  We show that certain lattice gauge theories exhibiting disorder-free
localization have a characteristic response in spatially averaged spectral
functions: a few sharp peaks combined with vanishing response in the zero
frequency limit. This reflects the discrete spectra of small clusters of
kinetically active regions formed in such gauge theories when they fragment
into spatially finite clusters in the localized phase due to the presence of
static charges. We obtain the transverse component of the dynamic structure
factor, which is probed by neutron scattering experiments, deep in this phase
from a combination of analytical estimates and a numerical cluster expansion.
We also show that local spectral functions of large finite clusters host
discrete peaks whose positions agree with our analytical estimates. Further,
information spreading, diagnosed by an unequal time commutator, halts due to
real space fragmentation. Our results can be used to distinguish the
disorder-free localized phase from conventional paramagnetic counterparts in
those frustrated magnets which might realize such an emergent gauge theory.
