On the Completion Problem: Toward a Formal Reconstruction of Closure

Abstract This paper formulates the Completion Problem. Rather than assuming a closure operator, it asks whether a mathematically canonical notion of completion can be reconstructed from a primitive constraint structure. The paper introduces completion systems, distinguishes existence, minimality, uniqueness, and canonicality of completions, formulates the Completion Conjecture, and identifies the conditions under which closure may emerge as a derived operator rather than a primitive assumption. No identity, distinguishability, boundary, entity, or physical interpretation is assumed. The aim is deliberately narrow: to determine whether closure is derivable at all within the Reconstruction Program.