Assumption: Finite Ward-insertion contract

For each $r$:

1. ${{\mathcal{T}}^r}_{\mu \nu }={{\mathcal{T}}^r}_{\nu \mu }$;

2. all displayed distributions, derivatives, and products with smooth functions are defined on $U_r$;

3. the improved insertion is declared by

   $${t^r}_{\mu \nu }={{\mathcal{T}}^r}_{\mu \nu }+\frac{1}{3}({\partial }_{\mu }{\partial }_{\nu }-{\eta }_{\mu \nu }{\square }_x){\mathcal{L}}^r;$$

4. a finite list of conformal Killing fields $X_a$ and compact joint tests $f_{raj}\in {C_c}^{\infty }(U_r)$ is fixed;

5. the intended Ward-contact assignment $X\mapsto {{\mathcal{C}}^r}_X$ is specified, and is linear on the tested real span of the $X_a$.