  Entropy and energy are found to be closely tied on our quest for quantum
gravity. We point out an interesting connection between the recently proposed
outer entropy, a coarse-grained entropy defined for a compact spacetime domain
motivated by the holographic duality, and the Bartnik-Bray quasilocal mass long
known in the mathematics community. In both scenarios, one seeks an optimal
spacetime fill-in of a given closed, connected, spacelike, codimension-two
boundary. We show that for an outer-minimizing mean-convex surface, the
Bartnik-Bray inner mass matches exactly with the irreducible mass corresponding
to the outer entropy. The equivalence implies that the area laws derived from
the outer entropy are mathematically equivalent as the monotonicity property of
the quasilocal mass. It also gives rise to new bounds between entropy and the
gravitational energy, which naturally gives the gravitational counterpart to
Wall's ant conjecture. We also observe that the equality can be achieved in a
conformal flow of metrics, which is structurally similar to the Ceyhan-Faulkner
proof of the ant conjecture. We compute the small sphere limit of the outer
entropy and it is proportional to the bulk stress tensor as one would expect
for a quasilocal mass. Lastly, we discuss some implications of taking quantum
matter into consideration in the semiclassical setting.
