  We present a mathematical proof of Einstein's formula for the effective
viscosity of a dilute suspension of rigid neutrally--buoyant spheres when the
spheres are centered on the vertices of a cubic lattice. We keep the size of
the container finite in the dilute limit and consider boundary effects.
Einstein's formula is recovered as a first-order asymptotic expansion of the
effective viscosity in the volume fraction. To rigorously justify this
expansion, we obtain an explicit upper and lower bound on the effective
viscosity. A lower bound is found using energy methods reminiscent of the work
of Keller et al. An upper bound follows by obtaining an explicit estimate for
the tractions, the normal component of the stress on the fluid boundary, in
terms of the velocity on the fluid boundary. This estimate, in turn, is
established using a boundary integral formulation for the Stokes equation. Our
proof admits a generalization to other particle shapes and the inclusion of
point forces to model self-propelled particles.
