Why Cognitive Refinement Must Converge: A Limit Form of Cognitive Boundary

If a system can continually refine its distinctions, why does cognition not diverge into infinite fragmentation, but instead converge into a stable boundary? This paper shows that cognitive boundary is not arbitrarily fixed, but necessarily emerges as the limit of a constrained partition process. Under finite structure space, indistinguishability relations, and structure-preserving mappings, any refinement of partitions must satisfy both equivalence constraints and invariance conditions. We prove that infinite refinement is impossible under these constraints. Any descending sequence of partitions must terminate in finite steps and converge to a unique minimal invariant partition. This limit partition defines the cognitive boundary. The central result is: Cognitive boundary is the convergence limit enforced jointly by indistinguishability and structural invariance. Thus, cognitive systems do not expand indefinitely through finer distinctions; they are forced to converge to a stable partition that cannot be further refined without breaking structural consistency.