Proof

Proof. A metric variation supported outside $P_F$ cannot change $Lh$ on $F$. This follows from locality of $C$, the fixed zero Cauchy data, and finite propagation. A scalar variation outside $F$ cannot change $\partial \varphi$ on $F$. These facts prove the bounds for $B_f$. The bound for $W_g$ is immediate.

In (eq:Q)–(eq:QR), the covariance is fixed; only the displayed residual fields vary. This gives the partial-contraction bounds. The residual fields in ${\mathcal{T}}_{f,f^{\prime }}$ are $w$ on $F,F^{\prime }$, giving the two past sets.

The support of a classical product is contained in the union of the supports of its factors: a variation outside both supports changes neither factor. Apply this to (eq:oneedges) and the classical word.

For (eq:G1223), the fixed kernels $K,\overline{K}$ introduce no metric configuration dependence. Hence its metric support is empty, while its scalar dependence is confined to $F\cup F^{\prime }$. The other two two-edge outputs contain only $h_{00}$ on $G$ and one residual $w$ slot. This gives their displayed metric bounds and empty scalar support.

The inherited distributional pairings introduce no additional configuration variables.