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Hey Eliezer, your explanations are incredible but I don't understand why we are folding the amplitude distribution over that diagonal line. I do understand that the folded distribution is unfactorable, but what is the motivation behind the folding operation?
I was also having a problem with this, but I think it can be resolved by saying that every operator is symmetric in A and B. Therefore, whenever we take a measurement, we will fail to distinguish between a blob at (1,0) and a blob at (0,1). More fundamentally, any interaction with a third particle will be the same whether the blob is at (1,0) or (0,1).
Even more important is that a blob at (0,1) plus a blob with the opposite phase at (1,0) can have a subtractive effect, even cancelling out entirely if both have the same amplitude.
In a sense, then, factoring the topology of configuration-space by the identity A==B causes the symmetry of operators to be an unavoidable consequence of the topology, rather than some freakish coincidence.