This paper asks how explicit representations can support reusable cognitive schemas in knowledge-based problem solving. We develop a structural framework in which schemas are organized by the information and relations required for their use, rather than introduced as unrelated primitives.

The framework also distinguishes context-dependent relations from more stable structures that can be reused across different representations. Tasks are described through the information available, the unknowns to be determined, and the constraints that admissible solutions must satisfy.

This makes it possible to separate limitations of the representation from limitations of the solving procedure. In particular, we distinguish inconsistency, underdetermination, and contextual insufficiency, where the current representation lacks distinctions or relations required by the external task meaning.

We also show formally when a reduction of representation preserves the task-relevant solution structure. The resulting task--schema interface offers a structured way to describe representational conditions relevant to problem solving.

It supports the reuse and stabilization of derived knowledge while remaining independent of the particular mechanism used to generate candidate solutions. This may provide a useful component for future solver architectures that combine structured knowledge, verification, and learned proposal mechanisms.