Theorem: The classical normalization determines

For every $h\in \mathcal{B}$, the map $R_1(V_{1,g}[h];\cdot )$ of (eq:Rone) satisfies (eq:sheet), and it is the only map ${\mathcal{L}}_{\mathrm{off}}\to {{\mathfrak{A}}_0}^{(1)}$ that does.

Proof. Existence is Lemma lem:transpose. If $\mathsf{R}$ satisfies (eq:sheet), write $\mathsf{R}(F_c)=\hat{\varphi }({d^{\prime }}_c)$ with ${d^{\prime }}_c\in \mathcal{D}$. Then $⟨{d^{\prime }}_c-d_c[h],u⟩=0$ for all $u\in {\Delta }_S\mathcal{D}$, so ${d^{\prime }}_c-d_c[h]\in P_m\mathcal{D}$ by Lemma lem:detect, and $\hat{\varphi }({d^{\prime }}_c)=\hat{\varphi }(d_c[h])$ by Lemma lem:graded.