G'day Ken
That's a flawed analysis. You've fallen into a trap that is often the stimulus for perpetual motion theorists.
A chain link follows a circular path while going around the sprocket, and a linear path between sprockets. Neither circular nor linear motion absorb energy. The linear bit is easy to understand - it's Newton's First Law. The circular bit is not so straight-forward, but have a rummage around in your memory for the circular motion physics you learned in high school. You may recall that to move a a chain link in a circular path around a sprocket requires a force on the link radial to the sprocket. A radial force doesn't alter the torque input or output at the sprocket, so can't absorb energy. If the force had a component in the direction of the chain motion, then yes, there would be an energy input required simply to turn it. But all mass-related forces are perpendicular to the chain. In addition, the rotational or mass-related forces at one sprocket are balanced by those same (opposite) forces at the other, regardless of relative sprocket diameters.
But what about the transition form linear to circular motion. Well, nothing happens there either. It simply ceases to be linear and commences to be circular. There is no transition phase (ie, the time duration to change from one to the other is zero, so any power lost is also zero).
If you're not convinced, have a think about energy. How does a chain link travelling within a closed path (whether circular, elliptical, or any other shape) absorb energy? The answer is that it doesn't. There's a thermodynamic principle here, related to the First Law: "If a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) is zero". The only energy input required is simply that necessary to overcome friction.
It's similar to an orbiting satellite, which is changing direction all the time, but it doesn't absorb energy. If it did, the orbit would decay very rapidly. Reciprocal motion is just a particular case of orbital motion where the minor axis of the orbital ellipse is zero. A mass orbiting in such a way will travel back and forth, changing direction by 180 deg at each end, and it still doesn't absorb energy.
Cam