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

Abstract

An ordered Pontecorvo–Maki–Nakagawa–Sakata (PMNS) construction with prescribed tribimaximal matrix-element splits fixes the second-column squared moduli to the nontrimaximal rational values \((\tfrac{11}{36},\tfrac{25}{64},\tfrac{175}{576})\), yielding the exact solar–reactor sum rule (SR-I), \(\sin^2\theta'_{12}\cos^2\theta'_{13}=\frac{11}{36}\). Adopted Koide–Brannen mass ratios define two auxiliary square-root configurations with identical positive masses but opposite signs for the lightest coordinate. A rotation leaving the second coordinate fixed connects these configurations. A prescribed sine-component lift (B31) and split (A12) map its angle to the reactor angle without an additional continuous mixing parameter. The rotation plane, use of its angle in mixing, lift, splits, orientation, and block order remain inputs. Assumed real rotations give a vanishing Jarlskog invariant; Majorana phases remain unspecified. For the principal orientation, \(\delta'_{\rm CP}=\pi\); the atmospheric–reactor sum rule (SR-II) maps the NuFIT~6.1 normal-ordering reactor \(3\sigma\) interval to 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 range. 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 principal B31 point fixed by the adopted mass ratios gives \(\Delta\chi^2\simeq3.68\). One coordinate is optimized per curve, none at this fixed point; no calibrated statistical preference follows. A 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 the principal orientation, and mass-splitting data the 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 precision of the Jiangmen Underground Neutrino Observatory; this separation is not an exclusion significance.