The smooth four-dimensional boundary

The h-cobordism bridge used on this page requires a connected smooth h-cobordism of dimension at least six between closed simply connected manifolds: for a homotopy $n$-sphere the two-disk complement is an $n$-dimensional cobordism with $S^{n-1}$ faces, so the argument applies in the range $n\ge 6$ and in particular to the seven-dimensional Milnor spheres. In dimension four the corresponding two-disk complement is a four-dimensional cobordism with three-dimensional spherical boundary components, and the h-cobordism theorem is not available there: the bridge used here simply does not apply.

Nothing above proves or refutes a smooth four-dimensional Poincaré theorem, and no smooth four-dimensional exotic sphere is constructed or excluded. The existence of exotic smooth structures in dimension four is a separate question that this page neither uses nor settles; the seven-dimensional examples and their homeomorphism-to-$S^7$ proof say nothing about it.