Originally Posted by Brian Ratliff
Actually, this is not a single inverted pendulum. It is actually a double inverted pendulum; one pendulum being the bicycle, the other being the rider, with a complex system of interconnects between them. Redistributing the weight of the rider moves the pivot point of the higher pendulum closer to that of the lower pendulum, effectively shortening the lower pendulum, making the system more stable.
By "redistributing the weight of the rider" I presume that you mean by moving the masses of the rider's body into new positions. Certainly, that is the only explanation that would match the standard way of analyzing mechanics. Reducing the vertical force placed on the saddle by increasing the vertical forces applied to the pedals does nothing at all about moving masses. Your false explanation of moving the weight has been exploded innumerable times over the years. I suggest that you review your first semester course in the university curriculum for physics or engineering, assuming that you have taken such.