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Can You Prove Two Particles Are Identical?

Ok...

So we aren't dealing with probability distributions because that implies that reality acknowledges individual identity in particles? And that can't be true because the experimental results show that the configurations are only concerned with having a particle in the right position, regardless of the particles origin? And therefore the "2" particles are "1" by existing on the same amplitude pattern? Is any of that remotely close?

I read the joint configuration post 3 times and I still feel like I'm missing something. The problem is probably on my end. =+)