Fine structure of fermion mixing and quark-mass hierarchies

Abstract

We investigate a semi-empirical flavour framework based on two linked hypotheses: a golden/pentagonal organisation of fermion mixing and the recurrence of its hierarchy maps in the quark-mass spectrum. The two large lepton angles are specified by the fifth-order Chebyshev conditions $T_5(θ_{12}^{\ell})=0$ and $T_5(θ_{23}^{\ell})=-1$, with angular arguments in radian measure, while the smaller reactor angle follows the approximate bilinear relation $θ_{13}^{\ell}\simeqθ_{12}^{\ell}θ_{23}^{\ell}/π$. An alternative golden-partition construction yields nearly identical large-angle predictions and an exact circular–hyperbolic duality. The golden ratio $φ$ also enters two distinct geometrical prescriptions for the Cabibbo angle, while the heptadecagonal seed $α_C=π/17$ governs the suppressed mixing hierarchy. The near-equality $\tanα_C\simeq\cos54^\circ/π$ further connects the two geometries. The resulting quark hierarchy obeys the quadratic map $θ_{13}^q=\sqrt{θ_C^2+(θ_{23}^q)^2}-θ_C\simeq(θ_{23}^q)^2/(2θ_C)$. Transferring the quadratic and bilinear maps to quark masses yields the crossed relations $m_d\simeq m_c^2/(2m_t)$ and $m_u\simeq m_b m_s/(m_F/φ)$, where $m_F=G_F^{-1/2}$. The complementary triples $(t,c,d)$ and $(b,s,u)$ reproduce both mass targets within $1\%$ for the adopted reference values. Gatto–Sartori–Tonin-type correspondences and Froggatt–Nielsen scaling provide compatible phenomenological interpretations. These results suggest a common golden/geometrical organising structure for flavour mixing and mass hierarchies, whose underlying dynamical origin remains to be established.