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 alpha³ + alpha² - 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 Mᵣef^ (f) = (v/sqrt (2) ) exp (-Delta^ (f) ), with Delta^ (f) = k0 - phi rf - Qf²/sqrt (2) and k0 = 6 + 7u - 2u², 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 final manuscript and a reference implementation used to generate the reported numerical results.
Andreas K. E. Mueller (2026) studied this question.