P_ₑₑᵣ(t): A Lyapunov-like Function for Non-Autonomous Operational Stability in Complex Systems with applications to AI Safety

I introduce P_ₑₑᵣ(t)™, a scalar function that quantifies the instantaneous probability of operational
failure in non-autonomous complex systems. I analyze that Pₑₑᵣ(t) is a valid Lyapunov candidate
function for the dynamics of operational degradation. Positive definiteness is established from its
multiplicative construction under explicit assumptions on the stability factors as state-dependent
functions. I derive its time derivative along system trajectories and show it is negative semi-definite
under a stated stability-improving assumption. Using Barbalat’s Lemma, which is the appropriate
tool for non-autonomous settings, I establish sufficient conditions for asymptotic stability and
exponential decay of the error measure under a class-Κ bound condition.

The framework is instantiated as NT-AutonomyGuard, an external pre-execution safety layer for
artificial general intelligence systems that estimates operational risk through externally observable
signals, including context drift, residual risk, and inferred self-preservation tendencies, without
requiring internal model access. I discuss relationships with control barrier functions, scalable
oversight, and alignment research, and show how Pₑₑᵣ(t) bridges classical nonlinear control with
modern AI safety and operational governance.

The proposed construction extends multiplicative Lyapunov-like barrier formulations to
operational safety contexts and suggests practical applications across AI governance and other
safety-critical domains.