Universal pattern in fermion mass ratios and implications for flavor models

We show that a single analytic expression using only the labels that distinguish Standard Model mass eigenstates—the ordering index d∈ \1,2,3\ d ∈ { 1 , 2 , 3 } and electric charge q q —fits all fermion mass ratios with high precision and stable O(1) O ( 1 ) parameters. The functional form ln (m₁/m₂)= d₁ζ ₁ d₂ζ ₂ κ ^[(d₁ |q₁|)γ (μ )-(d₂ |q₂|)γ (μ )] ln ( m 1 / m 2 ) = d 1 ζ 1 d 2 ζ 2 κ [ ( d 1 | q 1 | ) γ ( μ ) - ( d 2 | q 2 | ) γ ( μ ) ] reproduces all 30 independent quark ratios from MZ M Z to 10¹²\,GeV 10 12 GeV with χ ²/dof 10⁻⁶ χ 2 / dof ≲ 10 - 6 using per-scale fits (24 parameters), fixed- κ κ fits (18), or global fits (5–6). Parameters remain near κ 2.3 κ ≃ 2.3 , ζ ₁ 1.16 ζ 1 ≃ 1.16 , ζ ₂ -0.8 ζ 2 ≃ - 0.8 , γ 1.1 γ ≃ 1.1 . Three orthogonal checks establish specificity and robustness: (i) a 120-permutation test where only the physical mapping of ratios to $$(d,|q|)$$ ( d , | q | ) fits; (ii) a 720-basis scan in which the adjacent-ratio basis is uniquely preferred by AIC/BIC and many alternatives fail to converge; (iii) alternative functional forms are decisively disfavored by AIC/BIC. The same form, with natural parameters, describes charged-lepton and neutrino ratios; for $$q=0$$ q = 0 the charge term vanishes, leaving pure dζ d ζ scaling. We sketch three minimal mechanisms—modified Froggatt–Nielsen, partial compositeness, and a Randall–Sundrum overlap—that generate the additive d d – $$|q|$$ | q | dependence at the exponent level and reduce to the neutrino limit. The pattern predicts percent-level deviations from standard running near 10¹⁴\,GeV 10 14 GeV and, with meV meV -scale anchors, m_ν 0.06\,eV Σ m ν ≃ 0.06 eV . To our knowledge, no previous empirical relation achieves simultaneous cross-sector consistency with such parameter stability; the result offers a compact target for flavor model building.