If showing you a formal proof that you will do a particular action doesn’t result in you doing that action, then the supposed “proof” was simply incorrect.
Yes, that’s the point, you can make it necessarily incorrect, your decision to act differently determines the incorrectness of the proof, regardless of its provenance. When the proof was formally correct, your decision turns the whole possible world where this takes place counterfactual. (This is called playing chicken with the universe or the chicken rule, a technique that’s occasionally useful for getting an agent to have nicer properties, by not letting the formal system that generates the proofs know too much early on about what the agent is going to decide.)
Yes, that’s the point, you can make it necessarily incorrect, your decision to act differently determines the incorrectness of the proof, regardless of its provenance. When the proof was formally correct, your decision turns the whole possible world where this takes place counterfactual. (This is called playing chicken with the universe or the chicken rule, a technique that’s occasionally useful for getting an agent to have nicer properties, by not letting the formal system that generates the proofs know too much early on about what the agent is going to decide.)