Lemma: Preservation of the fixed backgrounds

Every admitted $F_{\kappa }$ is the identity near $\Sigma$ and outside a compact subset of $U$, preserves the slab, and satisfies

The truncated pullbacks form a group.

Proof. A compactly supported smooth vector field has a complete flow, which fixes the complement of its support. Here the flow is supported where both $g$ and $\sigma$ are one. It is the identity near the slab endpoints. Closure of the truncated pullbacks follows by expanding their product: the order-two bracket contribution is $\frac{1}{2}[\xi ,{\xi }^{\prime }]$, again supported in $U$.