Proposition: The auxiliary row is admitted and nonzero

For fixed $\xi ,f,g$, ${\mathfrak{B}}_{\xi }[b,\varphi ]$ is an even, ghost-number-zero compact microcausal polynomial of degree $(1,1)$ in $(b,\varphi )$. For the tuple of Theorem thm:obstruction,

Also the named lift ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}$ belongs to ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and

for that tuple. These conclusions persist at classical-symbol order under every admissible smooth change of ordering.

Proof. XII-D's compact root kernel $B_{f,g}(y,x)$ satisfies $\mathrm{WF}(B_{f,g})\subset Z_2$. On $U$, its collar factor is one, so ${\mathcal{L}}_{\xi }{\beta }_{f,g}$ is obtained from that kernel by differential operators and multiplication by fixed smooth compact parameter coefficients. Applying $D$ does not enlarge its wavefront set. Its $y$ support is in $S_{\xi }$ and its scalar projection is in the compact set

The resulting bilinear kernel is compact with wavefront set in $Z_2$, which excludes the two all-causal cones. Thus it belongs to XII-A's domain.

The initial configuration in Theorem thm:obstruction is on ${\mathcal{C}}_{\mathrm{sol}}$ and gives a nonzero value. Therefore ${\mathfrak{B}}_{\xi }\notin {\mathcal{N}}_{\mathrm{sol}}$.

The functional ${{\Phi }^{(2)}}_{\mathrm{D}}$ is already admitted. Substitution of the smooth configuration $K\xi$ into some of its metric slots, followed by subtraction, preserves compactness and microcausality. Hence its named finite-difference lift is admitted. Its nonzero shell evaluation proves it also lies outside ${\mathcal{N}}_{\mathrm{sol}}$.

For the present switch, (UCP) holds independently of $\gamma$. Thus the companion vanishes for every free scalar and metric, and the two lifts ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}$ and ${\mathfrak{B}}_{\xi }$ have the same restriction to ${\mathcal{C}}_{\mathrm{sol}}$.

Finally, the inherited ordering maps are the identity modulo $\hslash$. A nonzero classical-symbol class cannot be annihilated by such transport XIIA.