Dynamical phase selection controls compute scaling in looped transformers

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 $τ(\varepsilon)[1-λ_{\max}(-\varepsilon)]\toπ$. Composed with a regular distribution of problem difficulty, the same critical slowing down produces the workload-level tail $P(τ>N)\sim N^{-2}$. 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.

The Transformer Revolution, Part 1: Dynamic Processing through Output- Weight Interconnections

This paper offers a new interpretation of the Transformer during inference. Against the "stochastic parrot" view that large language models merely reproduce statistical regularities learned in training, we argue that Transformers construct and apply prompt-dependent transformations whose parameters are generated during inference. We call this form of computation SIDPP: Sequence-level Interactive Dynamic Parallel Processing. The Transformer is interpreted as a system that transforms concepts by means of concepts. Token vectors are the concepts to be transformed; parameterized transformations defined by matrices and vectors are the transforming concepts. These may be static, when fixed through training, or dynamic, when generated from the input sequence. Mechanically, they correspond to groups of simple neural networks. The Transformer's architectural novelty lies in output-weight interconnections, through which the outputs of some networks determine the weights of others, alongside ordinary output-input interconnections. By means of these interconnections, the system constructs transformations from the prompt and uses them to modify token representations. The contribution of dynamic processing grows with prompt length and may equal or exceed that of static processing, a phenomenon we call strong prompt sensitivity. This account bears on interpretability, predictability, control, and the design of smaller, more sustainable systems. Finally, since the human neural system possesses the mechanisms required to implement SIDPP, we argue that a form of SIDPP may, in principle, be neurally realized in the cerebral cortex. We therefore conjecture that human language processing may itself be a form of SIDPP produced by a functional architecture relevantly similar to that of the Transformer.