Boundary charges in gauge theories (such as the ADM mass in general relativity) can be understood as integrals of linear conserved n − 2 forms of the free theory obtained by linearization around the background. These forms are associated one-to-one with reducibility parameters of this background (such as the time-like Killing vector of Minkowski spacetime). In this paper, closed n − 2 forms in the full interacting theory are constructed in terms of a one-parameter family of solutions to the full equations of motion that admits a reducibility parameter. These forms thus allow one to apply Stokes theorem without bulk contributions and, provided appropriate fall-off conditions are satisfied, they reduce asymptotically near the boundary to the conserved n − 2 forms of the linearized theory. As an application, the first law of black-hole mechanics in asymptotically anti-de Sitter spacetimes is derived.