Koide-Type Fermion Mass Relations with Charge-Assigned Dyadic Exponents

This work proposes a phenomenological framework for the three-family spectra of light neutrinos, charged leptons, down-type quarks, and up-type quarks. For a positive mass triple m, define the escort weights p_i(q) = m_i^q / Σ_j m_j^q and the quadratic moment R_q = Σ_i p_i(q)^2. Two postulates assign from the charges q_ν = 1, q_e = 1/2, q_d = −1/2, and q_u = −1/4, and impose R_{q_f} = 2/3 in all four sectors. The quadratic-cone geometry, its complete parametrization, the cubic invariant, the third-mass formula, and a hierarchy bound are derived. With B − L = 1/N_c for quarks, the anomaly equations produce the ratios 1, 2, N_c − 1, and N_c + 1; requiring a dyadic scale selects N_c = 3. Within the stated mass prescription, the three charged spectra are compatible with the constraint. For neutrinos, the subsystem defined by the two postulates admits one normal-ordering solution when both measured mass-squared differences are assigned; inverted ordering is excluded. An empirical extension introduces a complete logarithmic shape coordinate and two relations among the four sectors, followed by a quadratic constraint on three positive variables associated with charge-difference vectors. The closure equation has four real roots; the branch λ₁ = −1.3227907… < −1 is selected empirically by the charged-sector shapes. Assuming the additional relations to be exact, five input observables m_e, m_μ, Δm²₃₁, m_b, and m_t determine the twelve masses. The six measured closure outputs agree with their comparison values within 1.5σ. Once the branch is fixed, the neutrino sector depends only on m_e, m_μ, and Δm²₃₁ and gives m₁ = 3.227 ± 0.013 meV, Σ_i m_i = 62.65 ± 0.26 meV, and Δm²₂₁ = (7.439 ± 0.062) × 10⁻⁵ eV². The additional relations were formulated from the same spectra, so their numerical agreement is an internal check. Replacing the charm pole-mass estimate by m_c(m_c) excludes the first shape relation at about 10σ, demonstrating the dependence of the closure on the mass prescription.