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On closest approach to the moons, the inertial capacity of the system changes significantly, and this is as if the mass of each atom increases slightly. As a result, the moment of inertia of the gyroscopes spinning at angles neither perpendicular nor tangent to the plane of the moon suffer differential inertia effects.
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Jerry, this is total bafflegab. It has no basis in reality or science. You have not the slightest basis for this supposition or any experimental evidence to support your claims. It is not consistent with observed results and there is no reason to think that the universe works in the way you suggest.
To counter your suppositions only requires that a single part of your house of cards be invalidated. It then falls down. Regardless of how a gyroscope is oriented, regardless of the mass of the rotating component, regardless of the rpms of that component, regardless of the method used to sense the position of the rotating component and regardless of the actual construction of the gyroscope it will not be affected in any way by a change of an "inertial field" or the amount of gravity present or any other force.
Unless the gyroscope is subject to a distance varying field of immense proportions such as found near the Swarchchild radius of a quantum black hole there simply is no effect that can influence the spinning of a rotating mass in a differential manner that would change how a gyroscope operates, regardless of orientation.