Structural Properties of the Evaluation Operator

This paper derives the formal properties of the system-level evaluation operator used in the Modal-Dependence Calculus. The operator determines structural admissibility by assigning a value to a set based on the states of its elements, where a structure is admissible only if all elements satisfy the required dependence condition. The analysis establishes key properties of the operator, including idempotence, monotonicity, minimality, and invariance under reindexing. These results show that admissibility is determined entirely by element-level closure and is unaffected by changes in size, ordering, or representation of the structure. This provides a complete characterization of the evaluation mechanism governing system-level admissibility.