  In a series of recent reports, doped lead apatite (LK-99) has been proposed
as a candidate ambient temperature and pressure superconductor. However, from
both an experimental and theoretical perspective, these claims are largely
unsubstantiated. To this end, our synthesis and subsequent analysis of an LK-99
sample reveals a multiphase material that does not exhibit high-temperature
superconductivity. We study the structure of this phase with single-crystal
X-ray diffraction (SXRD) and find a structure consistent with doped
$\text{Pb}_{10}(\text{PO}_4)_6(\text{OH})_2$. However, the material is
transparent which rules out a superconducting nature. From ab initio defect
formation energy calculations, we find that the material likely hosts
$\text{OH}^-$ anions, rather than divalent $\text{O}^{2-}$ anions, within the
hexagonal channels and that Cu substitution is highly thermodynamically
disfavored. Phonon spectra on the equilibrium structures reveal numerous
unstable phonon modes. Together, these calculations suggest it is doubtful that
Cu enters the structure in meaningful concentrations, despite initial attempts
to model LK-99 in this way. However for the sake of completeness, we perform ab
initio calculations of the topology, quantum geometry, and Wannier function
localization in the Cu-dominated flat bands of four separate doped structures.
In all cases, we find they are atomically localized by irreps, Wilson loops,
and the Fubini-Study metric. It is unlikely that such bands can support strong
superfluidity, and instead are susceptible to ferromagnetism (or out-of-plane
antiferromagnetism) at low temperatures, which we find in ab initio studies. In
sum, $\text{Pb}_{9}\text{Cu}(\text{PO}_4)_6(\text{OH})_2$ could more likely be
a magnet, rather than an ambient temperature and pressure superconductor.
