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 Walls 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.
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