The Fermion Mass & Charge Spectrum from Unified Equations

The circulant parameterization √mk ∝ 1 + √2·kS·cos(δS + 2πk/3) organises each fermion family by one clock, one amplitude and one phase [1–5, 18]. A previously proposed quantum-number law fixes the phase from gauge charges alone: δS = εS·uS(1−uS) with 1/uS = Nc3/2+T₃·C₂(3)q, giving 2/9, 3/16, 8/81 and 11/144 — equivalently 1/u ∈ {3, 4, 9, 12} — for the charged-lepton, down, neutrino and up sectors [19]. This paper reports the scheme in which that law is testable, the result of the test, the law's inversion into a charge dictionary, its group-theoretic origin, and the clock structure beneath it. In the own-scale scheme — every mass at its own scale, in the scheme of the interactions it feels, the light quarks floored at the condensate boundary — the measured values are 1/u = 2.9999(2), 4.004(57), 8.868(47) and 11.911(40): the two odd-isospin sectors sit exactly on the law (0.4σ and 0.07σ), while both even-isospin sectors fall short of theirs by a common one per cent (up 2.2σ, neutrinos 2.8σ; a shared factor 0.9916 ± 0.0028 on the Nc, χ² = 0.3/1). That strain lands on the one step of the derivation that is provably open: the binary tetrahedral group generates the third-turn, the quarter-turn and the twelfth on the rotor, but contains no element of order 9 — the neutrino's rational is a composite carried across the seesaw, and the even-isospin factor of Nc is a threefold cover no group element supplies. The up phase discriminates at 3.8σ against the earlier proposal δU = 2/27 [4, 23] and lies within 1 % of the value pre-registered in [19]; its sharp falsifier is the inverse prediction mc(mc) = 1.2573 ± 0.0055 GeV, presently 2.2σ from the PDG determination. The scheme is a phenomenological prescription, not a derived result; what removes the freedom is that two internal discriminators select it from the data: the top-mass definition (0.6σ against 2.9σ) and the floor scan. Inverted, the law reads 1/u = 32+Q·(4/35/3)q and returns the electric charges from the measured masses to 10⁻⁴ in the lepton sector, one unit of charge being one factor of Nc because Nc is odd; on the Clifford occupation number of the C⊗O construction, where charge is a number operator [22], its colour flag and isospin become the triplet condition and the grading of the ideal. With the charged leptons preserving the Z₃ of 2T and the neutrino mass matrix a Z₂, the leading-order PMNS matrix is tri-bimaximal [33–36]; one rotation accommodates θ₁₃ with θ₁₂ then at 0.9σ [37]; and a polar-decomposition theorem shows the mixing and the neutrino phase residual are independent parameters of the Dirac texture. The five sector clocks close into a two-branch seesaw: on each branch, (direct clock) × (inverse clock) = mD² with mD the branch's own condensate decay constant — exactly (v/√2)² on the electroweak branch, and fπ² on the QCD branch at 0.5σ (a candidate that predicts md(2 GeV) = 4.75 MeV against the lattice 4.70 ± 0.05 and would reduce the spectrum's dimensionful content to two condensates, one unification scale and the top Yukawa). A common-scale consistency fit gives μ* = 3.3×10⁶ GeV at χ²/dof = 0.12/2, with ΛQCD the one irreducible anchor by transmutation. Live predictions: mτ = 1776.97 MeV (0.4σ from the 2024/25 world average [9, 12]); mc(mc) = 1.2573(55) GeV; normal neutrino ordering, required; Δm²₂₁ = 7.88×10⁻⁵ eV² if the heavy phase is exactly 9, disfavoured at 3.2σ and decidable by JUNO [13]; and mu/md = 0.4728 against the surviving Clebsch candidate's 0.4536 [15]. Eighteen verification scripts reproduce every number, and five retracted intermediate claims are preserved in a provenance appendix.