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The Cumulant Expansion for the Anderson Lattice with Finite U: The Completeness Problem

``Completeness'' (i.e. probability conservation) is not usually satisfied in the cumulant expansion of the Anderson lattice when a reduced state space is employed for $U\to \infty $. To understand this result, the well known ``Chain'' approximation is first calculated for finite $U$, followed by taking $U\to \infty $. Completeness is recovered by this procedure, but this result hides a serious inconsistency that causes completeness failure in the reduced space calculation. Completeness is satisfied and the inconsistency is removed by choosing an adequate family of diagrams. The main result of this work is that using a reduced space of relevant states is as good as using the whole space.