Hamming-weight obstruction to Fredkin-only Toffoli synthesis

Every Fredkin gate preserves the Hamming weight of a computational-basis string, so any network of these gates does too. A Toffoli gate maps $110$ to $111$ and hence cannot equal such a network on the same three wires. Fixed ancillas returned unchanged do not remove the obstruction. Extra encoded outputs or input-dependent garbage change the resource model and do not by themselves implement the same clean coherent three-qubit gate.