  We carry out the first rigorous numerical proof based on Evans function
computations of stability of viscous shock profiles, for the system of
isentropic gas dynamics with monatomic equation of state. We treat a selection
of shock strengths ranging from the lower stability boundary of Mach number
$\approx 1.86 $, below which profiles are known by energy estimates to be
stable, to the upper stability boundary of $\approx 1669$, above which profiles
are expected to be provable by rigorous asymptotic analysis to be stable. These
results open the possibilities of: (i) automatic rigorous verification of
stability or instability of individual shocks of general systems, and (ii)
rigorous proof of stability of all shocks of particular systems.
