  Local diffusion of strictly hyperbolic higher-order PDE's with constant
coefficients at all simple singularities of corresponding wavefronts can be
explained and recognized by only two local geometrical features of these
wavefronts. We radically disprove the obvious conjecture extending this fact to
arbitrary singularities: namely, we present examples of diffusion at all
non-simple singularity classes of generic wavefronts in odd-dimensional spaces,
which are not reducible to diffusion at simple singular points.
