Will graduating a semester early hurt applications to grad schools?
Your application will be essentially complete by the end of fall term, but it should very pointedly mention what further work you will be doing in the spring term, whether or not you are paying tuition for the privilege of studying and thinking about mathematics.

There really needn't be a formal structure imposed on your work/study/research, and it needn't have any official label "research" versus "study". But do write what you plan to do, and do mention the guidance you hope/expect to have from more-experienced mathematicians (as opposed to just doing whatever strikes your fancy off in some closet). That is, do be sure to make the point that you will be engaged with actual contemporary mathematics, arguably more intensely, and at a more serious level, than the usual homework-exam model would encourage or allow.

If you can describe your plans for "spring term" in vivid and enthusiastic detail, you can make it sound far better than "taking classes". I'd not worry about hype-ing "research" too much, although, yes, it would be intellectually dishonest to not follow one's curiosity. But, of course, one's personal discoveries, however novel to oneself, may be old news to experienced professionals, so one should not presume.

Just tell your plans!