Lemma: Existence of the required exact Feynman inverses

The choices above exist. For $\mathcal{B}$, they are obtained from the normally hyperbolic operator $\mathcal{L}$ by

They can be symmetrized under the real formal transpose without changing their exact inverse equations or their Feynman wavefront class.

Proof. The operators are smooth, normally hyperbolic after the displayed fiber normalization, and formally self-adjoint for nondegenerate Hermitian bundle pairings. The spacetime is globally hyperbolic. The exact Feynman construction in IS (Theorems 1.2 and 1.4) therefore applies. The required operator-sign convention is obtained by multiplying the operator and its inverses by $-1$ if needed.

The formula for $F_{\mathcal{B}}$ follows from $\mathcal{B}=\mathcal{A}\mathcal{L}$. Its transpose is also a left and right inverse of $\mathcal{B}$. The Feynman relation is transpose-invariant, so averaging the inverse and its transpose preserves the required microlocal class.