Assumption: Imported root

The root $B_{f,g,\mu \nu }(y,x)$ satisfies:

1. Its designated composition exists as a tensor-valued distribution on $M_y\times M_x$.

2. Its components have compact support: $B_{f,g,\mu \nu }\in {\mathcal{E}}^{\prime }(M_y\times M_x)$.

3. Componentwise, in the fixed Cartesian trivialization,

   $$\mathrm{WF}(B_{f,g,\mu \nu })\subset Z_2:=\{(y,x;\xi ,-\xi ):\xi \ne 0\}.$$

These are the root-composition hypotheses imported from XII-D, version 1, lem:root-kernel, as retained by XVI-J. They are assumed, not derived in this manuscript. The root's singular support is not removed.