  We perform two distinct spatio-spectral reconstructions of the gamma-ray sky
in the range of 0.56-316 GeV based on Fermi Large Area Telescope (LAT) data.
Both describe the sky brightness to be composed of a diffuse-emission and a
point-source component. The first model requires minimal assumptions and
provides a template-free reconstruction as a reference. It makes use of spatial
and spectral correlations to distinguish between the different components. The
second model is physics-informed and further differentiates between diffuse
emission of hadronic and leptonic origin. For this, we assume parametric, but
spatially varying energy spectra to distinguish between the processes and use
thermal Galactic dust observations to indicate the preferred sites of hadronic
interactions. To account for instrumental effects we model the point-spread,
the energy dispersion, and the exposure of the telescope throughout the
observation. The reconstruction problem is formulated as a Bayesian inference
task, that is solved by variational inference. We show decompositions of the
Gamma-ray flux into diffuse and point-like emissions, and of the diffuse
emissions into multiple physically motivated components. The diffuse
decomposition provides an unprecedented view of the Galactic leptonic diffuse
emission. It shows the Fermi bubbles and their spectral variations in high
fidelity and other areas exhibiting strong cosmic ray electron contents, such
as a thick disk in the inner Galaxy and outflow regions. Furthermore, we report
a hard spectrum gamma ray arc in the northern outer bubble co-spatial with the
reported X-ray arc by the eROSITA collaboration. All our spatio-spectral sky
reconstructions and their uncertainty quantification are publicly available.
