Discussion on question by Idiot: What should I do if I lack fundamental knowledge and only got into a PhD program due to luck?

In my opinion, cramming for tests rather than “truly studying” is an indication that you’re simply in the wrong field. If you were in the right field, you would find it utterly fascinating and want to learn as much about it as you possibly could. There is nothing wrong with deciding that you’ve made a mistake and withdrawing from your Ph.D. program.

"This is not imposter syndrome I'm describing." People who have imposter syndrome don't know they have imposter syndrome.

You’re asking for advice, but when everybody gives you the obvious advice of just improving your own knowledge, you dismiss it as impossible. What are you looking to hear? Obviously, if your plan is to never learn basic undergrad material, you are doomed and should quit immediately to avoid wasting more of your life. Happy now?

Of all the people I know of, math majors have by far the lowest self-esteem when in fact they're more or less considered geniuses by everyone else. Being able to synthesize a new proof for a theorem is exceedingly non-trivial, a guestimate says 99.9% of the world's population won't even understand what the theorem is about, let alone being able to write a paper on it. Your goals and interests aside, stop questioning your aptitude.

@Ghoster You've got a point here. There have been things (not math related though) when I was a teen which were so fascinating to me that I didn't feel like studying while effectively studying them. Right now I'm 27 and the spark is gone, I don't really have the fascination to things I used to have, and I just do things because they're either useful or easy enough to do them. It's a shame, but this led me to choose a major which I obviously am not interested in (although talented enough to learn it). My PhD friends talk about mathematics the same as if you'd be talking about a hobby. I don't.