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Recent LHC results suggest a standard model (SM)-like Higgs boson in the vicinity of 125 GeV with no clear indications yet of physics beyond the SM. At the same time, the SM is incomplete, since additional dynamics are required to accommodate cosmological dark matter (DM). In this paper we show that interactions between weak scale DM and the Higgs which are strong enough to yield a thermal relic abundance consistent with observation can easily destabilize the electroweak vacuum or drive the theory into a non-perturbative regime at a low scale. As a consequence, new physics - beyond the DM itself - must enter at a cutoff well below the Planck scale and in some cases as low as O (10-1000TeV), a range relevant to indirect probes of flavor and CP violation. In addition, this cutoff is correlated with the DM mass and scattering cross-section in a parameter space which will be probed experimentally in the near term. Specifically, we consider the SM plus additional spin 0 or 1/2 states with singlet, triplet, or doublet electroweak quantum numbers and quartic or Yukawa couplings to the Higgs boson. We derive explicit expressions for the full two-loop RGEs and one-loop threshold corrections for these theories.