# Dynamical phase selection controls compute scaling in looped transformers

arXiv: https://arxiv.org/abs/2608.26556 · HTML: https://mlsanity.com/paper/2608.26556/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 3 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Built on by (most cited first, 2 of 2)

- [Improving Test-Time Scaling with Adaptive Looped Transformers](https://mlsanity.com/paper/2609.35748/) (2026)
- [Looped Reasoning Finishes Earlier Than You Think](https://mlsanity.com/paper/2610.18174/) (2026)

## Abstract (arXiv)

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-λmax(-varepsilon)]→π. Composed with a regular distribution of problem difficulty, the same critical slowing down produces the workload-level tail P(τ>N)~ 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.