The Galactic Center (GC) has been long known to host gamma-ray emission detected to >10 TeV. HESS data now points to two plausible origins: the supermassive black hole (perhaps with >PeV cosmic rays and neutrinos) or high-energy electrons from the putative X-ray pulsar wind nebula G359.95-0.04 observed by Chandra and NuSTAR. We show that if the magnetic field experienced by PWN electrons is near the several mG ambient field strength suggested by radio observations of the nearby GC magnetar SGR J1745-29, synchrotron losses constrain the TeV gamma-ray output to be far below the data. Accounting for the peculiar geometry of GC infrared emission, we also find that the requisite TeV flux could be reached if the PWN is ~1 pc from Sgr A* and the magnetic field is two orders of magnitude weaker, a scenario that we discuss in relation to recent data and theoretical developments. Otherwise, Sgr A* is left, which would then be a PeV link to other AGN.
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