CLOSURE MATHEMATICS AND THE CONSTRUCTION OF WORLDS — V From Agency to Cognitive Worldhood

Cognition is often characterized through memory, prediction, control, internal models, abstraction, or flexible behavior, but these capacities need not coincide. A system can retain information without constructing a world model, predict without preserving causal or counterfactual structure, or control successfully without representing what is absent. This paper develops a target-relative mathematical framework for distinguishing these cases and for formalizing the transition from operational agency to cognitive worldhood. Given a reachable-history space , a family of cognitive targets , and a continuation horizon , we define a continuation-sensitive target signature that records both target consequences and continuation admissibility. Equality of these signatures induces a canonical equivalence relation and quotient , representing the exact history distinctions required by the specified targets and horizon. Approximate implementations are evaluated by a target-and-horizon representation defect . Exact distinction, however, may exceed the distinction required by an implemented cognitive architecture. We therefore introduce an acceptable-realization map , construct a cognitive incompatibility hypergraph , and show that the minimum number of mutually compatible representational classes is its weak chromatic number. This yields a requisite cognitive variety , a lower bound on the state capacity of any deterministic representation satisfying the cognitive admissibility criterion. We further relate this compatible variety to the exact quotient variety and establish recursive realizability of the full continuation quotient using its right-congruence structure. On this foundation, cognitive worldhood is defined as the capacity to preserve target-relevant distinctions beyond immediate perceptual coupling, integrate them relationally, maintain them through recursive update, and restore representational adequacy when that organization fails. Modal simulation and cross-context abstraction are treated as stronger extensions rather than prerequisites of minimal cognitive worldhood. A six-part operational profile—detachment, relational integration, recursive updateability, modal reach, transfer, and cognitive reclosure—connects the formal theory to controlled perturbation tests while remaining explicitly protocol-relative. The framework does not identify closure with cognition, prediction with understanding, compression with abstraction, generative modeling with consciousness, or cognitive worldhood with a complete internal replica of reality. Its central proposal is instead that cognition carries target-relevant distinction across forms of absence. Keywords: Closure Mathematics; cognitive worldhood; world models; representation; state aggregation; predictive state; recursive representation; hypergraphs; requisite variety; abstraction; counterfactual representation; cognitive reclosure.