Fixed-Point Reasoners: Stable and Adaptive Deep Looped Transformers

Looped Transformers offer a promising alternative to purely feed-forward computation by iteratively refining latent representations, improving language modeling and reasoning. Yet recurrent architectures remain unstable to train, costly to optimize and deploy, and constrained to small, fixed recurrence depths. We introduce Attractor Models, in which a backbone module first proposes output embeddings, then an attractor module refines them by solving for the fixed point, with gradients obtained through implicit differentiation. Thus, training memory remains constant in effective depth, and iterations are chosen adaptively by convergence. Empirically, Attractor Models outperform existing models across two regimes, large-scale language-model pretraining and reasoning with tiny models. In language modeling, Attractor Models deliver a Pareto improvement over standard Transformers and stable looped models across sizes, improving perplexity by up to 46.6% and downstream accuracy by up to 19.7% while reducing training cost. Notably, a 770M Attractor Model outperforms a 1.3B Transformer trained on twice as many tokens. On challenging reasoning tasks, we show that our model with only 27M parameters and approximately 1000 examples achieves 91.4% accuracy on Sudoku-Extreme and 93.1% on Maze-Hard, scaling favorably where frontier models like Claude and GPT o3, fail completely, and specialized recursive reasoners collapse at larger sizes. Lastly, we show that Attractor Models exhibit a novel phenomenon, which we call equilibrium internalization: fixed-point training places the model's initial output embedding near equilibrium, allowing the solver to be removed at inference time with little degradation. Together, these results suggest that Attractor Models make iterative refinement scalable by turning recurrence into a computation the model can learn to internalize.

Chain-of-thought prompting and looped Transformers both give a fixed model more test-time computation, but they differ in what they remember. Chain-of-thought stores intermediate state in generated tokens that remain in the context, whereas a looped Transformer carries state through recurrent hidden activations. We argue that this persistent mutable memory is a central resource for test-time reasoning. We compare three memory regimes, the compressed latent loop, the full sequence-state loop, and the chain-of-thought scratchpad. Our main result shows that a compressed loop is limited by the size of its recurrent state. Running the loop longer adds computation but does not by itself create a growing scratchpad, so a loop with a small recurrent state remains a small-space reasoner even when run for many steps. Under a standard complexity assumption, such loops cannot decide problems that are P-complete under logspace reductions, whereas polynomial-length chain-of-thought can. The separation is specific to compressed loops, as full sequence-state loops carry state at every input position and live in a memory-rich regime closer to explicit scratchpads. Controlled pointer-chasing and associative-recall sweeps illustrate this memory-budget view, with performance sensitive to whether the persistent-state budget matches the task's working-memory demand.

Looped transformers promise test-time compute scaling by spending more iterations on harder problems, but it remains unclear which architectural choices let them extrapolate to harder problems at test time rather than memorize training-specific solutions. We introduce a fixed-point based framework for analyzing looped architectures along three axes of stability -- reachability, input-dependence, and geometry -- and use it to characterize when fixed-point iteration yields meaningful predictions. Theoretically, we prove that looped networks without recall have countable fixed points and cannot achieve strong input-dependence at any spectral regime, while recall combined with outer normalization reliably produces a regime in which fixed points are simultaneously reachable, locally smooth in the input, and supported by stable backpropagation. Empirically, we train single-layer looped transformers on chess, sudoku, and prefix-sums and find that downstream performance tracks the framework's predictions across tasks and architectural configurations. We additionally introduce internal recall, a novel recall placement variant, and show that it becomes competitive with -- and on sudoku, substantially better than -- standard recall placement once outer normalization is applied.