<p>The exponential characterization of a partially dissipative Timoshenko system with memory effects acting on the shear component is addressed in this paper. A wider class of admissible kernels that may exhibit flat zones and jump discontinuities is explored. Then, by the introduction of a novel condition, referred to as the shape condition, it is proven that the previous classical criteria for the system’s exponential characterization are no longer sufficient in our extended setting. Here, the geometric structure of the kernel, particularly its flatness and the distribution of discontinuities, is seen to play a crucial role in determining the characterization of exponential stability. Hence, a refined characterization is established in terms of three simultaneous conditions: dissipation, equal wave speeds, and the shape condition. The results not only generalize previous findings based on smooth kernels but also offer a deeper understanding of the interplay between the geometric aspects of the memory kernel and the system’s asymptotic behavior.</p>