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

In an ordered Pontecorvo–Maki–Nakagawa–Sakata (PMNS) construction, prescribed tribimaximal matrix-element splits (A12/A23) and their order fix the second-column squared moduli to the non-tribimaximal rational values \((\tfrac{11}{36},\tfrac{25}{64},\tfrac{175}{576})\). This yields the exact solar–reactor sum rule (SR-I), \(\sin^2\theta'_{12}\cos^2\theta'_{13}=\frac{11}{36}\). The assumed real rotations imply a vanishing Jarlskog invariant; Majorana phases remain unspecified. For the principal (all-positive) orientation, \(\delta'_{\rm CP}=\pi\), and the fixed column yields an exact atmospheric–reactor sum rule (SR-II). Both sum rules are mass-independent. SR-II maps the NuFIT~6.1 normal-ordering reactor \(3\sigma\) interval into the upper octant, \(\sin^2\theta'_{23}\simeq0.534\text{–}0.542\). Reversing one or all three orientation signs gives \(\delta'_{\rm CP}=0\) and an atmospheric angle below its \(3\sigma\) interval throughout that reactor range. Adopted Koide–Brannen mass ratios determine an auxiliary geometric angle; its prescribed sine-component lift (B31) and the A12 split set the reactor angle, selecting a point on both sum rules without fitting a continuous mixing parameter. These inputs fix all four oscillation mixing parameters. Mass ratios and structural prescriptions remain explicit assumptions. On the NuFIT~6.1 solar–reactor profile, first-row curve minima are \(\Delta\chi^2\simeq0.33\), \(1.84\), and \(20.71\) for SR-I and the trimaximal patterns TM1 and TM2, respectively; the fixed B31 point gives \(\Delta\chi^2\simeq3.68\). One reactor coordinate is optimized per curve; both fixed-point coordinates are held constant. These diagnostics establish no calibrated statistical preference. A separate mass-splitting diagnostic gives \(\Delta\chi^2\simeq3.36\) after profiling the overall mass scale. Solar data with reactor constraints test SR-I; reactor data test the fixed point; atmospheric and CP data with reactor constraints test SR-II and the principal orientation; mass-splitting data test the splitting ratio implied by the adopted mass ratios. At the NuFIT~6.1 best-fit reactor value \(\sin^2\theta'_{13}=0.02248\), the SR-I and TM1 solar predictions differ by \(3.39\) times the projected six-year solar-angle precision of the Jiangmen Underground Neutrino Observatory; this separation is not an exclusion significance.