  A coarse-grained system of one-dimensional (1D) hard spheres (HSs) is created
using the Delaunay tessellation, which enables one to define the quasi-0D
state. It is found from comparing the quasi-0D and 1D free energy densities
that a frozen state due to the emergence of quasi-0D HSs is thermodynamically
more favorable than fluidity with a large-scale heterogeneity above crossover
volume fraction of $\phi_c=e/(1+e)=0.731\cdots$, at which the total entropy of
the 1D state vanishes. The Delaunay-based lattice mapping further provides a
similarity between the dense HS system above $\phi_c$ and the jamming limit in
the car parking problem.

Disordered Systems and Neural Networks;Mesoscale and Nanoscale Physics;Soft Condensed Matter;Statistical Mechanics