Although UChicago is test optional, not submitting a test score may hurt your application because being homeschooled outside the US, you don’t have a GPA nor a class rank and you don’t have to abide by any homeschooling rules in the US. All of these would have helped the school evaluate you in the context of peers in a day-to-day setting and lessened their risk of taking you.
This concern is partially alleviated by you having taken a couple of AP exams, but a 4 in Calculus BC (albeit with a subscore of 5 in AB) and a 3 in Physics C Mechanics do not bode well for a math major. Many top students who would be your competitors for UChicago admission, and who do not wish to major in math, got 5 in both and several other AP classes. You would be at a disadvantage compared to them.
It is great that you have learned group theory online and are able to write a 10-page paper yourself. However, I would be hesitant to call your role an “independent researcher” because symmetric group and automorphism are standard math topics. Are you reporting any new, unpublished result in your paper?
If so, you would want to really highlight that and perhaps submit your paper to a professional math journal. If not, the paper could just be a regurgitation of standard definitions, graphical illustrations of groups and automorphisms, and well-known facts about them. Which is still amazing for a high school student but it’s not research per se.
Another issue is budget that earlier posters have pointed out. UChicago is need-blind for domestic students but not for internationals. And because you are an international, the net-price calculator likely won’t give you an accurate estimate of the cost. Thus, you would want this part straightened out before applying ED.