Having read Bell's theorem and proofs. It seems it builds upon the assumption that we measure identical particles. Usually these thought experiments involve giving A and B an identical scratchcard. If they scratch the same box they will find the same symbol. If they scratch a different, then we can calculate the minimum coincidence rate thus set up the Bell's inequality.
But does this inequality hold if you give A and B a different scratchcard (aka measuring different particles)?
It seems the question is a bit misunderstood.
Okay let me clarify. Let's imagine an extreme case. I'm an evil scratchcard dealer and deal cards to A and B. A and B don't know anything about the cards. They may think they are identical or entangled but they won't find out anything till they measure it anyway. And I don't know which boxes they will scratch on which card. I have full control of the variables I give to A and B and there are no constraints other than I must conserve the quantities at least statistically. While reserving the right that each card in a particular pair can be completely different and there is no computable dependency between them.
Can I deal cards such way that I can fool A and B, so they find that their measurements violate the inequality, just becuase I dealed the cards to them in a clever pattern without entanglement or other? Or does Bell's theorem explicitly proves the such shenanigans are not possible at all?