  If low-energy supersymmetry is realized in nature, a seemingly contrived
hierarchy in the squark mass spectrum appears to be required. We show that
composite supersymmetric theories at the bottom of the conformal window can
automatically yield the spectrum that is suggested by experimental data and
naturalness. With a non-tuned choice of parameters, the only superpartners
below one TeV will be the partners of the Higgs, the electroweak gauge bosons,
the left-handed top and bottom, and the right-handed top, which are precisely
the particles needed to make weak scale supersymmetry breaking natural. In the
model considered here, these correspond to composite (or partially composite)
degrees of freedom via Seiberg duality, while the other MSSM fields, with their
heavier superpartners, are elementary. The key observation is that at or near
the edge of the conformal window, soft supersymmetry breaking scalar and
gaugino masses are transmitted only to fundamental particles at leading order.
With the potential that arises from the duality, a Higgs with a 125 GeV mass,
with nearly SM production rates, is naturally accommodated without tuning. The
lightest ordinary superpartner is either the lightest stop or the lightest
neutralino. If it is the stop, it is natural for it to be almost degenerate
with the top, in which case it decays to top by emitting a very soft gravitino,
making it quite difficult to find this mode at the LHC and more challenging to
find SUSY in general, yielding a simple realization of the stealth
supersymmetry idea. We analyze four benchmark spectra in detail.
