A non zero $θ_{13}$ from a minimal and predictive neutrino mass matrix

We propose a new, minimal Majorana neutrino mass matrix texture and study its predictions within a partial tri-bimaximal (TBM) mixing scheme, where $\sinθ_{12} = 1/\sqrt{3}$ and $\sinθ_{23} = 1/\sqrt{2}$, with $θ_{13}$ and $δ$ treated as free parameters. The texture forbids $θ_{13} = 0$ and does not correspond to a $μ-τ$ symmetric structure. We find that the predictions show distinct behavior depending on the sign of a single real texture parameter. The texture is realized through a hybrid framework of one Type-I seesaw and two Type-II seesaw mechanisms under an extended symmetry group, $SU(2)_L \otimes U(1)_Y \otimes A_4 \otimes Z_{10} \otimes Z_{7}$ with properly chosen model parameters. The texture is seen to favor the normal hierarchy for neutrino masses.