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Flavor unification, dark matter, proton decay and other observable predictions with low-scale $S_4$ symmetry

We show how gauge coupling unification is successfully implemented through non-supersymmetric grand unified theory, $SO(10)\times G_f (~G_f=S_4, SO(3)_f, SU(3)_f)$, using low-scale flavor symmetric model of the type $SU(2)_L\times U(1)_Y$ $ \times SU(3)_C \times S_4$ recently proposed by Hagedorn, Lindner, and Mohapatra, while assigning matter-parity discrete symmetry for the dark matter stability. For gauge coupling unification in the single-step breaking case, we show that a color-octet fermion and a hyperchargeless weak-triplet fermionic dark matter are the missing particles needed to complete its MSSM-equivalent degrees of freedom. When these are included the model automatically predicts the nonsupersymmetric grand unification with a scale identical to the minimal supersymmetric standard model/grand unified theory scale. We also find a two-step breaking model with Pati-Salam intermediate symmetry where the dark matter and a low-mass color-octet scalar or the fermion are signaled by grand unification. The proton-lifetime predictions are found to be accessible to ongoing or planned searches in a number of models. We discuss grand unified origin of the light fermionic triplet dark matter, the color-octet fermion, and their phenomenology.