Emergence via Phase Transitions: Mechanism Landscapes and Universal Convergence Across Complex Systems

Abstract:Across machine learning, biology, and physics, independently evolving systems often converge toward strikingly similar high-level structures despite radically different microscopic details. Grokking circuits converge across random seeds, evolutionary lineages rediscover similar metabolic solutions, and renormalization flows approach common fixed points. We propose the Hierarchical Emergence Framework (HEF) as a candidate universality framework for such convergence phenomena. HEF models emergence as a phase transition in a mechanism landscape constrained by thermodynamic and information-theoretic laws. The framework introduces a critical energy threshold Ec separating an exploration regime with competing mechanisms from a convergence regime governed by a unique minimum-cost mechanism. Under structural assumptions, we prove physical feasibility, derive strict metric contraction, and establish convergence toward a unique fixed-point representation independent of initial conditions. We further connect this convergence structure to causal emergence through Effective Information and mechanism competition entropy. To test the framework, we study delayed generalization ("grokking") in modular arithmetic transformers across 111 experiments. We identify a reproducible empirical fingerprint of the Ec transition: the weight norm peaks systematically before grokking in 92% of runs. Normalized accuracy curves collapse onto a tanh kink (R^2=0.93) consistent with a Landau-Ginzburg universality class, and all grokked models converge to 0.9745+/-0.014 regardless of initialization, weight decay, or training fraction (ANOVA p>0.13). HEF is not presented as a universal theory of emergence, but as a falsifiable mathematical scaffold for studying convergence phenomena across complex systems.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2606.07563 [cs.LG]

(or arXiv:2606.07563v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.07563 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.