One could argue that he appears intelligent only because he's spent his life so far learning effectively.
Rationalist!Harry is calibrated to match the knowledge and recall of a 34-year-old autodidact. Even presuming a very friendly environment and that said 34-year-old autodidact's training was not optimal, I just don't think there's enough time.
I can buy a 10 year old reading Ender's Game and The Lord of the Rings and maybe even Lensmen. It's a bit harder to imagine one that would consider wanting to want the math behind proving N=NP, nevermind going further than that.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.