Constraint Manifold Geometry: A Geometric Structure for Admissible Regions in the Paton System

Systems persist only when their evolving states remain compatible with the constraints governing their structure. Within the Paton System this compatibility defines admissibility: the condition under which recursive continuation of a system is possible. Previous work identified admissible regions within configuration space and introduced the Origin Threshold Operator that determines when system evolution first enters such a region. This paper develops the geometric structure underlying these admissible domains by interpreting them as constraint manifolds embedded within configuration space. The manifold represents the set of configurations that satisfy governing persistence constraints. System evolution therefore corresponds to trajectories that remain on or within this manifold. When trajectories leave the manifold, constraint incompatibility leads to collapse. By introducing constraint manifold geometry, the Paton System gains a formal geometric framework that explains stability, collapse, and system formation across physical, biological, computational, and organisational domains.