Authority--Limited Delay Margins and Two--Ceiling Phase Structure in Nonlinear Delayed Feedback Systems}

This preprint analyzes the effect of bounded actuator authority on delay margins in a nonlinear inverted pendulum with delayed state feedback. Systematic numerical sweeps over delay and saturation limits reveal a three-regime phase structure: asymptotic stability, bounded limit cycles, and large-angle collapse due to basin escape. A refined near-critical sweep shows inward deformation of the stability boundary at low authority, captured by a rational delay-margin correction law. At larger delays, the minimum authority required for survivability follows a hyperbolic envelope, defining a hard nonlinear delay ceiling. Sustained actuator saturation correlates strongly with stability loss, supporting an authority-deficit mechanism for boundary deformation. The resulting two-ceiling topology quantifies how bounded actuation distorts classical linear delay-margin predictions.