(As was shown in the Loophole Hypothesis) Newton's laws (classical laws) allow the Loophole, wherein a satellite in UCM (uniform curvature motion) is trapped in two states, in motion (in relation to the tangent and with respect to the perpendicular) and at rest (in relation to the parent body curvature and with respect to the perpendicular), and (as will be shown here) the Loophole does not exclude quantization (because the in motion state does not exclude the satellite traveling a curvature matching vibratory path, and the combined states of UCM, where the satellite travels a curvature matching vibratory path for the in motion state, is analogous to a standing matter wave).

NOTE 1
By the Loophole Hypothesis, in order for a satellite to be in UCM (uniform curvature motion) about a parent body, the satellite's tangential velocity and inward acceleration must be such that there is no change in perpendicular distance between the satellite and parent body curvature.

Reference: Loophole Hypothesis, see statement 10.

NOTE 2
By the Loophole Hypothesis, once a satellite is in UCM about a parent body, the satellite is in motion (in relation to the tangent and with respect to the perpendicular) and at rest (in relation to the parent body curvature and with respect to the perpendicular).
Reference: Loophole Hypothesis, see statement 12.

NOTE 3
UCM requires two reference systems for its complete description, one reference system for each of the two UCM states: the tangent reference system for the in motion state of UCM, and the curvature reference system for the at rest state of UCM.
Reference: Loophole Hypothesis, see statements 8 and 11.

NOTE 4
By the Loophole Hypothesis, the at rest state of UCM occurs when the satellite is in the in motion state of UCM, because it is then that the satellite remains parallel to the parent body curvature, i.e. there is no change in perpendicular distance between the satellite and parent body.
Reference: see Loophole Hypothesis, statement 11.

Statement 1
The in motion state of UCM does not exclude the satellite traveling a vibratory path that matches the parent body curvature --- called a curvature matching vibratory path.

Statement 2
For the in motion state of UCM, as long as the satellite's path does not result in a change in perpendicular distance between the satellite and parent body (i.e. remains parallel to the parent body curvature), the satellite can travel a curvature matching vibratory path.

Statement 3
A satellite traveling a curvature matching vibratory path, by De Broglie's Matter Wave Hypothesis, has a wave properity, called wavelength, given by

equation 1

Statement 4 --- The Crux
The combined states (at rest and in motion) of UCM, where the satellite is traveling a curvature matching vibratory path for the in motion state of UCM, is analogius to a standing matter wave.

Statement 5 --- curvature matching region
The satellite can be anywhere within a curvature matching region surrounding the parent body, as long as there is no change in perpendicular distance between the satellite and parent body.

Statement 6 --- circumference
The circumference of a curvature matching region is equal to its diameter times pi.

equation 2

statement 7 --- diameter
The radius of a curvature matching region is equal to half its diameter, which means the diameter of a curvature matching region is also equal to twice its radius.

equation 3

statement 8
Substituting for D in equation 2 gives

equation 4

Statement 9 --- quantization
Using the vibrating string analogy, the curvature matching regions in which the satellite can be are those into which an integral number of matter waves can fit.

equation 5

statement 10
a rearrangment gives

equation 6

statement 11
Equating equation 1 and equation 6 in order to eliminate wavelength gives

equation 7

satement 12
A rearrangment gives the quantization of the satellite

equation 8

Statement 13
Using equation 4 and substituting for C in equation 8 gives

equation 8A

Statement 14
A rearrangement gives the quantization of angular momentum

equation 8B