  We study diffraction catastrophes of wave functions in diffeomorphism
invariant quantum theories, for which $\hat H\Psi=0$. These wave functions can
be represented in terms of integrations over cycles in a complexified lapse
variable $N$. The integrand $\exp(i{\mathbb S}(N))$ may have multiple essential
singularities at finite values of $N$ and at infinity. A basis set for Greens
functions and solutions of the wave equation is represented by Lefschetz
thimbles connecting these singularities. The finite $N$ singularities are shown
to be directly related to $A_{n\ge 3}$ caustics. We give an example similar to
a minisuperspace cosmological model constructed by Halliwell and Myers, to
which we add a scalar field. We show that caustics with codimension $d\ge 2$
exhibit strong entanglement with respect to partitions of their unfolding
degrees of freedom. If an unfolding direction corresponds to a physical clock
in a solution of the Wheeler-DeWitt equation, the caustic bears some
resemblance to a quantum measurement. The R\'enyi entanglement entropy ${\cal
R}_n$ is expressed in terms of integrals over $2n$ lapse variables $N_i$.
Writing the integrand as $\exp(i\Gamma)$, we find that the finite $N$ essential
singularities of $\exp(i{\mathbb S})$ are replaced with non-essential
singularities of $\exp(i\Gamma)$ at cyclically related $N_i = N_j$ , which the
Lefschetz thimbles evade. The relative homology classes to which the
integration cycles belong are higher dimensional variants of links.
