Abstract

We extend the constant-slot construction of XVI-J by retaining the full second variation of the declared unintegrated cubic density, including derivatives on both contracted tensor slots. The source remains restricted to the polarization of $2C_2\partial \partial {\sigma }_{\epsilon }$; the plateau source and the $M_2$ summand are not included.

The density's jet structure proves that the numerical edge products contain at most two derivatives in total. Their punctured distributions are homogeneous of degrees $-4,-5,-6$, and admit extensions with subtraction orders $0,1,2$. A single subtraction function and grade-dependent zero moments define a linear extension map on their finite-dimensional span. Constant-coefficient relations are preserved, while the possible defects of differentiation are retained as explicit contact distributions.

Under the imported compactness and $Z_2$ wavefront hypotheses for the root, pullback, multiplication, and proper pushforward construct a jointly continuous bilinear selected-sector coefficient. Its constant-slot truncation agrees with XVI-J, including the chosen extension at coincidence. Changes of prescription propagate derivatives of diagonal contact terms through the root.

The analytic existence proof does not depend on the supplied numerical coefficient tables. Those tables are documented separately as exact-arithmetic computational verification. The result is not a complete interaction coefficient, a retarded interacting field, or a BRST/BV observable.