A boundary-marked five-bordism gives a common middle two-handle trace

Apply [F1] to the supplied $V$ in dimension five, keeping the full marked vertical collar. Remove its zero and five handles and order by index. The horizontal attaching face starts at connected $W_0$; a one-handle on a connected four-face leaves it connected. Join its feet by an embedded arc in the interior of that face, off the protected collar. The orientation-compatible attachment makes the closed core's normal bundle orientable. Thus [F2] trades it to a three-handle while retaining the same outgoing face, and induction trades all one-handles. Reorder, reverse, and repeat from connected $W_1$ for the reversed one-handles; these trades replace original index four by index two. No zero, one, four or five handle is reintroduced. Sort the remaining two- and three-handles by [F1] and choose the regular middle four-manifold $W_{\ast }$ between them.

From $W_0$ to $W_{\ast }$ only five-dimensional two-handles occur. From $W_1$ the reversed three-handles have complementary index $5-3=2$, so the same is true. Every such handle changes its four-dimensional level by interior framed circle surgery. Interior surgery preserves connectedness: the tube boundary $S^1\times S^2$ is connected, so gluing the old tube back could not join different components of its complement; since the old level is connected, that complement is connected, and the replacement attaches along the same connected interface. Thus every level in this construction stays connected, as required to iterate the base-circle argument.

Start with either original link presentation of $W_i$. By [F3] move the first five-dimensional handle's attaching circle to the fixed base, retaining its rank-three framing through an integral level frame. Its base representative avoids the old link, so choose its small tube disjoint from every old attaching thickening. By [F2] the actual four-dimensional surgery is two added two-handles on the corresponding $K$ and disk-framed zero meridian $U$. By [F4] its induced O3 boundary map is identity on the old link and the external base collar. Hence every previous handle and both end parametrizations are carried correctly. The new level again has a finite two-handle presentation from the same base.

Repeat step 3.1 for the finitely many five-dimensional two-handles from each end. At every stage the preceding conclusion gives exactly the hypothesis needed for [F3], so there is no circular assumption that an arbitrary four-manifold already has such a presentation. Both finite sequences terminate at the same actual level $W_{\ast }$ chosen in step 1.1, with its original full marked boundary, and give the stated two link descriptions. If no handles occur, the empty sequence is allowed.

If an actual five-bordism was not initially supplied, [F4] first corrects the signature seam by interior connected sums with signed projective planes, represented by split $+1$ or $-1$ framed unknot insertions. It produces the required connected fixed-collar $V$, without a fundamental-group hypothesis. Apply step 1.1, step 2.1, step 3.1, step 4.1 to these corrected traces. This adds precisely the asserted O1 preparation to the O3 common-middle construction. The conclusion leaves two descriptions of $W_{\ast }$; comparing them by an unrestricted trace-changing movie is a further theorem and is not inferred from their common diffeomorphism type.