The complete gate-list expansion rule

For each row $m=2,3,\dots ,15$, perform the following block, in the displayed order:

1. Apply $X_j$ for $j\in Z_m$, in increasing order.

2. Apply

   $$\mathrm{CCX}(0,1;u),\mathrm{CCX}(u,2;v),\mathrm{CCX}(v,3;f).$$

3. Apply $\mathrm{CX}(f;t)$ for $t\in T_m$, in increasing order.

4. Apply

   $$\mathrm{CCX}(v,3;f),\mathrm{CCX}(u,2;v),\mathrm{CCX}(0,1;u).$$

5. Apply $X_j$ for $j\in Z_m$, in decreasing order.

Concatenate all fourteen blocks in increasing $m$. This defines $U_4$.

Every instruction is now determined by a finite table and a fixed expansion rule. There is no instruction of the form “compare four bits” counted as one gate; the comparison is the explicit three-Toffoli conjunction chain. Every gate acts on distinct appropriate wires.