Note: Nonuniqueness of generalized quantum master equations for a single observable

When deriving exact generalized master equations for the evolution of a reduced set of degrees of freedom, one is free to choose what quantities are relevant by specifying projection operators. Different choices of projectors can lead to master equations that are structurally dissimilar and have different memory kernels, despite the resulting dynamics being necessarily the same. This owes to the interplay between projections and explicit imposition of dynamical constraints. We give a simple example to show this point and prove how different memory kernels can yield the same dynamics.