Remark: What the background notation means

Later $g_{\mathrm{cl}}$ denotes the prescribed metric one-jet

The scalar preparation must also be specified: a relational scalar coefficient is not a functional of the metric alone. We therefore write the classical target honestly as

If ${\mathcal{O}}_f[g_{\mathrm{cl}}]$ is used as shorthand, the fixed scalar preparation, switch, and initial surface are suppressed inputs. None is reconstructed from quantum data.