Infrared Organization and Critical Cognitive Field Formation in Transformer Dynamics
Authors: Byung Gyu Chae
Organizations: Electronics and Telecommunications Research Institute, 218 Gajeong-ro, Yuseong-gu, Daejeon 34129, Republic of Korea
Abstract
Large language models exhibit remarkable emergent behaviors, yet the physical mechanism governing their collective dynamics remains poorly understood. Cognitive Field Theory predicts that learning reorganizes the time-scale density of states (TDOS) through the infrared accumulation of slow relaxation modes, thereby enhancing the memory self-energy, reducing the cognitive forgetting gap, and strengthening the collective susceptibility. Using publicly available Pythia language models, we extract relaxation spectra directly from Transformer layer Jacobians throughout training, network depth, and model scale, allowing the TDOS, memory self-energy, forgetting gap, memory kernel, and infrared critical exponent to be measured quantitatively. The measurements reveal progressive infrared accumulation of slow relaxation modes, producing an approximately flat infrared TDOS with ρ(λ)∼λ−0.1 and scale-free memory kernels K(t)∼t−1. The memory self-energy exhibits a pronounced transient maximum during early optimization before relaxing toward a metastable near-critical regime, corresponding to the smallest cognitive forgetting gap and the largest collective susceptibility predicted by Cognitive Field Theory. These observations provide quantitative experimental evidence that Transformer dynamics are governed by infrared collective organization. The reproducibility of the same dynamical behavior across training, network depth, and model scales suggests that infrared slow-mode organization represents a universal collective principle of Transformer dynamics.
We present a theoretical framework to explain the emergence of inductive reasoning abilities in Transformer language models. While previous works on Transformer learning dynamics have so far been mostly tied to specific tasks, we study a generalized class of inductive tasks that unifies several synthetic tasks known in the literature, including in-context n-grams and multi-hop reasoning. In this class, we theoretically prove that the training dynamics of attention models can be confined to a highly interpretable, low-dimensional invariant manifold. On this manifold, the learning dynamics are captured by a handful of interpretable coordinates rather than millions of parameters, making both theoretical and empirical analysis more tractable. Using this framework, we characterize how data statistics govern the competition between in-context and in-weights learning, we study how random initializations determine the `winning' circuit when multiple solutions are possible, and we demonstrate that the coordinate frame associated with the manifold can be used to automatically detect which circuits have been learned in trained models. By casting circuit formation as a low-dimensional dynamical phenomenon, we take a step toward a predictive theory of how Transformers learn.
We study the inference-time behavior of deep linear encoder-only transformers through the lens of interacting particle systems. In this perspective, tokens are modeled as particles that interact dynamically through successive linear self-attention layers. We show that in embedding dimension two, for any key, query, and value matrices, the dynamics can be reformulated as a generalized Kuramoto-type model with pure second-harmonic coupling. This formulation is amenable to Watanabe--Strogatz theory which reveals the dynamics are intrinsically low-dimensional regardless of the parameter matrices. For a class of token initializations associated with the Ott--Antonsen (OA) manifold, we show that the parameter matrices induce a diverse variety of long-time behaviors in linear transformers, including clustering, oscillations, and bifurcations. The oscillations and bifurcations are characterized by uncovering a hidden Hamiltonian structure in the dynamics. By establishing a structural stability result, we further show that dynamics initialized near the OA manifold exhibit the same long-time behavior as those initialized exactly on the manifold. Motivated by our theory in dimension two, we conduct numerical experiments for analogous parameter regimes in higher-dimensional transformers. Our numerical experiments suggest that the long-time behaviors characterized in our theoretical results persist in higher dimensions.
Transformers perform inference by iteratively transforming token representations across layers. This layerwise computation has been studied empirically, and recent mean-field theories of Transformer dynamics explain how attention can drive token distributions toward clustering. However, existing mean-field analyses largely treat model parameters as prescribed, leaving open how training reshapes this clustering picture. We study this question in a noisy mean-field Transformer in which only a parameter-linear FFN is trained under L2 regularization. We find and analyze a training-induced phase in the dynamics: after initially following attention-driven clustering, the token distribution can leave the clustered regime near the final layers. Our mathematical analysis is based on an entropy-regularized interaction energy that captures the clustering bias of attention. More broadly, our results point toward a training-aware mean-field theory of Transformer dynamics, in which training and inference dynamics are treated together.