Structural Origin of Quark Masses Hierarchy

This work presents a zero-parameter structural formulation of the quark-mass hierarchy.Using two fixed rational ratios,\[R=13/6, S=31/24,\]the six quark masses are represented by the unified relation\[m_q/m_e=K_q B_q A_q\,n,_q=3R^2S.\]The amplification factors are defined by\[A_d=8R/S,_u=8π R^2/S.\]The structural coefficients are written directly as\[(K_u,K_d,K_c,K_s,K_t,K_b)=(1/2R,1/2,1+1/2,1-1/4,2+1/2R,2+1/2).\]The generation index is\[n ={cases}0 & (u,d), \\1 & (c,s), \\2 & (t,b).{cases}\]The resulting structural mass values are\[m_u ≈ 2.145\,MeV,_d ≈ 4.648\,MeV,_c ≈ 1273.6\,MeV,\]\[m_s ≈ 93.56\,MeV,_t ≈ 173.0\,GeV,_b ≈ 4.185\,GeV.\]These values are compared directly with reference quark-mass values across all three generations. The work is intended as a compact numerical correspondence, not as a replacement for the Standard Model or its Yukawa mechanism.No individual quark-mass parameters are fitted, and no tuning is performed. A minimal Python implementation is provided for full reproducibility:https://github.com/yasuotanakaresearch/zero-parameter-structure