@causative I am not sure the photon rocket has much to do with this. But suppose you gather the photons from the sun - say a photon rocket in the inner solar system? Or photons beamed up by a laser beam? Constant refueling. So what if we do not yet know how to do it. Except, well, there are other factors that limit the kinetic energy/input energyequation, just like I am sure therre are with this engine. The enginre can work without it having to be a perpetual motion machine.

Obviously a photon rocket in the inner solar system cannot reach unbounded speeds because it would first reach escape velocity for the solar system. So say instead your rocket goes in a straight line and harvests starlight - as the starlight gets increasingly blueshifted, the starlight from in front of the rocket exerts photon pressure slowing the rocket down.

@causative Indeed, even though one can imagine something exceeding the kinetic energy/energy input boundary if taken in isolation, there are always other factors in play that prevent any engine from crossing it. Just because the limit exists, and crossing it produces a perpetual motion machine, does not automatically mean that this engine does indeed cross that limit. The engine might work, but it would also be subject to all of these other factors which have not yet been considered.

The engine could work, and still not be a perpetual motion machine.