Locally everywhere does not imply everywhere

A couple days ago, Levent Alpöge, a mathematician working at Anthropic, discovered a counterexample to the Jacobian conjecture using Claude Fable 5. I was curious whether most mathematicians were trying to prove or disprove the conjecture, so I asked Claude. Before a counterexample to the Jacobian conjecture was found, did most mathematicians believe it was true or false? Claude’s response was The premise of this question isn’t quite right — no counterexample to the Jacobian conjecture has been found. It remains an open problem in mathematics: no one has proven it true, and no one has found a counterexample disproving it. … If you encountered a claim that a counterexample was found, do you have a source for that? I’d be happy to look into it, since that would actually be a major result in algebraic geometry if true. Of course Claude doesn’t know that it solved the conjecture. It didn’t even solve the conjecture. It was an inanimate tool in the hand of a mathematician, just like a…