If there's something that you can prompt with e.g. "here's the proof for Fermat's last theorem" or "here is how you crack Satoshi's private key on a laptop in under an hour" and get a useful response, that's AGI.

Just to be clear, we are nowhere near that point with our current LLMs, and it's possible that we'll never get there, but in principle, if such a thing existed, it would be a next-word predictor while still being AGI.

I’m pretty sure you’re committing a logical fallacy there. Like someone in antiquity claiming “I get annoyed when experienced folks say thunderstorms aren’t the gods getting angry, it’s nature and physical phenomena. But we don’t know how the weather works”. Your lack of understanding in one area does not give you the authority to make a claim in another.