A continuum temperature singularity for a radial pair potential

We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying $\varphi (r)=o(r^{-3})$. Its free-cube canonical free energy is finite at every positive inverse temperature and density.