  We use the framework of asymptotic safety above the Planck scale to constrain
the parameter space of simple models of new physics that can accommodate the
measured value of the anomalous magnetic moment of the muon and the relic
density of dark matter. We couple parametrically to the trans-Planckian quantum
physics a set of SU(2)$\times$U(1) invariant extensions of the Standard Model,
each comprising an inert scalar field and one pair of colorless fermions that
communicate to the muons through Yukawa-type interactions. The presence of an
interactive UV fixed point in the system of gauge and Yukawa couplings imposes
a set of boundary conditions at the Planck scale, which allow one to derive
unique phenomenological predictions in each case and distinguish the different
representations of the gauge group from one another. We apply to the models
constraints from the measured branching ratio of $h\to \mu\mu$, direct LHC
searches for electroweak production with leptons and missing energy in the
final state, and the dark matter relic density. We find that they further
restrict the available parameter space.
