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The full, radio to $\gamma$-ray spectrum of the Fermi bubbles is shown to beconsistent with standard strong-shock electron acceleration at the bubble edge,without ad-hoc energy cutoffs, 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 apronounced downstream dust component, which masked the anticipated $\sim35$ GHzspectral break, and the same overall radio softening consistent with Kraichnandiffusion previously reported in $\gamma$-rays.

Fermi-bubble bulk and edge analysis reveals dust, cooling breaks, and cosmic-ray diffusion, facilitating a self-consistent model