It may very well be possible to build such an AI. However there are several issues with it:
The AI can be adapted for other, less restricted, domains if knowledge on how it works spreads. There would be a large incentive to since such an oracle would only be of limited utility.
The AI adds code that will evolve into another AI into it's output. It's remotely possible, depending on what kind of problems you have it working on. If you were using it to design more efficient algorithms, in some cases an AI of some form might be the optimal solution.
Even if you 100% trust the AI to provide the optimal output, you can't trust that the optimal output to the problem you've specified is what you actually want.
The AI could self-modify incorrectly and result in unfriendly AI. In order to be provably friendly/restricted, it would have to be 100% certain of any modification. That's a very tall order, especially in AI where everything has to be approximations or probabilistic.
It might not be as safe as you think it is. The AI runs some code and gets an unexpected result. Possibly because of a bug in the environment itself. Look up how difficult it is to sandbox untrusted code and you will get some appreciation for how a superintelligence could figure a way out of it's box.
But it can't do anything with any exploits it finds because it is restricted to hard-coded axioms? Well, maybe. If it's using probabilities and some form of machine learning, it might be able to learn that "executing this code give me this result" and then learn to take advantage of that. I don't believe that a system can work only in formal proofs. However I might be completely wrong about this one, it's just a thought.
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I am confused about how Philosopher's stone could help with reviving Hermione. Does QQ mean to permanently transfigure her dead body into a living Hermione? But then, would it not mean that Harry could do it now, albeit temporarily? And, he wouldn't even need a body. He could then just temporarily transfigure any object into a living Hermione. Also, now that I think of it, he could transfigure himself a Feynman and a couple of Einsteins...