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

Let 𝒢_(m,n)(J) be the Gluck twist of the m-twist n-roll spin of a classical knot J. For fixed n, we prove that its oriented diffeomorphism type is independent of m and invariant under four-moves on J. The key local result is a framed compression disk for each crossing-circle product torus, whose push-off is the preferred longitude; even surgery in this direction is trivial relative to the complement. As an application, every longitudinally twisted double of Gompf's cork C(r, s;h), r, s>0>h, is diffeomorphic to S⁴ for every iterate, proving the conjecture attributed to Gompf by Tange. The same conclusion holds for simultaneous meridional and longitudinal twists and finite boundary sums. For each fixed cork, we classify the resulting two-end decompositions of S⁴ by the absolute difference of their exponents, allowing orientation reversal and exchange of sides. We also obtain relative extension results for crossing-torus exteriors.