Jamming as a topological satisfiability transition with contact number hyperuniformity and criticality

The jamming transition between flow and amorphous-solid states exhibits paradoxical properties characterized by hyperuniformity (suppressed spatial fluctuations) and criticality (hyperfluctuations), whose origin remains unclear. Here, we model the jamming transition by a topological satisfiability transition in a minimum network model with simultaneously hyperuniform distributions of contacts, diverging length scales and scale-free clusters. We show that these phenomena stem from isostaticity and mechanical stability: The former imposes a global equality, and the latter local inequalities on arbitrary subsystems. This dual constraint bounds contact number fluctuations from both above and below, limiting them to scale with the surface area. The hyperuniform and critical exponents of the network model align with those of frictionless jamming, suggesting a universality class of nonequilibrium phase transitions. Our results provide a minimal, dynamics-independent framework for jamming criticality and hyperuniformity in disordered systems.