A NEGATIVE LAW OF ROBUST PERSISTENCE A Necessary Retention–Loss Bound for Dissipative Systems under Antagonistic Forcing

We establish a negative law of robust persistence for dissipative systems subject to antagonistic forcing. Rather than providing a sufficient condition for stability or invariance, the result identifies a necessary bound that no control strategy can overcome. For a minimal mechanical template admitting an energy storage function and distinct channels for actuation and disturbance, we prove that robust persistence inside a bounded energy region is impossible whenever the antagonistic channel possesses strictly greater boundary leverage than the combined effect of dissipation and actuation. The result is formulated as a retention–loss inequality, derived from robust viability theory and expressed entirely in physically interpretable quantities. A tight two–dimensional mechanical example demonstrates that the bound is not an artifact of analysis but a real operational limit, saturable up to an arbitrarily small margin.