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