  Using new large area maps of the cold neutral medium (CNM) fraction, $f_{\rm
CNM}$, we investigate the relationship between the CNM, the abundance of
polycyclic aromatic hydrocarbons (PAHs), and the anomalous microwave emission
(AME). We first present our $f_{\rm CNM}$ map based on full-sky HI4PI data,
using a convolutional neural network to covert the spectroscopic HI data to
$f_{\rm CNM}$. We demonstrate that $f_{\rm CNM}$ is strongly correlated with
the fraction of dust in PAHs as estimated from mid- and far-infrared dust
emission. In contrast, we find no correlation between $f_{\rm CNM}$ and the
amount of AME per dust emission, nor between $f_{\rm CNM}$ and the AME peak
frequency. These results suggest PAHs preferentially reside in cold, relatively
dense gas, perhaps owing to enhanced destruction in more diffuse media. The
lack of correlation between $f_{\rm CNM}$ and AME peak frequency is in tension
with expectations from theoretical models positing different spectral energy
distributions of AME in the cold versus warm neutral medium. We suggest that
different PAH abundances and emission physics in different interstellar
environments may explain the weaker than expected correlation between 12$\mu$m
PAH emission and AME even if PAHs are the AME carriers.
