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

We formulate an analytical, non-calibrated benchmark for threshold-safe staging of a fixed external load in a reduced compartmental voter-flow model. The state $$A(t)$$ A ( t ) is an effective stock of stress, backlog, or alienation, and $$S(t)$$ S ( t ) is a local mobilisation intensity near the mainstream baseline. The term “safe” refers only to local non-amplification of S S , not to a normative or empirical assessment of a political system. Near the baseline, the nonlinear dynamics have a local stability threshold Δ c Δ c : below it mobilisation contracts, whereas above it transient amplification is possible. The main contribution is a comparison theorem showing that a scalar leaky reservoir is a conservative envelope of the full two-dimensional dynamics. Consequently, a schedule that remains below Δ c Δ c in the scalar model is a sufficient, but not necessary, safety certificate for the nonlinear system. The affine use of A A outside the local simplex is explicitly treated as an effective stress-coordinate assumption. The envelope yields closed-form benchmarks for single-release exposure, complete-relaxation splitting with fixed per-release overhead, the finite-recovery constant-peak profile, and the fixed-horizon safe-capacity supremum Δ c(1+ρ T) Δ c ( 1 + ρ T ) , which is not attained by any finite schedule at equality. The scalar formulas are classical; their role here is to make a stability-derived threshold mechanism analytically transparent.