Tadpoles and the background-dependent boundary

In the declared convention, a one-vertex tadpole is absent. A background-dependent graviton, ghost, or scalar one-point graph requires a second vertex. Feeding that order-${\kappa }^2$ metric one-point response through the order-$\kappa$ relational metric insertion first contributes at ${\kappa }^3$.

For example, the possible pairs are

The two legs between the vertices form the loop, while the metric leg reaches the relational insertion.

A different normal-ordering convention can generate one-vertex tadpole normalization terms. Such terms must be included explicitly, rather than declaring the two-graph inventory universal. In particular, the word “one-point function” must not silently mean an expectation value in an unspecified state.

The main lesson is narrower than the original plan: ghosts are necessary for the BRST carrier, but the non-tadpole scalar coefficient at ${\kappa }^2\hslash$ does not contain a ghost loop.