Route 3: the held-fixed sign cancels

Let ${{\mathcal{A}}_2}^{\mathrm{sc}}[H,k;\varphi ]$ be the complete classical scalar-only second coefficient from X-A: it includes the second scalar response, mixed response–coordinate term, inverse-coordinate term, and quadratic Taylor term XA. Its dependence on the second metric coefficient is

Indeed, the $k$ change in ${\mathsf{X}}_2$ is ${\mathsf{X}}_1[r]$, and the $k$ change in the second scalar response is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{V}}_{1,g}[r]\varphi$.

Set $H=h+\gamma$ and

For a free scalar, XI-C's forward held-fixed difference is

where