Let S be a K3 surface. We study the reduced Donaldson–Thomas theory of the cap (S × ℙ1)/S∞ by a second cosection argument. We obtain four main results: (i) A multiple cover formula for the rank 1 Donaldson–Thomas theory of S × E, leading to a complete solution of this theory. (ii) Evaluation of the wall-crossing term in Nesterov’s quasimap wall-crossing between the punctual Hilbert schemes and Donaldson–Thomas theory of S × Curve. (iii) A multiple cover formula for the genus 0 Gromov–Witten theory of punctual Hilbert schemes. (iv) Explicit evaluations of virtual Euler numbers of Quot schemes of stable sheaves on K3 surfaces.