Theorem: Selected-sector coefficient and continuity

Assume Input input:all, choose the ambient propagator (eq:DF), and fix ${\mathfrak{R}}_{\varpi }$. For the $C_2$/constant-slot sector only,

is defined for every real smooth residual metric $h$ and real smooth scalar $\phi$.

It is jointly continuous and bilinear for their $C^{\infty }$ topologies. Its values are generally complex. There are compact sets $K_h,K_{\phi }$, a finite integer $m$, and a constant $C$ such that

The integer $m$ is not evaluated.

The coefficient agrees with the original punctured sector when localized away from $y=w$. Nonvanishing of (eq:C) is not asserted.

Proof. Existence and the pairing with noncompact smooth functions follow from Theorem thm:attachment and compact support. Constants may equivalently be replaced by cutoffs equal to one on that support.

Fix the compactly supported component distributions

They have some common finite distributional order $m$ on a compact neighborhood of their supports. Lemma lem:Llinear shows that $h\mapsto \mathcal{K}(y,w;h)$ is a linear differential operator of order two with fixed smooth coefficients. Applying the order-$m$ estimate to $\mathcal{K}(y,w;h)\phi (x)$, and using the product rule, gives (eq:continuity). The compact sets may be taken as the corresponding $w$- and $x$-projections of that neighborhood.

The finite-order estimate proves joint continuity and bilinearity. It does not identify distributional order with the pointwise order-two residual jet: the additional $m$ comes from the fixed attached distributions.

Finally, the extension restricts to $q^{\circ }$ away from the diagonal, so the localized pairings agree.