Proposition: Relative free differential equals the auxiliary pairing

Assume {(UCP${}_W$)}. For every $c\in {\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}$ and every $u=(\gamma ,\varphi ,b)\in {{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$,

coefficientwise, where ${\mathfrak{B}}_{c/2}$ is defined by linearity from ${\mathfrak{B}}_{\xi }$. It vanishes on ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$. For $c=2\vartheta {\xi }_{\ast }$ on the detecting tuple, $s_0\mathcal{O}(u_{\ast })=-8I\vartheta \ne 0$. Moreover

Proof. Fix $\xi \in {\mathfrak{g}}_W$ and $u\in {{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$. Since ${{\Phi }^{(2)}}_{\mathrm{D}}$ is a polynomial of degree two in $\gamma$ and the scalar-sourced slot has no metric argument, $\lambda \mapsto {\Delta }_{\lambda \xi }\mathcal{O}(u)$ is a real polynomial of degree at most two:

For every real $\lambda$, $\lambda \xi \in {\mathfrak{g}}_W$, so Theorem thm:eval gives ${\Delta }_{\lambda \xi }\mathcal{O}(u)={\mathfrak{B}}_{\lambda \xi }[b,\varphi ]=\lambda {\mathfrak{B}}_{\xi }[b,\varphi ]$, the pairing being linear in the parameter through ${\mathcal{L}}_{\xi }\beta$. Comparing coefficients of $\lambda$ and ${\lambda }^2$ gives $⟨{{\mathcal{O}}^{\prime }}_{\gamma },K\xi ⟩(u)={\mathfrak{B}}_{\xi }[b,\varphi ]$ and (eq:quadzero). For $c={\sum }_i{\vartheta }_i{\xi }_i$ with odd generators ${\vartheta }_i$ and ${\xi }_i\in {\mathfrak{g}}_W$, linearity of $\mathcal{R}c=K(c/2)$ and of ${{\mathcal{O}}^{\prime }}_{\gamma }$ gives $s_0\mathcal{O}(u)={\sum }_i{\vartheta }_i{\mathfrak{B}}_{{\xi }_i/2}[b,\varphi ]={\mathfrak{B}}_{c/2}[b,\varphi ]$. The three forms in (eq:relO) are XII-F's shell rewriting $A\gamma =-2D^{\ast }b$ on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$ and the flat adjoint. Vanishing on ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$ and the witness value follow from Theorem thm:eval with $\xi ={\xi }_{\ast }$ and $c=2\vartheta {\xi }_{\ast }$.