Theorem: Cancellation in the complete coefficient expansion

In

restore the order-$\kappa$ shift $\gamma \mapsto \gamma +\kappa {\mathcal{L}}_{\xi }\gamma$ while applying the stated background transformation. The order-${\kappa }^2$ held-fixed defect cancels exactly.

Proof. Linearity gives

The first coefficient's $K\xi$ difference vanishes for a free scalar by (eq:KT). Changes of the second coefficient caused by this additional order-$\kappa$ shift start at ${\kappa }^3$. Its remaining difference is (eq:held), which has exactly the opposite sign. The tree has no external metric argument to vary here.