The concepts of backward consistency and minimality are shown to be the essential tools in studying error propagation in fast least-squares adaptive filters. A conceptual framework identifying the principal error propagation mechanisms is developed, allowing the numerical stability of new or existing algorithms to be ascertained withoutthe technical labor previously thought necessary. The key findings show that arithmetic instability is not a fundamental feature of fast least-squares algorithms. Rather, unstable propagation can only derive from the violation of backward consistency constraints. The absence of minimality (in the system theory sense) is identified as the principal culprit in this regard, although the issue of stability for predictable sequences also plays an intriguing role. The concepts are illustrated for some popularly used algorithms, and order-recursive algorithms are confirmed to have intrinsic advantages over their transversal counterparts.