Mass hierarchies and flavor mixing in the Standard Model (SM) depend on many free parameters, and their origin remains unresolved. We propose a hybrid framework in which a discrete generation rule with minimal step $\sqrt{2}$ is combined with micro-corrections (discrete labels), enabling a chained reconstruction of charged-lepton and quark mass hierarchies with a small parameter set (relative RMS error $\simeq 4.03\times10^{-3}$). The same mixing rule then provides a unified description of the CKM/PMNS mixing-angle hierarchies, and a data-driven identification shows that, in the quark sector, the necessary correction is localized in the $13$ component. In other words, the other components are already consistent with the rule alone, and only the $13$ entry demands an extra degree of freedom, so absorbing the $13$ residual as the minimal correction is sufficient. Indeed, using only the $13$ residual improves CKM consistency from $\mathrm{RMS}(|V|)=1.79\times10^{-2}$ to $1.09\times10^{-4}$. Mapping the residual to a dimension-six effective operator yields a new-physics scale $\Lambda\simeq 14.8~\mathrm{TeV}$ (95\% CI: $[14.796,14.817]~\mathrm{TeV}$). On the lepton side, the analogous distortion parameter is evaluated via Gaussian Monte Carlo, providing its distribution and confidence intervals, while the correlation with $\sum m_\nu$ is found to be extremely small, $|{\rm corr}|\sim10^{-3}$. In addition, embedding the geometric input into a Type-I seesaw and adopting the normal ordering (NO) as the main branch gives $\sum m_\nu=0.12067~\mathrm{eV}$, $m_\beta\simeq 0.03171~\mathrm{eV}$, and, from a phase scan, $m_{\beta\beta}\in[0.01007,0.03144]~\mathrm{eV}$ (68\% CI: $(0.01346,0.02906)~\mathrm{eV}$). Thus the framework ties masses, mixing, absolute neutrino masses, and $0\nu\beta\beta$ to a single generation rule with minimal corrections, yielding falsifiable predictions in terms of $\Lambda$, $\sum m_\nu$, $m_\beta$, and $m_{\beta\beta}$. Since the cosmological upper bound on $\sum m_\nu$ depends on the data combination, our value is allowed by conservative CMB-centered bounds and sits near the boundary for tighter sets including BAO (e.g., Planck 2018 and DESI BAO).\citep{Planck2018,DESI2024_BAO,PDG2025_SumMnu}