Soft spheres at Point J — φ ≈ 0.64, Z→6, anomalous vibrational modes
At jamming point J (φ_J ≈ 0.64 for 2D bidisperse), the coordination number Z jumps discontinuously to 2d=4 (2D) or 6 (3D) — the isostatic value. The density of states D(ω) has a plateau at low frequencies (boson peak) unlike crystals, and contact force distribution P(f) ~ f⁻⁰·² for small forces (force chains). Pressure scales as P ~ (φ−φ_J)^1 just above jamming (O'Hern 2003).