Kirby moves classify integral framed-link presentations

If $D_0,D_1$ are related by the stated finite sequence, apply [F2] to each move and compose its orientation-preserving diffeomorphisms. The result is an orientation-preserving diffeomorphism $M_0\to M_1$.

Conversely, let $f:M_0\to M_1$ be an orientation-preserving diffeomorphism. Use the two given diagrams as undecorated presentations based in the same standard oriented $S^3$. Take empty external boundary, empty ribbon graph and arbitrary equal auxiliary weights; the global supplier's undecorated comparison is independent of its later decoration and evaluation clauses. Every component framing is its actual normal two-disk framing, as in [F1]. Thus [F3] applies to this specified $f$ and yields a finite O1/O2/O3 and framed-isotopy sequence from $D_0$ to $D_1$. In particular it compares the given endpoint diagrams; it does not merely produce some new presentations of their output manifolds.

The O1 and O2 operations in this sequence are precisely the split signed-unit-unknot moves and fully framed slides of [F1]. Replace each O3 insertion or deletion by the finite sequence supplied by [F4]. That lemma has no restriction on the companion knot's linking with the rest of the diagram, so it applies at every intermediate stage. Its integer normal framings agree with [F1]. This replacement need not preserve the particular marked map $f$, which the present claim does not require, but it preserves the exact endpoint framed diagrams. A finite list with finitely many finite replacements is finite.

The resulting sequence uses only ambient isotopy, fully framed handle slides and split $\pm 1$ unknot stabilization or destabilization. It joins $D_0$ to $D_1$, proving the converse. Together with step 1.1 this establishes the equivalence with all stated orientation, smoothness and finiteness conventions.