Conservation-Limited Instability and Fold Bifurcation in Finite-Capacity Transport Systems

This paper proves a structural theorem for a class of dynamical systems governed by conservation form, finite saturating export capacity, and monotone parameter forcing. The central result is that any scalar system satisfying these structural conditions must lose its attracting equilibrium at a finite critical parameter value, and that this loss occurs generically through a codimension-one saddle-node bifurcation. The argument proceeds from system architecture rather than from model specification: the structural constraints alone determine the bifurcation class without assuming it in advance. The theorem is formulated through five hypotheses. H0 requires a compact admissible domain, excluding escape to infinity as a termination mechanism. H1 requires conservation form as a balance between inward load and export. H2 requires finite saturating export, imposing a ceiling on the maximal transport rate. H3 requires monotone parameter forcing of the load-capacity gap, ensuring that the feasibility boundary is approached irreversibly. H4 imposes standard non-degeneracy conditions at the terminal equilibrium. A genericity proposition establishes that H4 holds for an open dense set of systems satisfying H0 to H3 in the C² topology. The conjunction of H0 to H4 is designated the Constraint Property (Property C). Four consequences follow from the theorem. An attracting equilibrium branch exists below a critical parameter value and disappears above it, with the transition occurring at the feasibility ratio σ equal to one. The terminal singularity is a codimension-one saddle-node fold. The attracting branch persists as a slow manifold away from the fold. Recovery time diverges as T ∼ K(1 − σ)^(−1/2) as the feasibility boundary is approached. A conditional corollary extends the fold classification and the recovery-time scaling to vector-valued fast states via centre manifold reduction, provided a terminating branch exists with a simple zero eigenvalue at the terminal point. Lemma 5 introduces exact scalar closure as the additional structural condition under which the conserved coordinate is guaranteed to be dynamically observable. Under this condition the gradient of the conserved quantity is shown to be a left zero-eigenvector of the Jacobian at the fold, with nontrivial pairing following from the closure identity and the simplicity of the eigenvalue. The feasibility ratio σ and the reserve margin R therefore carry the fold signature unconditionally within the class. A minimality section provides counterexamples showing that each of the five hypotheses is individually necessary. A numerical demonstration using a minimal scalar model verifies the predicted fold structure analytically and confirms the inverse-square-root recovery-time scaling numerically. Spreading depolarisation in excitable tissue is presented as a concrete physiological instantiation, with a direct mapping of theorem variables to measurable experimental quantities and a set of falsifiable predictions. This theorem provides the structural foundation for a research programme on finite-capacity transport systems spanning electrophysiology, biochemistry, ecology, and chemical engineering.