Crunching hard grains together leads to their rigidification, irrespective of how the system is prepared. While certain features of the resulting jammed materials -- e.g. their contact force distribution -- are also remarkably invariant, others -- e.g. their density -- can vary quite a bit. The underlying physics at play behind either, however, remains unclear. Such a purely geometric system begs for an explanation of the same nature. For instance, deterministic optimization algorithms unequivocally partition a complex energy landscape in inherent structures (ISs) and their respective basins of attraction. Can these basins be defined solely through geometric principles? We here address the issue by proposing a geometric class of gradient descent--like algorithms, which we use to study key model of disordered matter in the hard-sphere universality class, the random Lorentz gas. The statistics of the resulting ISs is found to be strictly inherited from those of Poisson--Voronoi tessellations. The landscape roughness is further found to give rise to a hierarchical organization of ISs, which various algorithms explore differently. In particular, greedy and reluctant schemes tend to favor ISs of markedly different densities. The resulting ISs nevertheless robustly exhibit a universal force distribution, thus confirming the geometric nature of the jamming universality class. Along the way, the physical origin of a dynamical Gardner transition is identified.
