The objective function is the problem. The objective function is given in all problems.
In "find an X such that Y" the objective function is "is this an X? is Y satisfied?". In induction you have P and n to ratchet up the proof by increasing n such that P(n) holds.
Combining this reply with your previous one, it sounds like you're setting up a situation where the LLM can neither know the problem nor whether it's making progress on the problem. With those constraints no human could do mathematics either or really anything else.
In "find an X such that Y" the objective function is "is this an X? is Y satisfied?". In induction you have P and n to ratchet up the proof by increasing n such that P(n) holds.
Combining this reply with your previous one, it sounds like you're setting up a situation where the LLM can neither know the problem nor whether it's making progress on the problem. With those constraints no human could do mathematics either or really anything else.