Theorem: Faithfulness and classical presentation independence

The construction has the following properties:

1. no two distinct bulk polynomial functionals are identified;

2. if two admissible presentations define the same polynomial functional on every smooth off-shell configuration, they define the same element of ${\mathcal{P}}_{\Sigma }$, with the same topology coordinates;

3. kernel permutations, tensor symmetries, compact integration by parts with all boundary terms retained, and bulk/surface decompositions that preserve the actual functional do not change the admitted element.

Equation-of-motion equivalence is not included.

Proof. The first assertion is the injectivity in Proposition prop:bulk.

For the second, Lemma lem:kernelunique identifies the symmetric coefficient kernels. Equality of the functionals also identifies all their directional derivatives and metric derivatives. Lemma lem:charge then identifies every boundary coordinate.

The listed rewritings preserve the distributional pairing when all terms are retained. For example, the decomposition

is equality of compact distributions. Dropping its delta term is not one of the admitted rewritings.