From Fixed Points to Scaling Laws: A Parameter-Free Prediction of Grokking Phase Transitions in Large Language Models Using the Functional Renormalization Group

The empirical success of Large Language Models (LLMs) is currently governed by phenomenological scaling laws and heuristic observations of phase transitions, such as "Grokking". However, a rigorous, first-principles mathematical framework guaranteeing the stability and universality of these dynamics remains absent. In this work, we bridge this gap by formulating the training dynamics of overparameterized neural networks as a statistical field theory on the Fisher Information Metric (FIM) manifold. We derive the exact Wetterich flow equation for the effective average loss, demonstrating that macroscopic scaling laws emerge parameter-free as critical exponents of a non-trivial infrared (IR) fixed point. Furthermore, we prove that the Grokking transition is not an optimization artifact, but a strict Non-Equilibrium Steady State (NESS) Lyapunov attractor, driven by the discontinuous collapse of the Hessian's spectral entropy. By establishing the completeness of the O(∂²) truncation, we quantify finite-size corrections to scaling laws. Finally, we provide three concrete, falsifiable predictions regarding the Hessian spectral gap, entropy collapse, and RG data collapse, offering a mathematically closed foundation for AI safety and core mechanism research.