Refining the In-situ and Ex-situ Designation of Galactic Globular Clusters: A Statistical Re-evaluation in the [Mg/Mn] vs. [Al/Fe] Plane with APOGEE DR16 & DR17.

High-resolution near-infrared spectroscopy from the APOGEE survey has revolutionized Galactic archaeology, enabling detailed chemical abundance studies of stars in dust-obscured regions. The [Mg/Mn] vs. [Al/Fe] plane has emerged as a powerful diagnostic for distinguishing stars formed in situ within the Milky Way from those accreted ex situ. However, literature classifications often rely on simplified, linear divisions in this chemical space. In this work, we critically reassess the in-situ and ex-situ status of member stars in selected Galactic globular clusters using APOGEE data. By applying a robust statistical framework to the high-precision abundances from DR16 and DR17, we demonstrate that the separation between these populations is not well represented by sharp, linear boundaries. Instead, we find that in-situ and ex-situ stars occupy overlapping regions characterized by probabilistic boundaries with intrinsic curvature. Our comparative analysis of the two data releases reveals that the precise location and shape of these separation regions are sensitive to the underlying sample selection, abundance calibration, and reported uncertainties. This finding underscores a significant caveat: origin predictions for individual stars or clusters can vary substantially depending on the adopted APOGEE data release and the specific selection criteria used to define the separating region in the chemical plane. We therefore urge caution in the application of rigid chemical dividers and highlight the necessity of statistically informed, probabilistic approaches for future classification in Galactic archaeology.