A non-ergodic framework for understanding emergent capabilities in Large Language Models

Abstract: LLMs have emergent capabilities that come unexpectedly at scale, but we need a theoretical framework to explain why and how they emerge. We prove that LLMs are actually non-ergodic systems while providing a mathematical framework based on Stuart Kauffman's theory of the adjacent possible (TAP) to explain capability emergence. Our resource-constrained TAP equation demonstrates how architectural, training, and contextual constraints interact to shape model capabilities through phase transitions in semantic space. We prove through experiments with three different LLMs that capacities emerge through discrete transitions guided by constraint interactions and path-dependent exploration. This framework provides a theoretical basis for understanding emergence in LLMs and guides the development of architectures that can guide capability emergence.