Remark: Why this is conditional

To apply Theorem thm:compatibility to VII-B and V, one must first prove that the actual $q=L^{\ast }a$, after the stated bundle and duality identification, is the tensor $c$ in (eq:bridge-wave-relation); that the relevant wave operator is this componentwise $\square$; and that V's selected solution space is identified with the comparison space being used.

Only after these identifications does the theorem reduce the remaining full-space representability test to conservation. The supplied passages do not perform those identifications or prove $K^{\ast }q=0$. The theorem therefore supplies a complete conditional bridge, not an unconditional bridge for the registered observable.