New Semantic Unit Operations: A Graph-Theoretical Metamodel for Sequential Fallback and Spatial Branching in Valuation Networks.

Abstract

In classical first-order model theory, semantic valuations are traditionally restricted to static, univalent functional mappings (v: V to M). This abstraction fails to account for the discrete topological state transitions that occur during recursive evaluation and quantifier saturation. This paper establishes a novel computational metamodel that formalizes semantic valuations as directed network unit operations using structural graph theory. By mapping valuation architectures as Directed Acyclic Graphs (DAGs), we prove a strict dimensional coherence between the spatial density of multivariable arrays (indexed via horizontally structured multivariable arrays) and the exact scaling dimensions of their corresponding strictly upper-triangular adjacency matrices. We formalize an exhaustive taxonomy consisting of 4 atomic operators and 16 hierarchical two-stage cascades. This framework stabilizes the topological routing of semantic registers and delineates the exact algebraic boundary between classical deterministic closure and parallel state branching.