Within the Standard Model, the fundamental origin of the mass-force hierarchy remains an unresolved problem, characterized by arbitrary empirical parameters and the lack of a predictive framework for mass force values. We address this limitation by hypothesizing that all fundamental constants, irrespective of their physical dimensions, manifest universal natural-unit frequency wave characteristics. The calculations are done dimensionless then the dimensions are added later. This scaling architecture is governed by the first principles of number theory and the in the dimensionless domain, and quantum parsimony and resonance in the dimensioned domain. In the dimensionless domain the exponent, expref(target : ref) base reference frequency, ref, of the ratio of a frequency target, vtarget divided by the vref plus 1. The difference of this irrational number exponent and possible integer i/j values associated with the first local minimum, a Diophantine residual, Dr defines the mass force architecture from the perspective of the reference. There is a Dr manifold with wave properties. There are an infinite number of valid isomorphic power laws. We evaluated the scaling between the proton in hydrogen, p, vp, 2.26873181(7.01) 1023 Hz and the gravitational binding energy of the electron vGBEe, 2.90025 10-24 Hz. The emerged i/j equals -1/1, Dr equals -7.78314705 10-3, and an associated coupling frequency v(GBEe:p, -1/1) equals 8.111650 10-1 Hz. The ratio of vp divided by v(GBEe:p, -1/1) squared equals 7.8225404 1046, the ratio of mass equivalent of the proton to the gravitational force. Planck time is calculated from the v(GBEe:p, -1/1), the proton, and the Bohr model constants. This is a totally new number theory method outside of the Standard Model that defines an infinite number of valid power laws associating the mass force hierarchy between any two constants.
Donald Chakeres (2026) studied this question.