  We present a theoretical foundation for the Index theorem in naive and
minimally doubled lattice fermions by studying the spectral flow of a Hermitean
version of Dirac operators. We utilize the point splitting method to implement
flavored mass terms, which play an important role in constructing proper
Hermitean operators. We show the spectral flow correctly detects the index of
the would-be zero modes which is determined by gauge field topology. Using the
flavored mass terms, we present new types of overlap fermions from the naive
fermion kernels, with a number of flavors that depends on the choice of the
mass terms. We succeed to obtain a single-flavor naive overlap fermion which
maintains hypercubic symmetry.
