Square-Root Mass Mixing and a Split-and-Recombined Tribimaximal Ansatz for the PMNS Matrix

We propose a deterministic phenomenological ansatz for the Pontecorvo--Maki--Nakagawa--Sakata (PMNS) matrix that combines a Koide--Brannen square-root mass-space 3–1 rotation \(U_{\nu31}\) with a phase-dressed, split-and-recombined tribimaximal product. The construction fixes the Koide--Brannen mass ratios, a third-generation phase assignment represented through a rephased basis, the sign convention \(\theta_{eij}=-\theta_{\nu ij}\), and the noncommuting ordering 3–1 \(\to\) 1–2 \(\to\) 2–3. The main candidate is selected by a uniform matrix-element splitting rule: in the TBM-like 1–2 and 2–3 sectors one halves the off-diagonal rotation-matrix element, rather than the geometric angle. The geometric half-angle branches are retained as controls, giving four candidate matrices in total and no continuously fitted mixing parameter. Candidate~1, the uniformly A-type candidate, gives\(\sin^{2}\theta^{\prime}_{12}\simeq0.312011\),\(\sin^{2}\theta^{\prime}_{13}\simeq0.020690\),\(\sin^{2}\theta^{\prime}_{23}\simeq0.533876\), and\(\delta^{\prime}_{\mathrm{CP}}=\pi\).Relative to the phase-dressed tribimaximal baseline it improves the solar and reactor angles simultaneously and yields an upper-octant atmospheric prediction; the reactor angle remains the principal residual tension. We also give the charged-lepton-basis Majorana mass matrix, a CP-real representative, and the residual Klein generators of Candidate~1, so that the construction is reproducible as an effective-field-theory input. The ansatz is phenomenological, not a derived flavor theory; its sharp falsifiable features are CP conservation \((\delta^{\prime}_{\mathrm{CP}}=\pi)\), an upper-octant \(\sin^{2}\theta^{\prime}_{23}\simeq0.534\), and a low reactor-angle prediction.