Structural Dimensionality and the Conditions of Set Admissibility

This paper extends the admissibility framework of the Modal–Dependence Calculus (MDC) from sequences to arbitrary sets. A set S is admissible if and only if every element x in S satisfies the definability condition D*(x, c), corresponding to tau(x) = 1. We introduce structural parameters of breadth (beta), defined as the cardinality of admissible elements, and depth (delta), defined as the supremum of dependence chain lengths terminating at the invariant core c. These parameters formalize the dimensional configuration of dependence structures without altering the admissibility condition. Set-level admissibility is therefore determined solely by universal grounding: the existence of any element with undefined dependence yields tau(S) = 0. This establishes that admissibility is a global structural property independent of scale, and that structural failure is non-compensatory across the set.