  We provide a rigorous proof of the fact that the level density of all su(m)
spin chains of Haldane-Shastry type associated with the A_{N-1} root system
approaches a Gaussian distribution as the number of spins N tends to infinity.
Our approach is based on the study of the large N limit of the characteristic
function of the level density, using the description of the spectrum in terms
of motifs and the asymptotic behavior of the dispersion relation.
