Universal Jamming Criticality and Self-Organizing Principles from Disorder to the Limit of Perfect Crystalline Order.

While crystals are defined by periodic order, the nature of amorphous solids remains elusive due to their disordered, diverse, and nonequilibrium structures. Here, we focus on jammed elastic packings and systematically tune structure from crystalline to fully disordered to unveil the universal underlying characteristics. We demonstrate that their mechanical properties are universally governed by jamming criticality, featuring characteristic scaling behaviors near the jamming transition, except for the singular close-packed point. This is facilitated by random nonaffine elasticity arising from contact-level disorder. Consequently, the jamming density can approach close packing, suggesting a fundamental decoupling between jamming criticality and the glass transition physics. Moreover, we uncover a universal coordination-number distribution and contact hyperuniformity in marginally jammed states, independent of particle-level structure. These findings suggest a general organizing mechanism for emergent rigidity in disordered solids, underscore the broad relevance of jamming physics, and complement principles of mechanical self-organization.