The Rigidity Trap: Differentiation, Reintegration Failure, and Collapse Dynamics in Scaling Adaptive Systems

This paper formalizes the Rigidity Trap — a dynamical pathway through which high-performing adaptive systems systematically dismantle their own alarm mechanisms, creating brittle architectures that fail catastrophically upon environmental shift. We identify three interconnected phenomena: (1) scaling-induced differentiation, where growing system complexity produces O(n²) conflict channels and governance saturation at a provable threshold n* = O(C_M); (2) the Transition–Collapse Fork, a sigmoid-gated bifurcation where high rigidity suppresses alarm sensitivity and closes the transition path, making collapse the default trajectory under sustained success; and (3) the Rigidity Trap proper, a self-reinforcing feedback loop of success → rigidity → alarm suppression → undetected mismatch → catastrophic failure. We derive conditions under which endogenous micro-instability maintenance prevents trap entry, demonstrate the mechanism through toy computational experiments showing accelerated endurance collapse in success-exposed systems, and connect the trap to Recovery Theory's Coherence Maximization Paradox across individual, team, and organizational scales. The framework bridges organizational theory, continual learning, and self-organized criticality, providing quantitative predictions for when and how successful systems become vulnerable to sudden failure.