Lemma

The formulas in this appendix give the coefficients ${\mathcal{H}}_1,{\mathcal{H}}_2,{\mathcal{H}}_3$ of (eq:Hdef).

Proof. Expand $q^{-1}=u_0+\kappa u_1+{\kappa }^2{u}_2+\cdots$ in the exact Christoffel formula. Since $\partial q=\kappa \partial \gamma$, the order-$n$ Christoffel coefficient is (eq:Gamma-n). Substitution into (eq:ricci) yields (eq:R-n). Contracting the Ricci expansion with the inverse metric yields (eq:r-n); multiplying $q_{\mu \nu }R$ gives (eq:C-n). Finally, multiplying by $\sqrt{-q}$ and by two inverse metrics gives (eq:H-n).