  The shell model is the standard tool for addressing the canonical nuclear
many-body problem of nonrelativistic nucleons interacting through a static
potential. We discuss several of the uncontrolled approximations that are made
in this model to motivate a different approach, one based on an exact solution
of the Bloch-Horowitz equation. We argue that the necessary self-consistent
solutions of this equation can be obtained efficiently by a Green's function
expansion based on the Lanczos algorithm. The resulting effective theory is
carried out for the simplest nuclei, d and 3He, using realistic NN interactions
such as the Argonne v18 and Reid93 potentials, in order to contrast the results
with the shell model. We discuss the wave function normalization, the evolution
of the wave function as the "shell model" space is varied, and the magnetic
elastic effective operator. The numerical results show a simple renormalization
group behavior that differs from effective field theory treatments of the two-
and three-body problems. The likely origin of this scaling is discussed.
