Turbulence-like 5/3 spectral scaling in contextual representations of language as a complex system

Understanding neural scaling laws is increasingly important for theory and practice, yet first-principles explanations remain limited even in simple settings. In this work, we study capacity-limited scaling in a bottlenecked bigram model with power-law (Zipf) token statistics, viewed as an analytically tractable toy model of a large language model. Fixed random embeddings serve as a proxy for representational capacity and induce an explicit bottleneck of width, allowing us to derive scaling laws for prediction regret as a function of model width, vocabulary size, and the Zipf exponent. Our results complement recent theoretical accounts based on optimization dynamics by isolating scaling due purely to representational constraints. In the appropriate regime, the theory recovers the functional form of classical neural scaling laws and suggests that heavy-tailed language statistics may underlie both optimization- and capacity-driven scaling. Experiments natural language datasets (WikiText2, OpenWebText) closely match the predicted curves.

Larger language models become simultaneously better and worse at handling contextual information -- better at ignoring false claims, worse at ignoring irrelevant tokens. We formalize this apparent paradox through the first scaling laws for contextual entrainment, the tendency of models to favor tokens that appeared in context regardless of relevance. Analyzing the Cerebras-GPT (111M-13B) and Pythia (410M-12B) model families, we find entrainment follows predictable power-law scaling, but with opposite trends depending on context type: semantic contexts show decreasing entrainment with scale, while non-semantic contexts show increasing entrainment. Concretely, the largest models are four times more resistant to counterfactual misinformation than the smallest, yet simultaneously twice as prone to copying arbitrary tokens. These diverging trends, which replicate across model families, suggest that semantic filtering and mechanical copying are functionally distinct behaviors that scale in opposition -- scaling alone does not resolve context sensitivity, it reshapes it.

Language model loss follows remarkably regular scaling laws over model and data size, yet it remains unclear why the aggregate loss should exhibit a power-law form. Existing explanations often attribute this regularity to a heavy-tailed spectrum of pattern difficulty in natural language, but this view has not been directly validated at token-level granularity in large-scale real-data training. We present a token-level framework that decomposes scaling laws into localized learning events of individual contextualized tokens. By fitting token loss trajectories with sigmoids, we show that token learning is concentrated in localized transitions, giving rise to a learning-time spectrum that dominates the scaling-law shape. Across more than one hundred pre-training runs on large and diverse real-language corpora with modern LLM architectures, scaling up to 6B parameters and 300B training tokens, the measured learning-time spectrum quantitatively reconstructs the validation loss derivative along the training-step $T$, data-scale $D$, and model-scale $M$ axes. We further show that the same signal is actionable: by reshaping the training distribution according to when tokens become learnable, we alter the optimization trajectory and achieve 11% faster validation-loss reduction. These results provide direct empirical evidence that scaling laws are governed primarily by the distribution of token-level learning times, and that this distribution can be used not only to explain scaling behavior but also to improve training performance.