Eur. Phys. J. C (2025) 85:1082 https://doi.org/10.1140/epjc/s10052-025-14771-0 Regular Article - Theoretical Physics Universal pattern in fermion mass ratios and implications for flavor models Samir Varmaa VS Asset Management, LLC, Cos Cob, CT 06807, USA Received: 10 July 2025 / Accepted: 9 September 2025 © The Author(s) 2025 Abstract We show that a single analytic expression using Froggatt–Nielsen textures [3] and many others—describe at only the labels that distinguish Standard Model mass most one sector at a time and fail when extended unchanged to eigenstates—the ordering index d ∈ {1, 2, 3} and electric the remaining sectors. Consequently, the literature contains charge q—fits all fermion mass ratios with high precision and no single analytic expression that fits all quark, chargedstable O(1) parameters. The functional form ln(m 1 /m 2 ) = lepton, and neutrino mass ratios simultaneously.1 ζ ζ γ (μ) γ (μ) d11 d22 κ [(d1 |q1 |) −(d2 |q2 |) ] reproduces all 30 independent We close this gap by exploiting a minimal and modelquark ratios from M Z to 1012 GeV with χ 2 /dof  10−6 agnostic choice of labels for mass eigenstates: an ordering using per-scale fits (24 parameters), fixed-κ fits (18), or index d ∈ {1, 2, 3} that ranks the states by mass within a global fits (5–6). Parameters remain near κ ≃ 2.3, ζ1 ≃ 1.16, sector, and the electric charge q in units of |e|. Only q is a ζ2 ≃ −0.8, γ ≃ 1.1. Three orthogonal checks establish conserved gauge quantum number; d is a bookkeeping index specificity and robustness: (i) a 120-permutation test where used to organize the data and fits (we return to this point in only the physical mapping of ratios to (d, |q|) fits; (ii) a 720- Sects. 4–5). If an underlying organizing principle governs basis scan in which the adjacent-ratio basis is uniquely pre- fermion masses, it may act through these two labels. Guided ferred by AIC/BIC and many alternatives fail to converge; by this constraint, we propose the minimal ansatz: (iii) alternative functional forms are decisively disfavored by     AIC/BIC. The same form, with natural parameters, describes m1 ζ ζ γ (μ) γ (μ) ln = d11 d22 κ (d1 q1 ) −(d2 q2 ) , (1) charged-lepton and neutrino ratios; for q = 0 the charge term m2 vanishes, leaving pure d ζ scaling. We sketch three minimal where m 1 > m 2 ,2 all masses are MS—six reference scales mechanisms—modified Froggatt–Nielsen, partial composand uncertainties (across ten orders of magnitude) taken iteness, and a Randall–Sundrum overlap—that generate the from [5]3 —and with a mildly running exponent,4 additive d–|q| dependence at the exponent level and reduce to the neutrino limit. The pattern predicts percent-level deviations from standard running near 1014 GeV and, with meV- 1 While a rare intersecting-brane Pati-Salam compactification has been scale anchors, Σm ν ≃ 0.06 eV. To our knowledge, no pre- shown to reproduce all Standard Model masses and mixings [4], that vious empirical relation achieves simultaneous cross-sector construction requires 12 adjoint Higgs fields and significant flux tuning, placing it well outside the simple texture, Koide-type, GUT, or seesaw consistency with such parameter stability; the result offers a ansatz considered here. compact target for flavor model building. 2 Here the subscripts 1/2 denote the heavier/lighter mass within that pair; they are not the global indices used later in the basis scan. 3 All masses up to 1012 GeV are taken from [5], which are derived 1 Introduction from PDG world averages and evolved using five-loop quantum chromodynamics (QCD) renormalization-group equations; they coincide The masses of the Standard Model fermions span more than with the latest PDG inputs at their respective renormalization scales 12 orders of magnitude, yet no accepted principle explains within quoted uncertainties. For comparison with independent lattice the observed pattern. Existing empirical relations—Koide’s determinations, see the FLAG 24 averages [6]. 4 This running exponent plays a role similar to an anomalous dimencharged-lepton formula [1], the Georgi–Jarlskog relation [2], sion in quantum field theory: it governs the scale dependence of effective mass ratios, even though the formula itself is not derived from a a e-mail: samir@vsasset.com (corresponding author) renormalizable Lagrangian. 0123456789().: V,-vol 123 1082 Page 2 of 22 Eur. Phys. J. C (2025) 85:1082 γ (μ) = a + b ln(μ/M Z ) [5 param.], or – Section 5 treats the q = 0 neutrino limit, derives γ (μ) = a + b ln(μ/M Z ) + c ln (μ/M Z ) [6 param.]. 2 (2) absolute-mass predictions, and explains why the ratios are expected not to run. Equation (1) reproduces – Section 6 collects falsifiable consequences: quark ratios at 1014 GeV via deviations from RunDec v3.1 [9] at 1. all 30 independent quark mass ratios between M Z and five-loop QCD and two-loop electroweak orders, and val1012 GeV, ues for Σm ν , m β , and m ββ relevant to upcoming surveys. 2. both charged-lepton ratios over the same scales,5 and – Section 7 identifies specific theoretical modifications 3. the two neutrino ratios implied by NuFIT-6.0 [7] oscilla- required for flavor models to reproduce the empirical pattion data. tern, including constraints on Froggatt–Nielsen, warped extra dimensions, and discrete symmetry frameworks. No structural modification is required as we move from one – Section 8 discusses the hierarchy of evidence strength sector to the next, and all fitted parameters remain O(1). across sectors, compares Eq. (1) with texture and grand The empirical strength of Eq. (1) varies with data rich- unified theory (GUT) relations, and outlines possible theness. For quarks, 30 measurements confront at most five oretical origins. or six parameters,6 yielding χ 2 /dof ∼ 10−5 and residuals – Change-of-basis scan (information criteria). Among all below the quoted experimental uncertainties. Charged lep- 720 independent ordered chained five-ratio bases (adjatons offer a consistency test with under-determined param- cent basis plus 719 alternatives), the adjacent basis is eters; nonetheless, the fitted values stay natural in size. For uniquely preferred: it is the only basis whose fiveneutrinos, where q = 0 removes the κ term entirely, the parameter global fit attains a negative Akaike information formula reduces to d ζ scaling and predicts absolute masses criterion (AIC; same conclusion with the Bayesian inforonce the lightest eigenvalue is anchored. In each section, we mation criterion [BIC]). Of the 719 non-adjacent bases, contrast these results with the best-known alternatives and only 343 even admit a finite-likelihood fit; the remaining show explicitly why those alternatives cannot describe all 376 fail to converge (effectively “+∞” AIC). See Fig. 2 three sectors at once. and Table 1. The decision to parametrize mass ratios using only generation index and electric charge—the two labels that universally distinguish Standard Model fermions—was motivated 2 A puzzle in plain sight by seeking maximum empirical coverage with minimal theoretical bias. The surprising success of this ansatz, particuFigure 1 plots the five quark-mass ratios at the six reference larly the cross-sector universality and the permutation analscales used in this study. Neither a linear (shown) nor the ysis, suggests these labels may play a more fundamental role logarithmic view (not shown) reveals a monotonic or polyin flavor physics than previously appreciated [8]. Section 7 nomial trend; the points resemble uncorrelated scatter. What translates these empirical findings into concrete guidance for follows shows that the “scatter” is in fact threaded by a simmodel builders. ple expression whose four shape parameters are surprisingly Our main findings are summarized here and detailed in stable under every refitting scheme we could devise. the following sections: – Section 3 demonstrates the quark fits across multiple approaches, establishes parameter stability when fitting with or without experimental uncertainties, shows that even constraining κ to be scale-independent preserves the empirical pattern, and proves through comprehensive permutation analysis that only the physical assignment of quantum numbers to mass ratios yields acceptable fits. – Section 4 applies the same structure to charged leptons, highlighting natural parameter sizes despite underdetermination. 5 Charged-fermion ratios drift by (1.0 ± 1.4) × 10−3 from M to Z 1012 GeV, consistent with zero, so a constant γ suffices. 6 We test both a five-parameter γ (μ) = a + b ln(μ/μ0 ) and six- Fig. 1 Quark-mass ratios at six reference scales. No simple trend is parameter γ (μ) = a + b ln(μ/μ0 ) + c ln2 (μ/μ0 ), with μ0 = M Z . apparent, motivating the search for an empirical organizing principle 123 Eur. Phys. J. C (2025) 85:1082 Page 3 of 22 1082 Throughout the paper, we underscore three empirical facts: (i) there is no a priori reason why the same functional form should succeed across all three sectors, each with different levels of empirical constraint; (ii) alternative relations fail to do so without sector-specific amendments; and (iii) all fitted parameters remain natural in magnitude. Taken together, these points argue that Eq. (1) captures a genuine organizing principle behind the fermion mass spectrum, independent of any ultraviolet model. 3 Quark sector Six reference scales—91.1876 GeV (M Z ), 125.1 GeV, Fig. 2 Information-criterion scan over all five-ratio bases. Each point is one basis; ordinate: AIC from the five-parameter global fit; abscissa: 173.1 GeV, 105 GeV, 108 GeV, and 1012 GeV—are provided dynamic range of that basis (max/min across its five ratios pooled over by the running-mass compilation of Huang [5]. At each scale, the six reference scales). Red dot: adjacent basis, the only one with we construct the five successive ratios, negative AIC. Blue points: other bases that converge (all with large, positive AIC). Non-convergent bases (376 of the 719 non-adjacent) are   not plotted and correspond to effectively infinite AIC. BIC shows the m d /m u , m s /m d , m c /m s , m b /m c , m t /m b , same qualitative separation. AICs > 500 are not shown so that all 30 independent quark ratios enter the analysis. in total, including the adjacent basis). For each basis we comWhy consecutive ratios suffice The five adjacent-generation puted the Akaike information criterion (AIC) under a comratios form an algebraically independent basis for all quark mon Gaussian least-squares likelihood with the same N and k mass ratios. Writing rk ≡ ln(m k+1 /m k ) after fixing a across bases.7 Figure 2 shows AIC versus a proxy for numerglobal ascending enumeration (u, d, s, c, b, t) ≡ (1, . . . , 6) ical conditioning (“dynamic range”; see Appendix C). (indices increase with mass; this enumeration is independent Specifically, we enumerated all 720 ordered chained fiveof the pairwise convention m 1 > m 2 used in Eq. (1)), ratio bases (Hamiltonian paths), of the form m i1 /m i2 , m i2 /   m i3 , m i3 /m i4 , m i4 /m i5 , m i5 /m i6 over all 6! permutations mi  i−1 (i 1 , . . . , i 6 ). These are a distinguished subset of the admissiln = rk (i > j). mj ble spanning-tree bases (any five ratios form a basis iff their k= j numerator–denominator graph is a connected tree). We focus Thus, in log space the vector of adjacent ratios r = on chained trees because—while not well conditioned—they (r1 , . . . , r5 )T spans the Z-module of all ratio logs; any are less badly pathological: they compress the dynamic range other independent five-tuple of ratios is obtained by a left of the five basis ratios (typically smaller | ln rk | than nonmultiplication r = A r with an integer unimodular matrix chained choices). With only diagonal uncertainties propaA ∈ GL(5, Z). In a properly propagated (co)variance treat- gated, the larger dynamic ranges of non-chained trees make ment this change of basis leaves the weighted least-squares the normal equations more ill-conditioned, leading to freoptimum invariant up to numerical rounding, so the choice of quent non-convergence and decisively worse information criadjacent ratios is a convenience, not a bias (see Appendix B teria.8 for a compact proof). 7 Because N and k are fixed across the basis scan, BIC shifts all AIC Practical basis choice: information content and numer- values by a constant and yields the same ordering; the adjacent basis is ical stability Appendix C proves that, with full covariance also the only one with negative BIC. We therefore show AIC only. R 2 propagation, changing the basis of ratios is a unimodular is not informative for this non-linear, cross-basis comparison because it does not penalize model complexity and is not invariant under basis transformation that leaves the weighted least-squares opti- rescalings. mum invariant. In practice, however, one often works with 8 There are 66−2 = 1296 labeled spanning trees on six masses (Caycentral values or with incomplete covariance, and then the ley). Accounting for orientation of each ratio (numerator/denominator) basis choice can affect both convergence and model selec- gives 1296 × 25 = 41,472 unordered, oriented bases; allowing arbition metrics. To test robustness we evaluated the same five- trary ordering of the five ratios yields 1296 × 25 × 5! = 4,976,640 ordered, oriented bases. We restrict attention to the 6! = 720 chained parameter global fit (Sect. 3.3) over all integer, full-rank (Hamiltonian-path) bases because, under diagonal-only uncertainties, five-tuple ordered chained “ratio bases” built from pairwise they are less badly pathological (smaller dynamic-range spreads) than quark-mass ratios at each reference scale (720 ordered bases generic trees, which typically suffer markedly worse conditioning. 123 1082 Page 4 of 22 Eur. Phys. J. C (2025) 85:1082 Table 1 Outcome of the Category Count AIC sign Notes change-of-basis scan for the five-parameter global fit Adjacent basis 1 Negative Unique best (also negative BIC) Other bases: converged 343 Positive All decisively worse Other bases: no fit 376 +∞ Optimizer fails to find a finite likelihood Total non-adjacent 719 More bases fail than succeed Table 2 Four-parameter fit to the five quark ratios at each scale. Errors rescale all errors uniformly and do not affect the central values. The are statistical 1σ values propagated from Huang [5]; current lattice- same convention is used in Tables 3 and 4. All quoted masses are MS QCD systematic uncertainties, estimated at ∼ 0.5% for bottom-quark values at the scales indicated. Corresponding p-values range from ∼0.01 masses—based on N f = 2 + 1 + 1 lattice determinations [6]—would to 0.03 for individual parameters μ [GeV] Fit type κ ζ1 ζ2 γ χ 2 /dof 91.2 With uncertainties 2.367(5) 1.1564(12) −0.7671(17) 1.0949(24) 2.40 × 10−6 125 With uncertainties 2.368(5) 1.1562(12) −0.7691(18) 1.0975(25) 2.80 × 10−6 173 With uncertainties 2.364(5) 1.1558(11) −0.7699(17) 1.1011(23) 2.42 × 10−6 105 With uncertainties 2.314(5) 1.1634(13) −0.8019(19) 1.1565(26) 4.54 × 10−6 108 With uncertainties 2.278(6) 1.1697(14) −0.8241(21) 1.1945(29) 7.40 × 10−6 1012 With uncertainties 2.239(5) 1.1766(14) −0.8448(22) 1.2310(29) 9.04 × 10−6 91.2 Central values 2.340 1.1609 −0.7719 1.1061 8.31 × 10−4 125 Central values 2.339 1.1611 −0.7743 1.1096 9.59 × 10−4 173 Central values 2.338 1.1602 −0.7745 1.1119 7.99 × 10−4 105 Central values 2.287 1.1681 −0.8067 1.1682 1.05 × 10−3 108 Central values 2.249 1.1749 −0.8292 1.2074 1.38 × 10−3 1012 Central values 2.212 1.1819 −0.8499 1.2443 1.50 × 10−3 Two empirical facts stand out. (i) The adjacent basis is lattice uncertainties) from 0.25% (top/bottom at 1012 GeV) to the only basis with negative AIC (same for BIC), and it lies 24% (down/up at M Z ), the resulting χ 2 /dof ∼ 10−5 reflects at the extreme left-minimal dynamic range. (ii) Many bases the formula’s accuracy in reproducing central values rather simply do not fit: excluding the adjacent basis, only 343 out than over-fitting.9 of the remaining 719 bases yielded any convergent fit at all; Crucially, fitting to central values alone (ignoring exper376 failed to converge or returned ill-posed parameters, i.e., imental uncertainties entirely) yields virtually identical effectively infinite AIC/BIC. Thus, more bases fail than suc- parameter values and only modest increases in χ 2 /dof to ceed. These observations support our use of adjacent ratios in ∼ 10−3 . This demonstrates that the empirical pattern is not the main analysis: beyond the formal change-of-basis invari- an artifact of the uncertainty weighting scheme but reflects ance, the adjacent basis is also the information-optimal and genuine structure in the mass ratios themselves. The paramnumerically stable choice in practice. See Fig. 2 where the eter stability across both fitting approaches, both with and abscissa uses the observed ratio dynamic range across scales; without uncertainties, as well as the quality of the fit as shown Appendix C also defines a purely algebraic proxy D(A) for in Fig. 3, is remarkable. basis conditioning. The two are highly correlated across the 720 bases and lead to the same separation. 3.2 (3 × 6) = 18-parameter per-scale fit (fixed κ) 3.1 (4 × 6) = 24-parameter per-scale fit Freezing κ at its M Z value, κ = 2.36707, reduces the fit to three parameters per scale. Table 3 shows that χ 2 /dof Equation (1) was fitted independently at each scale with four decreases for every scale up to 105 GeV and then increases free parameters (κ, ζ1 , ζ2 , γ ). Table 2 lists the best-fit values, only slightly up to 1012 GeV, a strong indication that κ is a one-sigma errors, and the resulting χ 2 /dof. Residuals never exceed ∼ 4% and χ 2 /dof ≤ 9×10−6 . All parameters remain 9 Because experimental uncertainties in quark mass evolution are domin the natural range 2.2  κ  2.4, 1.15  ζ1  1.18, inated by correlated renormalization scale uncertainties that affect all −0.85  ζ2  −0.77, and 1.09  γ  1.23. With residuals ratios similarly, the effective degrees of freedom are reduced, leading  4% confronting experimental errors (which dominate the to χ 2 /dof 1; see Appendix A for details. 123 Eur. Phys. J. C (2025) 85:1082 Page 5 of 22 1082 Fig. 3 Comparison of all 24-parameter fits (κ free) across the six energy scales. Even when κ is allowed to vary, the parameter values remain remarkably stable genuine scale-independent factor rather than a tuning knob. Again, as Table 3 shows, fitting to central values alone proOverall parameter shifts are  5.5 % and residuals increase duces nearly identical parameters with χ 2 /dof rising only slightly with the maximum residual being ∼ 5%. Figure 4 modestly to ∼ 10−3 . The parameter stability across both illustrates the high accuracy of the fit. uncertainty-weighted and central-value-only fits confirms 123 1082 Page 6 of 22 Eur. Phys. J. C (2025) 85:1082 Table 3 Three-parameter refit with κ fixed to its M Z value. Corresponding p-values range from ∼ 10−6 to ∼ 10−3 μ [GeV] Fit type ζ1 ζ2 γ χ 2 /dof 91.2 With uncertainties 1.15643(6) −0.76707(9) 1.09492(3) 1.20 × 10−6 125 With uncertainties 1.15645(6) −0.76944(9) 1.09811(3) 1.40 × 10−6 173 With uncertainties 1.15539(6) −0.76930(9) 1.09991(3) 1.22 × 10−6 105 With uncertainties 1.15513(9) −0.78898(14) 1.13158(5) 4.51 × 10−6 108 With uncertainties 1.15542(14) −0.80187(21) 1.15121(7) 1.25 × 10−5 1012 With uncertainties 1.15571(19) −0.81221(28) 1.16709(10) 2.76 × 10−5 91.2 Central values 1.1565 −0.7663 1.0943 4.45 × 10−4 125 Central values 1.1565 −0.7686 1.0974 5.10 × 10−4 173 Central values 1.1554 −0.7687 1.0994 4.33 × 10−4 105 Central values 1.1546 −0.7904 1.1326 8.18 × 10−4 108 Central values 1.1543 −0.8050 1.1537 1.39 × 10−3 1012 Central values 1.1540 −0.8177 1.1714 2.08 × 10−3 that the fixed-κ constraint captures genuine physical struc- formula’s ability to reproduce not just the pattern of mass ture rather than statistical fluctuations. ratios, but their experimental precision as well. Most remarkably, both global fits show parameter stabil3.3 Global fits ity when fitted to central values alone, completely ignoring experimental uncertainties. The quality of the central value fit Treating all 30 ratios simultaneously, we test both linear and is apparent from Fig. 6: almost every predicted value is coinquadratic parametrizations of the running exponent: cident with the central value being fitted. The five-parameter central-value fit yields κ = 2.287 (vs. 2.316 with uncertainties), ζ1 = 1.169 (vs. 1.164), ζ2 = −0.802 (vs. −0.795), γ (μ) = a + b ln μ/M Z [5 param.] and a = 1.149 (vs. 1.134)—variations of less than 2%. The six-parameter fit shows similar stability. This consistency across fitting methodologies demonstrates that our empirical γ (μ) = a + b ln μ/M Z + c ln2 μ/M Z [6 param.]. law captures genuine structure in the fermion mass spectrum Table 4 presents both fits. The five-parameter form rather than artifacts of the uncertainty weighting scheme. achieves χ 2 /dof = 9.29 × 10−6 , while adding the very Both central-value fits reproduce every data point with residmild curvature term reduces this to 3.86 × 10−6 . For the uals |Δ ln(m i /m j )| ≤ 0.04, approximately 4% in the ratios fits with uncertainties, there is a substantial improvement in themselves (see Fig. 6), so their χ 2 /dof 1 reflects the the information theoretic criteria when moving from the five- formula’s intrinsic accuracy, not over-fitting to experimental parameter fit to the six-parameter fit. Specifically, the Akaike errors. information criterion (AIC) and the Bayesian information criterion (BIC) go from −100.237 and −91.8299 to −126.26 3.4 Comprehensive stability analysis and −116.45, respectively (more negative is an improvement). However, the central value fits show only insignificant The parameter stability demonstrated in each individual fit changes in AIC and BIC when moving from a five-parameter above represents significant robustness. Figure 7 displays the fit to a six-parameter fit. This suggests that the information- residuals of the 18-parameter fit; every point lies within criterion pay-off is marginal. Thus, parsimony favors the five- |Δ ln(m i /m j )| ≤ 0.04 (4 % in the ratio itself). Similarly, parameter form; the six-parameter version merely confirms Fig. 8 displays the residuals of the two global fits. Table 5 that a small quadratic bend in γ (μ) is numerically allowed collects the χ 2 /dof values for all eight fits (with uncertainties but not required. and central values). Table 6 lists the corresponding paramFigure 5 displays both uncertainty-weighted global fits, eters, demonstrating their stability at the  5% level across showing the exceptional agreement between the formula and all fitting strategies. all 30 data points across six energy scales. Every residual lies The key empirical finding is that the same four O(1) numwell within the quoted experimental uncertainties, with the bers emerge regardless of fitting methodology: whether we formula tracking both the central values and their error bars use 24 parameters (4 × 6), 18 parameters (3 × 6), or global with remarkable precision. The χ 2 /dof ∼ 10−6 reflects the fits with 5–6 parameters; whether we weight by experimental 123 Eur. Phys. J. C (2025) 85:1082 Page 7 of 22 1082 Fig. 4 Comparison of all 18-parameter fits (κ fixed) across the six energy scales. The remarkable parameter stability across scales demonstrates the robustness of the empirical pattern uncertainties or fit central values only; whether we allow all tal organizing principle in the fermion mass spectrum, not a parameters to float or constrain κ to be scale-independent. statistical artifact. This consistency suggests we have identified a fundamen- The same four numbers reappear with consistent O(1) values in the lepton and neutrino sectors. 123 1082 Page 8 of 22 Eur. Phys. J. C (2025) 85:1082 Table 4 Five- and six-parameter global fits with γ (μ) = a +b ln(μ/M Z ) and γ (μ) = a +b ln(μ/M Z )+c ln2 (μ/M Z ), respectively. Corresponding p-values are extremely small, ranging from ∼ 10−30 to ∼ 10−6 for individual parameters, indicating highly significant fits Fit Type κ ζ1 ζ2 a b [10−3 ] c [10−5 ] χ 2 /dof 5-param w/unc. 2.316(8) 1.1638(19) −0.7949(29) 1.1340(40) 1.587(9) – 9.29 × 10−6 5-param central 2.287 1.1686 −0.8019 1.1494 1.440 – 6.26 × 10−4 6-param w/unc. 2.315(5) 1.1639(13) −0.7950(19) 1.1201(25) 3.130(26) −5.20(9) 3.86 × 10−6 6-param central 2.287 1.1686 −0.8019 1.1353 2.971 −4.948 5.98 × 10−4 3.5 Specificity test: permutation analysis 3.6 Change-of-basis and information criteria To address potential concerns about mathematical flexibility Equation (1) is written in terms of adjacent generation ratios, versus physical content, we performed a comprehensive test but any five ratios whose numerator–denominator graph is of Eq. (1)’s specificity. Given that, in our empirical setup, a spanning tree (i.e., a connected set of five ratios linkeach Standard Model mass eigenstate is uniquely labeled ing all six flavors) form a valid basis. In an ideal analysis by the pair {d, q}, we ask: does our functional form work with the full 5 × 5 covariance propagated for each basis, only when mass ratios are correctly assigned to their physical weighted least squares would be invariant under unimodular (d, q) labels, or could it fit any arbitrary assignment equally changes of basis (Appendix B). In practice, published inputs well? provide diagonal (or effectively diagonal) uncertainties for We systematically tested all 5! = 120 possible permu- the five ratios at each scale; with that pragmatic weighting tations of the five quark mass ratios at each energy scale, the invariance is not guaranteed. We therefore performed an keeping the {d, q} assignments fixed but reassigning which exhaustive, information-theoretic scan over all independent, measured mass ratio corresponds to which quantum number chained, five-ratio bases: pair. For each permutation, we performed the most unconstrained fit (four free parameters per scale) and computed the – for each basis we repeated the five-parameter global fit root-mean-square error (RMSE) relative to the unpermuted, (Sect. 3.3); physical assignment. – we recorded the Akaike information criterion, AIC = Figure 9 displays the results as a function of the normal- 2k − 2 ln L,  and the Bayesian information criterion, ized Euclidean distance from the physical assignment.10 The BIC = k ln N − 2 ln L,  with k = 5 parameters and pattern is striking and consistent across all six energy scales: N = 30 data points; only the physical assignment (distance = 0) yields accept- – to characterize conditioning, we computed the dynamic able fits with RMSE  0.01. Every alternative permutation range of a basis as the ratio of the largest to the smallest of produces dramatically worse results, with RMSE values typ- its five ratios pooled over the six reference scales (larger ically very substantially larger. No other assignment among values indicate a more ill-conditioned basis). the 119 alternatives approaches the quality of the physical mapping. Results Out of the 720 bases (adjacent + 719 alternaThis result demonstrates that Eq. (1) is not a flexible curve- tives), only the adjacent basis achieves a negative AIC (and fitting expression but rather encodes specific physical struc- BIC); every other basis that converges yields large, positive ture. The functional form requires the correct correspon- AIC/BIC values, i.e., decisively worse fits. Moreover, more dence between quantum numbers and mass ratios to achieve bases fail than succeed: among the 719 non-adjacent bases, its remarkable precision, providing strong evidence that it only 343 admit any finite-likelihood optimum, while 376 do captures genuine organizing principles in the fermion mass not converge at all under the same fitting setup (effectively spectrum rather than mathematical coincidence. AIC → +∞). Figure 2 visualizes AIC versus basis dynamic range. The adjacent basis (red dot) sits alone in the negativeAIC quadrant; all other fits (blue points) lie far away. Because AIC/BIC compare penalized likelihoods, these gaps correspond to ΔAIC = O(102 ) or larger—decisive on the usual 10 The normalized Euclidean distance between physical mass ratios m Jeffreys/Burnham–Anderson scale—and make separate R 2 2 i discussion unnecessary (Table 1).11 m i −m i and permuted ratios m i is defined as i (m i +m i )/2 , where each difference is normalized by the average of the corresponding ratios to 11 AIC and BIC are computed for the same data and parameter count ensure scale invariance. across bases; “more negative” indicates better support. The exact neg123 Eur. Phys. J. C (2025) 85:1082 Page 9 of 22 1082 Fig. 5 Global fits with experimental uncertainties included. Top: five- fitted value is in red. The excellent agreement between theory and data parameter fit with χ 2 /dof = 9.29 × 10−6 . Bottom: six-parameter fit demonstrates the robustness of the empirical pattern with χ 2 /dof = 3.86 × 10−6 . The empirical value is in blue and the 123 1082 Page 10 of 22 Eur. Phys. J. C (2025) 85:1082 Fig. 6 Global fits to central values only (no error bars). Top: five-parameter fit. Bottom: six-parameter fit. The empirical value is in blue and the fitted value is in red. The excellent agreement demonstrates that the pattern is not an artifact of experimental uncertainties 123 Eur. Phys. J. C (2025) 85:1082 Page 11 of 22 1082 Fig. 7 Residuals of the 18-parameter (3×6) fit This scan complements the permutation test of Sect. 3.5. degrades the information criteria by ΔAIC ∼ O(102 ) relaThe permutation test fixes the quantum-number assignment tive to the five-parameter baseline in Table 4. Second, electric and shows that only the physical mapping of (d, q) to mea- charge must enter through the combined structure (d|q|)γ (μ) : sured ratios is viable. The present scan fixes the mapping and using |q|γ (μ) alone, or forcing ζ2 = −ζ1 (“Symmetric ζ ”), varies the ratio basis; it shows that, under realistic weighting, similarly worsens AIC/BIC by large margins. Allowing a the adjacent basis is uniquely well posed and information- trivial overall scale ρ inside the κ exponent does not help theoretically preferred. and effectively reparameterizes the baseline. The only variant that improves the information criteria is a very mild quadratic 3.7 Alternative functional forms bend in γ (μ), consistent with the global six-parameter fit already reported in Table 4. Overall, the five-parameter baseTo check whether simpler or differently factorized expres- line is the most parsimonious form that succeeds. For the sions can perform as well as Eq. (1), we repeated the global basis scan in Sect. 3, we therefore report AIC only. Because quark fit with uncertainties for a representative set of alter- k and N are fixed across that scan, BIC differs from AIC by an natives. In all fits we evaluate (d|q|)γ (μ) with q → |q| so additive constant and yields the same ordering; the adjacent that non-integer γ (μ) yields real powers; the signed-power basis is also the only one with negative BIC. variant (q without | · |) fails numerically. Table 7 summarizes These functional-form checks are orthogonal to the the results. change-of-basis scan in Sect. 3.6: the former holds the basis Two robust conclusions emerge. First, the generation pref- fixed and varies the ansatz, while the latter fixes the ansatz ζ ζ actor d11 d22 is required: removing it (“No-d prefactor”) and varies the basis; both favor the baseline expression in the adjacent basis. ative value for the adjacent basis depends slightly on whether one uses These checks, together with the permutation analysis in the five- or six-parameter global form; both yield the same qualitative Sect. 3.5, show that the multiplicative structure linking generconclusion. 123 1082 Page 12 of 22 Eur. Phys. J. C (2025) 85:1082 Fig. 8 Residuals from all global fits, showing consistency between uncertainty-weighted and central-value-only approaches ation and charge is genuinely specific. Simpler or factorized 4 Charged lepton sector alternatives are either decisively disfavored by the information criteria (typically by O(102 )) or not mathematically well At each reference scale, we work with the two ascending defined for the data. Apart from a very mild quadratic running ratios m μ /m e and m τ /m μ .12 Because these ratios vary by in γ (μ), the five-parameter baseline is the most economical less than 0.0001±0.0014 (i.e., consistent with zero) between successful form. M Z and 1012 GeV, the 12 data points provide only a modest lever arm against the four parameters in Eq. (1). The fit is 3.8 Comparison with texture approaches therefore a consistency test: success is not guaranteed even though the system is formally under-determined.13 Standard texture models, such as two-charge Froggatt– Nielsen schemes [3], fit at most one ratio per generation and typically require ten or more parameters. Koide’s [1] relation 4.1 Four-parameter fit does not extend to quarks; the Georgi–Jarlskog relation [2] matches a single ratio at a unification scale but fails else- Table 8 shows the best-fit values. Residuals are below 2.5 × where. In contrast, Eq. (1) reproduces all 30 quark ratios 10−4 , giving χ 2 /dof = 9.6 × 10−11 . All parameters remain with at most six parameters and χ 2 /dof 10−5 . O(1) and of the same magnitude as in the quark sector. That The same analytic form is now applied, unchanged, to the a single analytic form reproduces both charged leptons and charged lepton and neutrino sectors. quarks with natural coefficients is, by itself, non-trivial. 12 For charged leptons, we use |q| = 1; the sign of q does not enter Eq. (1). 13 Over the same ten-decade range as the quarks, the ratios drift by (0.0 ± 0.8) × 10−3 for m μ /m e and (0.1 ± 1.4) × 10−3 for m τ /m μ , completely consistent with scale-independence. 123 Eur. Phys. J. C (2025) 85:1082 Page 13 of 22 1082 Table 5 χ 2 /ν for the quark sector. The first four fits are performed scale by scale; the five- and six-parameter rows are global fits to all 30 ratios Fit and weighting 91 GeV 125 GeV 173 GeV 105 GeV 108 GeV 1012 GeV (4 × 6) w/unc. 2.40 × 10−6 2.80 × 10−6 2.42 × 10−6 4.54 × 10−6 7.40 × 10−6 9.04 × 10−6 (4 × 6) central 8.31 × 10−4 9.59 × 10−4 7.99 × 10−4 1.05 × 10−3 1.38 × 10−3 1.50 × 10−3 (3 × 6) w/unc. 1.20 × 10−6 1.40 × 10−6 1.22 × 10−6 4.51 × 10−6 1.25 × 10−5 2.76 × 10−5 (3 × 6) central 4.45 × 10−4 5.10 × 10−4 4.33 × 10−4 8.18 × 10−4 1.39 × 10−3 2.08 × 10−3 6-par w/unc. 3.86 × 10−6 (global) 6-par central 5.98 × 10−4 (global) 5-par w/unc. 9.29 × 10−6 (global) 5-par central 6.26 × 10−4 (global) Table 6 Parameter values for Fit and weighting κ ζ1 ζ2 γ (M Z ) all eight quark-sector fits. For the per-scale fits, the quoted ± is (4 × 6) w/unc. 2.30 ± 0.06 1.163 ± 0.009 −0.797 ± 0.033 1.147 ± 0.059 the scatter of the six independent determinations; for global fits, it (4 × 6) central 2.29 ± 0.06 1.168 ± 0.009 −0.802 ± 0.033 1.158 ± 0.059 is the fit uncertainty (3 × 6) w/unc. 2.367 (fixed) 1.156 ± 0.001 −0.785 ± 0.020 1.124 ± 0.031 (3 × 6) central 2.367 (fixed) 1.155 ± 0.001 −0.786 ± 0.022 1.125 ± 0.033 6-par w/unc. 2.315 ± 0.015 1.164 ± 0.004 −0.795 ± 0.004 1.120 ± 0.002 6-par central 2.287 1.169 −0.802 1.135 5-par w/unc. 2.316 ± 0.015 1.164 ± 0.004 −0.795 ± 0.004 1.134 ± 0.002 5-par central 2.287 1.170 −0.802 1.149 Running γ (μ) 5 Neutrino sector Allowing γ (μ) = a + b ln(μ/M Z ) or a + b ln(μ/M Z ) + In the neutrino sector, we normally discuss mass eigenstates c ln2 (μ/M Z ) produces χ 2 /dof ∼ 10−9 but returns order- rather than flavor eigenstates. On the other hand, our label, d, unity or larger standard errors for some parameters. The represents the generation number. In many flavor construcdegeneracy simply reflects the near scale-independence of tions, the labels that diagonalize the mass matrix need not the lepton ratios: current data cannot meaningfully distin- coincide with the weak-interaction basis, so one often works guish a constant from a slowly varying exponent. For leptons, with “effective” generation indices. A well-known example we therefore keep the constant-γ fit for the rest of the paper. is in A4 models of lepton flavor (e.g., [10]) where the mass eigenstates arise from non-trivial mixtures of the symmetry triplet, and one effectively re-labels them by descending mass rather than gauge quantum numbers. In our ansatz, we similarly take di ∈ {1, 2, 3} to index the mass eigenstates in order of magnitude, understanding that these are shorthand 4.2 Context for the actual mixed states in the weak basis. Therefore, for mass eigenstates, d represents an effective generation index Koide’s [1] relation provides a single constraint on the three reflecting the average generation content of each mass eigencharged-lepton masses using a different functional form that state rather than integer labels. fails for quarks and neutrinos. Texture models fit the lepton In normal ordering for neutrinos, m 1 < m 2 < m 3 . ratios only at the cost of abandoning the quark hierarchy or However, our convention for quarks and leptons has been of introducing sector-specific matrices. Equation (1) achieves m 1 > m 2 in the ansatz. For neutrinos, therefore, we write simultaneous agreement with the quark fits (Sect. 3) without the master ansatz as: any additional sector-dependent factor and with O(1) coefficients. This cross-sector coherence strengthens the empirical     m i+1 ζ1 ζ γ γ claim that Eq. (1) captures a genuine organizing principle in ln = di+1 di 2 κ (di+1 qi+1 ) −(di qi ) , the fermion spectrum. mi The neutrino sector, where q = 0 eliminates the κ term where we label the heavier mass as i + 1 and the lighter entirely, provides an even more severe test and is treated next. mass as i. However, the ansatz simplifies radically for neu123 1082 Page 14 of 22 Eur. Phys. J. C (2025) 85:1082 Fig. 9 RMSE as a function of normalized Euclidean distance for all energy scales. Alternative assignments produce dramatically larger all 120 possible permutations of quark mass ratio assignments. Only residuals, demonstrating the specificity of the empirical pattern the physical assignment (distance = 0) yields acceptable fits across Table 7 Functional-form checks (global quark fit with uncertainties; cate better fits. “Ill-posed” means the optimizer fails or returns complex same data as Table 4). The five-parameter baseline in Table 4 has AIC values due to non-integer powers of signed charges ≈ −100.24 and BIC ≈ −91.83. Smaller (more negative) AIC/BIC indiID Functional form Params AIC BIC ζ ζ γ (μ) −(d |q |)γ (μ) Baseline m1 ln m 2 = d11 d22 κ (d1 |q1 |) 2 2 5 −100.24 −91.83 γ (μ) γ (μ) No-d prefactor m1 ln m 2 = κ (d1 |q1 |) −(d2 |q2 |) 3 80.64 86.25 ζ ζ γ (μ) γ (μ) |q| only m1 ln m 2 = d11 d22 κ |q1 | −|q2 | 5 3.16 11.57 ζ −ζ γ (μ) γ (μ) Symmetric ζ m1 ln m 2 = d1 d2 κ (d1 |q1 |) −(d2 |q2 |) 4 12.73 19.73 Log-additive split m1 ln m = c0 + α1 ln d1 + α2 ln d2 + ρ |q1 |γ (μ) − |q2 |γ (μ) 6 95.61 105.42 2     m1 ζ ζ c1 (d1 |q1 |)−(d2 |q2 |) +c2 (d1 |q1 |)2 −(d2 |q2 |)2 Poly. in (d|q|) ln m 2 = d11 d22 κ 6 Ill-posed Ill-posed No | · | on q Same as baseline with q → q (signed power) 5 Ill-posed Ill-posed ζ ζ γ (μ) −(d |q |)γ (μ) ] Scale in exponent m1 ln m 2 = d11 d22 κ ρ[(d1 |q1 |) 2 2 6 −95.97 −87.56 γ (μ) quadratic Baseline with γ (μ) = a + b ln(μ/M Z ) + c ln2 (μ/M Z) 6 −126.26 −116.45 trinos because qi = 0. As we said earlier, however, we do generation indices, though the actual effective values may not know the precise values for di+1 and di for the neutrino differ depending on the flavor-mass eigenstate mixing patsector since we are talking about mass eigenstates. For illus- tern. The key point is not the specific ζ values, which depend tration, we take the d values to be 1, 2, 3 as representative on the choice of effective generation indices, but rather that 123 Eur. Phys. J. C (2025) 85:1082 Page 15 of 22 1082 Table 8 Charged-lepton fit with Parameter Estimate Std. error [10−6 ] t-stat. [105 ] χ 2 /dof four free parameters. Quoted errors are 1σ statistical κ 2.91835 2.94 9.9 9.6 × 10−11 uncertainties propagated from Ref. [5] γ 0.885797 5.33 1.7 ζ1 1.10965 1.76 6.3 ζ2 −1.49050 13.3 −1.1 natural O(1) values emerge for any reasonable assignment Table 9 Neutrino observables at two illustrative anchors that satisfy of d1 , d2 , d3 . Because d1 , d2 , d3 are each constrained to lie ζ1 > 0, ζ2 < 0 and |ζi | ∼ 1. Errors are 1σ uncertainties from NuFIT6.0 [7] between 1 and 3, other reasonable assignments, e.g., taking di weighted by Pontecorvo–Maki–Nakagawa–Sakata (PMNS) Quantity m 1 = 1.0 meV m 1 = 1.5 meV mixing angles, would yield qualitatively similar O(1) natu- m 2 (meV) 8.71 ± 0.11 8.78 ± 0.11 ral parameter values, confirming the robustness of the crossm 3 (meV) 50.35 ± 0.25 50.36 ± 0.25 sector consistency (for a full renormalization-group analysis Σm ν (eV) 0.0601 ± 0.0003 0.0606 ± 0.0003 including mixing effects, see [11]). m β (meV) 8.95 ± 0.11 9.02 ± 0.11 Furthermore, because q = 0, the κ term vanishes entirely. ζ1 1.114 ± 0.008 0.822 ± 0.010 Using 1, 2, 3 for the generation index, we see that the ansatz ζ2 −0.955 ± 0.017 −0.498 ± 0.019 simplifies radically14 :     m2 m3 ln = 2 ζ1 , ln = 3ζ1 2ζ2 . (3) Hence, m1 m2 L 21 L 32 − ζ1 ln 3 ζ1 = , ζ2 = . There are only two unknown exponents and exactly two mea- ln 2 ln 2 sured mass ratios, so any choice of a lightest mass m 1 fixes Setting a minimum mass for m 1 = 0.0001 eV and scanning (ζ1 , ζ2 ) algebraically. The neutrino sector therefore serves as the physically allowed window m 1 ≤ 0.03 eV (NuFIT 6.0 a self-consistency check rather than a genuine fit.15 No matsplittings and 1σ errors [7]) gives the results summarized in ter what the actual values of d1 , d2 , d3 are, the key point is Table 9. that the ζ values would remain reasonably O(1) regardless of what non-integer values of di we used since those values Cross-sector consistency Quarks and charged leptons favor are constrained to be between 1 and 3. We continue therefore ζ1 ≈ +1 and ζ2 ≈ −1. Demanding the same sign pattern with d1 , d2 , d3 = 1, 2, 3 and proceed as follows. and O(1) size restricts the anchor to16 Log-log form and anchor scan With qi = 0, the κ-term m 1  1.5 meV. vanishes, so we work entirely in double logs. Denoting L 21 ≡ ln(ln(m 2 /m 1 )) and L 32 ≡ ln(ln(m 3 /m 2 )), Eq. (1) reduces Anchors near 1 meV yield ζ1 = 1.11 ± 0.01, ζ2 = −0.96 ± to 0.02, remarkably close to the quark and lepton values. Numerical predictions We adopt the standard observables L 21 = ζ1 ln 2, L 32 = ζ1 ln 3 + ζ2 ln 2. (4) Σm ≡ m + m 2 + m 3 , m 2β ≡ i |Uei |2 m i2 , and m ββ ≡  ν 2 1   i Uei m i , with Uei the PMNS matrix elements. For the preferred meV-scale anchors, we obtain 14 Any reasonable assignment of effective generation indices d i ∈ [1, 3] for neutrino mass eigenstates yields O(1) values for ζ1,2 , confirming the robustness of cross-sector consistency. The specific choice Σm ν ≃ 0.061 eV, m β ≃ 9 meV, di = i is illustrative; other assignments (e.g., weighted by PMNS mix- (5) ββ ≃ 0.46 meV, m ββ ≃ 4.76 meV. m min max ing angles) produce qualitatively similar natural parameter values. A complete treatment would require solving the full mixing problem, which is beyond our empirical scope. These lie just below the current Planck+BAO limit (Σm ν < 15 If the lightest mass were exactly zero, the successive ratio m /m 2 1 0.09 eV) and beneath KATRIN’s 0.2 eV reach, but sit (or m 3 /m 2 in inverted ordering) would be undefined, and Eq. (3) would simply not apply rather than be falsified; present limits still allow that 16 The sign of ζ possibility, so throughout this paper we assume a small but non-zero 1 flips from positive to negative at m 1 ≃ 3.426 meV smallest mass. and the sign of ζ2 flips from negative to positive at m 1 ≃ 2.17 meV. 123 1082 Page 16 of 22 Eur. Phys. J. C (2025) 85:1082 Table 10 Predicted high-scale log quark mass ratios with one-σ uncertainties propagated from low-scale inputs via RunDec v3.1 [9]. Numbers in parentheses denote one standard deviation Ratio d/u s/d c/s b/c t/b 5-param. fit ln ratio pred. 0.748 (8) 2.996 (31) 2.493 (27) 1.475 (9) 4.333 (11) 6-param. fit ln ratio pred. 0.748 (5) 2.996 (20) 2.496 (10) 1.474 (6) 4.345 (8) RunDec 3.1 0.77 (18) 2.99 (11) 2.46 (8) 1.515 (30) 4.094 (11) 5-param. fit, dev. from RunDec −3(23)% 0.0(4)% 2.0(4)% −2.6(2.0)% 5.8(0.4)% 6-param. fit, dev. from RunDec −3(23)% 0.0(4)% 2.0(4)% −2.7(2.0)% 6.1(0.4)% squarely in the target ranges of next-generation CMB-S4 sur- 3. Neutrinos Assuming d1 , d2 , d3 = 1, 2, 3, cross-sector veys and β-decay experiments such as the Project 8 [12] and naturalness (ζ1 ≈ +1, ζ2 ≈ −1) confines the lightest HOLMES [13]. mass to m 1  1.5 meV. In that window, the model predicts Σm ν ≃ 0.06 eV and m β ≃ 9 meV. Stage-IV CMB Outlook If a future measurement pins down any one of Σm ν , surveys and sub-40 meV β-decay experiments (Project m β , or m ββ and the effective generation indices are estab8 [12], HOLMES [13]) will probe this range directly. lished, the log-log relations in Eq. (4) lock ζ1,2 and make the neutrino sector as predictive—and falsifiable—as the quark – No unconditional prediction can be made until both and charged-lepton sectors. the effective generation indices and an anchor observable are determined. Once a β-decay experiment fixes m β , Eq. (4) determines (ζ1 , ζ2 ); the model can then 6 Predictions and tests be cross-checked against Σm ν or m ββ . – Tritium β decay. KATRIN reaches only m β ≃ 0.2 eV, The empirical pattern established above yields specific, falsi- well above our 9 meV target, but the Project 8 [12] fiable predictions across all three fermion sectors that can be and HOLMES [13] aim for sub-40 meV sensitivity tested by near-term experimental and theoretical advances: and could directly test the prediction. 1. Quark sector Using the a +b ln μ and a +b ln(μ/M Z )+ c ln2 (μ/M Z ) fits of Sect. 3, we evolve the five successive 7 Theoretical implications and model-building guidance ratios from M Z up to 1014 GeV. The relative departure from a straight RunDec v3.1 prediction is shown in The empirical pattern uncovered in Sects. 3–5 imposes strinTable 10.17 gent constraints on flavor model construction. Rather than High-precision lattice QCD could, with continued algo- broadly categorizing models as “favored” or “disfavored,” rithmic progress, be able to test the (t/b) prediction at we outline specific modifications required for representative the ∼ 5% level, provided bottom-quark mass uncertain- theoretical frameworks to reproduce our ansatz. ties improve from the current ∼ 0.5%, as summarized in Because Eq. (1) is multiplicative in X i j ≡ ln(m i /m j ), FLAG’s latest review [6], toward sub-percent accuracy. taking a logarithm renders it additive in {ln d, (d|q|)γ }. In Since m t is already tightly constrained by collider data, what follows we therefore work at the exponent level, i.e., a lattice determination of m b at this level would enable a directly with X i j = ln(m i /m j ), which the mechanisms direct, sharp test of our model. below naturally generate as sums. 2. Charged leptons The fixed-κ fit implies that the two observed ratios are essentially scale-independent; any 7.1 Universal constraints from the empirical pattern future improvement in the tau mass immediately tightens the allowed (ζ1 , ζ2 ) window. Any viable flavor model must explain: 17 To produce a definitive high-scale prediction, we evolved all six 1. The multiplicative structure for the exponent: X i j ≡ quark masses to μ∗ = 1014 GeV with RunDec v3.1 [9], using its ζ ζ five-loop QCD β-function and four-loop heavy-quark decoupling in a ln(m i /m j ) scales as di 1 d j 2 (equivalently, m i /m j ∝ (5) ζ ζ six-flavor scheme. We took αs (M Z ) = 0.1179±0.0009, included two- exp[di 1 d j 2 ] when q = 0). loop electroweak corrections, and applied standard threshold matching γ 2. The electric-charge dependence enters through κ (d|q|) at each heavy-quark mass. At μ∗ (in GeV), we find m u = 0.00049(8), m d = 0.00107(8), m s = 0.0213(17), m c = 0.248(7), m b = with γ ≈ 1.1, where q is the electric charge in units of 1.127(11), m t = 67.6(4). |e|. 123 Eur. Phys. J. C (2025) 85:1082 Page 17 of 22 1082 3. Cross-sector universality: similar O(1) κ, ζ1 , ζ2 values which matches the RS result in Eq. (15) up to notation. The for quarks and charged leptons q=0 neutrino limit is automatic since the charge-generation 4. The vanishing of the κ term for electrically neutral term vanishes. Our empirical template Eq. (1) encodes the fermions (q = 0) same combination via a simple non-linearity; we do not claim 5. Permutation specificity: among 120 possible quantum a UV-complete derivation here. number assignments, only the physical one yields acceptable fits. 7.2.2 Minimal example: partial compositeness These features suggest a deep connection between flavor In composite Higgs models with partial compositeness, structure and gauge quantum numbers that goes beyond con- fermion masses arise from linear mixings y L i q̄ L i O Ri and ventional model constructions. y Ri Ō L i ψ Ri with a strong sector, giving m i ∼ Y∗ y L i y Ri v where Y∗ is an O(1) strong-sector Yukawa. Parametrize the 7.2 Illustrative modifications for model classes mixings as   ζ /2 We present three minimal worked examples that reproduce y L i ∝ di 1 exp − ρ2 (di |qi |)γ (μ) , the same additive building blocks underlying Eq. (1): (i) a   (8) ζ /2 two-spurion Froggatt–Nielsen (FN) map (Sect. 7.2.1); (ii) y Ri ∝ di 2 exp − ρ2 (di |qi |)γ (μ) , a partial-compositeness (PC) parametrization (Sect. 7.2.2); and (iii) a warped-overlap construction in Randall–Sundrum encoding mild generation dependence and a small charge- (RS) geometry (Sect. 7.2.3). generation deformation. Then 7.2.1 Minimal example: two-spurion Froggatt–Nielsen mi   ln = ζ1 ln di +ζ2 ln d j +ρ (d j |q j |)γ (μ) − (di |qi |)γ (μ) , mj A horizontal U (1) F with two spurions reproduces the addi- (9) tive (exponent-level) dependence on {ln d, (d|q|)γ } without committing to a UV completion. Let matching the same additive structure as Eqs. (15) and (7); for θd  θq  q=0 the charge term vanishes. This is a minimal parametriza- d ≡ , q ≡ , 0 < d,q < 1, tion (no UV claim), included to illustrate that the required Λ Λ d–|q| dependence can also arise in PC-type settings. and collect O(1) coefficients into a structured prefactor ζ /2 ζ /2 Ci j ≡ di 1 d j 2 , reflecting the fitted generation depen- 7.2.3 Minimal example: warped extra dimensions dence. Assign effective exponents In Randall–Sundrum models [14–16], fermion masses arise Ni ≡ α1 ln di + α0 , Mi ≡ β (di |qi |)γ (μ) , from wavefunction overlaps. Standard implementations use constant bulk masses. Our pattern suggests a more complex with di ∈ {1, 2, 3} and |qi | ∈ {2/3, 1/3, 1, 0} (qi = 0 for structure. We show that a standard Randall–Sundrum (RS) neutrinos). Then diagonal Yukawas take the schematic form setup produces the two key ingredients of Eq. (1)—a multiplicative generation factor and a charge-weighted (d|q|)γ Yii = Cii dNi qMi , m i ∝ Yii . term—under mild, explicit assumptions. Taking ratios and logarithms gives Setup Consider RS1 with metric ds 2 = e−2ky ημν dx μ dx ν + dy 2 , y ∈ [0, π R], and bulk Dirac masses m 5,L i = c L i k, mi m 5,R j = c R j k for the ith left- and jth right-handed zero ln = ζ1 ln di + ζ2 ln d j mj modes. The canonically normalized zero-mode wavefunc-   (1/2−c R j )ky tions scale as f L i (y) ∝ e(1/2−c L i )ky , f R j (y) ∝ e . + (ln q−1 ) (d j |q j |)γ (μ) − (di |qi |)γ (μ) (6) The Yukawa entry is the overlap + (ln d−1 ) ΔNi j .  πR [1−(c L i +c R j )]kπ R Choosing the basis so that ΔNi j is absorbed into the gen- Yi j ∝ dy e−4ky f L i (y) f R j (y) ≃ N L i N R j e , 0 eration term (fixing α1,0 accordingly) and defining κ ≡ q−1 (10) yields the canonical form mi   where N L i and N R j are the normalization factors (explicit ln = ζ1 ln di +ζ2 ln d j + (ln κ) (d j |q j |)γ (μ) − (di |qi |)γ (μ) , (7) forms not needed below). Hence the diagonal masses scale mj 123 1082 Page 18 of 22 Eur. Phys. J. C (2025) 85:1082 as takes its usual RS value. (ii) Allowing a small left–right split c L i = c Ri only shifts ζ1,2 without affecting the (d|q|)γ strucm i ∝ N L i N Ri e [1−(c L i +c Ri )]kπ R . (11) ture. (iii) Because q → 0 kills the second term, the q = 0 constraint from the neutrino sector is automatic. A more elabAssumption 1 (generation prefactor) Empirically the fit orate UV origin for the deformation can be provided (e.g., ζ ζ prefers multiplicative factors di 1 and d j 2 in Eq. (1). In the brane kinetic terms or gauged horizontal symmetries), but is RS overlap, such factors arise from the smooth c-dependence not required for this minimal example. We do not claim a UVof the zero-mode normalizations when the bulk masses vary complete derivation here; the FN and RS constructions are mildly with a generation label d (mass ordering); cf. Eq. (11). explicit, economical frameworks that reproduce the observed We parametrize this by multiplicative d–|q| structure with O(1) parameters. Buildζ /2 ζ /2 ing a fully UV-complete realization is beyond the scope of N L i N R j ∝ di 1 d j2 , (12) this empirical paper. We note that both FN-type U (1) F settings and RS geomewith signs ζ1 > 0, ζ2 < 0 matching the quark and charged- tries admit UV-complete embeddings in explicit compactifilepton fits (Sects. 3, 4). cations; our goal here is only to show that the required mulAssumption 2 (charge-generation coupling) To capture the tiplicative d − |q| structure arises from minimal, standard observed (d|q|)γ dependence (and the q → 0 neutrino limit), ingredients. take the sum of bulk masses to acquire a small, universal deformation that couples electric charge to generation: 7.2.4 Discrete flavor symmetries c L i + c Ri = c0 + λ(di |qi |)γ , (13) Models based on A4 [10,17], S4 [18], etc., typically yield fixed Clebsch–Gordan coefficients. Reproducing our pattern with λ dimensionless and γ the same exponent that appears requires: empirically. This preserves standard RS localization while linking flavor and gauge structure in the minimal way sug- – Multiple flavon fields with hierarchical VEVs: gested by the data. φ1 /Λ1 φ2 /Λ2 1. Result Using (11), the ratio of diagonal masses is – Effective operators at different orders: (L i φ1n 1 φ2n 2 E j H )/Λn 1 +n 2 . mi ζ1 ζ2   – The powers n 1 , n 2 must be generation- and chargeln = ln di + ln d j + kπ R (c L j + c R j ) − (c L i + c Ri ) dependent to yield the observed structure. mj 2 2   ζ ζ – Note: The generation-dependent power-law di 1 d j 2 = ζ1 ln di + ζ2 ln d j + (2λkπ R) (d j |q j |)γ − (di |qi |)γ ,       requires going beyond minimal discrete implementageneration factor charge-generation term tions, possibly through sequential symmetry breaking or (14) additional continuous symmetries. after absorbing the factor 1/2 into ζ1,2 (pure convention). These modifications enable discrete symmetry models to Defining ρ ≡ 2λkπ R and recalling |q| = 0 for neutrinos, approximate the ansatz’s generation- and charge-dependent Eq. (14) implies structure, though achieving the exact form may require addimi   tional theoretical mechanisms. ln = ζ1 ln di +ζ2 ln d j +ρ (d j |q j |)γ − (di |qi |)γ , (15) mj 7.3 The neutrino constraint: implications of q = 0 i.e., the same additive dependence on {ln d, (d|q|)γ } that underlies our empirical pattern. Our template Eq. (1) uses A particularly stringent constraint comes from the neutrino a non-linear (exponentiated) encoding of this linear com- sector where q = 0 causes the κ term to vanish entirely. This bination at the level of ln(m i /m j ); empirically this captures implies: the data most parsimoniously (Sect. 3.7). The q = 0 neutrino limit is immediate since the charge-generation term vanishes. – Any UV completion must explain why electric charge We therefore view Eq. (15) as a minimal RS mechanism that enters through (d|q|)γ rather than q γ alone. yields the required d–|q| dependence, without claiming iden- – Models attempting to unify quark and lepton sectors must tity to Eq. (1). naturally produce this vanishing.  ζ1 ζ2 Comments (i) Nothing in Eq. (15) is tuned: a mild defor- – The residual pattern mmi+1 i = exp di+1 di for neutrinos mation of the bulk masses yields ρ = O(1), while kπ R provides an independent test. 123 Eur. Phys. J. C (2025) 85:1082 Page 19 of 22 1082 – See Eq. (3) for the explicit q = 0 limit. 8.2 Relation to other empirical mass relations – Koide-type relations. Koide’s [1] celebrated lepton forThis feature severely constrains grand unified theories where mula provides a striking mass relation that remains very quarks and leptons sit in common multiplets—the mecha- stable under renormalization-group evolution, but cannot γ nism generating the κ (d|q|) term must respect the q → 0 be extended to quarks where RG running is significant, limit. nor to neutrinos without sector-specific modifications. More fundamentally, such relations work beautifully in the charged-lepton sector but cannot be applied across all 7.4 Common themes and future directions three fermion sectors with a single functional form. All successful modifications share key features: – Texture and GUT relations (e.g., Georgi–Jarlskog [2], Froggatt–Nielsen [3]). These link selected Yukawa eigen1. Flavor and gauge structures must be intrinsically linked, values at the unification scale, typically m d = m e or not independent, as required by the (d|q|)γ term and the m b = m τ , but fail when run to low energy or across all neutrino sector’s q = 0 constraint. generations simultaneously. By contrast, Eq. (1) fits suc2. The generation index must enter through approximate cessive ratios at every scale and works unchanged in all continuous symmetries or geometric factors. three sectors. 3. The electric charge dependence suggests electroweak – Democratic/seesaw models. Approaches based on ransymmetry breaking may play a role in flavor. dom O(1) Yukawa entries reproduce hierarchies statistically, not algebraically, and make no precise ratio predictions. The purely multiplicative structure of Eq. (1) sugThese modifications are illustrative rather than exhaustive. gests a broken horizontal U (1) or discrete flavor symmeThe empirical pattern provides a clear target [10,19] for try acting on the ordered pair (d, q) [8]. Whether realized model builders: any complete theory of flavor should explain through Froggatt–Nielsen charges or wavefunction localnot just the existence of hierarchies, but their specific func- ization in warped space (see Sect. 7), any viable theory tional form across all fermion sectors. should now reproduce both the running exponent γ (μ) and the O(1) coefficients inferred here. 8 Discussion and outlook 8.3 Near-term tests 8.1 Hierarchy of empirical support – High-precision lattice QCD can confront the (t/b) prediction at the ∼ 5% level, provided bottom-quark mass 1. Quarks: over-constrained success Thirty independent uncertainties improve from the current ∼ 0.5%—as ratios across six scales confront four parameters per scale reported in recent N f = 2 + 1 + 1 heavy-quark sim- (or five/six in the global fit). The resulting χ 2 /dof 1 ulations [6,20]—toward sub-percent accuracy. and the modest,  6%, deviations from RunDec v3.1 – Improved m τ measurements at Belle II will shrink the evolution (Sect. 6) make the quark sector the most com- allowed charged-lepton parameter space and test the pelling evidence for Eq. (1). scale-independence implied by a constant γ . 2. Charged leptons: under-constrained but natural Effec- – Cosmology or 0νββ that pins any neutrino anchor observtively two ratios versus four parameters render the fit able will fix (ζ1 , ζ2 ); the ansatz then predicts the remaintautological, yet the extracted (κ, ζ1 , ζ2 , γ ) are all O(1) ing observables and becomes falsifiable in the neutrino and the ratios stay flat across six decades in energy. Any sector. future detection of significant running not captured by γ (μ) would immediately falsify the ansatz. 3. Neutrinos: consistency only (for now) With q = 0, the 9 Conclusions formula collapses to Eq. (3); two exponents are fixed by the two measured mass splittings. The test is merely that The pattern revealed in this work is empirically robust. A some (ζ1 , ζ2 ) of natural size exist for the allowed anchor single functional form, constructed from the only two labels window. Once an external observable pins m 1 and the that distinguish Standard Model fermions, fits all measured effective generation indices are established, the sector will fermion mass ratios with stable parameters. become predictive and directly comparable to quarks and For the quark sector, whether fitted scale-by-scale (24 or leptons. 18 parameters), globally (five or six parameters), with or 123 1082 Page 20 of 22 Eur. Phys. J. C (2025) 85:1082 without experimental uncertainties, the same four O(1) num- do generation index and electric charge enter multiplicabers emerge: κ ∼ 2.3, ζ1 ∼ 1.16, ζ2 ∼ −0.8, γ ∼ 1.1. tively? What dynamics produce the specific values κ ∼ 2.3, This stability is remarkable: parameter variations for ζ1 ∼ 1.16, ζ2 ∼ −0.8? Why does the exponent γ run logaquarks remain below 5% across eight distinct fitting strate- rithmically for quarks but remain constant for leptons? And gies spanning ten orders of magnitude in energy scale. The most fundamentally, what is the origin of the (d|q|)γ strucpattern extends beyond quarks—charged leptons and neutri- ture that respects the q → 0 limit for neutrinos? The thenos respect the identical functional form with equally natural oretical modifications outlined in Sect. 7 provide necessary parameter values. Thus, the quark sector provides strong evi- conditions, but a fully satisfactory theory must derive these dence, the charged-lepton sector supplies a non-trivial con- features from first principles. sistency check, and the neutrino sector offers a clear path We therefore propose this empirical relation as an importo future falsification. This cross-sector universality, com- tant constraint that realistic flavor models may wish to conbined with the minimal input requirements (generation index sider. Any viable theory of fermion masses should not only and electric charge only), suggests a fundamental organizing explain individual sector patterns, but also account for why principle in flavor physics. the same four natural numbers appear to govern all fermion The specificity of this empirical pattern is further demon- mass ratios across the entire Standard Model spectrum. strated by comprehensive permutation analysis: among all Whether this pattern is a clue to flavor dynamics or a 120 possible assignments of mass ratios to quantum number numerical accident will be decided by the next generation pairs, only the physical assignment yields acceptable fits (see of precision mass measurements. Fig. 9). Alternative mappings produce residuals from 3 to 100 Data Availability Statement This manuscript has no associated data. times larger, indicating that Eq. (1) encodes specific physical [Author’s comment: Data sharing not applicable to this article as no structure rather than mathematical flexibility. This combina- new datasets were generated or analyzed; all inputs are taken from pubtion of parameter stability and assignment specificity pro- lished sources cited in the manuscript. Derived numbers that underlie vides compelling evidence for genuine organizing principles the figures are reproducible from those sources with the code described below.] in the fermion mass spectrum. A complementary change-ofbasis scan over all 720 chained (Hamiltonian-path) indepen- Code Availability Statement Code/software will be made available on dent five-ratio bases (Sect. 3.6) shows that the adjacent basis reasonable request. [Author’s comment: The code/software generated is the only one with negative AIC/BIC, and that more alterna- during and/or analysed during the current study is available from the corresponding author on reasonable request.] tive bases fail to fit than succeed (376 vs. 343), underscoring both specificity and well-posedness. Open Access This article is licensed under a Creative Commons AttriPrevious empirical relations have generally not achieved bution 4.0 International License, which permits use, sharing, adaptation, both a comprehensive scope and parameter stability. Koide- distribution and reproduction in any medium or format, as long as you type relations work beautifully within single sectors but can- give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes not be extended. Texture and GUT models require sector- were made. The images or other third party material in this article specific modifications and many more parameters. Our four- are included in the article’s Creative Commons licence, unless indiparameter expression simultaneously describes all three cated otherwise in a credit line to the material. If material is not fermion sectors with residuals below 4%. included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitThe theoretical implications outlined in Sect. 7 demon- ted use, you will need to obtain permission directly from the copystrate that our empirical pattern places non-trivial constraints right holder. To view a copy of this licence, visit http://creativecomm on model building. The required modifications—linking fla- ons.org/licenses/by/4.0/. vor structure to electric charge—point toward a more uni- Funded by SCOAP3 . fied understanding of these supposedly independent parameters. Whether this reflects a fundamental symmetry principle or emerges from dynamics remains an open question A Expected χ 2 for correlated uncertainties for future investigation. For completeness, Sect. 7.2 provides three minimal worked examples that reproduce the multi- When experimental errors are highly correlated, the expected plicative d − |q| structure. χ 2 /dof for a correct model can be significantly less than 1. Our analysis provides empirical targets rather than theo- This arises because correlated uncertainties reduce the effecretical explanations. The remarkable fit quality and cross- tive number of independent measurements, leading to a supsector universality suggest these patterns reflect genuine pressed χ 2 value. Mass running studies often exhibit such organizing principles, but identifying the underlying physics correlations since systematic uncertainties in renormalizaremains an open challenge. tion scale choices affect all energy scales similarly. A complete theoretical explanation of this empirical pat- For N observables with a single fully correlated scale error tern would need to address several key questions: Why σ , the covariance matrix is Ci j = σ 2 . Its Moore–Penrose 123 Eur. Phys. J. C (2025) 85:1082 Page 21 of 22 1082 inverse is Ci+j = 1/(N σ 2 ), so for residuals δi we have ln(m t /m b ))T ( i δi ) 2 denote the adjacent-ratio log vector. Any other independent χ 2 = δT C +δ = ∼ χ12 , five-tuple of pairwise ratio logs can be written as r = A r N σ2 with an integer unimodular matrix A ∈ GL(5, Z) whose whose expectation value is χ 2  = 1, i.e., χ 2 /dof = rows each represent a difference of two log masses (a sin1/(N − k) 1 for N  10 and k ≤ 5. This general gle pairwise ratio).18 We enumerate all 720 ordered chained effect explains why excellent fits to correlated data can yield (Hamiltonian-path) integer bases of this type used in Fig. 2; χ 2 /dof 1 without indicating overfitting. the adjacent basis is the identity A = . Conditioning and dynamic range. Although every spanning tree yields a valid Z-basis, their numerical behaviour difB Adjacent-ratio basis and change-of-basis invariance fers when only diagonal uncertainties are used. Chained (adjacent) trees keep the five basis ratios closer to unity Let (m 1 , . . . , m 6 ) ≡ (m u , m d , m s , m c , m b , m t ) at a fixed in magnitude (smaller | ln rk |), whereas non-chained trees— scale and define the five adjacent log-ratios rk ≡ ln(m k+1 / especially those linking the lightest to the heaviest masses— m k ), k = 1, . . . , 5. For any i > j, produce basis ratios with widely varying | ln rk |. In that set-    i−1 ting the effective weighting becomes strongly anisotropic and mi ln = rk , the normal equations are more ill-conditioned. Empirically mj this manifests as higher failure rates and worse AIC/BIC k= j for non-chained trees. Thus chained bases, though not well so the mapping from adjacent ratios to arbitrary pairwise conditioned, are less badly pathological and comparatively ratios is linear with integer coefficients. Any other inde- robust in our scans. pendent choice of five ratios corresponds to r = A r with A ∈ GL(5, Z). If the full covariance C of r is propagated to Remark (when do five ratios form a basis?) Let rk = r as C  = AC A T , then the weighted least-squares objective ln(m ak /m bk ) and let G be the undirected graph on {1, . . . , 6} with edges {ak , bk }. With five edges on six vertices, the rows  T   χ 2 (θ ) = r − f(θ ) C −1 r − f(θ ) {eak − ebk } span the five-dimensional log-ratio space iff G is connected, i.e., a spanning tree. In that case, for any u, v the is unchanged under the simultaneous transformation (r, f, C) unique path P(u → v) in G yields → (r , Af, C  ), and the normal equations for θ are iden-  mu ma tical up to floating-point rounding. Hence, using adjacent- ln = σab ln , σab ∈ {±1}, generation ratios is a basis choice that does not bias parameter mv mb (a,b)∈P(u→v) estimation. so all 30 pairwise ratios are reconstructible. Our scan conPractical note The invariance proven above requires carsiders the 6! = 720 Hamiltonian-path (“chained”) spanningrying through the full 5 × 5 covariance for each basis. tree bases; other spanning trees are related by unimodular Public inputs for quark running provide diagonal (or effectransformations in GL(5, Z). tively diagonal) errors for the five ratios at each scale; with that pragmatic weighting, the log-likelihood is not strictly Dynamic range (conditioning proxy) For each basis we invariant under A ∈ GL(5, Z), and numerical condition- compute a proxy for numerical conditioning, ing becomes basis dependent. This is precisely what the information-criterion scan in Sect. 3.6 quantifies: the adja- maxi, j |Ai j | cent basis is uniquely well posed (negative AIC/BIC), many D(A) ≡ . (16) mini, j: Ai j =0 |Ai j | alternative bases converge but are decisively worse (large positive AIC/BIC), and a majority fail to admit a finite- This scales like a condition number for the linear map r → likelihood optimum under identical settings. r and correlates with estimator instability. Any monotone proxy for conditioning gives the same qualitative picture; the adjacent basis minimizes D by construction.19 C Basis enumeration, dynamic range and AIC/BIC 18 Appendix B gives the change-of-basis invariance when the full What we call a “basis” At a fixed scale, let covariance is propagated. 19 Equivalently, one may multiply by cond (A) to penalize near2 r = (ln(m d /m u ), ln(m s /m d ), ln(m c /m s ), ln(m b /m c ), singular maps; both choices pick out the adjacent basis in practice. 123 1082 Page 22 of 22 Eur. Phys. J. C (2025) 85:1082 Likelihood and criteria For each basis we refit the same five- 8. F. Feruglio, A. Romanino, Rev. Mod. Phys. 93, 015007 (2021). parameter global model to the 30 data points expressed in that https://doi.org/10.1103/RevModPhys.93.015007 9. F. Herren, M. Steinhauser, Comput. Phys. Commun. 224, 333 basis and evaluate the Gaussian least-squares likelihood (2018). https://doi.org/10.1016/j.cpc.2017.11.014 1   10. G. Altarelli, F. Feruglio, Nucl. Phys. B 741(1–2), 215–235 (2006). L(θ ) = exp − 21 Δ(θ ) C −1 Δ(θ ) , https://doi.org/10.1016/j.nuclphysb.2006.02.015 (2π ) N /2 |C|1/2 (17) 11. T. Ohlsson, S. Zhou, Nat. Commun. 5, 5153 (2014). https://doi. org/10.1038/ncomms6153 Δ(θ ) ≡ r − f (θ ), 12. Project 8 Collaboration, A.A. Esfahani, S. Böser, N. Buzinsky, M.C. Carmona-Benitez, C. Claessens, L. de Viveiros, P.J. Doe, S. where r collects the five ratio logs at each of the six reference Enomoto, M. Fertl, J.A. Formaggio, J.K. Gaison, M. Grando, K.M. scales (see Appendix B), f (θ ) are the model predictions, and Heeger, X. Huyan, A.M. Jones, K. Kazkaz, M. Li, A. Lindman, C. C is the (block-)diagonal covariance built from the quoted Matthé, R. Mohiuddin, B. Monreal, R. Mueller, J.A. Nikkel, E. Novitski, N.S. Oblath, J.I. Peña, W. Pettus, R. Reimann, R.G.H. per-ratio uncertainties at those scales. The information crite- Robertson, G. Rybka, L. Saldaña, M. Schram, P.L. Slocum, J. ria are then Stachurska, Y.H. Sun, P.T. Surukuchi, J.R. Tedeschi, A.B. Telles, F. Thomas, M. Thomas, L.A. Thorne, T. Thümmler, W.V.D. AIC = 2k − 2 ln L̂, BIC = k ln N − 2 ln L̂, (18) Pontseele, B.A. VanDevender, T.E. Weiss, T. Wendler, A. Ziegler, The project 8 neutrino mass experiment (2022). arXiv:2203.07349 13. M. Borghesi, B. Alpert, M. Balata, D. Becker, D. Bennet, E. with k = 5 and N = 30 fixed across the scan and L̂ = L(θ̂ ) Celasco, N. Cerboni, M. De Gerone, R. Dressler, M. Faverzani, the maximized Gaussian likelihood evaluated at the best-fit M. Fedkevych, E. Ferri, J. Fowler, G. Gallucci, J. Gard, F. Gatti, parameters for that basis. Because k and N are constant, A. Giachero, G. Hilton, U. Koster, D. Labranca, M. Lusignoli, J. Mates, E. Maugeri, S. Nisi, A. Nucciotti, L. Origo, G. Pessina, S. BIC differs from AIC by an additive constant and yields the Ragazzi, C. Reintsema, D. Schmidt, D. Schumann, D. Swetz, J. same ranking; hence only AIC is shown in Fig. 2. Bases for Ullom, L. Vale, Nucl. Instrum. Methods Phys. Res. Sect. A Accel. which the optimizer fails to converge or return well-posed Spectrom. Detect. Assoc. Equip. 1051 (2023). https://doi.org/10. parameters correspond to effectively infinite AIC/BIC; we 1016/j.nima.2023.168205 14. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999). https:// report these as “no fit” in the main text (376 of 719 non- doi.org/10.1103/PhysRevLett.83.3370 adjacent bases). 15. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999). https:// doi.org/10.1103/PhysRevLett.83.4690 16. T. Gherghetta, A. Pomarol, Nucl. Phys. B 586, 141 (2000). https:// doi.org/10.1016/S0550-3213(00)00392-8 References 17. E. Ma, Phys. Rev. D 70, 031901 (2004). https://doi.org/10.1103/ PhysRevD.70.031901 1. Y. Koide, Phys. Lett. B 120(1–3), 161 (1983). https://doi.org/10. 18. C.S. Lam, Phys. Rev. D 78, 073015 (2008). https://doi.org/10. 1016/0370-2693(83)90644-5 1103/PhysRevD.78.073015 2. H. Georgi, C. Jarlskog, Phys. Lett. B 86, 297 (1979). https://doi. 19. C.S. Lam, Phys. Rev. D 87, 013001 (2013). https://doi.org/10. org/10.1016/0370-2693(79)90842-6 1103/PhysRevD.87.013001 3. C. Froggatt, H. Nielsen, Nucl. Phys. B 147, 277 (1979). https://doi. 20. J.T. Tsang, Status and outlook of quark flavour physics (2025). org/10.1016/0550-3213(79)90316-X https://arxiv.org/abs/2503.05554 4. M. Sabir, A. Mansha, T. Li, Z.W. Wang, J. High Energy Phys. 2025(1) (2025). https://doi.org/10.1007/jhep01(2025)201 5. G.Y. Huang, S. Zhou, Phys. Rev. D 103(1), 016010 (2021). https:// doi.org/10.1103/PhysRevD.103.016010 6. Y. Aoki, T. Blum, S. Collins, L.D. Debbio, M.D. Morte, P. Dimopoulos, X. Feng, M. Golterman, S. Gottlieb, R. Gupta, G. Herdoiza, P. Hernandez, A. Jüttner, T. Kaneko, E. Lunghi, S. Meinel, C. Monahan, A. Nicholson, T. Onogi, P. Petreczky, A. Portelli, A. Ramos, S.R. Sharpe, J.N. Simone, S. Sint, R. Sommer, N. Tantalo, R.V. de Water, A. Vaquero, U. Wenger, H. Wittig (2025). https:// arxiv.org/abs/2411.04268 7. I. Esteban, M.C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J.P. Pinheiro, T. Schwetz, J. High Energy Phys. 2024(12) (2024). https://doi.org/10.1007/jhep12(2024)216 123