I don't think manic depression fits with his paranoia (one of the primary features of his disorder, fueling his delusions, hallucinations and obsessive behavior), and he doesn't appear to cycle from one mood to another but rather grows progressively more disturbed over several days (in the spiral structure of the narrative).
I don't wanna get into a psychiatry discussion, because I don't know enough, but I'll take your word on the above. I was just wondering whether he really meant one disorder and not the other, and whether he meant a disorder at all. And you were very deliberate: it's a movie about schizo not about math, so I asked.
I also asked because, when I first watched it, I didn't really think all that much in terms of him being mentally ill, but rather as someone who's putting himself through hell for an impossibly difficult problem. Also there is that balance in the film, he is meant to have been well before embarking on the problem, a talented phd student of the russian guy who is trying to prevent him from taking things to a point of no return. Whereas the psychiatric illnesses we've mentioned to me have a connotation of something you're born with, that you need to control with medicine rather than by choice.
As far as my maths comment: it's not about maths in that aronofsky is not doing a documentary about number theory. It's also not close as far as the methods of the man, in the way he works, and so on. In some sense in the details. But it definitely could be about maths allegorically.
The other element is that of "false gods". There is a notion that he could be wrong, that it's all an accident, the russian guy tells him at some point that he's just "becoming a numerologist". For me that was where pride and paranoia came in. For this reason the religious parallel worked very well.