In your view, what exactly is the difference between 'proving ~P' and 'falsifying P'? Logically speaking they are the same, so if you can't prove a "negative", neither can you falsify a "positive". But again, I am sure you have a very special notion of "negative" in mind that makes these troubles go away.
Hmm, let us change it a bit. I can't prove the Theory of Gravity. If I pick up a rock and drop it, I can demonstrate the TOG, that is evidence that supports the TOG. but it is not "proof.' On the other hand, if I picked up a rock and it didn't fall when I released it that would disprove the TOG. I can only falsify the TOG.
Some things can be proved. I can prove the existence of the rock, I can't 'prove' the nonexistence (negative) of the rock, you might be hiding it in your fridge.
Let me put it another way, when scientists build a theory they don't try to prove a theory, they look for exceptions to the theory. Any exceptions disprove the theory.