Probability Signature: Bridging Data Semantics and Embedding Structure in Language Models

In recent years, deep neural network-based large language models (LLMs) have demonstrated remarkable performance (Comanici et al., 2025; OpenAI et al., 2024; DeepSeek-AI et al., 2025). The development of these models has largely followed what Richard Sutton termed "the bitter lesson"-that the most effective approach to improving AI performance has historically been to leverage greater computational resources, larger models, and more data, rather than incorporating human knowledge or specialized architectures (Sutton, 2019). This trend has been formalized through scaling laws, which quantify the relationship between model performance and factors such as model size, dataset size, and computational budget through power law relationships (Kaplan et al., 2020). While these scaling laws provide valuable quantitative predictions for model performance, they also reveal a concerning limitation: the power law relationship suggests that achieving further significant improvements may require prohibitively large increases in model and data size, making continued scaling increasingly impractical and resource-intensive. One promising approach to address these limitations is to develop a deeper understanding of the underlying mechanisms that drive transformer models' success in natural language processing (NLP). The No Free Lunch theorem establishes that no single algorithm can perform optimally across all problem domains, highlighting the fundamental importance of understanding both the characteristics of the data and the properties of the algorithms that process it (Wolpert & Macready, 1997).