Flavor from a √2‑Discrete Vacuum: 13‑Selective CKM Suppression and a Geometric Seesaw for PMNS(NO)

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 √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 4.03×10⁻³). 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 RMS(|V|)=1.79×10⁻² to 1.09×10⁻⁴. Mapping the residual to a dimension-six effective operator yields a new-physics scale Λ 14.8~TeV (95% CI: [14.796,14.817]~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 ∑ m_ν is found to be extremely small, | corr|~10⁻³. In addition, embedding the geometric input into a Type-I seesaw and adopting the normal ordering (NO) as the main branch gives ∑ m_ν=0.12067~eV, m_β 0.03171~eV, and, from a phase scan, mββ∈[0.01007,0.03144]~eV (68% CI: (0.01346,0.02906)~eV). Thus the framework ties masses, mixing, absolute neutrino masses, and 0νββ to a single generation rule with minimal corrections, yielding falsifiable predictions in terms of Λ, ∑ m_ν, m_β, and mββ. Since the cosmological upper bound on ∑ m_ν 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).{Planck2018,DESI2024_BAO,PDG2025_SumMnu}