Couldn't it be that your measurement for a proton being at A' is simply measuring the wrong feature of the protons?
edit: I have no idea if protons are indistinguishable from one another, but this experiment doesn't seem compelling.
In my statistical mechanics course, we went through an illuminating exercise where we started by trying to take account of every atom in a gas cloud. We started taking limits and making assumptions. One of them was that all atoms are indistinguishable from each other. This decreases the possible states of the system by N! (factorial, not surprise). After making that assumption, out pops the ideal gas law.
For protons, they may come with a serial number, but we're not allowed to read it because of the constraints built into God's programming language.
Protons don't behave like cats.