Scope.

.{ Here “full” means the full second variation of the fixed unintegrated cubic density, not the complete interaction. All derivative placements on its two contracted slots are retained. The overall sector still contains only the $C_2$ source and two graviton edges.

The preparation width $0<\epsilon <T$, preparation data $f,g$, and ramp ${\sigma }_{\epsilon }$ are fixed. The independent interaction switch is ${\chi }_{\mathrm{I}}\in {C_c}^{\infty }(M;\mathbb{R})$.

The five levels are the Hessian coefficients, punctured scalar kernels, their selected extensions, smooth source-decorated coefficients, and the root-attached functional. Statements at one level do not imply statements at another.

No sharp-width limit, state, all-orders time ordering, ultraviolet completion, or interacting gauge identity is constructed. }