Proposition: Existence of the declared evaluations

Every admissible task yields a finite real number. The dependence on its supplied smooth coefficient fields is continuous on compact supports. For any finite collection of tasks, the resulting report dictionary is finite.

This is not an algorithm for evaluating arbitrary smooth descriptions or a finite certificate of every continuum identity.

Proof. Each retained action coefficient has finite jet order and polynomial field degree. Its brackets and specified Euler derivatives have the same finiteness property. The compact tests or compact interaction profile make the integral finite. Expanding in the finite exterior basis produces finitely many ordinary smooth coefficient integrals. Finite seminorm estimates give continuity. The last assertion follows from choosing finitely many tasks and finite report alphabets.