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.

Two AI Metrics Diverged: Will it Make All the Difference?

As exponential compute scaling continues, will the capabilities of frontier AI models outstrip what is accessible to developers on a small fixed budget? Or will capabilities converge, with "meek models inheriting the earth"? Building on Gundlach et al. (2025b), we show that the answer depends on how we value and measure AI capabilities. We discuss conventional performance measures and show that, while validation loss shows a shrinking gap, on other metrics frontier models grow their lead forever. Classifying performance metrics by their functional forms in relation to training (and inference) compute, we provide tight mathematical conditions for determining which metrics favor meek models, and show that bounded performance metrics always do. But careful interpretation of performance metrics is essential: we show that many common bounded metrics have closely-related counterpart metrics that are unbounded (and vice versa). Determining the apt metric in a domain is a prerequisite for policy, since bounded and unbounded metrics may suggest opposing policy responses. If a particular capability -- like software engineering, synthetic biology, or rhetorical persuasiveness -- is unbounded when measured in the terms we care about, frontier-level capability will likely be concentrated in the hands of a few wealthy actors. Conversely, if that capability is instead bounded, frontier-level capabilities proliferate through meek models into the hands of the many.