Proposition: Actual-functional presentation independence

The partial product respects all finite relations between actual presentations in its declared domain. Kernel permutation, tensor symmetry, and compact integration by parts preserving the actual off-shell functional do not change the result. Boundary terms must be retained. Equation-of-motion identification is not included.

Proof. The four presentation maps are injective. The six new domains form a direct sum, and their whole old overlap is zero. Thus every coefficient is a map on actual tensors, independent of both test presentation and domain decomposition. The inherited old coefficients already have that property. Valid functional rewritings represent the same actual argument; dropping a term in (eq:sharpadjoint) is not such a rewriting.