Convergence Without Identity: Structural Similarity and Recurrence Under Constraint in the Paton System

This paper formalises a structural principle within the Paton System: that independently formed systems may converge toward high similarity under shared constraints but cannot achieve exact identity due to irreducible differences in formation timing, microstate configuration, and interaction history. The work further demonstrates that constraint-consistent environments produce recurring structural forms across scale, establishing recurrence as a natural consequence of admissibility filtering rather than duplication. By unifying convergence limits and structural recurrence, the paper defines a Tier-6 constraint condition governing similarity, identity, and repeatability across physical, biological, cognitive, and engineered systems. The result is a reframing of “sameness” as constraint-aligned convergence on a continuous spectrum, where identity remains structurally excluded.