Proposition: Actual-functional presentation independence

The constructed coefficient maps preserve every rewriting that leaves the declared tensor of actual off-shell functionals unchanged. This includes finite presentation relations, polarization, kernel permutations, tensor symmetries, and integration by parts with all boundary terms retained.

Proof. The coefficient maps are linear maps on a subspace of the actual tensor product ${\mathcal{P}}_{\Sigma }\otimes {\mathcal{P}}_{\Sigma }$. Within the presentation blocks, actual equality is respected by the inherited degree-two descent and Theorem thm:degree4. Across blocks it is respected by simultaneous gluing. Hence equal actual tensors have equal coefficients.