Homeostatic Basins and Dissipative Traps: A Substrate-Independent Topology of Dissipative Dynamics with Mathematical Proofs Across Three Canonical Systems

Abstract - We propose that dissipative open systems admitting a timescale separation into fast ob-servable variables and slow accumulating variables generically exhibit two classes of conver-gent behaviour: homeostatic basins (Lyapunov-stable with decreasing maintenance costand convergent slow variables) and dissipative traps (locally Lyapunov-stable in fast vari-ables but with monotonically accumulating slow-variable debt leading to threshold collapse).We formalise these definitions using Lyapunov functions with fast–slow decomposition, thencompute the distinction explicitly in three canonical systems: a single qubit under Lindbla-dian dynamics, the Scheffer lake eutrophication model, and a damped double-well Duffingoscillator. We identify three universal mathematical features (fast–slow decomposition withaccumulation, threshold bifurcation, non-monotonic Kullback-Leibler divergence) and threenon-universal features (spectral gap scaling, early warning signal type, physical meaning of“debt”) that depend on bifurcation structure. This is an interpretive reframing generatingtestable predictions, not a claim of new physics.Keywords: Lindbladian dynamics · dissipative systems · Lyapunov stability · fast–slow decom-position · critical transitions · early warning signals · attractor topology · homeostasis