Independent studies during a gap year between high school and undergrad?

I’m considering taking a gap year in between my senior year of high school and college.

I’m taking many advanced math courses as of now, and I’m planning on utilizing a gap year to take even more.

I am a very self-driven and motivated student, and I can assure you of the academic rigor I put myself through and the quality of my self-learning (you’ll have to trust me on this one).

My concern: I am afraid that the schools I apply to at the end of the gap year will not see my independent studies as legitimate, nor worth their time considering. Are my worries correct? Is there any way I can have my studies “accredited” somehow, or seen as carrying the same weight as a “normal” course?

You’ll just have to do a very good job of describing what you did and learned.

How do you know?