I introduce P_ₑₑᵣ(t)™, a scalar function that quantifies the instantaneous probability of operationalfailure in non-autonomous complex systems. I analyze that Pₑₑᵣ(t) is a valid Lyapunov candidatefunction for the dynamics of operational degradation. Positive definiteness is established from itsmultiplicative construction under explicit assumptions on the stability factors as state-dependentfunctions. I derive its time derivative along system trajectories and show it is negative semi-definiteunder a stated stability-improving assumption. Using Barbalat’s Lemma, which is the appropriatetool for non-autonomous settings, I establish sufficient conditions for asymptotic stability andexponential decay of the error measure under a class-Κ bound condition.The framework is instantiated as NT-AutonomyGuard, an external pre-execution safety layer forartificial general intelligence systems that estimates operational risk through externally observablesignals, including context drift, residual risk, and inferred self-preservation tendencies, withoutrequiring internal model access. I discuss relationships with control barrier functions, scalableoversight, and alignment research, and show how Pₑₑᵣ(t) bridges classical nonlinear control withmodern AI safety and operational governance.The proposed construction extends multiplicative Lyapunov-like barrier formulations tooperational safety contexts and suggests practical applications across AI governance and othersafety-critical domains.