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Abstract We lay the foundation for a version of $r$-spin theory in genus zero Riemann surfaces with boundary. In particular, we define notion disks, their moduli space, and Witten bundle; show that space is compact smooth orientable orbifold corners, prove bundle canonically relatively oriented relative to space. sequel this paper, use these constructions open intersection relate it Gelfand–Dickey hierarchy, thus providing an analog Witten’s conjecture setting.