  In this paper we analyze the fundamental solution of the \textit{quantum
difference equation} (qde) for the moduli space of instantons on
two-dimensional projective space. The qde is a $K$-theoretic generalization of
the quantum differential equation in quantum cohomology. As in the quantum
cohomology case, the fundamental solution of qde provides the capping operator
in $K$-theory (the rubber part of the capped vertex). We study the dependence
of the capping operator on the equivariant parameters $a_i$ of the torus acting
on the instanton moduli space by changing the framing. We prove that the
capping operator factorizes at $a_i\to 0$. The rationality of the $K$-theoretic
1-leg capped descendent vertex follows from factorization of the capping
operator as a simple corollary.
