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Before turning to HCSCO, we benchmarked the methodology on HgBa$_2$CuO$_4$ itself. We find that at fixed oxygen stoichiometry, HBCO lies 74meV/atom above the hull. Upon modeling the doped compound using via a $3 \times 3$ supercell calculation (11\% doping), we find HBCO to be even more unstable at 130meV/atom above the hull. As noted above in our work with LSCO, metastable structures theoretically predicted to be $\sim$100meV/atom above the hull do indeed exist. Thus, we propose that $\sim$100meV/atom uncertainty is a reasonable confidence interval.  With that caveat in mind, we find that HCSCO lies 170meV/atom above the hull at fixed oxygen stoichiometry, and 240meV/atom for fixed oxygen chemical potential, which means the compound is likely unstable, but not out of the realm of possibility for successful synthesis.  % Outcome : proposed compound with more confidence.  % Discuss the notion of confidence.  % Accuracy of LDA.  \emph{Electronic structure} -- In order to evaluate the prospects of superconductivity, we examine the orbitally-resolved density of states (Fig.~\ref{fig:hcsco-bands}). There is indeed a single band crossing the Fermi level Hg(CaS)$_2$CuO$_2$, similar to the cuprates. Additionally, we find the charge-transfer energy of HCSCO to be 1.94eV, identical to that of the other Hg-based cuprates, so we expect similar superconducting transition temperatures. The jury is still out on this material as it has not been successfully synthesized.