Simulation Based on Michel Fodje's epr-simple simulation translated from 
Python to Mathematica by John Reed 13 Nov 2013 and Quaternions
Modified by Fred Diether for Completely Local-Realistic Sep 2021
Some parts by Bill Nelson.  Includes Joy’s (S^3) Quaternion Model.

Load Quaternion Package, Set Run Time Parameters, Initialize Arrays and Tables

<<Quaternions`;
β0=Quaternion[1,0,0,0];
β1=Quaternion[0,1,0,0];
β2=Quaternion[0,0,1,0];
β3=Quaternion[0,0,0,1];
Qcoordinates={β1,β2,β3};
m=100000;
trialDeg=721;
Ls1=ConstantArray[0,m];
Ls2=ConstantArray[0,m];
λ1=ConstantArray[0,m];
λ2=ConstantArray[0,m];
outAa=Table[{0,0,0,0,0},m];
outBb=Table[{0,0,0,0,0},m];
outA12=Table[{0,0,0,0,0},m];
outA22=Table[{0,0,0,0,0},m];
outB12=Table[{0,0,0,0,0},m];
outB22=Table[{0,0,0,0,0},m];
plotq=Table[{0,0},m];
a1=ConstantArray[0,m];
b1=ConstantArray[0,m];
A=ConstantArray[0, m];
B=ConstantArray[0, m];
nPP=ConstantArray[0,trialDeg];
nNN=ConstantArray[0,trialDeg];
nPN=ConstantArray[0,trialDeg];
nNP=ConstantArray[0,trialDeg];
nAP=ConstantArray[0,trialDeg];
nBP=ConstantArray[0,trialDeg];
nAN=ConstantArray[0,trialDeg];
nBN=ConstantArray[0,trialDeg];
ϕ=3.1; β=0.32; ξ=-((4π)/45);  (*Adustable parameters for fine tuning*)

Generating Particle Data with Three Independent Do-Loops

Do[θ=RandomReal[{-179,180}];  (*Singlet vector angle*)   (*Hidden Variable*)
λ1[[i]]=β(Cos[(θ/ϕ) ]^2);
λ2[[i]]=Sign[θ];  (*Orientation Hidden Variable*)
θθ=N[Flatten[{FromPolarCoordinates[{1,θ *π/180}],0}]];
Ls1[[i]]=λ2[[i]]*θθ.Qcoordinates;  (*Quaternion particle for the A side*)
Ls2[[i]]=-λ2[[i]]*θθ.Qcoordinates,{i,m}]          (*Quaternion particle for the B side*)(*Conservation of angular momentum*)

Do[a=RandomInteger[{-179,180}];  (*Detector vector angle 1 degree increments*)
aa=N[Flatten[{FromPolarCoordinates[{1,a *π/180}],0}]];
Da=aa.Qcoordinates;  (*Convert to quaternion coordinates*)
qa=Da**Ls1[[i]];
If[Abs[Re[qa]]>λ1[[i]],Aa=Re[Da**Limit[ Ls1[[i]], Ls1[[i]]->Sign[Re[Da**Ls1[[i]]]]Da]],
Aa=λ2[[i]]*Sign[qa[[4]]+ξ]];
A0=λ2[[i]]*Sign[qa[[4]]+ξ];
outAa[[i]]={a,Aa,i,A0,qa};
If[Abs[Re[qa]]>λ1[[i]],outA12[[i]]=outAa[[i]],outA22[[i]]=outAa[[i]]],{i,m}]
outA1=DeleteCases[outA12,{0,0,0,0,0}];  (*Split outAa into outA1 and outA2*)
outA2=DeleteCases[outA22,{0,0,0,0,0}];

Do[b=RandomInteger[{-179,180}];  (*Detector vector angle 1 degree increments*)
bb=N[Flatten[{FromPolarCoordinates[{1,b *π/180}],0}]];
Db=bb.Qcoordinates;  (*Convert to quaternion coordinates*)
qb=Ls2[[i]]**Db;
If[Abs[Re[qb]]>λ1[[i]],B=Re[Db**Limit[ Ls2[[i]], Ls2[[i]]->Sign[Re[Db**Ls2[[i]]]]Db]],
B=-λ2[[i]]*Sign[qb[[4]]+ξ]];
B0=-λ2[[i]]*Sign[qb[[4]]+ξ];
outBb[[i]]={b,B,i,B0,qb};
If[Abs[Re[qb]]>λ1[[i]],outB12[[i]]=outBb[[i]],outB22[[i]]=outBb[[i]]],{i,m}]
outB1=DeleteCases[outB12,{0,0,0,0,0}];      (*Split outBb into outB1 and outB2*)
outB2=DeleteCases[outB22,{0,0,0,0,0}];

Matching Events Observed by Alice and Bob using Trial Numbers

list13=outA1[[All,3]];  (*Two lists of only trial numbers used for matching*)
list23=outB1[[All,3]];

Local Detection Analysis of the Events Observed by Alice

listA3=Select[outA1,Intersection[{#[[3]]},list23]!={#[[3]]}&];
M1=Length[listA3];
tna=listA3[[All,3]];
outA4=Select[outA1,Intersection[{#[[3]]},tna]!={#[[3]]}&];
outA5=Table[{0,0,0,0,0},M1];
ssca=ConstantArray[0,M1];

For the spinorial sign changes we will need eq. (12).

Do[If[listA3[[i]][[2]]==listA3[[i]][[4]],δ=0,δ=1];
If[δ==0,ssca[[i]]=1,ssca[[i]]=-1];  (*spinorial sign change*)
outA5[[i]]={listA3[[i]][[1]],ssca[[i]]*listA3[[i]][[2]],listA3[[i]][[3]],listA3[[i]][[4]],listA3[[i]][[5]]},{i,M1}]  
outA=Sort[Catenate[{outA2,outA4,outA5}],#1[[3]]<#2[[3]]&];  (*Combine lists and sort*)
a1= outA[[All,1]];  (*These results are what Alice observes as defined in Eq.(??)*)
A =  outA[[All,2]];

Local Detection Analysis of the Events Observed by Bob

listB3=Select[outB1,Intersection[{#[[3]]},list13]!={#[[3]]}&];
M2=Length[listB3];
tnb=listB3[[All,3]];
outB4=Select[outB1,Intersection[{#[[3]]},tnb]!={#[[3]]}&];
outB5=Table[{0,0,0,0,0},M2];
sscb=ConstantArray[0,M2];

Do[If[listB3[[i]][[2]]==listB3[[i]][[4]],δ=0,δ=1];
If[δ==0,sscb[[i]]=1,sscb[[i]]=-1];  (*spinorial sign change*)outB5[[i]]={listB3[[i]][[1]],sscb[[i]]*listB3[[i]][[2]],listB3[[i]][[3]],listB3[[i]][[4]],listB3[[i]][[5]]},{i,M2}]  
outB=Sort[Catenate[{outB2,outB4,outB5}],#1[[3]]<#2[[3]]&];  (*Combine lists and sort*)
b1= outB[[All,1]];  (*These results are what Bob observes as defined in Eq.(??)*)
B =  outB[[All,2]];

Statistical Analysis of the Particle Data Received from Alice and Bob

theta=ConstantArray[0,m];
Do[θ1=a1[[j]]-b1[[j]]+361;  (*All angles are shifted by 361 degrees since θ1 is an index*)
theta[[j]]=θ1;
aliceD=A[[j]];bobD=B[[j]];
If[aliceD==1,nAP[[θ1]]++];
If[bobD==1,nBP[[θ1]]++];
If[aliceD==-1,nAN[[θ1]]++];
If[bobD==-1,nBN[[θ1]]++];
If[aliceD==1&&bobD==1,nPP[[θ1]]++];
If[aliceD==1&&bobD==-1,nPN[[θ1]]++];
If[aliceD==-1&&bobD==1,nNP[[θ1]]++];
If[aliceD==-1&&bobD==-1,nNN[[θ1]]++],{j,m}]

Calculating Mean Values of A, B, and AB, and Plotting the Results

pPP=0; pPN=0; pNP=0;pNN=0; 
mean=ConstantArray[0,trialDeg];
Do[sum=nPP[[i]]+nPN[[i]]+nNP[[i]]+nNN[[i]];
If[sum==0,Goto[jump],
{pPP=nPP[[i]]/sum;
       pNP=nNP[[i]]/sum;
       pPN=nPN[[i]]/sum;
       pNN=nNN[[i]]/sum;
mean[[i]]=pPP+pNN-pPN-pNP}];
Label[jump],{i,trialDeg}]

simulation=ListPlot[mean,PlotMarkers->{Automatic,Tiny}];
negcos=Plot[-Cos[x Degree],{x,0,720},PlotStyle->{Magenta},AspectRatio->7/16, Ticks->{{{0,-360°},{90,-270°},{180,-180°},{270,-90°},{360,0°},{450,90°},{540,180°},{630,270°},{720,360°}},Automatic},GridLines->Automatic];
p1=Plot[-1+2x Degree/π,{x,0,180},PlotStyle->{Gray,Dashed}];
p2=Plot[3-2x Degree/π,{x,180,360},PlotStyle->{Gray,Dashed}];
p3=Plot[-5+2x Degree/π,{x,360,540},PlotStyle->{Gray,Dashed}];
p4=Plot[7-2x Degree/π,{x,540,720},PlotStyle->{Gray,Dashed}];

Comparing Mean Values with -Cosine Function and Computing Averages

Show[negcos,p1,p2,p3,p4,simulation]

AveA=N[Sum[A[[i]], {i, m}]/m];
AveB=N[Sum[B[[i]], {i, m}]/m];
Print["AveA = ",AveA]
Print["AveB = ",AveB]
PAP=N[Sum[nAP[[i]], {i, trialDeg}]];
PBP=N[Sum[nBP[[i]], {i, trialDeg}]];
PAN=N[Sum[nAN[[i]], {i, trialDeg}]];
PBN=N[Sum[nBN[[i]], {i, trialDeg}]];
PA1=PAP/(PAP+PAN);
PB1=PBP/(PBP+PBN);
Print["P(A+)= ", PA1]
Print["P(B+)= ",PB1]
totAB=Sum[nPP[[i]]+nNN[[i]]+nPN[[i]]+nNP[[i]], {i, trialDeg}]
PP=N[Sum[nPP[[i]], {i, trialDeg}]/totAB]
NN=N[Sum[nNN[[i]], {i, trialDeg}]/totAB]
PN=N[Sum[nPN[[i]], {i, trialDeg}]/totAB]
NP=N[Sum[nNP[[i]], {i, trialDeg}]/totAB]
CHSH=Abs[N[mean[[315]]]-N[mean[[225]]]+N[mean[[405]]]+N[mean[[45]]]]

Product Calculation

q=0;
q2a=outA[[All,5]];
q2b=outB[[All,5]];
plotq=Table[{0,0},m];
angle=ConstantArray[0,m];
Do[If[λ2[[i]]==1,q=q2a[[i]]**q2b[[i]],q=q2b[[i]]**q2a[[i]]];
angle=theta[[i]]-361;
plotq[[i]]={angle,Re[q]},{i,m}]
sim=ListPlot[plotq,PlotMarkers->{Automatic,Small},AspectRatio->7/16, Ticks->{{{-360,-360°},{-270,-270°},{-180,-180°},{-90,-90°},{0,0°},{90,90°},{180,180°},{270,270°},{360,360°}},Automatic},GridLines->Automatic];
negcos1=Plot[-Cos[x Degree],{x,-360,360},PlotStyle->{Magenta}];
Show[sim,negcos1]

Blue is the correlation data and magenta is the -cosine curve for an exact match.

Deviation from negative cosine curve

dev1=ConstantArray[2,720];
dev2=ConstantArray[2,720];
dev3=ConstantArray[2,720];
Do[dev1=mean[[i]];dev2[[i]]={dev1,i},{i,720}]
devang=dev2[[All,2]]-361;
Do[dev3[[i]]=mean[[i]]+Cos[devang[[i]] Degree],{i,720}]
ListPlot[N[dev3],PlotMarkers->{Automatic,Tiny},Joined->True,AspectRatio->1/2,Ticks->{{{0,-360°},{90,-270°},{180,-180°},{270,-90°},{360,0°},{450,90°},{540,180°},{630,270°},{720,360°}},Automatic},GridLines->Automatic]

Mean[N[dev3]]
Mean[N[Abs[dev3]]]