  The full, radio to $\gamma$-ray spectrum of the Fermi bubbles is shown to be
consistent with standard strong-shock electron acceleration at the bubble edge,
without the unnatural energy cutoffs and unrealistic electron cooling of
previous studies, if the ambient interstellar radiation is strong; the
$\gamma$-ray cooling break should then have a microwave counterpart, undetected
until now. Indeed, a broadband bubble-edge analysis uncovers a pronounced
downstream dust component, which masked the anticipated $\sim35$ GHz spectral
break, and the same overall radio softening consistent with Kraichnan diffusion
previously reported in $\gamma$-rays.
