A dictionary for distributional observations

This section gives three linear inference rules for a separate distributional problem. It assumes no particular graph kernel and proves no finite-part existence theorem for a graph.

Let $X,Y$ be smooth manifolds, with a fixed test-function convention. Write $\mathcal{D}(X)={C_c}^{\infty }(X)$ and ${\mathcal{D}}^{\prime }(X)$ for complex distributions, using the complex-bilinear distribution–test pairing. Convergence below means distributional convergence.

The names VII-C and VII-D refer only to the separate Wick/graph track. Nothing from that track is needed to state or prove the results here.