Rank and Scale Tests of Shell-Motivated Fermion-Mass Ansatze with a Constrained Neutrino Extension

Abstract

We study whether the observed hierarchy of quark and lepton masses can be described by a small set of simple relations between the three fermion generations. Using the nine charged-fermion Yukawa couplings at the common scale M_Z, we distinguish genuine mass relations from those that arise simply because a model has enough freedom to reproduce the data. A systematic scan of projector-based models identifies a particularly accurate relation between the first- and third-generation mass slopes. We then construct more restrictive models that also constrain the second generation and test them by leave-one-out reconstruction. Several six-parameter models predict all nine charged masses with errors of only a few percent; the best fixed models reduce the worst leave-one-out error to about 4% without adding continuous parameters. We also give a possible microscopic interpretation of the most successful integer relations. Two mass-independent combinations of Standard-Model quantum numbers reproduce the required charged-sector pattern, and simple twofold or threefold matching classes fix the relative coefficient before the masses are used. In a separate two-loop running test, a down-sector ratio close to 1/15 near 6 x 10^6 GeV evolves to a value close to 1/14 at M_Z. The same framework gives neutrino-mass sums near 0.059-0.061 eV for normal ordering, while one family also allows inverted ordering near 0.103 eV. These results do not prove preons or fermion compositeness, but provide concrete targets for future microscopic models.