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

In three linesMathematical framework (HEF) modeling emergence as phase transition in mechanism landscape. Empirical study across 111 grokking experiments in modular arithmetic transformers: convergence to 0.9745±0.014 regardless of initialization, weight norm peaks before grokking in 92% of runs, accuracy curves collapse onto tanh kink (R²=0.93).

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