Phase-Transitional Scaling

We introduce \emph{Phase-Transitional Scaling} (PTS), a falsifiable framework that treats certain emergent abilities of large language models (LLMs) as phase transitions characterized by a sigmoidal response with threshold (TK) and sharpness (\gammaK). We connect this phenomenology to three complementary theoretical perspectives: finite-size mean-field theory, percolation on representational graphs, and noise-activated barrier crossing in training dynamics. Comprehensive experiments across 12 diverse capabilities validate the sigmoid form (47/48 comparisons), demonstrate that (TK) is controlled by data complexity ((R^2=0.89)) while (\gammaK) is controlled by training dynamics ((R^2=0.76)), and establish universal curve collapse across different architectures (94\% variance explained). PTS provides quantitative, predictive scaling laws that outperform power-law baselines by 4(\times) in out-of-sample prediction accuracy.