Input statement: XIII-A and XIII-D: two closed doors

For every nonempty $W$ with $\overline{W}\subset \mathrm{int}\{g=1\}\cap \{\epsilon <t<T\}$, the coefficientwise vanishing space ${\mathcal{N}}_W=\{F\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}:F{|}_{{\mathcal{C}}_W}=0\}$ satisfies: with $L_q=\int q^{\mu \nu }(D^{\ast }b{)}_{\mu \nu }$, $q\in {C_c}^{\infty }(W;{\mathrm{Sym}}^{2}TM)$, and $G_h=\int h{\mathrm{tr}}_{\eta }\gamma$, there are compact $q,h$ with

${\mathcal{N}}_W[[\hslash ,\kappa ]]$ is not a two-sided ideal of ${\star }_H$, no associative product on $({\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}/{\mathcal{N}}_W)[[\hslash ,\kappa ]]$ makes restriction a unital homomorphism, a unital homomorphism killing every $L_q$ while preserving the brackets forces $\hslash 1=0$, and the admissible rank-one ordering change gives ${\alpha }_d({L_q}^2)={L_q}^2+\hslash C(q,h{)}^2$ XIIIA. Completion of the auxiliary pairing inside ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ is Proved obstructed; the cochain ${\mathcal{C}}_{\xi }=-{\mathfrak{B}}_{\xi }[D\gamma ,\varphi ]$ is admitted and is not an integrated completion XIIID.