You would have proven or ‘know’ that it is invalid but not that it is false. For example, you could say that the sky is blue because humans have wings. False premise but perhaps not false conclusion though the statement is invalid.

In any case, you’re talking about a structure of ideas rather than observations coming from an external stream. Even in that case you encounter Goedel’s theorem as a hard limit. Any set of premises or axioms that are ‘non-trivial’ are either inconsistent or incomplete. And one of the things that is a part of the incompleteness is that you can never prove consistency. Therefore, you may prove as many theorems for any axiomatic system that you like but there will always be doubt because you can never prove that the system you started with is consistent.

Anyway, most of what you wrote seems strange to me so I will end this here.