In lemma4.5, of PlonK: Permutations over Lagrange-bases for Oecumenical Noninteractive arguments of Knowledge they claim that we can construct a polynomial protocol $P^*$ with an $S$-ranged polynomial protocol $P$. However, in my opinion, I think it constructed $P$ using $P^*$ in the proof.
Specifically, in the last step of the construction, the verifier queries the identity of a polynomial(not in a range $S$), so it is invoking a polynomial protocol to construct an S-ranged polynomial protocol, not the opposite. In the construction of the proof of lemma4.5, I think the prover can prove that $F_i(x)$ equals $0$ in range $S$.
Can anyone explain if my thoughts are right? Thanks.