  The two leading hypotheses for the Galactic Center Excess (GCE) in the
$\textit{Fermi}$ data are an unresolved population of faint millisecond pulsars
(MSPs) and dark-matter (DM) annihilation. The dichotomy between these
explanations is typically reflected by modeling them as two separate emission
components. However, point-sources (PSs) such as MSPs become statistically
degenerate with smooth Poisson emission in the ultra-faint limit (formally
where each source is expected to contribute much less than one photon on
average), leading to an ambiguity that can render questions such as whether the
emission is PS-like or Poissonian in nature ill-defined. We present a
conceptually new approach that describes the PS and Poisson emission in a
unified manner and only afterwards derives constraints on the Poissonian
component from the so obtained results. For the implementation of this
approach, we leverage deep learning techniques, centered around a neural
network-based method for histogram regression that expresses uncertainties in
terms of quantiles. We demonstrate that our method is robust against a number
of systematics that have plagued previous approaches, in particular DM / PS
misattribution. In the $\textit{Fermi}$ data, we find a faint GCE described by
a median source-count distribution (SCD) peaked at a flux of $\sim4 \times
10^{-11} \ \text{counts} \ \text{cm}^{-2} \ \text{s}^{-1}$ (corresponding to
$\sim3 - 4$ expected counts per PS), which would require $N \sim
\mathcal{O}(10^4)$ sources to explain the entire excess (median value $N =
\text{29,300}$ across the sky). Although faint, this SCD allows us to derive
the constraint $\eta_P \leq 66\%$ for the Poissonian fraction of the GCE flux
$\eta_P$ at 95% confidence, suggesting that a substantial amount of the GCE
flux is due to PSs.

High Energy Astrophysical Phenomena;Cosmology and Nongalactic Astrophysics;Instrumentation and Methods for Astrophysics;Machine Learning;High Energy Physics - Phenomenology