  In the path integral formulation of causal set quantum gravity, the quantum
partition function is a phase-weighted sum over locally finite partially
ordered sets, which are viewed as discrete quantum spacetimes. It is known,
however, that the number of ``layered'' sets -- a class of causal sets that
look nothing like spacetime manifolds -- grows superexponentially with the
cardinality $n$, giving an entropic contribution that can potentially dominate
that of the action. We show here that in any dimension, the discrete
Einstein-Hilbert action for a typical $K$-layered causal set reduces to the
simple link action to leading order in $n$. Combined with earlier work, this
completes the proof that the layered sets, although entropically dominant, are
very strongly suppressed in the path sum of causal set quantum gravity whenever
the discreteness scale is greater than or equal to a (mildly
dimension-dependent) order one multiple of the Planck scale.
