Minimal Quark-Lepton Complementarity from a Rigid Deformation of Tri-Bimaximal Mixing

Abstract

We propose a minimal three-family realization of quark--lepton complementarity (QLC) in which the non-trivial correlation matrix is an exactly unitary, rigid deformation of tri-bimaximal mixing. In a canonical TBM convention the deformation is a rotation about the normalized axis $n_{gg}=(2,1,2{)}^T/3$, with rotation angle fixed by the Cabibbo angle, ${\alpha }_{gg}=-3{\theta }_C/4$. The resulting ansatz, $\text{UPMNS}={\text{VCKM}}^{†}\text{UTBM}{R}_{n_{gg}}(-3{\theta }_C/4)$, contains no new continuous parameter in the Dirac lepton-mixing sector once the structural choice is fixed. With the 2025 Particle Data Group CKM fit it predicts ${\theta }_{12}=33.29^{\circ }$, ${\theta }_{13}=8.46^{\circ }$, ${\theta }_{23}=49.19^{\circ }$, and $\text{dcp}=180.79^{\circ }$, with CKM-induced uncertainties much smaller than present oscillation errors. We derive leading Wolfenstein sum rules; in particular, the real deformation preserves the CKM-induced suppression $\text{Jcp}=-A\eta {\lambda }^3/6+\mathcal{O}({\lambda }^4)$ and hence a near-CP-conserving phase. A retrospective simplicity scan shows that the pair $(2,1,2),3/4$ ranks first among 5510 oriented primitive-integer/rational candidates when tested on epoch-matched 2016 and 2018 oscillation data. In an enlarged 49,622-candidate 2018 scan it is second in raw score but remains first among candidates no more complex than itself under several independent simplicity measures; profiling the rotation strength for the fixed $(2,1,2)$ axis gives $r_{\mathrm{BF}}=0.74941$. We formulate the deformation by a convention-covariant flavor-space generator and give an exact effective eigenframe realization $M_{\nu }=AI+B\text{Sgg}+C{\text{Sgg}}^2$. The construction is presently allowed in normal ordering, with the atmospheric sector providing its strongest pressure; its upper-octant atmospheric angle and near-$\pi$ Dirac phase provide sharp falsifiability targets.