  From a stable vector of a stable grading on a simple Lie algebra, Yun defined
a rigid automorphic datum that encodes a epipelagic representation, and also an
irregular connection on the projective line called $\theta$-connection. We show
that under geometric Langlands correspondence, $\theta$-connection corresponds
to the Hecke eigensheaf attached the rigid automorphic datum, assuming the
stable grading is inner and its Kac coordinate $s_0$ is positive. We provide
numerous applications of the main result including physical and cohomological
rigidity of $\theta$-connections, global oper structures, and a de Rham analog
of Reeder-Yu's predictions on epipelagic Langlands parameters.
