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\ (₃₁\) 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 \ (₄₈₉\) = -\ (₈₉\), 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 \ (^2^₁₂0. 312011\), \ (^2^₁₃0. 020690\), \ (^2^₂₃0. 533876\), and \ (^₂₏=\). 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 \ ( (^2^fit₂₃, ^fit₂₏) \) projection; the favorable \ (^2 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\) below the NuFIT~6. 1 normal-ordering best fit in \ (^2^fit₁₃\), while the direct Daya Bay comparison in \ (^2 (2^DB₁₃) \) gives a milder discrepancy of about \ (-1. 7\) ; 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 \ (^₂₏=-1\) branch) and upper-octant atmospheric prediction \ (^2^₂₃ 0. 534\) are falsifiable features.