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This study investigates how the dynamical phases of looped transformers, influenced by initialization, affect test-time compute scaling despite identical architectures and training objectives. By identifying distinct bifurcation mechanisms, such as saddle-node folds and Neimark-Sacker transitions, the authors reveal that these phases dictate the computational cost during inference. The findings highlight that the dynamical phase of a trained model, rather than its architecture, plays a crucial role in determining compute efficiency, with implications for workload distribution and performance scaling.
Test-time compute in looped transformers is governed more by the model's dynamical phase than by its architecture, revealing a surprising dependency on initialization.
A looped transformer performs inference by iterating a weight-tied map, making its computation a dynamical process whose cost is set by the resulting inference dynamics. Here we show that networks with identical architecture and objective, trained to identical accuracy, nevertheless realize distinct dynamical phases depending strongly on initialization, and that the bifurcation defining each phase determines how test-time compute scales. The phases are distinguished by their bifurcation mechanisms, including a saddle-node fold and a Neimark-Sacker-type transition to bounded nonstationary motion. In the fold phase, a one-dimensional normal-form reduction predicts both the relaxation-time and spectral-gap amplitudes from local derivatives of the trained map, yielding the parameter-free relation $\tau(\varepsilon)[1-\lambda_{\max}(-\varepsilon)]\to\pi$. Composed with a regular distribution of problem difficulty, the same critical slowing down produces the workload-level tail $P(\tau>N)\sim N^{-2}$. In the Neimark--Sacker phase, the fold scaling law disappears rather than merely changing its prefactor. Thus, test-time compute is not determined by architecture alone. It is governed by the dynamical phase of the solution found by training.