Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning — part 7/7

Cleaned-text code points [89652, 93243). Bibliography omitted here; source.md includes it.

The third question is mechanistic: what structure appears in the latent states themselves? Controlled implicit-reasoning experiments show systematic generalization, depth extrapolation, and overthinking as the number of iterations changes ( Kohli et al., 2026 ) . Mechanistic analyses of weight-tied reasoning language models find cyclic trajectories approaching distinct fixed points, with attention behavior stabilizing across iterations ( Blayney et al., 2026 ) . Preference probes show that pairwise differences between iteration states can encode relational signals ( Kirin, 2026 ) , while latent-chain-of-thought probes caution that hidden states need not be directly readable as natural-language reasoning traces ( Lu et al., 2025 ) . Closest to our vocabulary, fixed-point analyses characterize stability through reachability, input-dependence, and geometry ( Labovich, 2026 ) , and two-scale trajectory studies distinguish small within-block refinements from larger cross-block drift ( Pappone et al., 2025 ) . Concurrent HRM analysis also links failures to fixed-point violations and multiple fixed points, but its strongest Sudoku gains rely on data augmentation, input perturbation, and model bootstrapping ( Ren and Liu, 2026 ) . Our work takes the next step: we directly measure the attractor landscape of finite-iteration, fixed-state weight-tied models, relate residual-state updates to iteration steps, and explain failure modes as trajectories entering unstable or wrong basins. We then use this landscape view to ask which training interventions reshape the basins, why those changes raise the inference-time upper bound, and how learned stopping can save compute without discarding the iterations that actually move a trajectory toward a correct attractor.

#### Flat minima.

The connection between the geometry of the loss landscape and a model’s generalization ability is a long-standing area of research. A central hypothesis, dating back to the work of Hochreiter and Schmidhuber (1997) , posits that optimizers that converge to “flat” minima in the parameter space tend to produce models that generalize better than those that converge to “sharp” minima. The intuition is that a flat region of the loss landscape is more robust to small perturbations in the model’s weights, such as those caused by shifts between the training and test data distributions. This idea has been revitalized by modern optimization methods like Sharpness-Aware Minimization (SAM) ( Foret et al., 2021 ) . SAM formalizes this intuition into a min-max optimization objective that explicitly searches for parameters residing in neighborhoods with uniformly low loss values. By doing so, SAM encourages convergence to flatter regions of the parameter landscape, leading to improvements in generalization across various benchmarks.

Our work translates this concept of robustness from parameter space to state space. While SAM seeks solutions that are stable with respect to perturbations of the model weights θ \theta , our framework seeks attractors that are stable with respect to perturbations of the latent state 𝐳 \mathbf{z} . The path stochasticity we introduce during training (section 5.2 ) serves a similar purpose to the perturbation step in SAM: it forces the model to learn an update function f θ f_{\theta} that defines a smooth and robust attractor landscape, where trajectories reliably converge to the correct solution despite noise. In essence, we are searching for “flat minima” in the landscape of the state-space dynamics, not just in the landscape of the training loss.