Einstein coefficients and species

Fix the curvature sign and coefficients by the first-derivative density

whose integrated expansion, modulo divergences, is

For homogeneous $E_n$, set

Thus $v_n(u_1,\dots ,u_n)$ is the coefficient of $z_1\cdots z_n$ in $E_n[{\sum }_i{z}_i{u}_i]$. There is no additional factorial.

Let $d_{IJK}$ be real and completely symmetric. Define

and

The reference quartic is

With an interaction parameter $\lambda$, the locked action is

Every ${\theta }_{\ell }$ is constant on $\mathcal{X}$.

Einstein self-coupling and local deformation theory provide context Deser,BarnichHenneaux,BoulangerEtAl; no classification theorem from those works replaces the calculation below.