Cancellation of quantum mechanical higher loop contributions to the gravitational chiral anomaly

We give an explicit demonstration, using the rigorous Feynman rules developed in~$\0^{1}$, that the regularized trace $\tr \gamma_5 e^{-\beta \Dslash^2}$ for the gravitational chiral anomaly expressed as an appropriate quantum mechanical path integral is $\beta$-independent up to two-loop level. Identities and diagrammatic notations are developed to facilitate rapid evaluation of graphs given by these rules.