Background: I am a senior at a (fairly prestigious) university in the United States, and I am applying to PhD programs in the country for pure math, aiming as high as I can. I am majoring in mathematical physics, and by the time I graduate, I will have taken over 10 graduate courses (mostly in pure math, some in theoretical physics); for some topics, I have taken every graduate course offered at my university on them. I mostly have As on my transcript with some Bs (including two on math grad courses) with a near perfect concentration GPA (only a few math courses I took count toward my concentration courses, however). I have more Bs overall in math than in physics. I have done some research/advanced reading courses in topics beyond what is covered in any graduate courses at my university with professors in both the math and physics departments (with whom I took upper-level/graduate courses), and I'm on very good relations with these professors. I will be asking them for recommendation letters. I do not have any publications or REU experience.
I have a long-running issue: I am terrible at in-class written exams. In every course I have taken so far, my homework average has been near perfect, and the same is true for take-home exams. I want to stress that this is true across the board for all of my courses, a huge portion of which includes custom problems assigned by the instructor, in case someone is worried that these results have been achieved through Internet searches. So in all the courses I didn't get an A in, I did terribly on at least one of the in-class exams with a performance in sharp contrast to other portions of coursework which brought my grade down.
I don't know for sure why this is the case. I suspect I am a slow worker and compensate by drawing as much time as necessary to solve untimed problems, and this seems to explain poor in-class exam performance. I also verge on a nervous breakdown/panic attack before exams, but I have never mentioned this to anyone at my university to get accommodated, as I believed it would be asking for special treatment and be merely an excuse for poor performance as I convince myself there is nothing wrong with me.
I recently took the math subject GRE, and my performance was abysmal (40th percentile). My mind went blank during the test and I had very bad pacing. I think the result is very incongruent to the rest of my application, and I am at a complete loss at what to do. I will be retaking the test in a couple of weeks, but I am not sure if I can turn things around. (I am good at proofs but not so good at computation.) Worse yet, the extra pressure that "this is my last chance or I submit my 40 percentile score" might make me much more prone to having a breakdown.
I initially considered taking a year off before applying after looking at my test result, but I run the risk of diluting my recommendation letters and being subject to a much more intense scrutiny of my new score and of what I did besides studying during that entire year off. I am terribly afraid my score puts me into an instant reject pile at top institutions. I might add that personally, taking a year off due to a low GRE score would be very torturous for me, as I would agonize everyday over studying things I frankly do not care so much about (as opposed to tackling proofs and potential research work, which draws me to a PhD program in the first place) and constantly be haunted by what looms ahead, but I would also try very hard to get a publication in that time to make it worth it if I did take one off.
Questions: Should I take a take a gap year before applying? If not, is it worth applying to a good number of top schools with this kind of score (say half of all programs I apply to, with the other half reach/safety)? Should I get a psychological evaluation for my test performance and mention my test panic/anxiety issues on the the applications? Should I talk about this with a professor I am asking a recommendation letter from?
The whole experience with the Subject GRE has been quite dejecting, to be honest, and now I am doubting my capability. Is there some other way I can (personally) determine if I am cut out for math grad school in my case? I really enjoy thinking about math problems and trying my hand at them, but I don't know if it is worth it if I can't go to a good school at the end of the day.
P.S. I understand that graduate schools need a form of standardization to compare applicants, and I might be heavily biased here, but I really don't see how the subject test, which (in my opinion) to quite an extent requires employing computation tricks, assesses research ability in math. How strongly is "I cannot solve these integrals at lightning speeds" correlated with "I cannot solve this problem for my dissertation"? I am perfectly happy and capable of doing category theory problems or talking in person in great depth about functional analysis topics I've seen, but I am bumbling about with a convoluted plane geometry question on paper with a 90 second timer ticking. It's confusing because I can't tell if the latter is what a math grad student rather ought to be capable of doing instead of the former if one had to choose between the two. When I talked about the rationale behind timed-exams with a professor who taught me a grad class, he echoed what I've said above: it is just to standardize and to ensure students solved the rest of the coursework themselves instead of relying on some other person/source. Maybe someone has some insight on this?
Sorry for the long post.
EDIT: Just a quick update - I retook, and my score bumped up by 20 points. I think this is still pretty average, but definitely better than a 40. Any thoughts, given this update? (I was waiting for the scores to update my answer.)