Fracture Density and Phase Boundaries in Sparse Random Systems: A Structural Map for Nested Critical Regimes

We identify a recurring structural feature across a range of sparse random systems, including percolation, rigidity/jamming models, and random constraint satisfaction problems: at criticality or modestly supercritical control parameters, the order parameter occupies a characteristic narrow fraction of the system (F ≈ 0.10–0.15). This fraction represents the minimal coherent core—spanning cluster, rigid backbone, or frozen-variable set—necessary to propagate global connectivity or constraint. By tracking this invariant via renormalisation-inspired coarse-graining, we delineate nested regimes and phase boundaries, showing where different effective laws govern system behaviour. We demonstrate that this framework clarifies the organization of seemingly disparate problems, suggests where different analytic tools are appropriate, and highlights the open opportunity for formalisation to rigorously map these boundaries. Initial text-only version of a conjectural preprint on structural invariants in sparse random systems. Illustrative figures planned for future revision.