Emergence as a Universal Phase Transition: From Mathematical Inevitability to Predictive Control Across AI, Physical, and Social Systems

Emergence—the phenomenon where complex behaviors arise from simple components—has re- mained a deeply puzzling concept across scientific disciplines. Here, we present a unified mathemat- ical framework that explains emergence as a predictable phase transition governed by the size of a system’s exploration space. We demonstrate that this framework not only explains the emergent capabilities of artificial intelligence but also provides a fundamental basis for the phase transitions of water and the evolution of social complexity in human societies. We posit that emergence is not a mystical phenomenon but a predictable consequence of a system’s exploration of a high-dimensional state space. By defining the relationship between the microscopic rules of a system and its macro- scopic properties through the lens of exploration capacity, we provide a model that explains why low-probability, ”miraculous” behaviors become reliable and regular in sufficiently large systems, such as large language models. The core of our thesis is the derivation of an inverse exponential relationship between the probability of an emergent behavior and the size of the exploration space, offering a quantitative basis for this previously qualitative concept. This revised framework introduces the concept of emergent areas as behavioral classes that undergo phase transitions based on system scale and dimensionality.