Finite formulas for the gravitational Euler coefficients

The Einstein action has bulk expansion

where the higher coefficients are fixed by (eq:EH). Their Euler derivatives are $2{\mathcal{H}}_2$ and $2{\mathcal{H}}_3$. This identifies their classical role without constructing a tree.

Because the Einstein Euler derivative contains $1/\kappa$, its order-two coefficient involves the order-three metric curvature coefficient. A pullback covariance check must likewise retain the corresponding metric Taylor coefficients before projection. Truncating the metric pullback prematurely at second order would lose required information. Genuine compact flows avoid that bookkeeping ambiguity.