Proposition: Finiteness and continuity

Every admissible task is a finite real number and is continuous in the supplied smooth data on compact supports. Every finite task collection admits a finite report alphabet.

Proof. The moments are compact linear integrals. The charges are compact first-jet integrals. Finite exterior expansion produces finitely many ordinary real integrals. Compact smooth seminorm estimates prove continuity. A finite experiment returns a finite list of labels, outcomes, flags, and permitted status words.