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> Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement.

No, people constantly prove statements of the form "if P=NP, then strange implication X". They do not consider it wasted effort at all, because of the contrapositive: if X is indeed very strange, they might be able to prove that it is false, and then they've settled P!=NP.

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If a counterexample to a conjecture is found then all work toward proving consequences of the conjecture will cease. No one is trying to discover consequences of the Jacobian Conjecture now.

At some point an AI will prove a result that is so long and complicated that no human will understand it. This should not preclude people from using that result. In general, whenever the body of knowledge is increased it is a good thing. Even if it isn’t increased by humans.

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