Theorem: Selected C 2 coefficient with full-Hessian vertex

Retain Assumptions ass:preparation, ass:root, and ass:normalization, the fixed density (eq:ell3), and the prescription ${\mathfrak{R}}_{\mathrm{K}}$. Here the full-Hessian vertex includes every derivative on both contracted slots of that density. The source remains restricted to $C_2$; neither the plateau source nor $M_2$ is included.

For every smooth residual metric $h$ and smooth scalar $\phi$, the selected-sector coefficient

is defined. The abbreviation $\mathrm{fv}$ denotes the full-Hessian vertex within this selected sector.

The functional is jointly continuous and bilinear for the $C^{\infty }$ configuration topologies. There exist a constant $C_0$, compact sets $K_h,K_{\phi }$, and a finite integer $m$ such that

The integer $m$ and sharp seminorm orders are not evaluated.

On compact tests supported away from $y=w$, the attached distribution agrees with the punctured expression (eq:QK). Nonvanishing of the final functional is not asserted.

Proof. Lemma lem:attachK supplies the products under (R3) and compact support under (R2) together with compactness of ${\chi }_{\mathrm{I}}$. The pushforward and smooth pairing therefore exist.

Choose $\zeta \in {C_c}^{\infty }(X)$ equal to one on a neighborhood of the fixed set $K_U$, independently of $h$. The last pairing in (eq:CK) means

Here $T_{\mathsf{k},\mu \nu }$ is exactly the unswitched distribution (eq:T). The test contains the one switch already present in $a_{\mathsf{k}}$. There is no division by ${\chi }_{\mathrm{I}}$ and no second factor of that switch.

On the fixed compact support of the test, the finitely many $T_{\mathsf{k},\mu \nu }$ have a common finite distributional order $m$. Each $a_{\mathsf{k}}(h)$ is a linear differential expression in $h(w)$ of order at most two, with fixed smooth coefficients depending on $y,w$. The distributional order-$m$ estimate and the ordinary product rule therefore give (eq:continuity); the compact sets may be chosen as projections of a compact neighborhood of $\mathrm{supp}\zeta$.

Linearity in $h$ follows from Proposition prop:hess, and linearity in $\phi$ follows from the pairing. The product seminorm estimate proves joint continuity. The additional $m$ is distributional order, not residual jet order: derivatives in the fixed attached distributions may differentiate the coefficient beyond its pointwise jet.

Finally, restriction of an extension recovers its punctured kernel. Pullback and the distributional product are local, so the attached expressions agree away from coincidence.