Proposition: Defined classical report values

Every task (R1) or (R2), at the released orders, has real extracted value zero. The evaluations are continuous in their smooth data on common compact support stages. A nonempty finite dictionary of these tasks exists.

Proof. Theorems thm:kinematic and thm:BV give the zero identities before evaluation. The compact evaluation rule respects the divergence identification.

At each formal order the expressions are finite local differential polynomials. Compact integration and a fixed coefficient extraction are continuous in the smooth seminorms of their finitely many input jets.

For nonemptiness choose a nonzero real bump $\rho \in {C_c}^{\infty }(U)$, any compact tensor profile for $r$, and, for example,

with compact antifield profiles and a finite exterior algebra containing the indicated generators. These are off-shell test profiles. At least one such task can be installed without requiring a nonzero compact free ghost.