Published February 5, 2026 | Version v1.2
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Flavor from a √2‑Discrete Vacuum: 13‑Selective CKM Suppression and a Geometric Seesaw for PMNS(NO)

Authors/Creators

Description

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}
 

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Additional details

Related works

Is described by
Preprint: 10.5281/zenodo.17774135 (DOI)
Preprint: 10.5281/zenodo.17774268 (DOI)
Preprint: 10.5281/zenodo.17792715 (DOI)

Software

Repository URL
https://github.com/Hiroto-Iwasaki/Particle-physics-paper-datasets/tree/main/F2DV
Programming language
Python
Development Status
Active

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