  Light stops consistent with the Higgs boson mass of $\sim126\,{\rm GeV}$ are
investigated within the framework of minimal supergravity. It is shown that
models with light stops which are also consistent with the thermal relic
density constraints require stop coannihilation with the neutralino LSP. The
analysis shows that the residual set of parameter points with light stops
satisfying both the Higgs mass and the relic density constraints lie within a
series of thin strips in the $m_0-m_{1/2}$ plane for different values of
$A_0/m_0$. Consequently, this region of minimal supergravity parameter space
makes a number of very precise predictions. It is found that light stops of
mass down to 400~GeV or lower can exist consistent with all constraints. A
signal analysis for this class of models at LHC RUN-II is carried out and the
dominant signals for their detection identified. Also computed is the minimum
integrated luminosity for $5\sigma$ discovery of the models analyzed. If
supersymmetry is realized in this manner, the stop masses can be as low as
400~GeV or lower, and the mass gap between the lightest neutralino and lightest
stop will be approximately 30-40~GeV. We have optimized the ATLAS signal
regions specifically for stop searches in the parameter space and find that a
stop with mass $\sim 375\,{\rm GeV}$ can be discovered with as little as $\sim$
60~fb$^{-1}$ of integrated luminosity at RUN-II of the LHC; the integrated
luminosity needed for discovery could be further reduced with more efficient
signature analyses. The direct detection of dark matter in this class of models
is also discussed. It is found that dark matter cross sections lie close to,
but above, coherent neutrino scattering and would require multi-ton detectors
such as LZ to see a signal of dark matter for this class of models.
