Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning

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reasoning attractors test-time-compute scalable dynamical-systems iterative-models

Summary

Equilibrium Reasoners (EqR) introduce a novel framework for scalable reasoning by learning task-conditioned attractors in latent dynamical systems, achieving over 99% accuracy on Sudoku-Extreme by unrolling up to 40,000 layers.

Scaling test-time compute by iteratively updating a latent state has emerged as a powerful paradigm for reasoning. Yet the internal mechanisms that enable these iterative models to generalize beyond memorized patterns remain unclear. We hypothesize that generalizable reasoning arises from learning task-conditioned attractors: latent dynamical systems whose stable fixed points correspond to valid solutions. We formalize this process through Equilibrium Reasoners (EqR), which enable test-time scaling without external verifiers or task-specific priors. EqR scales internal dynamics along two axes: depth, by running more iterations, and breadth, by aggregating stochastic trajectories from multiple initializations. Empirically, gains from test-time scaling are tightly coupled with stronger convergence toward solution-aligned attractors. This attractor perspective allows neural networks to adaptively allocate test-time compute based on task difficulty. While simple cases converge within 1 to 5 iteration steps, harder cases benefit from massive test-time scaling. By unrolling up to the equivalent of 40,000 layers, scalable latent reasoning boosts accuracy from 2.6% for feedforward models to over 99% on Sudoku-Extreme. These results suggest that learned attractor landscapes provide a useful mechanistic lens for understanding scalable reasoning in iterative latent models.
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Source: https://huggingface.co/papers/2605.21488

Abstract

Equilibrium Reasoners enable scalable reasoning through task-conditioned attractors that guide latent dynamical systems toward valid solutions, achieving significant accuracy improvements through iterative test-time computation.

Scaling test-time compute by iteratively updating a latent state has emerged as a powerful paradigm for reasoning. Yet the internal mechanisms that enable theseiterative modelsto generalize beyond memorized patterns remain unclear. We hypothesize that generalizable reasoning arises from learningtask-conditioned attractors:latent dynamical systemswhose stable fixed points correspond to valid solutions. We formalize this process throughEquilibrium Reasoners(EqR), which enabletest-time scalingwithout external verifiers or task-specific priors. EqR scales internal dynamics along two axes: depth, by running more iterations, and breadth, by aggregatingstochastic trajectoriesfrom multiple initializations. Empirically, gains fromtest-time scalingare tightly coupled with strongerconvergencetoward solution-alignedattractors. This attractor perspective allows neural networks to adaptively allocate test-time compute based on task difficulty. While simple cases converge within 1 to 5 iteration steps, harder cases benefit from massivetest-time scaling. By unrolling up to the equivalent of 40,000 layers, scalable latent reasoning boosts accuracy from 2.6% for feedforward models to over 99% onSudoku-Extreme. These results suggest that learned attractor landscapes provide a useful mechanistic lens for understanding scalable reasoning in iterative latent models.

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