Theorem: Classical presentation independence

If two admissible presentations agree on every smooth off-shell configuration, they define the same element of ${\mathcal{P}}_{\Sigma }$ and the same topology coordinates.

Kernel permutations, tensor identities, compact integration by parts with all surface terms retained, and valid bulk–surface decompositions preserve the admitted element. No equation-of-motion quotient is used.

Proof. Lemma lem:unique identifies their symmetric coefficient kernels. Equality of actual polynomials identifies every directional and metric derivative. Proposition prop:trace therefore identifies every boundary coordinate.

The stated operations preserve actual distributional pairings. In particular (eq:qa) is an equality, but deleting its delta term is not an allowed rewriting.