Proofs of the Law of Atomic Radii and the Atomic Numbers of the Noble Elements

In a recently published paper in OAJP this author proposed a model of the atom in which orbiting electrons are arranged in strings in such a way that they do not radiate when in steady state. The analysis assumed the following two laws: (a) the law of atomic radii and (b) the law of the atomic numbers of the noble elements. While it is clear these laws are correct as based on the derived analyses and experimental evidence, it is not otherwise intuitively clear why they must be valid. To this end they are formally proved herein.