The Tolerance Return Law: A Structural Constraint on Branching and Return in Admissible Systems

This paper formalizes the Tolerance Return Law within the Paton System. The law states that while exploratory branches may diverge freely from a stable datum, only branches that remain within the system’s tolerance boundaries can return and persist. Branches that exceed tolerance cannot reintegrate with the stable structure and naturally fade. The law stabilizes recursive exploration by separating unrestricted branching from structurally admissible return. This principle applies across multiple domains including dynamical systems, theoretical reasoning, biological evolution, computational search processes, and physical stability regimes. Within the Paton System architecture, the Tolerance Return Law operates between the Tier-3 Admissibility Gate and the Tier-4 Datum interface, functioning as a structural filter that permits exploration while preserving coherence.