Emergent cohesion via self-caging in maximally entangled rod packings

Random packings of disordered rigid rods exhibit emergent cohesion, as exemplified in a nest of twigs that is self-equilibrated, free-standing structures. We analyze the geometric motif underlying this cohesion using a rod packing that maximizes the average crossing number subject to non-penetration constraints. We show that this protocol leads to self-caging: collective geometric constraints that prevent rod escape even in finite systems with free boundaries, leading to packings that remain mechanically cohesive due to a combination of purely repulsive and frictional interactions. We show that self-caging is controlled by the available free-volume in translational and rotational configuration spaces, which is minimal when $N/(Z\alpha)=1/3$ where $N$ is the number of rods, $\alpha$ is the aspect ratio, and $Z$ is the average coordination number. Our results establish a minimal geometric motif for entanglement-induced cohesion in athermal rod packings, with implications for cohesive granular matter without attractive forces.