  The overlap hypercube fermion is constructed by inserting a lattice fermion
with hypercubic couplings into the overlap formula. One obtains an exact
Ginsparg-Wilson fermion, which is more complicated than the standard overlap
fermion, but which has improved practical properties and is of current interest
for use in numerical simulations. Here we deal with conceptual aspects of the
overlap hypercube Dirac operator. Specifically, we evaluate the axial anomaly
and the index, demonstrating that the correct classical continuum limit is
recovered. Our derivation is non-perturbative and therefore valid in all
topological sectors. At the non-perturbative level this result had previously
only been shown for the standard overlap Dirac operator with Wilson kernel. The
new techniques which we develop to accomplish this are of a general nature and
have the potential to be extended to overlap Dirac operators with even more
general kernels.
