The Fermion Spectrum over Finite Relational Substrate
Abstract
The Standard-Model gauge group, one fermion generation as the spinor 16, the mass mechanism \(m=yv\), and the generation count is derived over a finite relational arithmetic substrate. We show that the unresolved residue is organised by the substrate's two structural numbers, the four-fold (\(4\mid\Omega-1\), the quarter-turn) and the cubic (\(3\mid\Omega+1\), the triality centre), the pair that fixes \(\Omega\equiv5\ (\mathrm{mod}\ 12)\). The charged-lepton Koide relation is derived exactly, \(Q=2/3\): generation universality forces the Yukawa amplitude matrix to a \(C_{3}\)-circulant, the quarter-turn fixes the amplitude \(\sqrt2\), and the lightest generation at the quarter-turn boundary fixes the leading phase \(\pi/12\), leaving the electron massless at leading order. The three generative roles supply the Froggatt-Nielsen charges \((0,1,2)\), giving the \(\lambda\)-texture and the Cabibbo angle \(V_{us}=\sqrt{m_{d}/m_{s}}\); the Georgi-Jarlskog factor is the colour rank \(N_c=3\); the up-quark Koide value is \(Q_u=5/6\). The mixing split is the lopsided \(M_{e}=M_{d}^{\mathsf T}\). For the neutrinos colourlessness fixes the signed (Takagi) amplitude invariant \(Q_\nu=2/3\); with the drive-invariant quarter-turn boundary branch this selects normal ordering and \(\sum m_\nu\simeq59\,\)meV, and the cube-root phase makes leptonic CP near-maximal, \(\delta_{\mathrm{CP}}\simeq-130^\circ\). Beyond the overall mass scale, the sector reduces to one carrier-scale phase…
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