Exact fixed-column sum rules and a mass-ratio test in a PMNS construction

An ordered Pontecorvo–Maki–Nakagawa–Sakata (PMNS) construction with specified tribimaximal (TBM) matrix-element splits fixes the second-column squared moduli to \((11/36,25/64,175/576)\), yielding \(sin^2θ'₁₂cos^2θ'₁₃=11/36\). The adopted Koide–Brannen mass ratios define two auxiliary square-root configurations that represent identical positive masses and differ in the sign of the coordinate associated with the lightest mass. A rotation that holds the second coordinate fixed connects them; a specified sine-component lift (B31) and matrix-element split (A12) map its angle to reactor mixing without an additional continuous fit parameter. The rotation plane, use of its angle in PMNS mixing, lift, splits, and block order remain inputs. Real rotations give \(JCP=0\). The principal orientation has \(δ'CP=π\) and maps the NuFIT~6.1 normal-ordering reactor \(3σ\) interval to the upper octant, \(sin^2θ'₂₃0.534–0.542\). The alternative real orientation, \(δ'CP=0\), predicts an atmospheric angle below the atmospheric \(3σ\) interval throughout that reactor range. Majorana phases remain unspecified. On the public NuFIT~6.1 solar–reactor profile, bilinear interpolation gives \(Δχ^2=3.683\) for the lifted fixed point and \(8.638\) for a direct-angle control with the same mass ratios and remaining prescriptions. The minimum along the exact solar–reactor relation is \(0.326\), compared with \(1.838\) and \(20.705\) for the first-row curves of trimaximal patterns TM1 and TM2, which preserve the first and second TBM columns, respectively. One coordinate is optimized along each curve; none is optimized at the fixed points. These diagnostics do not establish a calibrated model preference. Solar, reactor, atmospheric/CP, and mass-splitting tests remain distinct; projected precisions for the Jiangmen Underground Neutrino Observatory (JUNO) motivate a solar benchmark.