It is not. There are also rules about the type of shoe you can wear. But other than that, if you can cartwheel and backflip fast enough while staying completely in your lane, then you can do it in a race.

Because that's another rule. No leaving your lane.

A bicycle is a machine with mechanical advantages that propel you faster and further using the same energy input. So while it is technically locomotion powered entirely by you, it is not soley under your power because of said mechanical advantage. There have been similar trouble with amputee racers using blade and spring type prosthetics.