  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.
