Does the critical aspect ratio threshold for nematic phase formation in hard rod systems on Z^2 admit a sharp universal lower bound that depends only on dimension, and does this bound persist in the infinite-volume limit for systems with weak long-range interactions?

The rigorous aspect ratio bound for nematic formation in hard rod and hard rectangle systems on Z^2 provides a sufficient condition for the nematic phase, but the exact critical aspect ratio remains unknown: one does not know whether the sufficient bound is sharp, or how much room exists between the rigorous threshold and the true transition. The core difficulty is that proving phase transitions in lattice systems requires controlling entropy-energy competition at all scales simultaneously, and the relevant Peierls-type contour arguments become increasingly delicate as one moves toward the true critical aspect ratio. When long-range interactions are introduced, even of polynomial decay, the contour estimates break down because distant rods can correlate, invalidating the finite-range cluster expansion techniques that underpin current proofs. The KAM persistence results for infinite-dimensional Hamiltonian systems with long-range interactions suggest that certain ordered structures survive perturbation, but translating this dynamical stability into equilibrium statistical mechanics for a discrete lattice model requires new tools connecting KAM theory to Gibbs measure uniqueness or multiplicity.