Optimal Oversight Under Capacity

Confidence-gated human oversight is a central control in both of the governance frameworks that Mesbaul Haque Sazu introduced in 2023. The Governed Decision-Intelligence (GDI) framework makes remit-based human judgment a normative invariant and gates automation on a calibrated confidence signal [15]; the companion Full-Stack reference architecture makes the point, sharply, that reviewer capacity is a governance parameter rather than an implementation detail, because an escalation rate above the available review throughput does not produce oversight but a queue [14]. This paper takes that capacity observation as its starting axiom and develops the operations-research model it implies. We treat the escalation stream produced by a governed gate as the arrival process to a finite pool of reviewers, model the oversight station as a queue with a deadline imposed by the commit-time freshness invariant, and cast threshold selection as a constrained optimization that minimises expected decision cost subject to a service-level constraint on reviewer utilisation. We derive the escalation-rate function induced by a gate threshold, the queue's waiting-time and deadline-violation behaviour under a finite reviewer pool, and the Lagrangian conditions characterising the cost-optimal threshold. A central result formalises a warning the frameworks state qualitatively: below a critical threshold the oversight queue becomes unstable, so a gate that looks prudent on a per-decision basis voids human oversight in substance by delaying escalated decisions past their validity bound. An illustration on publicly available data in open-source platforms locates the stability boundary and traces the cost surface; it is meant to show the model's structure rather than to validate it in any single deployment. What the paper adds is the quantitative machinery underneath the oversight invariant; the invariant itself, and the governance reasoning behind it, remain the frameworks'.