A finite coefficient specification through second order

For $j=0,1,2$, define

Here $[{\kappa }^n]$ is Taylor coefficient extraction, equivalently the $n$-th derivative at zero divided by $n!$. The curvature, split, and fermion have all been specified. Thus (eq:Ij) is a finite definition of every vertex required at this order, not an unspecified choice of a cubic or quartic action.

In particular, $I_2$ retains the Einstein–Hilbert coefficient through fourth geometric order. Omitting that coefficient would generally invalidate the second-order master identity.

At order zero,

Indeed, the quadratic Einstein–Hilbert coefficient is $\frac{1}{2}⟨l,\mathsf{A}l⟩$, and the background-only part has been subtracted.