Abstract

We construct a selected-sector coefficient containing only the polarized source $2C_2\partial \partial {\sigma }_{\epsilon }$, the constant-slot part of the declared cubic Hessian, and two undifferentiated graviton edges. The plateau source, the $M_2$ source term, and vertex terms differentiating contracted fields are excluded.

The tensor contraction is derived from the inverse fiber pairing and is independent of the chosen symmetric-tensor basis. The Hessian and source-decorated coefficient on a specified pure-trace background are proved by explicit polynomial identities, without reliance on an unavailable script. An appendix supplies a second, exact-arithmetic verification through curvature atoms and two implementations of the source contraction.

The numerical square has scaling degree four in four relative dimensions. A subtraction function and a zero-moment condition select its extension; an explicit distributional action is given. Under the imported compactness and $Z_2$ wavefront bound for the root, pullback, multiplication, and proper pushforward construct a continuous bilinear functional of the residual metric and scalar test. Its prescription dependence is a diagonal contact term propagated through the root. This is not the complete two-edge interaction coefficient, a retarded interacting field, or a BRST/BV observable.