A Geometric Mass Formula for Charged Fermions from Heptagon Spectral Data

The note presents a compact geometric mass formula for the charged fermions based on regular-heptagon spectral data and a fixed cubic three-root spectrum. The three generations are identified with the roots of the polynomial alpha^3 + alpha^2 - 2 alpha - 1 = 0. Sector-dependent coefficients are generated from a common discrete label assignment for the fermion sectors. The charged-sector reference masses are not fitted independently, but are obtained from the Higgs vacuum expectation value v = 246.22 GeV through an exponential damping law M_ref^(f) = (v/sqrt(2)) exp(-Delta^(f)), with Delta^(f) = k0 - phi r_f - Q_f^2/sqrt(2) and k0 = 6 + 7u - 2u^2, where u = 2 sin(pi/7). For the nine charged fermions, the resulting predictions reproduce the observed hierarchy with a global root-mean-square deviation of 5.7%. This deposition contains the manuscript and a reference implementation used to generate the reported numerical results.