Exact fixed-input PMNS sum rules from a split-and-recombined tribimaximal construction

Abstract

The first solar result from the Jiangmen Underground Neutrino Observatory (JUNO) and the global neutrino-oscillation analysis NuFIT~6.1 sharpen tests of exact first-row lepton-mixing sum rules. We study a fixed-input construction of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix that combines a Koide–Brannen square-root neutrino-mass-ratio input with discrete split-and-recombined deformations of tribimaximal (TBM) mixing. The mass ratio fixes a geometric 3--1 seed; a prescribed lift (B31) converts it into a rotation defining an intermediate basis before the effective TBM-like blocks act. For each fixed branch, no continuous mixing parameter is fitted; the choice among branches remains a discrete phenomenological selection. The principal branch obeys two exact relations: \(\sin^{2}\theta_{12}\cos^{2}\theta_{13}=\tfrac{11}{36}\) and an atmospheric--reactor correlation, and it conserves the combined charge-conjugation and parity (CP) symmetry. An exact classification of the frozen two-order grammar yields 128 syntactic branches and 16 labeled squared-modulus classes. Once the B31 seed and positive orientations are fixed, \(\tfrac{11}{36}\) occurs in two of eight branches, whereas the full target second column occurs in one. A projection-by-projection analysis of public NuFIT 6.1 surfaces avoids combining overlapping profiled likelihoods and finds that a post-selection calibration can be reported only for the \(\theta_{13}\)–\(\theta_{12}\) projection. Using the published six-year JUNO precision as a parameter-level benchmark, the distances of the first-row rule from the trimaximal patterns preserving the first (TM1) or second (TM2) TBM column are \(3.39\sigma\) and \(17.76\sigma\); under an equal-prior two-rule decision these correspond to wrong-selection probabilities of \(4.5\%\) and \(3.4\times10^{-19}\), respectively. This is not a spectrum-likelihood forecast. A companion arithmetic no-go theorem excludes the specified finite residual-symmetry origins of the exact second column. We present the result as a reproducible phenomenological ansatz rather than a microscopic flavor model.