Please list in your next reply the “long-term extracurriculars in non-math (and non-academic) pursuits” so people here on cc can give their best advice.
It’s hard to advise when you don’t give all the relevant info.
You sound like a really impressive kid! Just please give more info here for a better idea of who you are, as this is such an important decision (repeat 10th or do 11th).
I am not at all as experienced as some others with BS, but I have learned enough to know that repeating a year when entering is incredibly common. Kids do it for a variety of reasons. So, if a lot of students have repeated, and if you have already skipped a grade at your local public school (LPS in CC talk), then if you go to BS as a non-repeating 11th grader, you will be not 1 year, but 2 years younger than good number of kids in your grade level. Since you are so accelerated in your math (for a US school system) I believe you when you say you are super comfortable being with kids several years older than you in class. But, at BS, you won’t just be in class with kids. You are LIVING there with them, round the clock. I think you would have a better experience entering as a 10th grader if possible.
You sound, by the fact that you have self learned some advanced math and the bit of academic research you listed (quantum field theory), like a what is often called a gifted student. I know a bit about that. Giftedness, if that’s you, is typically not acceleration in every of a child’s life - and I’m not talking about academics. You may have other areas of your life (maybe social development, is an example often give [I’m not judging that by your post, you seem to be a mature writer, just by the literature on giftedness] or executive functioning [planning, organization…]) that are not “advanced” but rather age-typical or even a little delayed from age peers. So if your academics are 12th (13th+) grade in some areas, 9th - 10th grade in others (non-math things like English, history, art), and your personal development is 9th-10th grade, and your body (you are a teen going through changes) is 9th-10th, I think you should repeat 10th.
Lastly, many of the BS that offer really strong research projects often try to have students do that in 11th grade not 12th (so the research project can show up on college applications, plus also 12th is so busy with applications, they don’t want kids to take on too much). And the BSs that I have looked into with special research emphasis have students prep for that experience (maybe a pre-research project class) in 10th grade. Others CCers can chime in who know more about this. I think your desire for research better fits a repeat 10th grade model. It would give you time in 10th grad to adjust to BS, and also to find out details about research options and see if it would be with a group/team, on your own with a faculty advisor, on your own with a local university professor plus a BS advisor, etc. Some of the schools the research is a summer program. And regardless of how it is, if you are going to work with a university, whether for their resources like their lab, or with a university professor mentoring, your BS will have to know you well to make that happen (they don’t want to pair up a kid who has really bad work habits with a university!) so, again, giving your BS 10th grade to get to know you, may open way more doors than just showing up in 11th and expecting to dive right in in 11th grade to research.
Some schools offer merit scholarships. If you do not want to repeat 10th due to the cost, you should research schools with merit scholarships. In the small pool we applied to, 1/3 had merit aide. And since you are already at a not well funded LPS and not going to a local private school even though you think you LPS is not very good, then you may be someone whose parent(s) are in the income bracket to qualify for Financial Aide (FA of CC) to reduce the cost of tuition. So you shouldn’t base the decision of 10th versus 11th on cost, because there are resources to help your family with that.