  We discuss how to define a kinematical horizon entropy on a causal set. We
extend a recent definition of horizon molecules to a setting with a null
hypersurface crossing the horizon. We argue that, as opposed to the spacelike
case, this extension fails to yield an entropy local to the
hypersurface-horizon intersection in the continuum limit when the causal set
approximates a curved spacetime. We then investigate the entropy defined via
the Spacetime Mutual Information between two regions of a causal diamond
truncated by a causal horizon, and find it does limit to the area of the
intersection.
