Distinguish Contextual ,Non Local,Non Realistic (Non CFD)

[Point Conception]

Case One : In Bell test violations can outcomes be they (1) contextual outcomes ( not pre - encoded in measured object but arise in interaction with object and measurement apparatus) be distinguished from (2) non realism /non counter factual. And (3) from non local effects in a space like separated test ? Case Two: When measurements are made at A and B on the same axis with perfect anti correlations then the assumption of CFD/realism includes locality : A (a,λ) = - B (b,λ) and non contextuality. It seems case Two assumptions can only apply when detectors are aligned. Then when detectors are not aligned case One question applies.

 

[rubi]

I don't really understand what you mean in your whole post, but the extra assumption $A(\lambda,\vec a)=-B(\lambda,\vec a)$ is only necessary in the original Bell inequality. In general, when you have 3 random variables $X$, $Y$, $Z$ with values in $\{-1,1\}$, they always satisfy the inequality
$$XY + XZ - YZ\leq 1\text{,}$$
which can be proved by checking all possible combinations. You then get an inequality between the correlations:
$$\left<XY\right>+\left<XZ\right>-\left<YZ\right>\leq 1$$
So far, this is only a purely mathematical inequality. In order to apply it to Bell tests, you need to plug in the variables that are measured in such a test ($A(\vec a)$, $A(\vec b)$, $A(\vec c)$, $B(\vec a)$, $B(\vec b)$, $B(\vec c)$). Non-contextuality and locality ensure that you have only 6 variables, instead of possibly infinitely many. However, for the original Bell inequality, this is not enough. You need 3 variables instead of 6 in order to plug them in the inequality above. Hence, you need an additional assumption, which is $A(\vec a)=-B(\vec a)$ (for all $\vec a$). Now you get:
$$\left<A(\vec a)A(\vec b)\right>+\left<A(\vec a)A(\vec c)\right>-\left<A(\vec b)A(\vec c)\right>\leq 1$$
Now you can use our third assumption and replace every second $A$ by a $-B$:
$$-\left<A(\vec a)B(\vec b)\right>-\left<A(\vec a)B(\vec c)\right>+\left<A(\vec b)B(\vec c)\right>\leq 1$$
With some algebraic manipulation, you can rearrange this into the original Bell inequality:
$$\left|\left<A(\vec a)B(\vec b)\right>-\left<A(\vec b)B(\vec c)\right>\right|\leq 1+\left<A(\vec a)B(\vec c)\right>$$
If you use an inequality with 4 variables (the CHSH inequality), you can drop the $A=-B$ assumption. Nevertheless, you still need non-contextuality and locality in order to break down the number of variables to 4.

 

[Point Conception]

I should say the perfect anti correlations when settings are aligned can lead to CFD, locality, and non contextuality as conclusions. These conclusions then applied as assumptions in the inequality. When inequality does not hold in tests with mis aligned detector settings, how can you distinguish what assumption failed: Was CFD non CFD, was locality non locality, was non contextual contextual ?