Remark: Why the fifth geometric coefficient is retained

The order-${\kappa }^3$ gravitational action coefficient requires $[{\kappa }^5]\mathcal{E}[q]$. Truncating the geometry at fourth order would generally invalidate the third-order master identity. The division by ${\kappa }^2$ in (eq:Iinv) is handled before quotienting by ${\kappa }^4$, not by inverting a nilpotent element.