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We introduce the concept of improvability of the dominant energy scalar, andwe derive strong consequences of non-improvability. In particular, we provethat a non-improvable initial data set without local symmetries must sit insidea null perfect fluid spacetime carrying a global Killing vector field. We alsoshow that the dominant energy scalar is always almost improvable in a precisesense. Using these main results, we provide a characterization of Bartnik massminimizing initial data sets which makes substantial progress toward Bartnik'sstationary conjecture.  Along the way we observe that in dimensions greater than eight there existpp-wave counterexamples (without the optimal decay rate for asymptoticallyflatness) to the equality case of the spacetime positive mass theorem. As aconsequence, there exist counterexamples to Bartnik's stationary and strictpositivity conjectures in those dimensions.

Bartnik mass minimizing initial data sets and improvability of the dominant energy scalar