  Stellar abundances and ages afford the means to link chemical enrichment to
galactic formation. In the Milky Way, individual element abundances show tight
correlations with age, which vary in slope across ([Fe/H]-[$\alpha$/Fe]). Here,
we step from characterising abundances as measures of age, to understanding how
abundances trace properties of stellar birth-environment in the disk over time.
Using measurements from $\sim$27,000 APOGEE stars (R=22,500, SNR$>$200), we
build simple local linear models to predict a sample of elements (X = Si, O,
Ca, Ti, Ni, Al, Mn, Cr) using (Fe, Mg) abundances alone, as fiducial tracers of
supernovae production channels. Given [Fe/H] and [Mg/H], we predict these
elements, [X/H], to about double the uncertainty of their measurements. The
intrinsic dispersion, after subtracting measurement errors in quadrature is
$\approx 0.015-0.04$~dex. The residuals of the prediction (measurement $-$
model) for each element demonstrate that each element has an individual link to
birth properties at fixed (Fe, Mg). Residuals from primarily massive-star
supernovae (i.e. Si, O, Al) partially correlate with guiding radius. Residuals
from primarily supernovae Ia (i.e. Mn, Ni) partially correlate with age. A
fraction of the intrinsic scatter that persists at fixed (Fe, Mg), however,
after accounting for correlations, does not appear to further discriminate
between birth properties that can be traced with present-day measurements.
Presumably, this is because the residuals are also, in part, a measure of the
typical (in)-homogeneity of the disk's stellar birth environments, previously
inferred only using open-cluster systems. Our study implies at fixed birth
radius and time, there is a median scatter of $\approx 0.01-0.015$ dex in
elements generated in supernovae sources.
