Structural Origin of Quark Masses Hierarchy

This work presents a zero-parameter structural formulation of the quark-mass hierarchy.Using two fixed rational ratios,\[R=\frac{13}{6}, \qquad S=\frac{31}{24},\]the six quark masses are represented by the unified relation\[\frac{m_q}{m_e}=K_q(3R^2S)\left(\frac{8\pi^{u_q}R^{u_q+1}}{S}\right)^n .\]The resulting structural mass values are\[m_u \approx 2.145\,\mathrm{MeV},\quadm_d \approx 4.648\,\mathrm{MeV},\quadm_c \approx 1273.6\,\mathrm{MeV},\]\[m_s \approx 93.56\,\mathrm{MeV},\quadm_t \approx 173.0\,\mathrm{GeV},\quadm_b \approx 4.185\,\mathrm{GeV}.\]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