The local fish does not supply the full residual

Under the specified flat Feynman conventions, the local scalar–metric fish components are

They are defined off zero and have scaling degree at most six. Finite-scaling-degree extensions exist, and their internal ambiguity has the form

This is the local extension interface retained from X-G and the standard extension theorem XG,XGST,BF.

For a separately chosen extension and finite coefficient list, the resulting local counterterm functional and its classical BV variation can be specified and evaluated on compact smooth inputs. All external differentiations and derivatives of test coefficients must be retained. The internal derivative bound is not automatically a bound of two on the total derivative order of the resulting local density.

None of these local data defines the missing nonlocal coordinate contraction or the entire one-loop master residual.