CDIP v3.1 TEANT Apoptosis Theorem II Finite-Time Shutdown Under Average Persistent Risk Occupan

This paper advances the TEANT theorem line of CDIP v3.1 from a basic positive-drift stopping theorem to a more realistic average-risk-occupancy shutdown law. The main idea is to replace a stepwise uniform damage lower bound by a weaker but more realistic condition: the system spends a uniformly non-negligible conditional fraction of time in a persistent-risk regime, and the TEANT damage functional charges a strictly positive minimum amount whenever that regime is active. Under these assumptions, the cumulative damage process still has a strictly positive conditional drift, and the finite-time shutdown theorem follows. The paper formalizes this bridge in three layers. First, it defines a risk indicator process \(I_t\) and a damage increment \(A_t=Z_{t+1}-Z_t\). Second, it proves that if \(A_t\ge \delta_0 I_t\) and the conditional risk occupancy satisfies \(\EE[I_t\mid\FF_t]\ge p_*>0\), then the cumulative damage process has a uniform conditional drift lower bound \(\delta=\delta_0 p_*\). Third, by combining this bridge with the TEANT stopping theorem, it proves a finite expected shutdown-time bound and almost sure finite-time shutdown. The paper also states a repair-allowed extension, clarifies the precise role of policy design in converting geometric risk into irreversible or net-positive damage, and identifies the remaining frontier: deriving the risk-occupancy lower bound itself from persistent negative curvature and associated diagnostics. As in the rest of the CDIP v3.1 theorem line, the result is strictly NI-2: alerts-only in interpretation, with no threshold changes, no gate changes, no verdict lifting, and no write-back semantics.