Within the Standard Model, the fundamental origin of the mass-force hierarchy remains an unresolved problem. We address this limitation by hypothesizing that the scaling architecture is governed by of principles of pure number theory, quantum parsimony and resonance, defined as the utilization of the smallest irreducible quantum number powers, i/j, to achieve the first local Diophantine residual, DR, in the dimensionless domain, and a logarithmic beat vk frequency in the dimensioned domain as resonances closest to 1 Hz, related to integer fitting of an irrational number, within a discrete integer power-law structure. While the hypothesis does not pre-determine specific powers that emerge inherently, it derives an infinite number of valid power laws that reveal significant inter-relationships across force-mass regimes. Utilizing a search algorithm to identify the first local minimum |Dr|. We evaluated the scaling between the ionization energy of the electron in hydrogen, the Rydberg constant, RH, vRH, 3. 2898421 (6. 54) 1015 Hz and the gravitational binding energy of the electron vGBEe, 2. 90025 10-24 Hz. The emergent relationship calculated has powers i/j equal to -3/2, the most parsimonious possible prime integer set, with the transformation residual vk (RH: GBEe, -3/2), 7. 857405564 10-1 Hz approaching 1 Hz. The ratio of vRH /vk (RH: GBEe, -3, 2) raised to the (5/2) power equals 1. 1343307656 1039, the ratio of electromagnetic force to gravitational force. We discover an infinite number of power laws that generate the identical results. These findings support the hypothesis and suggest a unified, predictive "Rydberg-like" hierarchy that integrates gravitational force as a “mirror” sign inversion and reciprocal of the electromagnetic force in the power domain.
Donald Chakeres (Tue,) studied this question.