  We define categories of Stokes filtered and Stokes graded $G$-local systems
for reductive groups $G$ and use the formalism of Tannakian categories to show
that they are equivalent to the category of $G$-connections. We then use the
interpretation of moduli spaces of Stokes filtered $G$-local systems as braid
varieties to prove physical rigidity of two well-known families of
cohomologically rigid connections, the Kloosterman and Airy connections. In the
Kloosterman case, our proof relies on Steinberg's cross-section.
