  The entropy of a half-BPS black hole in N=4 supersymmetric heterotic string
compactification is independent of the details of the charge vector and is a
function only of the norm of the charge vector calculated using the appropriate
Lorenzian metric. Thus in order for this to agree with the degeneracy of the
elementary string states, the latter must also be a function of the same
invariant norm. We show that this is true for generic CHL compactifications to
all orders in a power series expansion in the inverse charges, but there are
exponentially suppressed corrections which do depend on the details of the
charge vector. This is consistent with the hypothesis that the black hole
entropy reproduces the degeneracy of elementary string states to all orders in
a power series expansion in the inverse charges, and helps us extend the
earlier conjectured relation between black hole entropy and degeneracy of
elementary string states to generic half-BPS electrically charged states in
generic N=4 supersymmetric heterotic string compactification. Using this result
we can also relate the black hole entropy to the statistical entropy calculated
using an ensemble of elementary string states that contains all BPS states
along a fixed null line in the lattice of electric charges.
