In fact, it's just bloody hard to fundamentally increase your ability to solve math problems in a way that "no closed system can do" just by opening the system. So far as I can tell, it basically requires that the environment be magic and that you be born with faith in this fact.

As Wei mentioned, P≠NP is basically the conjecture that this isn't true: i.e., that you can exponentially increase your ability to solve math problems by your environment being magic and your not being born with faith in that fact. So for example, if your environment immediately inverted any one-way function, that would be evidence (no faith required) that your environment is not merely 'slightly' smarter than you are, but astoundingly smarter. In qualitative terms, I think it would be almost as astounding as if the environment solved the halting problem.