  In alloys cluster expansions (CE) are increasingly used to combine
first-principles electronic-structure and Monte Carlo methods to predict
thermodynamic properties. As a basis-set expansion in terms of lattice
geometrical clusters and effective cluster interactions, the CE is exact if
infinite, but is tractable only if truncated. Yet until now a truncation
procedure was not well-defined and did not guarantee a reliable truncated CE.
We present an optimal truncation procedure for CE basis sets that provides
reliable thermodynamics. We then exemplify its importance in Ni$_3$V, where the
CE has failed unpredictably, and now show agreement to a range of measured
values, predict new low-energy structures, and explain the cause of previous
failures.
