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, $\UPMNS=\VCKM^\dagger\UTBM R_{n_{gg}}(-3\theta_C/4)$, 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 $\dcp=180.79^\circ$, 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 $\Jcp=-A\eta\lambda^3/6+\mathcal O(\lambda^4)$ 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 $r_{\rm BF}=0.74941$. We formulate the deformation by a convention-covariant flavor-space generator and give an exact effective eigenframe realization $M_\nu=A I+B\Sgg+C\Sgg^2$. 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.