  We study how far the Index Theorem can be extrapolated from the continuum to
finite lattices with finite topological charge densities. To examine how the
Wilson action approximates the Index theorem, we specialize in the lattice
version of the Schwinger model. We propose a new criterion for solutions of the
Ginsparg-Wilson Relation constructed with the Wilson action. We conclude that
the Neuberger action is the simplest one that maximally complies with the Index
Theorem, and that its best parameter in d=2 is m0=1.1 +- 0.1 .
