Timelike CGC Translation Surfaces Generated by Timelike Curves in Minkowski $3-$Space

In this paper, we study timelike translation surfaces with constant Gaussian curvature (CGC) in the three-dimensional Minkowski space. Such surfaces are generated as the sum of two timelike space curves and naturally arise in the context of Lorentzian surface geometry. By employing a detailed analytic and geometric approach, we prove that any timelike translation surface with constant Gaussian curvature must be flat. As a consequence, we show that the only timelike translation surfaces satisfying this curvature condition are cylindrical surfaces. Furthermore, we establish that timelike translation surfaces with constant Gaussian curvature cannot be minimal everywhere. As a geometric characterization of the generating curves, we prove that one of the curves must necessarily be either a timelike hyperbola or a straight line. These results provide a complete local classification of timelike translation surfaces with constant Gaussian curvature in Minkowski 3-space and highlight a strong rigidity phenomenon in the timelike Lorentzian setting.