Rigorous demonstration of the sphere-packing phase transition

Prove rigorously that random sphere packings in three dimensions undergo a phase transition from gas-like behavior at low density to lattice-like behavior at high density.

Background

The paper describes a conjectured physical picture in which random sphere packings change character as density increases. It explicitly identifies rigorous verification of this transition as an unresolved major problem.

References

However, rigorously demonstrating that this phase transition occurs remains a beautiful and major open problem in the field (see and the references therein).

— Probabilistic combinatorics at exponentially small scales  (2512.15077 - Sahasrabudhe, 17 Dec 2025) in Section 3, subsection “Amorphous sphere packings in physics”