Threshold-Safe Shock Absorption in a Compartmental Voter-Flow Model:\ A Conservative Impulse-Control Benchmark

Abstract: We formulate a deterministic threshold-safety problem for a reduced compartmental voter-flow model. An exogenous load enters an alienation reservoir; between releases the reservoir recovers exponentially. Near the mainstream baseline the compartmental dynamics have a linear-stability threshold (Δ_c): below this level the mobilised component contracts, while above it transient amplification is possible. The paper introduces an impulse-control layer for this threshold mechanism. The threshold is obtained from the local stability boundary of the reduced dynamical system, the exposure functional is tied to positive logarithmic amplification, and the scalar reservoir model is proved to be a conservative envelope of the nonlinear voter-flow dynamics. This bridge yields explicit safety benchmarks: the single-release exposure and its zero buffer, the complete-relaxation splitting problem with fixed per-release overhead, the finite-recovery constant-peak profile, and the fixed-horizon capacity frontier (Δ_c(1+ρT)). The scalar recurrence used after the reduction is the familiar leaky-reservoir skeleton also found in multiple-dose pharmacokinetics, fractionated radiotherapy, reservoir operation, and setup-cost scheduling. Its role here is to make the threshold regimes of the compartmental shock-absorption model analytically transparent.