As weird as it may sound, this may help. You write yourself:
get rejected after an overly technical question
Advertising a PhD may come across as advertising that you think you're good. You may not mean it that way, but it will most certainly be received like that. I've performed many technical interviews and when I prepare myself for a candidate, I go over their resume (their ad). If the candidate advertises knowledge of a specific topic, he or she better know it.
The rejections you got may not have been because you didn't know a specific answer to a very technical question. Nobody knows everything. You may have been rejected because of the answer that you gave, and let me explain.
When I interview, I will make sure there is one topic with a couple of questions that I don't expect you to know from the top of your head. I will get online and get the answers if needed. I will ask the question (if we get to it) and see the response. If you get the answer right: well done, you will have my vote. If you don't then this is where the psychology comes in. I'll be looking for you to be honest. Don't make up answers, don't come up with a bullshit reply. If I get bullshit, no matter how good you were, you will fail the interview. If you bullshit me, you'll bullshit a customer, manager or anyone else when you're in the hot seat.
Don't underestimate the importance of attitude and honesty in an interview.