  Behind certain marginally trapped surfaces one can construct a geometry
containing an extremal surface of equal, but not larger area. This construction
underlies the Engelhardt-Wall proposal for explaining Bekenstein-Hawking
entropy as a coarse-grained entropy. The construction can be proven to exist
classically but fails if the Null Energy Condition is violated. Here we extend
the coarse-graining construction to semiclassical gravity. Its validity is
conjectural, but we are able to extract an interesting nongravitational limit.
Our proposal implies Wall's ant conjecture on the minimum energy of a
completion of a quantum field theory state on a half-space. It further
constrains the properties of the minimum energy state; for example, the minimum
completion energy must be localized as a shock at the cut. We verify that the
predicted properties hold in a recent explicit construction of Ceyhan and
Faulkner, which proves our conjecture in the nongravitational limit.
