  This paper contains two results of independent interest, the first being more
mathematical in nature whereas the second more physical. We first show that the
hierarchy of Higgs branch RG flows between the 6d $(1,0)$ SCFTs known as A-type
orbi-instantons is given by the Hasse diagram of certain strata and transverse
slices in the double affine Grassmannian of $E_8$. Secondly, we leverage the
partial order naturally defined on this Hasse diagram to prove the $a$-theorem
for orbi-instanton Higgs branch RG flows, thereby exhausting the list of
$c$-theorems in the even-dimensional (supersymmetric) setting.
