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In experiment, synthesis often is not performed at fixed stoichiometry, but rather in an atmosphere with, for example, tunable oxygen pressure. To capture this configuration, we model the La-Cu-S system  open to oxygen oxygen, which is  tuned by a chemical potential $\mu(\text{O}_2)$, using the formalism developed in Ref.~\cite{Ping_Ong_2008}. \cite{Ping_Ong_2008}. The stability energies of our two target compounds, along with LaCuSO, are shown as a function of the oxygen potential in Fig.~\ref{fig:mu-lcso}.  If we had access to this information during the original work, we would not have recommended that experimental groups attempt to synthesize the target compounds because instability energies of over 0.5eV/atom are quite large. If instead the energies lay up $\sim$100meV/atom above the hull, we would believe there is a reasonable probability the new composition would be stable. Thus, notwithstanding the systematic uncertainties in choice of $U$ and correction parameters, convex hulls are critical to evaluating potential new compounds.