Theorem: Reduced graph on the punctured domain

For every smooth real $h$, all blocks ${{\mathcal{V}}^{\alpha \beta }}_{IJ}[h]$ and all ${U_A}^{\circ }[h]$ are distributions on $\Omega$. They are linear in $h$. Their coefficients and derivative placements are exactly specified by (eq:L), (eq:Vblock), and (eq:U).

The intended compressed attachment is

on the full variables $(y,w,x)$, followed by integration in $y$. Its definition is tested separately in Section sec:attachment.

Proof. Using $r=y-w$,

Thus each summand is a constant tensor contraction of one of the products in Proposition prop:product, multiplied by smooth coefficients. Every sum is finite. The Hessian coefficients are linear in $h$.