The proof that I'll let the AI out is not something that's passively "plausible" or "implausible", it's something I control. I can make it wrong. If I do, it's false that the AI can make this proof valid. (It might be that the proof is correct, it's just unlikely, and the action of presenting the proof doesn't normally ensure its correctness.)

In other words, as far as I can see, your stipulation is that the AI can assert something that's actually unlikely. Here, I'm not referring to something that seems unlikely merely because of insufficient understanding of the data, and which AI can discover to be likely, but to something that will seem unlikely to AI as well. For example, most casinos going bankrupt last month because of bad luck, or an ice cube forming in a boiling kettle. If the AI is in the box and isn't performing actual magic tricks in the world, these events are unlikely. Permitting the game to stipulate that these events took place gives the AI supernatural powers of making anything at be true about the world, direct access to editing it, and at that point, in what sense is it "in the box"?