  We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar
unitary conjugation bringing them into a general relative position. We prove
that the eigenvalue density on the scale slightly above the local eigenvalue
spacing is asymptotically given by the free convolution of the laws of $A$ and
$B$ as the dimension of the matrix increases.
  This implies optimal rigidity of the eigenvalues and optimal rate of
convergence in Voiculescu's theorem. Our previous works [3,4] established these
results in the bulk spectrum, the current paper completely settles the problem
at the spectral edges provided they have the typical square-root behavior. The
key element of our proof is to compensate the deterioration of the stability of
the subordination equations by sharp error estimates that properly account for
the local density near the edge. Our results also hold if the Haar unitary
matrix is replaced by the Haar orthogonal matrix.
