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. The results are local and explicit: a saddle-node bifurcation of the fast subsystem; a conditional friction-scaling law F ∼ (Q − V)^{−1/2} along the attracting branch under the coupling F(x) = k/x; a barrier monotonicity of L(V) = −F(x*(V)) on the congestion regime of the reduced slow flow; a boundedness equivalence between bounded friction and bounded-away-from-fold trajectories on the reduced flow; a fold-repulsion and exit-time theorem for the full singular system on strict congestion subsets at every fixed fold distance δ > 0, under an explicit smallness threshold ε ≤ ε₀(δ, r); and an isolating-block certificate covering all initial conditions in the slow-manifold tube. Together these give a rated exit: for ε below the stated Lyapunov and isolating-block thresholds, every congestion-tube trajectory leaves the cooperative face without approaching the fold, in explicit time with leading term 16/(γ₀μ₀δε). Not proved: a global sufficiency result on the full singular system, a Krupa–Szmolyan classification, or uniform-in-δ behavior as δ → 0 at fixed ε. The blow-up regularisation of the 1/x singularity is incomplete. The inner problem is solved in closed form, the chart atlas and transition maps are derived, and the entry, passage and exit legs are proved by conventional invariant-manifold arguments with machine-checked computation, a subset carrying Lean 4 kernel proofs. The gluing of the legs and the time bookkeeping remain formal sketches. We derive four fold-consistent critical-slowing-down predictions for exogenous approach and one barrier prediction that tests the autonomous certificate directly, with explicit identifiability assumptions and competing mechanisms, and specify a falsification protocol.