Mass-Force Hierarchy Based on the Quantum Parsimony Diophantine Resonance Hypothesis: Gravitational and Electromagnetic

The origin of the mass–force hierarchy remains one of the major open questions in fundamental physics, since the Standard Model (SM) does not explain the observed pattern of particle masses or the significant differences in interaction strengths. In this work, we examine whether fundamental constants, written as natural-unit (NU) frequency scalars, display an underlying constrained integer-scaling power structure. To explore this idea, we use Hermann Minkowski to approximate ratios of logs of constants by simple rational exponents  on a two-dimensional integer lattice. The leading first approximations define scaling relations characterized by parsimonious integer ratios, Diophantine residuals, and a vcf conformal factor frequency , which measures departures from exact power laws. We apply this framework to hydrogen, using all as the references to the others, but focus on the Rydberg frequency to compare electromagnetic and gravitational quantities as an example. We find that all the exponents are near-perfect partial harmonic fractions in a highly structured form when the gravitational binding energy of the electron in hydrogen or the proton are the references. Any ratio of two constants are naturally encoded by the same integers i , j, vcf , and either constant scalar utilizing a standard algebraic power structure. Monte Carlo tests demonstrate that this ensemble’s patterns are unlikely to arise in randomized datasets. These results indicate that part of the hierarchy among fundamental constants may be captured by simple rational scaling relations based on resonant nodes, without altering established SM values.