Lemma: Retarded left inverse and transposition

For a smooth past-compact tensor $u$,

For compact smooth tensor-density tests $A$ and past-compact smooth sources $B$,

The same equalities hold distributionally on the admissible compact kernel stages used below.

Proof. Use ${{\mathsf{P}}_{\mathrm{tr}}}^2=1$, constancy of ${\mathsf{P}}_{\mathrm{tr}}$, and ${{\mathsf{E}}_0}^{\mathrm{R}}\square u=u$ on past-compact fields. The second identity is retarded–advanced transposition. Only the compact intersection $J^{-}(\mathrm{supp}A)\cap \mathrm{supp}B$ contributes. Distributional versions follow by transposition on the same proper-support stages.