Inventory and vertices (Constructed)

An independent enumerator (one-loop 1PI graphs as cycles of at most 3 vertices; each vertex of an $E=3$ graph carries at least one external leg) is run on the p14r loop-capable vertices plus the ${\kappa }^3$ vertices. It reproduces XVI-P P23's 15 loop graphs exactly (hhh 5, h$\varphi \varphi$ 4, $\bar{c}ch$ 2, $Kch$ 2, $cc{c}^{\ast }$ 1, ${\varphi }^{\ast }c\varphi$ 1). It adds three ${\kappa }^3$ tadpoles: V5_grav $\to$ hhh, and V_hhh$\varphi \varphi$ with an hh line $\to$ h$\varphi \varphi$ or a $\varphi \varphi$ line $\to$ hhh. All three are scaleless (massless self-contraction; $\int d^{d}l{l}^{2k}/l^2=0$, no log member), so they are derived 0.

Power counting: at one loop $I=V$, so ${\sum }_v({\mathrm{legs}}_v-2)=E$ and the $\kappa$ power equals $E$ (checked on every enumerated graph). Hence the one-loop ${\kappa }^3$ one-point and two-point sectors are empty.

${\kappa }^3$ vertices: the ghost, antifield and gauge-fixing vertices are exact at their order. $sh=\frac{1}{2}{L}_c(\eta +\kappa h)$ is affine in $h$, $\Psi$ is linear in $h$, and $sc$, $s\varphi$ are exact at $\kappa$. The only new vertices are therefore V5_grav and V_hhh$\varphi \varphi$. The cubic term of $\sqrt{-g}{g}^{\mu \nu }$ is derived in closed form and agrees with an independent exact determinant/inverse series; a mutation is detected. V5_grav is not constructed, because it enters only the scaleless tadpole.