  We establish rigorous bounds on dissipative transport in the disordered
Klein-Gordon chain with quartic on-site potential, governed by a parameter
$\lambda$. At $\lambda = 0$, the chain is harmonic and transport is fully
suppressed by Anderson localization. For the anharmonic system at $\lambda >
0$, our results show that dissipative effects set in on time scales that grow
faster than any polynomial in $1/\lambda$, as $\lambda\to 0$. From a technical
perspective, the main novelty of our work is that we do not restrict ourselves
to the atomic limit, i.e. we develop perturbation theory around the harmonic
system with a fixed hopping strength between near oscillators. This allows us
to compare our mathematical results with previous numerical work and contribute
to the resolution of an ongoing debate in the physics community, as we explain
in a companion paper arXiv:2308.10572.
