  We present a new reconstruction of the distribution of atomic hydrogen in the
inner Galaxy that is based on explicit radiation-transport modelling of line
and continuum emission and a gas-flow model in the barred Galaxy that provides
distance resolution for lines of sight toward the Galactic Center. The main
benefits of the new gas model are, a), the ability to reproduce the negative
line signals seen with the H$I$4PI survey and, b), the accounting for gas that
primarily manifests itself through absorption.
  We apply the new model of Galactic atomic hydrogen to an analysis of the
diffuse gamma-ray emission from the inner Galaxy, for which an excess at a few
GeV was reported that may be related to dark matter. We find with high
significance an improved fit to the diffuse gamma-ray emission observed with
the Fermi-LAT, if our new H$I$ model is used to estimate the cosmic-ray induced
diffuse gamma-ray emission. The fit still requires a nuclear bulge at high
significance. Once this is included there is no evidence for a dark-matter
signal, be it cuspy or cored. But an additional so-called boxy bulge is still
favoured by the data. This finding is robust under the variation of various
parameters, for example the excitation temperature of atomic hydrogen, and a
number of tests for systematic issues.
