Four-move invariance of Gluck twists and standard twisted cork doubles

We prove that even torus surgery along a framed compression direction is trivial relative to the complement of the compression neighborhood. For every crossing circle $C$ of a classical knot $J$, we construct such a disk for $C\times S^1$ in the Gluck manifold $\mathcal{G}_{m,n}(J)$ of the $m$-twist $n$-roll spin; its disk push-off is the preferred longitude of $C$. Together with a separate comparison of the twist parameter, this proves that, for fixed $n$, the oriented diffeomorphism type of $\mathcal{G}_{m,n}(J)$ is independent of $m$ and invariant under four-moves on $J$. Consequently, every longitudinally twisted double of Gompf's cork $C(r,s;h)$ is diffeomorphic to $S^4$ for all $r,s>0>h$ and all integer iterates, including every odd negative framing. This proves the conjecture attributed to Gompf by Tange. We also treat simultaneous meridional and longitudinal twists and the corresponding finite boundary sums. For each fixed cork, the resulting two-end decompositions of $S^4$ are classified by the absolute difference of their exponents, even when orientation reversal and exchange of sides are allowed.