New HEF theory explains universal convergence in AI, biology, and physics

A new paper from Truong Xuan Khanh introduces the Hierarchical Emergence Framework (HEF), a mathematical scaffold that unifies how complex systems—from machine learning models to biological networks—converge to similar high-level structures despite radically different microscopic details. The framework models emergence as a critical phase transition in a "mechanism landscape" constrained by thermodynamics and information theory. HEF defines a critical energy threshold Ec that separates an exploration regime from a convergence regime, and proves strict metric contraction toward a unique fixed point independent of initial conditions. This structure connects to causal emergence measured via Effective Information and mechanism competition entropy.

To validate HEF, the author conducted 111 experiments on delayed generalization (grokking) in modular arithmetic transformers. Results show a reproducible empirical fingerprint: weight norm peaks systematically before grokking in 92% of runs. Normalized accuracy curves collapse onto a tanh kink with R²=0.93, consistent with a Landau-Ginzburg universality class. All grokked models converge to final accuracy 0.9745±0.014 regardless of seed, weight decay, or training fraction (ANOVA p>0.13). The paper is not presented as a universal theory but as a falsifiable framework—complete with 15 pages of proofs—for studying convergence phenomena across ML, biology, and physics.