Monte Carlo Assessment of Natural Unit Base Power Law Mass Force Hierarchy of Hydrogen on the Minkowski ℤ2 Lattice

The origin of the mass–force hierarchy remains an open question in fundamental physics, since the Standard Model (SM) does not explain it. We examine whether fundamental constants, analyzed as natural-unit frequency scalars, display an underlying constrained integer-scaling power structure. We use Hermann Minkowski to approximate ratios of logs of constants by simple rational exponents, i/j, on a two-dimensional integer lattice. The first approximations define scaling characterized by parsimonious integer ratios, Diophantine residuals, and a conformal factor frequency vcf, a measure from exact power laws. We apply this to hydrogen, using all as the references to the others, but focus on the Rydberg frequency to compare electromagnetic and gravitational forces. All the exponents are near-perfect partial harmonic fractions in a structured form when the gravitational binding energy of the electron in hydrogen or the proton are the references. The ratio of two constants is encoded by the i, j, vcf, and either constant scalar. Monte Carlo tests demonstrate that the ensembles’ 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.