One cell observation confounds unrestricted interaction with residual error

Let $t,b\ge 2$ and observe one real response $Y_{ij}$ for each treatment $i$ and block $j$. In the additive mean model, the residual subspace consists of the double-centered arrays $E=\{\gamma \in {\mathbb{R}}^{t\times b}:{\sum }_j{\gamma }_{ij}=0\text{ for every }i,{\sum }_i{\gamma }_{ij}=0\text{ for every }j\},$ which has dimension $(t-1)(b-1)$ (Nested contrast models and projections in a balanced block design). If a treatment-by-block interaction $\gamma \in E$ is included in the mean, then the additive-fit residual is $\gamma +P_Eϵ$, where $P_Eϵ$ is the projection of the cell error onto $E$. For every $h\in E$ this same residual equals $(\gamma +h)+(P_Eϵ-h)$. Hence one observation per cell does not separately decompose the realized residual into unrestricted interaction and projected error, and fitting the unrestricted interaction leaves zero pure-error residual degrees of freedom. This is an algebraic design limitation; it does not assert nonidentifiability under every additional stochastic model.