I have a very, very long, not very likely, crazy theory. Here it goes.

If you recall, Harry made a number of TODOs when he started the year.

To-do 1. Research mind-alteration charms so you can test the Comed-Tea and see whether you actually did figure out a path to omnipotence. Actually, just research every kind of mind magic you can find. Mind is the foundation of our power as humans, any kind of magic that affects it is the most important sort of magic there is.

To-do 2. Actually this is To-do 1 and the other is To-do 2. Go through the bookshelves of the Hogwarts and Ravenclaw libraries, familiarising yourself with the system and making sure you've at least read all the book titles. Second pass: read all tables of contents. Coordinate with Hermione who has a much better memory than you. Find out if there's an interlibrary loan system at Hogwarts and see if the two of you, especially Hermione, can visit those libraries too. If other Houses have private libraries, figure out how to access legally or sneak in.

To-do 0: Check out what sort of information-search-and-retrieval spells exist, if any. Library magic isn't as ultimately important as mind magic but it has a much higher priority.

Did Harry forget about those TODOs?

Harry make a pass at reading through the table of contents in the Hogwarts main library in chapter 29. After Harry and Hermione finish gossiping about Pettigrew, Harry walks Hermione to Broomstick class. At this stage the Eliezer Yudkowsky in my head starts yelling "HEY GUYS! Table of contents! Broomstick class! Thursday, Thursday, Thursday! Come on! I'm putting all the pieces together for you!"

(Answer: on the first Thursday of the school year, during Broomstick class, Harry set off Neville's Remembrall. Harry had forgotten about his TODOs).

It would appear that Harry entirely forgot about the other two TODOs (study library magic, study mind magic). Indeed, in chapter 90, he asks Quirrell how he could study mind magic, and is suprised to learn that the books are to be found in the north-northwest stacks of the main Hogwarts library, filed under M.

But what if Harry had remembered his TODOs? What if Harry studied obliviation in secret, and obiviated himself during some emergency? What would we expect to see?

We might see something like this:

I guess that helped a little," said Harry. "Actually I think it helped more than playing Gobstones for an hour could've possibly helped, so thanks for being you. And no matter what, I'm not having anyone Obliviate everything I know about calculus. I'd sooner die."

"What?" said Hermione. "Why - why would you ever want to do that?"

Harry stood up from the table, and there was a rush of restored background noise as his rise broke the Quieting Charm. "I'm a tad sleepy so I'm going off to bed," Harry said...

(chapter 63)

Why would someone want to Obliviate everything they know about calculus? Well, Eliezer Yudkowsky wrote

I know many ways to prove the Pythagorean Theorem, including at least one proof that is purely visual and can be seen at a glance. But if you deleted the Pythagorean Theorem from my mind entirely, would I have enough math skills left to grow it back the next time I needed it? I hope so—certainly I've solved math problems that seem tougher than that, what with benefit of hindsight and all. But, as I'm not an AI, I can't actually switch off the memories and associations, and test myself in that way.

I imagine Harry might try to do just that.

Meta information makes me even less confident than before. If it were true that Harry rederived the Pythagorean theorem, in secret, etc., EY would have upvoted my comment asking for a bet. My comment received zero upvotes.