The frozen scalar density is not this graph

For a specified scalar two-point distribution ${\mathsf{H}}_2$,

is definable because $j_{\lambda }\otimes j_{\mu }$ is a compact smooth test density. That is a scalar pairing, not the tensor two-edge product, and it does not determine the interaction Hessian, normalization, or graph extension.

Here the tensor edge prescription has been explicitly chosen. It does not provide a positive state for the full gauge multiplet, ghost and scalar time-ordered blocks, or a complete family of time-ordered products.