On the basis of shell model simulations, it is conjectured that the Lanczos
construction at fixed quantum numbers defines-within fluctuations and beha
vior very near the origin-smooth canonical matrices whose forms depend on t
he rank of the Hamiltonian, dimensionality of the vector space. and second
and third moments. A framework emerges that amounts to a general Anderson m
odel capable of dealing with ground state properties and strength functions
. The smooth forms imply binomial level densities. A simplified approach to
canonical thermodynamics is proposed.