> The prompt for the Jacobian conjecture was obviously not random. the search space is too big to just try all the combinations of 3 variable polynomials.

Maybe the prompt contained a part like this: "the search space is too big to just try all the combinations of 3 variable polynomials, so be clever about it". Or maybe this part was omitted from the prompt, because modern LLMs are smart enough to figure this out without us having to mention it.