A Sharp Recoverability Boundary in Delayed Adaptive Feedback Systems

Adaptive feedback systems that correct errors from delayed sensory signals face an inherent vulnerability: corrective updates may no longer reflect the current state of the system. While classical stability analyses characterize asymptotic behavior near equilibrium, the conditions under which such systems can recover from finite perturbations remain poorly understood. Here we investigate the recoverability boundary in a minimal delayed adaptive feedback model, extending from a one-dimensional phase system to a two-dimensional phase–velocity system with inertia and stochastic drift. We demonstrate that recovery success does not degrade gradually with increasing delay; instead, it collapses abruptly at a critical threshold τ_c, within a transition interval of less than 0.25 s across all tested conditions. Notably, the linear stability bound τ_c^{lin} = π/(2K*) systematically overestimates the empirical τ_c, with the discrepancy reaching more than threefold at higher adaptation rates — establishing that linear analysis is not a reliable design criterion for recoverability in adaptive systems. A two-dimensional parameter sweep over feedback delay and damping (γ ∈ [0.3, 3.0]) reveals that τ_c varies by only 0.035 s across a tenfold range of damping, indicating that the boundary is governed by the loss of temporal correlation between delayed error signals and the instantaneous state — a mechanism we term information staleness — rather than by the physical timescale of the plant. Faster adaptation shifts the boundary but cannot eliminate it. As τ approaches τ_c, mean recovery time increases up to tenfold and its variance diverges, providing a prediction-free early-warning signal of imminent failure. These results delineate a structural constraint on prediction-free adaptive control and establish a quantitative baseline for evaluating when predictive architectures become necessary in delayed biological and engineered systems.