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

arXiv: https://arxiv.org/abs/2501.01638 · HTML: https://mlsanity.com/paper/2501.01638/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jan 2025.

## Built on by (most cited first, 1 of 1)

- [Beyond the Ergodic Wall: A Discrete Geometric Physics Sandbox for Analysing AI Scaling Limits and Complexity Collapse](https://mlsanity.com/paper/2610.10651/) (2026)

## Abstract (arXiv)

Large language models have emergent capabilities that come unexpectedly at scale, but we need a theoretical framework to explain why and how they emerge. We prove that language models 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 language models that capacities emerge through discrete transitions guided by constraint interactions and path-dependent exploration. This framework provides a theoretical basis for understanding emergence in language models and guides the development of architectures that can guide capability emergence.