Theorem: Held-fixed metric defect and its sign

For smooth free $\varphi$, hold $\gamma$ fixed and transform the classical coefficients by

Then the ordinary forward variation is

Equivalently, the backward defect has the production sheet's positive sign:

Proof. The full transformation of $H=h+\gamma$ would require

The supplied X-C-v3 background identity applies to these smooth coefficients and the free scalar XC. It identifies ${{\mathcal{A}}_2}^{\mathrm{sc}}[H^{\prime },{k^{\prime }}_{\mathrm{full}};\varphi ]$ with ${{\mathcal{A}}_2}^{\mathrm{sc}}[H,k;\varphi ]$. Subtracting the omitted second coefficient and using (eq:klinear) gives (eq:forwarddefect). The tree is independent of $h,k,\gamma$ and has zero variation under this particular background operation.