Closed extended $r$-spin theory and the Gelfand-Dickey wave function

We study a generalization of genus-zero $r$-spin theory in which exactly one insertion has a negative-one twist, which we refer to as the “closed extended” theory, and which is closely related to the open $r$-spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the genus-zero part of a special solution to the system of differential equations for the wave function of the $r$-th Gelfand-Dickey hierarchy. This parallels an analogous result for the open $r$-spin generating function in the companion paper to this work.