Collapse as a Catalyst of Adaptive Progress A Spectral-Stability Framework for Crisis-Driven Phase Transitions in Complex Systems

AbstractThis paper proposes a formal framework for understanding collapse as a structuralmechanism of regime transition in complex adaptive systems. The framework isdesigned to be domain-neutral and applicable across biological, institutional,technological, economic, and governance-level systems.The central claim is that continuity is not always a viable mode of adaptation. A systemmay preserve apparent operational continuity while internally accumulating unresolvedstructural burden. Once the effective instability acting on the system exceeds itsavailable adaptive reserve, smooth local correction may cease to remain dynamicallyaccessible. Under such conditions, discontinuous reorganization may emerge not as ananomaly external to adaptation, but as one of its admissible operators. The formalism is built around four core quantities: system state , structural instability ,adaptive capacity , and structural debt . Collapse is defined as a threshold eventinduced when producing a transition from a pre-collapse regime to a post-collapseregime . Progress is defined not by scale or expansion, but by post-transition reductionof structural debt and, in the stronger case, by increased adaptive capacity.A spectral extension is introduced by representing systemic interaction structurethrough a time-dependent operator and its dominant spectral value . Crisis proximity isthen linked to the condition under which destabilizing collective modes outpace thesystem’s adaptive response. This provides a compact bridge between local adaptationfailure and high-dimensional regime instability.The framework distinguishes catalytic collapse from destructive collapse and proposesa minimal criterion for adaptive reorganization. The manuscript concludes by outliningimplications for resilience theory, governance, systemic risk, adaptive intelligence, andthe design of systems capable of non-catastrophic reconfiguration.