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

Researchers propose the Hierarchical Emergence Framework (HEF), a mathematical model explaining why independently evolving complex systems converge toward similar structures despite different starting conditions. Testing on transformer networks shows reproducible phase transition signatures during grokking, with all models converging to identical accuracy levels regardless of initialization parameters.

The Hierarchical Emergence Framework addresses a fundamental puzzle in complexity science: why do disparate systems—from neural networks to biological evolution—independently rediscover similar solutions. This research bridges machine learning, physics, and information theory by modeling emergence as a phase transition governed by a critical energy threshold. The framework's predictive power emerges from thermodynamic constraints rather than explicit design, suggesting deep structural principles underlie convergence phenomena.

The empirical validation through grokking experiments provides compelling evidence for HEF's validity. Across 111 transformer training runs, 92% exhibited systematic weight norm peaks preceding generalization, while accuracy curves collapsed onto a universal curve matching Landau-Ginzburg physics predictions. Most strikingly, all grokked models converged to identical accuracy (0.9745±0.014) across diverse hyperparameter configurations, eliminating initialization dependence entirely. This reproducibility suggests phase transitions govern learning dynamics rather than random variation.

For the AI research community, HEF offers a falsifiable framework for understanding why neural networks display seemingly magical generalization phenomena. The connection between causal emergence and mechanism competition provides theoretical scaffolding for future complexity investigations. However, the framework's applicability remains limited to scenarios matching HEF's structural assumptions, and validation on non-arithmetic tasks remains pending.

The research strengthens theoretical foundations for deep learning rather than providing immediate practical applications. Future work should test whether HEF predictions hold across diverse architectures, datasets, and problem domains beyond modular arithmetic.

• →HEF models emergence as phase transitions constrained by thermodynamic laws, explaining convergence across independent complex systems.
• →Weight norm peaks predicted 92% of grokking transitions, with accuracy curves matching universal Landau-Ginzburg physics signatures.
• →All grokked transformer models converged to identical accuracy (0.9745±0.014) regardless of initialization or hyperparameters, eliminating initialization dependence.
• →Framework connects causal emergence to mechanism competition entropy, providing theoretical links between information theory and convergence phenomena.
• →Authors present HEF as a falsifiable mathematical scaffold rather than universal theory, enabling critical testing across complex systems.