Emergence of Countable Entities in Self-Organized Geometric Systems

We formalize the notion of "countable formation" in systems with internal organizational structure. By integrating Hausdorff geometry, Jensen-Shannon information metrics, and thermodynamic free-energy costs, we demonstrate that the emergence of discrete, countable entities is not merely combinatorial but a consequence of non-equilibrium phase transitions. We validate this framework using a unit disk graph realization, showing that the order parameter $\langle N \rangle$ exhibits power-law scaling near the critical formation strength $s_c \approx 0.39$.