
182   -1   -1
183   -1   -1
184   -1   -1
185   -1   -1
186   -1   -1
187   -1   -1
188   -1   -1
189   -1   -1
190   -1   -1
191   -1   -1
192   -1   -1
193   -1   -1
194   -1   -1
195   -1   -1
196   -1   -1
197   -1   -1
198   -1   -1
199   -1   -1
200   -1   -1
201   -1   -1
202   -1   -1
203   -1   -1
204   -1   -1
205   -1   -1
206   -1   -1
207   -1   -1
208   -1   -1
209   -1   -1
210   -1   -1
211   -1   -1
212   -1   -1
213   -1   -1
214   -1   -1
215   -1   -1
216   -1   -1
217   -1   -1
218   -1   -1
219   -1   -1
220   -1   -1
221   -1   -1
222   -1   -1
223   -1   -1
224   -1   -1
225   -1   -1
226   -1   -1
227   -1   -1
228   -1   -1
229   -1   -1
230   -1   -1
231   -1   -1
232   -1   -1
233   -1   -1
234   -1   -1
235   -1   -1
236   -1   -1
237   -1   -1
238   -1   -1
239   -1   -1
240   -1   -1
241   -1   -1
242   -1   -1
243   -1   -1
244   -1   -1
245   -1   -1
246   -1   -1
247   -1   -1
248   -1   -1
249   -1   -1
250   -1   -1
251   -1   -1
252   -1   -1
253   -1   -1
254   -1   -1
255   -1   -1
256   -1   -1
257   -1   -1
258   -1   -1
259   -1   -1
260   -1   -1
261   -1   -1
262   -1   -1
263   -1   -1
264   1   -1
265   1   -1
266   1   -1
267   1   -1
268   1   -1
269   1   -1
270   1   -1
271   1   -1
272   1   -1
273   1   -1
274   1   -1
275   1   -1
276   1   -1
277   1   -1
278   1   -1
279   1   -1
280   1   -1
281   1   -1
282   1   -1
283   1   -1
284   1   -1
285   1   -1
286   1   -1
287   1   -1
288   1   -1
289   1   -1
290   1   -1
291   1   -1
292   1   -1
293   1   -1
294   1   -1
295   1   -1
296   1   -1
297   1   -1
298   1   -1
299   1   -1
300   1   -1
301   1   -1
302   1   -1
303   1   -1
304   1   1
305   1   1
306   1   1
307   1   1
308   1   1
309   1   1
310   1   1
311   1   1
312   1   1
313   1   1
314   1   1
315   1   1
316   1   1
317   1   1
318   1   1
319   1   1
320   1   1
321   1   1
322   1   1
323   1   1
324   1   1
325   1   1
326   1   1
327   1   1
328   1   1
329   1   1
330   1   1
331   1   1
332   1   1
333   1   1
334   1   1
335   1   1
336   1   1
337   1   1
338   1   1
339   1   1
340   1   1
341   1   1
342   1   1
343   1   1
344   1   1
345   1   1
346   1   1
347   1   1
348   1   1
349   1   1
350   1   1
351   1   1
352   1   1
353   1   1
354   1   1
355   1   1
356   1   1
357   1   1
358   1   1
359   1   1
360   1   1

[FrediFizzx]

Hi Fred

Here are my results for A and B for 360 values of theta. Not sure how I can send an excel spreadsheet? I aimed for a positive correlation rather than a negative one. Correlation is 0.495650509

Austin

Code: Select all

θ   A   B
1   1   1
2   1   1
3   1   1
4   1   1
5   1   1
6   1   1
7   1   1
8   1   1
9   1   1
10   1   1
11   1   1
12   1   1
13   1   1
14   1   1
15   1   1
16   1   1
17   1   1
18   1   1
19   1   1
20   1   1
21   1   1
22   1   1
23   1   1
24   1   1
25   1   1
26   1   1
27   1   1
28   1   1
29   1   1
30   1   1
31   1   1
32   1   1
33   1   1
34   1   1
35   1   1
36   1   1
37   1   1
38   1   1
39   1   1
40   1   1
41   1   1
42   1   1
43   1   1
44   1   1
45   1   1
46   1   1
47   1   1
48   1   1
49   1   1
50   1   1
51   1   1
52   1   1
53   1   1
54   1   1
55   1   1
56   1   1
57   1   1
58   1   1
59   1   1
60   1   1
61   1   1
62   1   1
63   1   1
64   1   1
65   1   1
66   1   1
67   1   1
68   1   1
69   1   1
70   1   1
71   1   1
72   1   1
73   1   1
74   1   1
75   1   1
76   1   1
77   1   1
78   1   1
79   1   1
80   1   1
81   1   1
82   -1   1
83   -1   1
84   -1   1
85   -1   1
86   -1   1
87   -1   1
88   -1   1
89   -1   1
90   -1   1
91   -1   1
92   -1   1
93   -1   1
94   -1   1
95   -1   1
96   -1   1
97   -1   1
98   -1   1
99   -1   1
100   -1   1
101   -1   1
102   -1   1
103   -1   1
104   -1   1
105   -1   1
106   -1   1
107   -1   1
108   -1   1
109   -1   1
110   -1   1
111   -1   1
112   -1   1
113   -1   1
114   -1   1
115   -1   1
116   -1   1
117   -1   1
118   -1   1
119   -1   1
120   -1   1
121   -1   1
122   -1   1
123   -1   1
124   -1   1
125   -1   1
126   -1   1
127   -1   1
128   -1   1
129   -1   1
130   -1   1
131   -1   1
132   -1   1
133   -1   -1
134   -1   -1
135   -1   -1
136   -1   -1
137   -1   -1
138   -1   -1
139   -1   -1
140   -1   -1
141   -1   -1
142   -1   -1
143   -1   -1
144   -1   -1
145   -1   -1
146   -1   -1
147   -1   -1
148   -1   -1
149   -1   -1
150   -1   -1
151   -1   -1
152   -1   -1
153   -1   -1
154   -1   -1
155   -1   -1
156   -1   -1
157   -1   -1
158   -1   -1
159   -1   -1
160   -1   -1
161   -1   -1
162   -1   -1
163   -1   -1
164   -1   -1
165   -1   -1
166   -1   -1
167   -1   -1
168   -1   -1
169   -1   -1
170   -1   -1
171   -1   -1
172   -1   -1
173   -1   -1
174   -1   -1
175   -1   -1
176   -1   -1
177   -1   -1
178   -1   -1
179   -1   -1
180   -1   -1
181   -1   -1
182   -1   -1
183   -1   -1
184   -1   -1
185   -1   -1
186   -1   -1
187   -1   -1
188   -1   -1
189   -1   -1
190   -1   -1
191   -1   -1
192   -1   -1
193   -1   -1
194   -1   -1
195   -1   -1
196   -1   -1
197   -1   -1
198   -1   -1
199   -1   -1
200   -1   -1
201   -1   -1
202   -1   -1
203   -1   -1
204   -1   -1
205   -1   -1
206   -1   -1
207   -1   -1
208   -1   -1
209   -1   -1
210   -1   -1
211   -1   -1
212   -1   -1
213   -1   -1
214   -1   -1
215   -1   -1
216   -1   -1
217   -1   -1
218   -1   -1
219   -1   -1
220   -1   -1
221   -1   -1
222   -1   -1
223   -1   -1
224   -1   -1
225   -1   -1
226   -1   -1
227   -1   -1
228   -1   -1
229   -1   -1
230   -1   -1
231   -1   -1
232   -1   -1
233   -1   -1
234   -1   -1
235   -1   -1
236   -1   -1
237   -1   -1
238   -1   -1
239   -1   -1
240   -1   -1
241   -1   -1
242   -1   -1
243   -1   -1
244   -1   -1
245   -1   -1
246   -1   -1
247   -1   -1
248   -1   -1
249   -1   -1
250   -1   -1
251   -1   -1
252   -1   -1
253   -1   -1
254   -1   -1
255   -1   -1
256   -1   -1
257   -1   -1
258   -1   -1
259   -1   -1
260   -1   -1
261   -1   -1
262   -1   -1
263   -1   -1
264   1   -1
265   1   -1
266   1   -1
267   1   -1
268   1   -1
269   1   -1
270   1   -1
271   1   -1
272   1   -1
273   1   -1
274   1   -1
275   1   -1
276   1   -1
277   1   -1
278   1   -1
279   1   -1
280   1   -1
281   1   -1
282   1   -1
283   1   -1
284   1   -1
285   1   -1
286   1   -1
287   1   -1
288   1   -1
289   1   -1
290   1   -1
291   1   -1
292   1   -1
293   1   -1
294   1   -1
295   1   -1
296   1   -1
297   1   -1
298   1   -1
299   1   -1
300   1   -1
301   1   -1
302   1   -1
303   1   -1
304   1   1
305   1   1
306   1   1
307   1   1
308   1   1
309   1   1
310   1   1
311   1   1
312   1   1
313   1   1
314   1   1
315   1   1
316   1   1
317   1   1
318   1   1
319   1   1
320   1   1
321   1   1
322   1   1
323   1   1
324   1   1
325   1   1
326   1   1
327   1   1
328   1   1
329   1   1
330   1   1
331   1   1
332   1   1
333   1   1
334   1   1
335   1   1
336   1   1
337   1   1
338   1   1
339   1   1
340   1   1
341   1   1
342   1   1
343   1   1
344   1   1
345   1   1
346   1   1
347   1   1
348   1   1
349   1   1
350   1   1
351   1   1
352   1   1
353   1   1
354   1   1
355   1   1
356   1   1
357   1   1
358   1   1
359   1   1
360   1   1

I still have no idea of what you are doing. You can email the Excel file to admin@sciphysicsforums.com.
.

[Austin Fearnley]

Hi Fred

I will need to annotate the spreadsheet columms to make them more readable to you and others. Tomorrow.


In the meantime, what I have done is to ignore you code, but use your formulae to perform a simple simulation on an excel spreadsheet. I have only used 360 pairs of particles with thetas = 1, 2, 3, 4 degrees etc up to 360 degrees. So there only needs to be 360 rows on the spreadsheet. I have already posted the first column (theta) and the last two columnns (A and B values).

I have ony used a=0 degrees and b=45 degrees. The intermediary columns are just calculations using your formulae. For example, for theta = 1 deg,
theta = 0.017453293 radians
lambda = 0.249980962
cos(a-theta) = 0.999847695
abs(cos(a-theta)) = 0.999847695
abs(cos(a-theta)) - lambda = 0.749866733
sign(abs(cos(a-theta)) - lambda) = 1
A = +sign(cos(a-theta)) = +sign(0.999847695) = 1 {Note that I used a plus sign as my personal preference to make the correlation positive which is a slight change to your formulae}
b = 45 degrees
b = 0.785398163 radians
theta = 0.017453293 radians
lambda = 0.249980962
cos(b-theta) = 0.719339914
abs(cos(b-theta)) = 0.719339914
abs(cos(b-theta)) - lambda = 0.469358952
sign(abs(cos(b-theta)) - lambda) = 1
B = +sign(cos(b-theta)) = 1

For the above, lambda was exceeded and so makes the cosine formula appropriate.
Where lambda was not exceeded, for some other thetas, I used the sine formula as your formulae specify.
My calculations give the correlation to be just less than 0.5, which is the classic correlation.

I can send you a row with a measurement that uses the sine formula. Well, you will see that when I send the spreadsheet.

I am not sure why you are uncertain about what I have done. Maybe you are surprised that I used your formulae in a fixed spreadsheet rather than in a computer program. Anyway, no use discussing this further until you see my spreadsheet. I am trying to give up physics and especially computer programming. But I still quite enjoy filling in the occasional spreadsheet. For a disproof like this, it is easier and more enjoyable (for me) to make a one-off spreadsheet than to write a program.

Austin

[FrediFizzx]

@Austin Well, now that you finally posted more detail it is more understandable. What you are not realizing is that you are leaving orphans behind in A and/or B when the constraint is not happening so your trial numbers don't match up when you do the correlation.
.

[FrediFizzx]

Here is some slightly updated Mathematica code and with 300,000 trials at 1 degree resolution.



Here are the files,

EPRsims/newCS-12-forum.pdf
EPRsims/newCS-12-forum.nb

Note at the end of the PDF file there are two partial lists for outA5 and outB5. These are the original trial numbers all in order so all matching up.

Enjoy!!
.

[Joy Christian]

.
I have written out the correlation function in discrete form to make it clear what Fred means by trial numbers, k, matching in the computations:


.

[gill1109]

Code: Select all

set.seed(12072023)
a <- runif(1, 0, 2 * pi)
b <- runif(1, 0, 2 * pi)
N <- 10000
theta <- runif(N, 0, 2 * pi)
cs <- (cos(theta / 2)^2)/4
Ac <- cos(a - theta)
As <- sin(a - theta)
Bc <- cos(b - theta)
Bs <- sin(b - theta)
A <- (- sign(Ac)) * (abs(Ac) > cs) + (- sign(As)) * (abs(Ac) < cs )
B <- (+ sign(Bc)) * (abs(Bc) > cs) + (+ sign(Bs)) * (abs(Bc) < cs )
a
b
sum(A * B ) / N
- cos(a - b)


Results:

Code: Select all

> a
[1] 3.690828
> b
[1] 4.360243
> sum(A * B ) / N
[1] -0.573
> - cos(a - b)
[1] -0.7841853


[gill1109]

So Fred uses the value of theta to match trial numbers. This can only be done by communication between the two wings of the experiment. His code cannot be used to simulate two separate computers which output data which is analysed in the same way that experimenters do it. The experimenters would have to observe the hidden variable theta and use it to reorder the data coming out on each side of the experiment and only then compute their correlations.

He's very clever and I would like to see the mathematical formulas for the "matching" and a proof by calculus that it gives exactly the negative cosine, not just approximately.

Congratulations so far!

Sorry, no hat-eating, no 65 thousand Euro this time.

[Austin Fearnley]

Fred wrote:
@Austin Well, now that you finally posted more detail it is more understandable. :) What you are not realizing is that you are leaving orphans behind in A and/or B when the constraint is not happening so your trial numbers don't match up when you do the correlation.

I see no orphans :)
I generated 360 pairs of particles and calculated the correlation based on 360 pairs.

Joy wrote:
I have written out the correlation function in discrete form to make it clear what Fred means by trial numbers, k, matching in the computations:

Yes, I am fine with that. I also used k as an index number for the particles and summed over k = 1 to n, where n = 360. I see no way that I left orphans following your formulae. The formulae were very clearly presented.

But my correlation did not exceed the classical correlation.
I suspect the reason is that I used R3 statistics, otherwise known as normal arithmetic and trigonometry. Whereas Fred's code used quaternions. I am not sure why there is a difference but it equates to you claiming that you can exceed the classical correlation when using S3 but not when using R3. I did not look in detail at Fred's code so cannot replicate that quaternion method of calculation.

Best, Austin

[gill1109]

R code using the formulas provided by Joy Christian in the image title "Fred's Disproof of Bell's Theorem".

Code: Select all

A <- function(a, th) {
    ifelse(abs(cos(a - th)) > 0.25 * cos(th / 2)^2,
           -sign(cos(a - th)),
           -sign(sin(a - th)))
}

B <- function(b, th) -A(b, th)

N <- 10^6
th <- runif(N, 0, 2*pi)

a <- 0
b <- pi/4

mean(A(a, th) * B(b, th))
-cos(a - b)


I am not sure what this R code is showing? Is it agreeing or disagreeing with Fred's result?
.

This code does the same as my code in a more elegant way and gets the same result. In other words, it does not give Fred's result. As, obviously, it cannot, by Bell's theorem. Bell's theorem is confirmed yet again! All that work by Fred chasing a will-of-the-wisp! Fred's program works by violating locality (he has to violate locality to get the matching indices).

Code: Select all

> A <- function(a, th) {
+     ifelse(abs(cos(a - th)) > 0.25 * cos(th / 2)^2,
+            -sign(cos(a - th)),
+            -sign(sin(a - th)))
+ }
>
> B <- function(b, th) -A(b, th)
>
> N <- 10^6
> th <- runif(N, 0, 2*pi)
>
> a <- 0
> b <- pi/4
>
> mean(A(a, th) * B(b, th))
[1] -0.493868
> -cos(a - b)
[1] -0.7071068


[Joy Christian]

.
That is complete nonsense. The hidden variable is shared between Alice and Bob. That means the index k must be the same between Alice and Bob, as required by Bell and Physics.
.

[gill1109]

Hi Fred

I will need to annotate the spreadsheet columms to make them more readable to you and others. Tomorrow.


In the meantime, what I have done is to ignore you code, but use your formulae to perform a simple simulation on an excel spreadsheet. I have only used 360 pairs of particles with thetas = 1, 2, 3, 4 degrees etc up to 360 degrees. So there only needs to be 360 rows on the spreadsheet. I have already posted the first column (theta) and the last two columnns (A and B values).

I have ony used a=0 degrees and b=45 degrees. The intermediary columns are just calculations using your formulae. For example, for theta = 1 deg,
theta = 0.017453293 radians
lambda = 0.249980962
cos(a-theta) = 0.999847695
abs(cos(a-theta)) = 0.999847695
abs(cos(a-theta)) - lambda = 0.749866733
sign(abs(cos(a-theta)) - lambda) = 1
A = +sign(cos(a-theta)) = +sign(0.999847695) = 1 {Note that I used a plus sign as my personal preference to make the correlation positive which is a slight change to your formulae}
b = 45 degrees
b = 0.785398163 radians
theta = 0.017453293 radians
lambda = 0.249980962
cos(b-theta) = 0.719339914
abs(cos(b-theta)) = 0.719339914
abs(cos(b-theta)) - lambda = 0.469358952
sign(abs(cos(b-theta)) - lambda) = 1
B = +sign(cos(b-theta)) = 1

For the above, lambda was exceeded and so makes the cosine formula appropriate.
Where lambda was not exceeded, for some other thetas, I used the sine formula as your formulae specify.
My calculations give the correlation to be just less than 0.5, which is the classic correlation.

I can send you a row with a measurement that uses the sine formula. Well, you will see that when I send the spreadsheet.

I am not sure why you are uncertain about what I have done. Maybe you are surprised that I used your formulae in a fixed spreadsheet rather than in a computer program. Anyway, no use discussing this further until you see my spreadsheet. I am trying to give up physics and especially computer programming. But I still quite enjoy filling in the occasional spreadsheet. For a disproof like this, it is easier and more enjoyable (for me) to make a one-off spreadsheet than to write a program.

Austin

Well done Austin! Indeed, retrocausality is a nice way to do it.

[gill1109]

.
That is complete nonsense. The hidden variable is shared between Alice and Bob. That means the index k must be the same between Alice and Bob, as required by Bell and Physics.
.

Indeed. In my code, the hidden variable theta is a vector with an index, you could call it "k" if you like; it runs from 1 to N and it's the same between Alice and Bob as required by Bell and by physics and by my programming challenge. The R expression sum ( A * B ) actually means sum_{k = 1}^ N A(k) * B(k). There are N separate trials all with the same angles a and b. I believe I programmed your formulas correctly. I have checked that I get the same answer as Austin and as "guest", so I expect my code is just fine.

[Joy Christian]

Well done Austin! Indeed, retrocausality is a nice way to do it.

Retrocausality is a cop-out, and so is superdeterminism. None of those cop-outs are needed. We have a perfectly sensible local-realistic 3-sphere model in the market since 2007.
.

[gill1109]

Well done Austin! Indeed, retrocausality is a nice way to do it.

Retrocausality is a cop-out, and so is superdeterminism. None of those cop-outs are needed. We have a perfectly sensible local-realistic 3-sphere model in the market since 2007.
.

The 3-sphere is the surface of a ball in four-dimensional Euclidean space, often, in physics applications, taken to be three spatial dimensions and one temporal. By working on a smooth compact manifold in 4-d space-time you are ignoring the distinction between space and time, and what you do does not have much to do with experiments in laboratories in which information is transmitted back and forth sequentially in time, between a source and two distant locations.

[Joy Christian]

Well done Austin! Indeed, retrocausality is a nice way to do it.

Retrocausality is a cop-out, and so is superdeterminism. None of those cop-outs are needed. We have a perfectly sensible local-realistic 3-sphere model in the market since 2007.
.

The 3-sphere is the surface of a ball in four-dimensional Euclidean space, often, in physics applications, taken to be three spatial dimensions and one temporal. By working on a smooth compact manifold in 4-d space-time you are ignoring the distinction between space and time, and what you do does not have much to do with experiments in laboratories in which information is transmitted back and forth sequentially in time, between a source and two distant locations.

S^3 is a three-dimensional space, just as R^3 is. Both are a valid spatial part of one of the well-known solutions of Einstein's field equations of general relativity. S^3 is just better physics.
.

[gill1109]

Well done Austin! Indeed, retrocausality is a nice way to do it.

Retrocausality is a cop-out, and so is superdeterminism. None of those cop-outs are needed. We have a perfectly sensible local-realistic 3-sphere model in the market since 2007.
.

The 3-sphere is the surface of a ball in four-dimensional Euclidean space, often, in physics applications, taken to be three spatial dimensions and one temporal. By working on a smooth compact manifold in 4-d space-time you are ignoring the distinction between space and time, and what you do does not have much to do with experiments in laboratories in which information is transmitted back and forth sequentially in time, between a source and two distant locations.

S^3 is a three-dimensional space, just as R^3 is. Both are a valid spatial part of one of the well-known solutions of Einstein's field equations of general relativity. S^3 is just better physics.
.

I know that S^3 is three-dimensional in a mathematical sense and I know it is a valid spatial part of one of very many solutions of Einstein's field equations. The fact remains - Fred's simulation model does not contradict Bell's theorem. It can be seen as using retro-causality or non-locality. I haven't seen a formula for the matching part yet, so I can't say whether or not Fred's model could also be thought of as using "conspiracy". Many serious theoretical physicists are exploiting those three escape routes. It could be that Fred's model is physically very interesting. Not being a physicist I cannot judge.