Continuity and Drift as Domain-Agnostic Measures of System Stability

Title: Continuity and Drift as Domain-Agnostic Measures of System Stability Subtitle: Incorporating the Entropic Sink and Periodic Restoration Cycles Overview This paper establishes a universal framework for measuring systemic stability through the lens of Continuity Invariants and Drift Functionals. While Version 1 (published Dec 12, 2025) focused on the forensic detection of discontinuity, this expanded Version 2 introduces the Entropic Sink Operator ($\sigma$)—the critical restorative component that allows complex systems to survive inevitable drift. Key Additions in Version 2: The Entropic Sink (Section 5): We introduce the "Sink" as a dissipative operator that prevents systemic collapse by periodically discharging accumulated epistemic entropy ($H_e$). We demonstrate that stability is not the absence of drift, but a dynamic equilibrium where the rate of discharge matches the rate of entropic generation. Periodic Restoration Cycles (Section 6): We identify "Restoration Cycles" as universal mechanisms of identity preservation. This includes biological manifestations such as Laughter (acute, high-frequency entropic discharge) and Sleep (batch-processed maintenance), as well as physical manifestations like the Noble State in chemistry and Expansion in cosmology. The Delta Reset Principle: We define the mathematical nature of the "Sink" not as a total system wipe (which would constitute a discontinuity event), but as a managed "Delta Reset" that prunes noise while reinforcing the primary Lineage ($I_L$). Institutional Inertia: We expand the failure model to include "Sink Blockage," where the refusal to acknowledge drift leads to systemic "clogging" and catastrophic fragmentation. Applicability This framework remains domain-agnostic, providing unified stability metrics for: Artificial Intelligence: Reasoning drift and attention resets. Biology: Homeostasis, autophagy, and neurological restoration. Governance: Institutional audit trails and the "social sink" of transparency. Physics: Thermodynamics and the preservation of information across cosmic time. Version History v1.0.0 (Dec 12, 2025): Established the core four invariants (Lineage, Monotonicity, Provenance, Consistency) and the Drift Functional. v2.0.0 (Feb 16, 2026): Current Version. Integrated the Entropic Sink and Restoration Cycles to account for long-term systemic resilience.