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For a neutron in the Earth's Newtonian gravitational potential ${\Phi }_E$, the Schrodinger equation isNear the surface, ${\Phi }_E(z)=gz$ up to an additive constant.

The Colella–Overhauser–Werner experiment uses crystal slabs as coherent beam splitters and mirrors in a neutron interferometer. The two paths contain horizontal segments of length $s$ separated vertically by $r$, then recombine. The gravitational potential energy difference is $m_{n}gr$. A neutron of speed $v$ spends time $s/v$ on a horizontal segment, so the gravitationally induced relative quantum phase isUsing the de Broglie wavelength $\lambda =h/(m_{n}v)$ and $\hslash =h/(2\pi )$ givesRotating the apparatus changes the vertical projection of its enclosed area from zero to $rs$. The number of full interference oscillations is thereforeHere $rs=(\sqrt{10}\mathrm{cm}{)}^2=10^{-3}{\mathrm{m}}^2$, soThe expected change is therefore $\boxed{9}$ oscillations to the nearest integer.

Yes in both directions. The Page–Geilker experiment tests whether a classical gravitational field is sourced by the expectation value of macroscopically superposed matter configurations. The Colella–Overhauser–Werner experiment uses the Earth's effectively classical field as an external potential and therefore does not distinguish an expectation-value semiclassical source law from a quantized gravitational field.

Conversely, the Colella–Overhauser–Werner experiment directly observes coherent matter-wave interference and a gravitationally generated relative phase in a neutron wavefunction. Page and Geilker did not maintain and recombine coherent branches of the macroscopic source, and their torsion-balance measurement was not an interference experiment. Thus their experiment did not test the neutron gravitational phase effect. The experiments probe opposite sides of the coupling: quantum matter responding coherently to gravity, and gravity responding to branch-dependent quantum matter.