  The broad high-energy spectral component in blazars is usually attributed to
various inverse Compton scattering processes in the relativistic jet, but has
not been clearly identified in most cases due to degeneracies in physical
models. AP Librae, a low-synchrotron-peaking BL Lac object (LBL) detected in
2015 by H.E.S.S. at very high energies (VHE; $>$ 0.5 TeV), has an extremely
broad high-energy spectrum, covering $\sim$ 9 decades in energy. Standard
synchrotron self-Compton models generally fail to reproduce the VHE emission,
which has led to the suggestion that it might arise not from the blazar core,
but on kiloparsec scales from inverse Compton scattering of cosmic microwave
background (CMB) photons by a still-relativistic jet (IC/CMB). IC/CMB models
for the TeV emission of AP Librae in prior works have implied a high level of
infrared emission from the kpc-scale jet. With newly obtained Hubble Space
Telescope imaging, we obtain a deep upper limit on the kpc-scale jet emission
at 1.6 $\mu$m, well below the expected level. High-resolution ALMA imaging in
bands 3-9 reveals a residual dust disk signature after core subtraction, with a
clearly thermal spectrum, and an extent ($\sim$500 pc) which matches with a
non-jet residual emission seen after PSF subtraction in our 1.6 $\mu$m HST
imaging. We find that the unusually broad GeV and VHE emission in AP Librae can
be reproduced through the combined IC scattering of photons from the CMB and
the dust disk, respectively, by electrons in both the blazar core and sub-kpc
jet.
