The Atlantic Meridional Overturning Circulation (AMOC) is often analyzed using low-order box models to understand tipping points. Historically, these studies focus on freshwater flux as the primary bifurcation parameter, treating the temperature gradient as a fixed restoring target. However, the erosion of the equator-to-pole temperature contrast due to polar amplification suggests that thermal forcing should be treated as a dynamic control parameter. In this study, we use Cessi's reduced box model to map the global bifurcation structure of the thermohaline circulation. We relax the assumption of a fixed thermal background and analyze the system's behavior under joint thermal and haline forcing. We prove the existence of a cusp bifurcation, identifying the specific geometry of pitchfork and saddle-node bifurcations that bound the stable regime. This geometric characterization reveals that thermal erosion acts as a distinct mechanism for destabilization, capable of driving the system across critical thresholds even in the absence of anomalous freshwater forcing.