But it does not necessarily follow that they also have a maximum energy in the Earth's gravitational field. Nor would there be a maximum energy.
Thus, isn't the precise hypothesis ("the pull of gravity should make particles fall into discrete energy levels") still unproven?
The lowest energy levels, just like the energy levels of an electron in an atom, are the most widely spaced. As the particle assumes a higher and higher energy level, the spacing between the energy levels becomes ever smaller and less easily distinguished, until ultimately it becomes a continuum band. That is to say, there is an energy above which any amount of energy is permitted. But for slow enough neutrons, only certain energies are permitted.
Correct me if I'm wrong, but this experiment, standing on it's own, proves nothing. It shows only that the neutrons posessed a minimum amount of energy that caused them to rise more than 14 micrometers above the mirror. Let's say 10 neutrons reached the detector. To prove a quantum effect, there should be no increase in the number of neutrons detected until the mop is set at a greater, discreet height. So, there will be no increase in neutrons detected until the mop is set to 30 micrometers, for example. At that point, the number of neutrons detected suddenly rises. For example, it becomes 15. Then, the mop is elevated again. No more neutrons are detected until the mop reaches another discreet height. Maybe that is 35 micrometers. Then, 18 neutrons are detected. That would show a quantum effect.
This study also depends on the gravitational field of the Earth for it's results, and should be replicated in greater and lesser gravitational fields for confirmation. Is that right?