Many models currently exist which attempt to interpret the excess of gamma rays emanating from the Galactic Center in terms of annihilating or decaying dark matter. These models typically exhibit a variety of complicated cascade mechanisms for photon production, leading to a non-trivial kinematics which obscures the physics of the underlying dark sector. In this paper, by contrast, we observe that the spectrum of the gamma-ray excess may actually exhibit an intriguing energy-duality invariance under $E_gamma rightarrow E_ast^2/E_gamma$ for some $E_ast$. As we shall discuss, such an energy duality points back to a remarkably simple alternative kinematics which in turn is realized naturally within the Dynamical Dark Matter framework. Observation of this energy duality could therefore provide considerable information about the properties of the dark sector from which the Galactic-Center gamma-ray excess might arise, and highlights the importance of acquiring more complete data for the Galactic-Center excess in the energy range around 1 GeV.
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