Falsifying the logical inverse of X is identical to verifying X. There's nothing about negation that does anything here. You're making the same mistake people make when claiming "You can't prove a negative".

> The key though is that showing the inverse of X can not be true is much harder than showing that what X might be true.

This is nonsense modal logic. You're saying ¬◻¬X, which if necessity and possibility are duals, is equivalent to ◇X, and otherwise an irrelevant statement. The inverse of X is ¬X. ¬¬X is logically equivalent to X.