Proposition: Two contraction types under the stated convention

The connected ${\kappa }^2\hslash$ inventory of (eq:combinatorics) contains two types:

1. two lines between the $r\varphi$ part of $O_1$ and one $r\varphi \varphi$ vertex;

2. one scalar line from $F_f$ to a matter vertex, and a scalar line plus a metric line between that vertex and a second $r\varphi \varphi$ vertex.

Each has one external scalar leg and no explicit classical-background coefficient. There is no higher positive quantum power from these contraction graphs at order ${\kappa }^2$.

Proof. With $n$ interaction vertices and $E$ contraction lines, the retarded quantum power is $E-n$. For $n=1$, quantum power one requires $E=2$. Both lines must end on the two-leg order-one insertion. Only a matter vertex has the required metric and scalar legs.

For $n=2$, the insertion is $F_f$, with one scalar leg, and quantum power one requires $E=3$. After that root line is attached, two further lines must join the vertices. The attached vertex must retain a metric and scalar leg. Thus both vertices are matter vertices. A background quadratic vertex, a gravitational cubic, or a ghost vertex cannot match these species and leg counts.

A single $V_2$ attached to the one-leg root has at most one connecting line in this convention. An insertion without vertices has no self-contractions. The same degree bounds give $E\le 2$ for $n=1$ and $E\le 3$ for $n=2$; $n\ge 3$ requires higher coupling order.