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

Abstract

We test two eight-parameter logarithmic ansatze against the nine charged-fermion Yukawa eigenvalues evaluated in one renormalization scheme at the common scale mu = M_Z. A smooth log-linear trend is given a conditional renormalization-group motivation through dimensional transmutation in a hypothetical asymptotically free confining sector; this motivates, but does not microscopically derive, the mass formula. Model M1 has one parameter-independent rank constraint, R_1 = y_u y_mu^2 y_t/(y_e y_c^2 y_tau) = 1, whereas the inputs give R_1(M_Z) = 6.662. Its alternating residuals motivate the explicitly post-selected model M2, which assigns independent discrete curvatures to the charged sectors without increasing the parameter count. Its sole relation is R_2 = y_e y_mu^2 y_tau/(y_d y_s^2 y_b) = 1, while R_2(M_Z) = 0.4019. For an illustrative uncalibrated 25 percent relative model discrepancy per state, the exact weighted minima are chi^2_min = 6.003 for M1 and 1.389 for M2, with one nominal degree of freedom. A constrained neutral-sector extension M3 gives the neutrino mass-eigenstate triplet its own offset and slope but shares the M2 charged-lepton curvature. The resulting 12 x 10 design matrix yields a second left-null relation, m_1 m_3 m_mu^2/(m_2^2 m_e m_tau) = 1. Using NuFIT 6.0 central splittings, M3 predicts m_lightest = 1.2 x 10^-4 eV and sum m_nu = 0.05891 eV for normal ordering, or m_lightest = 4.09 x 10^-3 eV and sum m_nu = 0.10335 eV for inverted ordering. The sums lie only $0.21\%$ (NO) and $4.48\%$ (IO) above the oscillation floors; hence the lightest mass is the more model-specific output. M2 and M3 remain exploratory because they were selected after charged-sector diagnostics and the shared curvature has not been derived microscopically.