Negative-Entropy Construction Theory III

This paper continues the development of the Negative-Entropy Construction Theory (NECT) series. Part I (Yu, 2026a) established the logical-domain framework, distinguished internal formal consistency from global descriptive adequacy, and introduced structural residuals. Part II (Yu, 2026b) studied the passage from logical domains to geometric and dynamical realizations, including feasible state sets, local admissible variations, manifold realizations, and dynamical classification. The present paper, Part III, develops a mathematical framework for systems organized across multiple structural levels. A hierarchical logical domain is represented by a collection of objects, aggregation structures, relations, constraints, and a state space. The framework distinguishes object existence, relational structure, admissibility, and state realization, which are often conflated in informal descriptions of hierarchical systems. We first formalize aggregation, including quotient aggregation and overlapping aggregation, and distinguish both from relational and constraint structures. We then introduce inter-level transformations and study the information preserved or lost under state mappings. An observable factors through a higher-level representation precisely when it is constant on the fibers of the corresponding state map. For differentiable recursive maps, the subspaces of initially accessible directions that become locally invisible form a nested sequence, yielding an exact rank-based expression for the incremental loss of differential visibility. We extend the framework to hybrid continuous-discrete domains, in which each level consists of discrete branches carrying continuous state spaces. We define inter-level branch correspondences, same-level transition networks, feasible transition edges, and several distinct notions of closure. For finite directed graphs, terminal strongly connected components provide nonempty feasible closed subsets. We further construct a recursive closure expansion and establish its closure-operator properties under a fixed feasible graph. Finally, we examine the relations among branch cardinality, edge cardinality, graph organization, and continuous transition capacity. These descriptors are not mutually determining: branch and edge counts do not uniquely determine one another, and discrete graph invariance does not imply preservation of the continuous feasible regions associated with edges. Edge density is a derived quantity determined by branch and edge cardinalities.