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While we did not perform global structural prediction, we performed local checks for stability, which we acknowledged were by no means exhaustive. Using a $2\times2\times1$ unit cell, we performed full structural relaxation to check if the structure would be unstable towards distortion to the $T'$-type layered perovskite, knowing that substitution of the large La ion for the smaller Pr and Nd led to a rearrangement of the charge reservoir layer into the fluorite structure. We found that the $T$-type structure was indeed stable and there was no out-of-plane buckling, although the CuO$_6$ octahedra favored axial rotations ($a^0a^0c_p^-$ in Glazer notation).  \emph{Global stability} -- We checked the thermodynamic stability of the proposed compounds against competing phases by selecting commonly known reactants and computing the formation enthalpies of the synthesis pathways as shown in Table~\ref{tbl:pathways}. We computed the total energies of formation $\Delta E = E_\text{products} - E_\text{reactants}$, and find that all differentials are positive, indicating the reactions target phases are unfavorable. However, it is known that many functional materials are metastable, protected from decay by large energetic barriers. The parent cuprate La$_2$CuO$_4$ is an example: as shown on the last line of Table~\ref{tbl:pathways}, LCO is actually unstable by 28kJ/mol. We also examined the volume differentials $\Delta V = V_\text{products} - V_\text{reactants}$ with the knowledge that often high pressure synthesis allows otherwise unstable compounds to form. We indeed find that $\Delta V$ are overwhelming negative, meaning the application of high pressure may allow the formation of the target phases.  \begin{table}  \begin{tabular}{r|r|rl}  \hline  $\Delta E$ & $\Delta V$ & \multicolumn{2}{c}{Synthesis pathway} & \\  \hline  \hline  141 & -7.3 & La$_2$O$_2$S + CuS & $\rightarrow$ La$_2$CuO$_2$S$_2$\\  223 & -3.4 & Y$_2$O$_2$S + CuS & $\rightarrow$ Y$_2$CuO$_2$S$_2$\\  267 & -5.0 & Lu$_2$O$_2$S + CuS & $\rightarrow$ Lu$_2$CuO$_2$S$_2$\\  356 & -3.0 & Sc$_2$O$_2$S + CuS & $\rightarrow$ Sc$_2$CuO$_2$S$_2$\\  101 & -4.9 & La$_2$O$_2$S$_2$ + Cu & $\rightarrow$ La$_2$CuO$_2$S$_2$\\  \hline  148 & -3.3 & La$_2$O$_3$ + CuS & $\rightarrow$ La$_2$CuO$_3$S \\  454 & -0.7 & Sc$_2$O$_3$ + CuS & $\rightarrow$ Sc$_2$CuO$_3$S \\  97 & -4.9 & La$_2$O$_2$S + CuO & $\rightarrow$ La$_2$CuO$_3$S \\  269 & 2.8 & Sc$_2$O$_2$S + CuO & $\rightarrow$ Sc$_2$CuO$_3$S \\  \hline  28 & -5.1 & La$_2$O$_3$ + CuO & $\rightarrow$ La$_2$CuO$_4$ \\  \hline  \end{tabular}  \caption{Synthesis pathways for various cuprate oxysulfides based on  substitution of sulfur for both (top block) or only one (middle block) of  the apical oxygens in $R_2$CuO$_4$. Energies in kJ/mol and volumes in  kJ/mol/GPa. Since the energies of formation ($\Delta E = E_\text{products}  - E_\text{reactants}$) are positive, none of these pathways appear  favorable at ambient conditions. However, high-pressure synthesis will  help stabilize these pathways, since the majority of volume differentials  ($\Delta V = V_\text{products} - V_\text{reactants}$) are negative. We  benchmark our method against the standard synthesis pathway for  La$_2$CuO$_4$, shown on the last line. Surprisingly, $\Delta E$ is  +28~kJ/mol, so either DFT systemmatically overestimates enthalpies (which  means the actual enthalpies for our hypothetical compounds are  \emph{smaller}, in our favor), or we must add a bi-directional uncertainty  of $\pm 30$~kJ/mol to the computed enthalpies. Additionally, positional  entropy of the apical $S$ in the half-substituted $R_2$CuO$_3$S compounds  should also assist in synthesis.}  \label{tbl:pathways}  \end{table}  \emph{Reexamination} -- In the intervening years, the maturation of materials databases allowed us to revisit the question of global stability. Various databases have computed and tabulated the convex hulls of binary, ternary and some quaternary systems, and provided tools for researchers to apply their framework to novel chemical systems. In the following, we describe our new understanding of the global stability of La$_2$CuO$_2$S$_2$ and La$_2$CuO$_3$S$_S$ against all known competing phases in the La-Cu-S-O chemical system. Since the Cu site contains significant correlations, we must address the effect of $U$ on the energies provided by density functional theory.  Corrections to LDA/GGA energies for convex hull construction have been systematically investigated for transition metal oxides \cite{Wang_2006,Jain_2011}. The corrections arise from two sources: (a) the GGA overbinding of the anion (most commonly the $O_2$ molecule) and (b) correlations. The GGA overbinding differs based on the anion, and the corrections are tabulated in the Materials Project.  The correlation corrections are further divided into two components: (1) a contribution due to the U for atomic-like orbitals, treated by LDA+U, and (2) a correction in the energies required when comparing correlated (modeled using LDA+U) and uncorrelated (modeled using LDA) compounds. This is often the case in construction phase diagrams containing transition metal ions as their behavior can be considered “correlated” or “uncorrelated” depending on their valence and chemical environment.  In the above work, we had proposed several chemical reaction pathways to synthesize La$_2$CuS$_2$O$_2$ and La$_2$CuSO$_3$ which were subsequently tested by experiment [ref. Hua He, unpublished] and concluded that the two compounds were unstable (at least at high temperatures). Additionally, the phase with composition LaCuSO seemed to be quite stable at high temperatures, as it was the preferred quaternary composition in almost all the experimentally analyzed reactions.  With modern materials databases, we are able to reanalyze the entire La-Cu-S-O system to construct the convex hull (plotted in Fig. 1) and globally investigate stability. Notice that La$_2$CuS$_2$O$_2$ and La$_2$CuSO$_3$ are not among the stable compounds on the hull.