Clustered Attractor Manifolds and Dynamical Condensation in Self-Attention

We study the nonequilibrium dynamics of a minimal recurrent transformer with $N$ normalized tokens, $Q=K=I$, and a negative value map $V=-I$. Similarity-based attention selects nearby representations, while the negative value map drives tokens away from the selected field. This feedback can continually reorganize both the representation geometry and the attention network. For $d=2$, the tokens lie on a circle, where the regular polygon is an exact fixed point. As the attention feedback strength $γ$ is increased, the polygon loses stability through a flip bifurcation, giving rise to period-two motion, chaos, and cluster-exchange or cluster-flip states. Despite this temporal complexity, attention remains diffuse as $N\to\infty$ at finite fixed softmax sharpness $β$. Attention condensation instead emerges in the scaling regime $β\sim N^2$. In the hard-routing limit, repulsive updates amplify local perturbations and routing-partner switches transmit them ballistically, producing an emergent butterfly cone in representation space. High-dimensional geometry provides a distinct route to localization. For $d=N\to\infty$, simulations from Gaussian initial conditions provide evidence for a condensation transition at $β=O(1)$, driven by dynamically generated finite overlap gaps. Depending on $γ$, the resulting phases include diffuse simplex-like states, consensus flips, condensed active routing with signatures of chaos, and fragmented cluster flips. These results establish temporal activity, attention condensation, and geometric clustering as distinct collective phenomena, and show that sparse attention can sustain persistent dynamics rather than freeze it.

Transformer self-attention can be interpreted as a gradient flow on the unit sphere, in which tokens evolve under softmax interaction potentials and tend to form clusters. While prior work has established clustering behavior for single-head attention, the multi-head setting remains less understood due to geometric interference between heads, which invalidates standard monotonicity arguments. In this work, we develop a theoretical framework for multi-head self-attention dynamics and resolve several open questions. We show that, under suitable conditions on the score matrices, a natural multi-head energy functional is non-decreasing along both flat and spherical dynamics. We identify the key obstruction to per-head monotonicity as radial shadow terms, which are projections of each head's output onto token directions, persisting even under orthogonality assumptions. We introduce a sufficient condition ensuring monotonicity and establish robustness to approximate orthogonality. In a simplified scalar-head regime with equiangular token configurations, we derive a closed-form expression for the critical inverse temperature governing clustering behavior, and show that heterogeneous heads exhibit super-additive clustering rates. In this regime, we also prove a separation in clustering time between ReLU and softmax attention in the linearized dynamics. Finally, we establish an entropy production identity and show that attention entropy increases monotonically toward equilibrium as clustering progresses. Our results provide a unified perspective on the dynamics of multi-head attention and clarify the mechanisms underlying clustering and stability in transformer models.

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 $L^2$ 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.