Lemma: Intrinsic charge

The coefficient $v_{\Phi }$ in (eq:charge-decomposition) is unique. Write it as ${\mathsf{ch}}_{\Phi }(F)$. It is complex linear and satisfies

Every microcausal polynomial has zero charge.

Proof. The decomposition holds for the generators by Proposition prop:boundary. A smooth metric directional derivative preserves the same form, and a scalar directional derivative changes only the smooth source in that formula. The ordinary Leibniz rule preserves the decomposition under products.

If two decompositions existed, their difference would express $\delta (t)(v-v^{\prime })+\theta (r-r^{\prime })$ as a distribution without timelike wavefront covectors. A nonzero $v-v^{\prime }$ produces the normal Fourier order-zero term of Proposition prop:boundary; neither the step term nor a distribution microlocally smooth in the normal directions can cancel it. Hence $v=v^{\prime }$.

Uniqueness makes the charge intrinsic to the actual functional and proves linearity and (eq:charge-leibniz). For a microcausal polynomial its first derivative has no timelike covectors, so its charge is zero.