Graph variables and the admissible domain

Let $X_{\Gamma }=M^{n_{\Gamma }}$ be the numerical vertex space of one graph in the copied inventory. Let

Observable-only singular subgraphs keep their inherited definition.

Write $k_e(v_a-v_b)$ for each numerical line, including its specified derivatives and constant fiber maps. The lines are tempered distributions in relative coordinates. The two new cross-lines use the full $H_T$ kernels.

Let ${\mathcal{U}}_{\Gamma }$ be the open set on which the complete line product satisfies the Hörmander sum condition. We use

The inherited pinch configuration shows why this set cannot be identified with the entire collision complement by the supplied product argument.