Let me preface by saying that this may not be the right SE. I considered asking on the Physics SE, but I thought I might try here first. If it is wrong, I am not against it being migrated.
From basic principles of Physics, power is computed as Work/time. So consider a rider and bike system climbing a hill. The work done is the difference in potential from bottom to top, and obviously the time will be climb-time.
Now, my question is:
Given the same rider, same bike weight, and same climb-time, does your gearing affect power? Also assume that the climb is efficient, no slipping tires, normal pedaling, etc.
From a physical perspective, I expect the answer is no. Same difference in potentials, same time, same power. However, from the rider perspective, I know that it sure feels like a lot more power is being used to climb with a harder ratio.
I expect that the answer is that the disparity comes from idealizing the system. If we consider the bike a closed system, we expect that all the energy put into the bike carries it up the hill, but this is not the case. Moreover I think the inefficiency of the human body will be relevant. However, I still cannot resolve the question.