Harmonic Fractions and an Integer Power Law to Demonstrate a Relationship of the Neutron to the Properties of Hydrogen and Cosmic Observables

Power laws and harmonic oscillator systems represent a ubiquitous relationship among many physical phenomena. This study demonstrates a close power law relationship of the annihilation frequency of the neutron, approximately 2.27 \texttimes 10$^{\mathrm{23}}$ Hz, when used as a dimensionless base, to fundamental quantum properties of hydrogen and present-day cosmic observables. The following set of the three smallest integers: \textbraceleft -1, 0, 1\textbraceright , and the set of partial harmonic fractions: \textbraceleft 3/2, \textpm 1/2, \textpm 2/3, -3/4, \textpm 4/5\textbraceright , are associated with each physical entity investigated as a frequency equivalent. They are listed as follows: twice the maximum energy of a cosmic ray, 3/2; the base identity of the neutron, 1; the Bohr radius, 4/5; Rydberg's constant, 2/3; twice the peak spectral radiance of cosmic microwave background radiation, 1/2; Planck's constant, 0; the Sun's galactic radius, -1/2; the Sun's galactic period, -2/3; Hubble's constant, -3/4; the dimension of the observable universe, -4/5; and twice the gravitational binding energy of the electron in hydrogen, -1. When viewed in the physically equivalent frequency domain, the neutron partitions an abundance of physical constants from the very small to the very large.