A Singular Fold Model for Capacity-Constrained Dynamics

Abstract

We study a minimal fast-slow model for capacity-constrained accumulation: a smooth saddle-node fast subsystem coupled to a slow field g(x) = α − βk/x, singular at the fold boundary x = 0. Variables are dimensionless. Two are kept distinct: the capacity margin μ = Q − V, and the structural amplitude x, the latent fast order parameter with x2 ≈ μ on the critical branch. Friction couples to the amplitude, F = k/x, so on the branch F ∼ μ−1/2. In the capacity-margin variable the model is a Liénard system whose normalized energy has a strict minimum at the operating equilibrium. That equilibrium is globally asymptotically stable on {x > 0} for every ε > 0, with an explicit amplitude floor (Theorem 1). Write universal forward invariance for forward invariance of {x > 0} at every ε > 0 from every physical datum. It holds if and only if the friction pole is non-integrable, ∫0 h = ∞ (Proposition 3): an integrable coupling, including every regularisation B/(x+δ), loses it for above-capacity data at small ε, while data below an explicit energy level still converge. Away from the fold an isolating-block certificate (Proposition 4) and a two-term transit law (Proposition 5, with off-manifold data in Lemma 3) match measured exit times to 1.4×10−5 in T·ε. The passage depth −μ02/(2εB) + O(1) in ln x is a formal leading asymptotic with numerical support; deep-region matching, the chart gluing, and the quantitative stochastic theory are open; the chart analysis is provided in a companion note. Diagnostics P1–P4 are conditional on a declared frozen-parameter Ornstein–Uhlenbeck regime, whose validity condition D ≪ 4μ3/2 closes as the fold is approached; they test the fold geometry, and the friction coupling has its own protocol. Part of the finite certificate algebra carries Lean 4 kernel proofs (Appendix E), which attest the algebra. Version note (v85): the regularity hypothesis of Proposition 3 is strengthened to h ∈ C1(0, ∞); the pole equivalence is attached to the named property universal forward invariance, with both quantifiers stated; and a new part (iv) proves that an integrable coupling still converges from every datum below the fold energy c*, so integrability costs the universal property and no more. The LaTeX source is deposited alongside the PDF.