Exact parameter-free sum rules for the Pontecorvo–Maki–Nakagawa–Sakata matrix from a split-and-recombined tribimaximal construction

Abstract

We present an exact, fixed-input sum-rule construction for the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix: a deterministic ansatz combining a Koide–Brannen square-root mass-ratio input with a discrete split-and-recombined deformation of tribimaximal mixing (TBM), with every mixing parameter in closed form. For each fixed discrete branch, no continuous mixing parameter is fitted. The extracted angles obey two exact, parameter-free sum rules. The first-row relation \(\sin^{2}\theta_{12}\cos^{2}\theta_{13}=\tfrac{11}{36}\) is a discrete deformation of the trimaximal value \(\tfrac{1}{3}\), consistent with the first JUNO solar-angle measurement at \(\approx0.4\sigma\) and sharply testable at the full-exposure sub-percent JUNO precision; a second exact relation ties the atmospheric to the reactor angle. The construction is TBM-like yet its reactor angle is nonzero, with an explicit origin: the square-root input fixes the 3–1 seed through \(\tan(\tfrac{1}{2}\vartheta_{31})=\sqrt{m_{1}} / \sqrt{m_{3}}\), lifted by a discrete prescription. Candidate~1 is selected phenomenologically from four predefined discrete branches; this branch improves the solar-angle agreement relative to the TBM-limit baseline, predicts an upper-octant atmospheric angle, conserves CP by construction, and gives a reactor angle below the global-fit central value. For Majorana neutrinos, under one explicit Majorana-sector postulate — the constructed matrix is taken to be the full PMNS matrix, with no additional Majorana phase matrix inserted — that postulate together with the P\(_{3}\) convention fixes the conditional CP-parity assignment \((\alpha_{21},\alpha_{31})=(0,\pi)\); conditionally retaining the mass input then yields the point prediction \(m_{\beta\beta}\simeq1.9\,\mathrm{meV}\). A companion no-go theorem proves that the exact second column cannot arise, at any finite group order, from an exact residual flavor symmetry with a generation-distinguishing charged-lepton residual within the column-isolating residual classes analyzed there — for Majorana residual chains with three non-zero, non-degenerate masses, and for Dirac column-only isolating residuals — nor from the root-of-unity-\(\rho\) column-only antiunitary (generalized-CP) family analyzed there, including the reflection-type case, delimiting its possible ultraviolet origins. Reproduction scripts are provided.