A Singular Fold Model for Capacity-Constrained Dynamics

We study a minimal singular fast-slow dynamical model for capacity-constrained compounding. The fast subsystem takes the smooth saddle-node normal form, while the slow vector field g(x) = α − βk/x is singular at the boundary x = 0. Our results are local and explicit: a saddle-node bifurcation of the fast subsystem (Proposition 1); a conditional friction-scaling law F ~ (Q − V)^{−1/2} along the attracting branch under the coupling F(x) = k/x (Proposition 2); a Lyapunov monotonicity of L(V) = −F(x*(V)) on the congestion regime of the reduced slow flow (Proposition 3); a boundedness equivalence between bounded friction and bounded-away-from-fold trajectories on the reduced flow (Criterion 1); and a forward invariance theorem for the full singular system on strict congestion subsets (Theorem 1), conditional on an explicit minimum-distance threshold δ > δ_min(r) from the fold. We do not prove a global sufficiency result on the full singular system, a Krupa–Szmolyan classification, or behavior in the uncertified fold neighborhood {V > Q − δ²_min}. The blow-up regularisation of the 1/x singularity is presented as a formal heuristic in one chart only. We derive four fold-consistent empirical predictions with explicit identifiability assumptions and competing mechanisms, and specify a falsification protocol.