Homotopy $4$-spheres associated to an infinite order loose cork

We show the homotopy spheres ${\mathrm{\Sigma }}_n=-W{⌣}_{f^n}W$, formed by doubling the infinite order loose-cork $(W,f)$ by iterates of the cork diffeomorphism $f:\mathrm{\partial }W\to \mathrm{\partial }W$ is $S^4$. To do this we first show that ${\mathrm{\Sigma }}_n$ are obtained by Gluck twistings of $S^4$; then from this we show how to cancel $3$-handles of ${\mathrm{\Sigma }}_n$ and identify it by $S^4$.