By incorporating a holographically designed aperiodic photonic lattice within one of the arms of a Y-coupled Fabry-Perot quantum cascade laser architecture, it has been demonstrated that the multiband mode control exerted by the photonic lattice on emission spectra can, owing to the mutual optical coupling between the arms, be transferred to the second unpatterned arm. However, the underlying theoretical mechanism on how the lattice influences the threshold gain spectral properties of the Y architecture has, until now, remained unstudied. Here, we use the transfer matrix formalism, originally developed for studying aperiodic lattice lasers, to investigate this. A detailed threshold gain spectral study revealed that although the effects of facet feedback of the Y-coupled laser chip are present, due to the enhanced photonic density-of-states at user-specified frequencies, the aperiodic lattice has remarkable control over the Y architecture laser spectra, under the mutual optical coupling between the arms. Finally, indicated by the fringe patterns akin to double-slit interference, of the measured far-field beam profiles, phase-locked terahertz emissions from the Y architecture are demonstrated.