A Complete Derivation of the Fermion Spectrum from the Recognition Composition Law

We present a first-principles derivation of the masses of all twelve known fermions – three charged leptons, six quarks, and three neutrinos – and the fine-structure constant ${\alpha }^{-1}$, from a single discrete functional equation, the Recognition Composition Law (RCL), with $\mathbf{\text{zero}}\mathbf{\text{continuously}}\mathbf{\text{adjustable}}\mathbf{\text{parameters}}$. The mass spectrum follows from the RCL supplemented by four regularity conditions and eight structural theorems (T1–T8): the golden ratio $\phi =(1+\sqrt{5})\mathrm{/}2$ emerges as the unique hierarchy base (T6); an 8-step period is fixed by the 3-cube Hamiltonian cycle (T7); three spatial dimensions are selected by a unique combinatorial identity (T8). All integers entering the mass formula are the six combinatorial invariants of the 3-cube $Q_3$; none is fitted. The sole empirical input is the electron mass, which fixes an irreducible unit-conversion constant~${\tau }_0$. Predictions are confronted with PDG measurements. Charged-lepton masses are reproduced at sub-ppm accuracy for the muon and $\sim \text{\hspace{0.17em}⁣}{10}^{-4}$ for the tau (Table~\ref{tab:lepton_validation}). All six quark masses are predicted at integer level; first-generation quarks agree to better than $1\mathrm{\%}$, while second/third-generation residuals of $2$–$16\mathrm{\%}$ are expected integer-precision effects (Table~\ref{tab:quark_validation}). Neutrino mass-squared splittings agree with NuFIT~5.3 within $1$–$2\sigma$, normal ordering is predicted, and $\mathrm{\Sigma }{m}_{\nu }\approx 0.063$~eV satisfies cosmological bounds. All structural claims are machine-verified in Lean~4 (179 files, 0~$\mathtt{\text{sorry}}$; $\mathtt{\text{github.com/}}\mathtt{\text{allowbreak}}\mathtt{\text{jonwashburn/}}\mathtt{\text{allowbreak}}\mathtt{\text{recognition-science}}$).