Representation as Universal Completion: Realization Complexes, Fiberwise Consequence, and the Floor-Relative Image Problem

Abstract

Abstract The unrestricted representation problem for finite dependency systems is trivial: if a regime may freely declare its coordinate domains, admitted relation, and semantics, then every finite system is realized by its own tautological presentation. This paper isolates the exact nontrivial residue by classifying the external realization fiber relative to a finite set of regime axioms. For each dependency system, the jointly satisfiable axiom sets form a finite simplicial complex. Its minimal nonfaces provide a canonical irredundant obstruction theory; its maximal faces form the completion frontier; and a universal axiom completion exists exactly when the complex is a simplex, equivalently when every minimal obstruction is unary. Every finite simplicial complex occurs in a freely decorated realization frame, so feasibility alone imposes no universal representation restriction. The realization fiber also carries a consequence operator induced by the satisfaction Galois connection. Feasibility and consequence are classified jointly: a simplicial complex and closure operator arise from a finite realization-profile family exactly when feasible sets close to feasible sets and infeasible sets collapse to the full axiom set. Every compatible pair has a canonical realizing family consisting of its feasible closed sets and a unique reduced normal form consisting of its essential profiles. Over a fixed feasibility complex, compatible consequence closures correspond order-reversingly to admissible feasible Moore families, yielding a finite dual lattice with canonical least and greatest consequence theories. The paper further proves that safe record omission and representation feasibility are the same finite combinatorial theory under the omitted-sets-as-faces correspondence. The unrestricted external-lift theory is thereby closed horizontally across all finite feasibility complexes and vertically across every compatible consequence structure over each complex. The remaining representation problem is isolated precisely: characterize, from the dependency system alone, the subclass of compatible feasibility–consequence pairs induced by the substantive floor axioms.