Exact Critical Power Indices for Heterogeneous PRHR Series Systems Under Archimedean Dependence

For a series system with heterogeneous proportional reversed-hazard-rate components coupled through an Archimedean survival copula, we determine the exact admissible range of power-mean indices for a universal homogeneous stochastic upper envelope. The characterization reduces to the convexity of a single transformed inverse-generator function; under complete monotonicity, this condition is necessary already from the two-component case. The resulting critical power index is sharp: every order at or above the threshold is universally valid, whereas every smaller order fails for a suitable two-component perturbation. Under independence, the threshold is zero, making the geometric mean optimal. Within the Clayton family, negative critical indices occur, with an exact threshold for and a threshold strictly below for . Under additional generator concavity and monotonicity conditions, the same construction yields the stronger reversed hazard rate order, with exact thresholds for independence and for the Clayton model at . To address dependence uncertainty, we also derive a sharp classwise robust threshold and show that a negative Clayton-optimal order need not remain valid under an alternative Archimedean family. A deterministic sensitivity study confirms the computational scalability of the construction across system size, heterogeneity, dependence strength, and baseline choice.