CHSH inequality : renaming and absolute values

[jk22]

In some derivations of the CHSH inequality, https://en.m.wikipedia.org/wiki/CHSH_inequality, the following arises :

$$CHS=\int A(a,l1)B(b,l1)dl1-\int A(a,l2)B(b',l2)dl2+\int A(a',l3)B(b,l3)dl3+\int A(a',l4)B(b',l4)dl4\\
=\int A(a,l)B(b,l)dl1-A(a,l)B(b',l)+A(a',l)B(b,l)+A(a',l)B(b',l)dl$$

1) But is it safe to impose l1=l2=l3=l4=l ?

Because if the absolute value were taken, then we could calculate for the classical model with the 4 l's that cannot factorize :
$$p(AB=1)=\frac{b-a}{\pi}$$

From $$C(a,b)=-1+\frac{2(b-a)}{\pi}=p(AB=1)-p(AB=-1)$$

Hence :

2) $$p(|A_1B_1-A_2B_2|=2)=1/16+9/16$$
Then the average of the Bell operator is :

$$|A_1B_1-A_2B_2|+|A_3B_3+A_4B_4|=80/32=2.5>2$$

Thus, whereas the absolute value has no influence in the CHSH version where all the variables are renamed to the same, it changes the average if independent variables are taken.

So the main question is : does this renaming not influence the CHS value ?
Or in other words : can Bell theorem be proven for all classical models without the condition : $l1=l2=l3=l4$ ?

 

[PeterDonis]

In some derivations of the CHSH inequality

Which ones? Please give a reference.

 

[gill1109]

l1, l2, l3 and l4 are dummy variables. You certainly may give them all the same name "ell" if you like. These are integrals over "ell" lying in some space, of some function of "ell", with respect to "ell". The integral operation is additive. As long as you are integrating over the same space with respect to the same measure.

Source https://www.physicsforums.com/threads/chsh-inequality-renaming-and-absolute-values.985479/