Boundary, Distortion, and Persistence: A Viability‐Geometric Framework for Adaptive Systems

ABSTRACT

How do complex adaptive systems sustain their organization when collapse remains accessible? Prevailing frameworks—equilibrium stability and temporal continuity—do not capture persistence as an active systemic achievement. Drawing on viability theory, safe‐set control and stochastic dynamical analysis, this paper proposes a viability‐geometric framework grounded in three structural components: collapse boundaries, distortion fields and stabilization mechanisms. The Existence‐Under‐Accessible‐Collapse theorem shows that genuine persistence requires boundary exposure—formalized through the reachable set of an explicitly defined class of admissible perturbations—rather than mere continuation. A structural‐correspondence result identifies a common organizational equivalence class shared by boundary‐exposed, boundedly‐distorted and actively‐stabilized systems, while we delimit precisely what this correspondence does and does not preserve. The Boundary–Distortion Persistence Principle identifies minimal joint conditions for sustained viability. Our central substantive claim concerns distortion: within a two‐sided viable interval, a calibrated systematic perceptual bias can outperform veridical perception. We give a minimal boundary‐regulated diffusion model in which this claim is made precise and numerically verified , and we characterize how the two bounds of the viable interval depend on distinct system parameters—the upper bound on exogenous perturbation statistics, the lower bound on the cost of over‐correction. The framework yields discriminating, testable predictions and design guidance for resilient systems.