Minimal T opology of Perturbed Dynamic Systems Constraint, state space, and the emergence of geometry MAY 26 Abstract Complex systems across many domains often exhibit a similar behavioural pattern: sustained stability under load, progressive loss of recovery capacity, and abrupt transition once a threshold is crossed. Biological regulation, neural systems, engineered infrastructure, financial systems, materials, and cognitive systems can all display this structure, despite differing in their underlying mechanisms [1–4]. This paper proposes a constraint-first description of such behaviour. Instead of beginning with component-level mechanisms, systems are described as trajectories moving through constrained state spaces. Constraints define the reachable regions of the state space, while accumulated perturbation drives trajectories toward boundaries between regimes. When trajectories approach or cross these boundaries, abrupt transitions, hysteresis, and collapse-like behaviours may occur [2,3,5]. A minimal coupling topology is introduced for regulatory systems composed of interacting subsystems. A triadic subsystem model is insufficient when the interaction between subsystems produces an integrated system state that feeds back into the subsystems themselves. Adding this integrated state yields a four-node fully coupled structure, represented by the complete graph K4, geometrically equivalent to a tetrahedral simplex. Within this framework, geometry is not treated as a prior assumption. It emerges as the record of constrained interaction: attractor basins, transition surfaces, bottlenecks, and collapse trajectories appear from the relationship between state space, constraint, perturbation, and coupling. The framework is summarised as: (S, C, Π, K4) → G where S is system state space, C is constraint structure, Π is accumulated perturbation, K4 is minimal full coupling topology, and G is the emergent geometry governing system behaviour. The aim is not to replace existing dynamical systems theory, nor to derive new universal equations of motion. It is to identify a minimal structural grammar for describing how regulatory systems under constraint stabilise, drift, collapse, and reorganise. 1. Systems under pressure Across domains, complex systems often appear stable until they are not. A material can carry increasing load before yielding. A queue can absorb demand before backlog becomes unbounded. A physiological system can compensate for stress before dysregulation becomes visible. An AI conversation can remain coherent before drifting into unsupported claims. An infrastructure system can continue operating while internal recovery capacity is quietly being consumed. These transitions are often described after the fact as failures, collapses, breakdowns, or phase changes. The visible event may appear sudden, but the underlying trajectory is usually not instantaneous. A system can move gradually toward a boundary while continuing to produce acceptable outputs. This motivates a different starting point. Instead of asking only: which component failed? we can ask: how did the system’s trajectory move through its constrained state space until its prior regime was no longer recoverable? This is the core move. The minimal topology paper frames the same shift directly: collapse phenomena should not be understood only through component failure, but through the structural organisation of the system as a whole and the geometry of its possible trajectori