  Some studies of stars' multi-element abundance distributions suggest at least
5-7 significant dimensions, but other studies show that the abundances of many
elements can be predicted to high accuracy from [Fe/H] and [Mg/Fe] alone (or
from [Fe/H] and age). We show that both propositions can be, and are,
simultaneously true. We adopt a technique known as normalizing flow to
reconstruct the probability distribution of Milky Way disk stars in the space
of 15 elemental abundances measured by APOGEE. Conditioning on stellar
parameters $T_{\rm eff}$ and $\log g$ minimizes the differential systematics.
After conditioning on [Fe/H] and [Mg/H], the residual scatter for the best
measured APOGEE elements is $\sigma_{[X/{\rm H}]} \lesssim 0.02$ dex,
consistent with APOGEE's reported statistical uncertainties of $\sim 0.01 -
0.015$ dex and intrinsic scatter of $0.01-0.02$ dex. Despite the small scatter,
residual abundances display clear correlations between elements, which are too
large to be explained by measurement uncertainties or by the statistical noise
from our finite sample size. We must condition on at least seven elements
(e.g., Fe, Mg, O, Si, Ni, Ca, Al) to reduce residual correlations to a level
consistent with observational uncertainties, and higher measurement precision
for other elements would likely reveal additional dimensions. Our results
demonstrate that cross-element correlations are a much more sensitive and
robust probe of hidden structure than dispersion alone, and they can be
measured precisely in a large sample even if star-by-star measurement noise is
comparable to the intrinsic scatter. We conclude that many elements have an
independent story to tell, even for a "mundane" sample of disk stars and
elements produced mainly by core-collapse and Type Ia supernovae. The only way
to learn these lessons is to measure the abundances directly, and not merely
infer them.
