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

We propose a phenomenological ansatz for the Pontecorvo--Maki--Nakagawa--Sakata (PMNS) matrix in which a Koide--Brannen motivated square-root mass-space 3–1 rotation U\(_{\nu31}\) is combined with a phase-dressed, split-and-recombined tribimaximal structure. The construction uses fixed Koide--Brannen mass ratios, a third-generation phase assignment represented through a rephased basis, the sign convention \(\theta_{eij}\) = -\(\theta_{\nu ij}\), and a fixed non-commuting ordering 3–1 → 1–2 → 2–3. Two discrete prescriptions for each of the 1–2 and 2–3 splittings give four candidate matrices; no continuous mixing parameter is fitted. Here and below, primes denote the standard PDG-convention mixing parameters extracted from the constructed PMNS matrix, not the internal ordered-ansatz rotation angles. Among these four predefined candidates, Candidate~1 is the least-strained member under the diagnostic criteria used here and 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 TBM-limit baseline, this candidate improves the solar and reactor angles simultaneously and moves the atmospheric prediction from a high upper-octant value toward the shallow upper-octant region of the marginalized NuFIT~6.1 \((\sin^{2}\theta^{\mathrm{fit}}_{23},\delta^{\mathrm{fit}}_{\mathrm{CP}})\) projection; the favorable \(\Delta\chi^{2}\simeq 1.89\) at the Candidate~1 point refers to this marginalized two-parameter atmospheric projection only and is not a full global goodness-of-fit. The reactor angle remains the principal residual tension: the Candidate~1 value is about \(3\sigma\) below the NuFIT~6.1 normal-ordering best fit in \(\sin^{2}\theta^{\mathrm{fit}}_{13}\), while the direct Daya Bay comparison in \(\sin^{2}(2\theta^{\mathrm{DB}}_{13})\) gives a milder discrepancy of about \(-1.7\sigma\); these two diagnostics are not statistically independent and should not be summed. Mass-scale consequences are reported separately: once the Koide--Brannen absolute scale is retained, squared-mass-difference diagnostics introduce additional tension, so the PMNS mixing ansatz and the absolute mass-scale hypothesis should be assessed separately. The construction is therefore presented as a deterministic phenomenological ansatz, not as a derived flavor theory. Its sharp CP-conserving prediction (with the PDG reconstruction selecting the \(\cos\delta^{\prime}_{\mathrm{CP}}=-1\) branch) and upper-octant atmospheric prediction \(\sin^{2}\theta^{\prime}_{23}\simeq 0.534\) are falsifiable features.