A precise restricted second-order coefficient

Use X-C-v3's completed scalar-only coefficient

Here $c_2$ is the copied bare second-coordinate test, and ${\mathsf{L}}_{2,g}[H,k]$ uses $a_1[k]+a_2[H]$ and $b_1[k]+b_2[H]$. All entries are evaluated on actual smooth configurations. This is not a quantum admission assertion.

Its dependence on the second metric coefficient is

Indeed, the $k$ contribution to ${\mathsf{X}}_2$ is ${\mathsf{X}}_1[k]$, and the $k$ contribution to ${\mathsf{L}}_{2,g}$ is ${\mathsf{L}}_{1,g}[k]$.

Define the restricted candidate

It names the scalar-only completion together with the single graviton component constructed here. It is not declared to be the full interacting-gravity coefficient.