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. 0310^-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. 7910^-2 to 1. 0910^-4. Mapping the residual to a dimension-six effective operator yields a new-physics scale 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_ 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_=0. 12067~eV, m_ 0. 03171~eV, and, from a phase scan, m_0. 01007, 0. 03144~eV (68\% CI: (0. 01346, 0. 02906) ~eV). Thus the framework ties masses, mixing, absolute neutrino masses, and 0 to a single generation rule with minimal corrections, yielding falsifiable predictions in terms of, m_, m_, and m_. Since the cosmological upper bound on m_ 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, DESI2024BAO, PDG2025SumMnu