First-order phase transition in Onsager's model of hard rods in $\mathbb R^3$

We prove a first-order transition for hard rods in $\mathbb R^3$ whose orientations vary continuously over the sphere. For every sufficiently small positive ratio of diameter to length, the grand-canonical pressure has a nondifferentiable point, with a jump in number density at the intermediate Onsager scale. Following the phase-pressure strategy of Lebowitz, Mazel, and Presutti, we connect coexistence in the homogeneous Onsager functional to the microscopic pressure through an exact auxiliary-field representation, exponential decay of correlations in the rod gas at fixed auxiliary field, and estimates of the free-energy cost of interfaces between connected families of minimizing angular profiles.