Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning — part 2/7

Cleaned-text code points [15593, 30969). Bibliography omitted here; source.md includes it.

These two geometric properties map directly to two test-time scaling levers. Depth scaling increases the number of forward iterations D D , giving a single trajectory more opportunities to refine within the basin it has entered. Breadth scaling runs B B independent restarts from initial states { 𝐳 0 ( i ) } i = 1 B \{\mathbf{z}_{0}^{(i)}\}_{i=1}^{B} and aggregates their outputs, increasing coverage over possible basins. We use the number of function evaluations, NFE = D ⋅ B \mathrm{NFE}=D\cdot B , as a compact way to describe inference budgets throughout the scaling experiments.

### 4.2 Landscape Modes and Scaling Implications

The attractor landscape view becomes useful when it predicts how test-time compute should be allocated. Fig. 6 combines task alignment, reachability, and the two scaling levers into four qualitative regimes. Each regime identifies the dominant failure source and therefore predicts whether depth scaling, breadth scaling, or neither should help. 2 2 2 Task error is defined at the sequence level: any token mismatch counts as incorrect. Token-level losses therefore induce a qualitatively different and much noisier landscape than the task-level metric in this setting, so we visualize only the latter.

(a) No correct attractor: all reachable attractors decode to poor task outcomes. The failure is task misalignment rather than insufficient compute, so residual reduction does not translate into task improvement and neither depth nor breadth scaling helps.

(b) Correct and spurious attractors coexist: a correct attractor exists, but inference may converge to competing low-residual, high-error attractors. The failure is basin selection, so breadth scaling is most useful because additional restarts increase the chance of entering the correct basin; depth helps only after the trajectory enters that basin.

(c) Correct but hard to reach: the correct attractor is nearly unique but has a narrow or weak basin. The failure is reachability, so breadth increases the chance of entering the basin and depth can help weakly attracted trajectories settle once they do; gains are limited by basin mass and stability.

(d) Well-aligned landscape: the correct attractor is broad and stable, so residual decay is tightly coupled with task-error reduction. Depth reliably refines trajectories toward the solution, while breadth provides additional coverage but is no longer the main bottleneck.

Thus, depth and breadth are complementary: depth refines a trajectory after it reaches a useful basin, while breadth increases basin coverage, consistent with the depth–breadth interaction in Fig. 3 . Test-time scaling succeeds when correct attractors are both aligned and reachable, motivating the training interventions in Sec. 5 .

## 5 Shaping Attractor Landscapes

Sec. 4.2 shows that test-time scaling is effective when the learned landscape contains correct attractors and inference reaches them reliably. Attractor landscape shaping is therefore the guiding training principle: we want the iterative dynamics to (i) admit correct solutions as stable attractors and (ii) make their basins easy to reach from diverse initial states as test-time compute increases. We now describe how to move generic iterative models toward Equilibrium Reasoners .

We introduce two task-agnostic interventions that do not require external verifiers or hand-crafted search heuristics: (1) randomized state initialization ( RI ), which samples initial latent states rather than model weights to improve coverage under breadth scaling and reduce train–test mismatch, and (2) noise injection (NI), which implements path stochasticity by perturbing each iteration step to mitigate premature trapping and broaden exploration as the iteration budget grows. Algorithm 1 shows the instantiation of the procedure.

### 5.1 Randomized State Initialization

HRM and TRM ( Wang et al., 2025 ; Jolicoeur-Martineau, 2025 ) typically train with a fixed initial state 𝐳 0 \mathbf{z}_{0} shared across trajectories. In contrast, we sample 𝐳 0 ∼ μ 0 ( ⋅ ∣ 𝐱 ) \mathbf{z}_{0}\sim\mu_{0}(\cdot\mid\mathbf{x}) independently for each trajectory. This matches training to breadth scaling at test time, where multiple draws of 𝐳 0 \mathbf{z}_{0} probe different basins. The benefit is twofold: it broadens the regions shaped during training and encourages stable predictions across restarts.

(i) Coverage of correct attractors. With a fixed initializer, learning is constrained to a small state-space neighborhood and tends to shape trajectories only locally. This limits exposure to alternative basins. Randomizing 𝐳 0 \mathbf{z}_{0} expands the explored region during training and increases the likelihood that correct attractors are reachable at inference.

(ii) Stability and path independence. Randomizing 𝐳 0 \mathbf{z}_{0} also promotes consistency across restarts: the same ( 𝐱 , 𝐲 ) (\mathbf{x},\mathbf{y}) is observed under multiple initial states, so divergent predictions are penalized. This encourages path independence ( Anil et al., 2022 ) by aligning predictions across trajectories.

By default, we use a Gaussian μ 0 ( ⋅ ∣ 𝐱 ) \mu_{0}(\cdot\mid\mathbf{x}) with covariance σ 0 ​ I \sigma_{0}I . Appendix A.3 and Appendix A.3 study learnable initializers and initialization scale; we use a fixed Gaussian initializer in the main experiments to isolate stochastic coverage from learned-prior design.

### 5.2 Path Stochasticity via Noise Injection

Random initializations reduce the train–test gap induced by breadth scaling; path noise regularizes how trajectories evolve. This targets modes (b) and (c) in Sec. 4.2 : mild noise can help trajectories enter better basins and avoid premature convergence to incorrect stable states. Thus, RI and NI act on complementary parts of the trajectory: RI broadens where rollouts start, while NI smooths the local dynamics encountered along each rollout.

We augment the iteration with damping and additive noise: 𝐳 k + 1 = 𝐳 k + ( 1 − λ ) ​ r θ ​ ( 𝐳 k , 𝐱 ) + β ​ ε k , \mathbf{z}_{k+1}=\mathbf{z}_{k}+(1-\lambda)\,r_{\theta}(\mathbf{z}_{k};\mathbf{x})+\beta\,\varepsilon_{k}, (2) where ε k ∼ 𝒩 ⁡ ( 0 , I ) \varepsilon_{k}\sim\mathcal{N}(0,I) . Here λ ∈ [ 0 , 1 ) \lambda\in[0,1) controls damping and β ≥ 0 \beta\geq 0 controls the noise magnitude. In other words, we inject isotropic Gaussian noise at each step and use β \beta to control its strength. This preserves the same update architecture while allowing controlled local exploration around the deterministic trajectory. We consider variants with different β \beta and a learnable noise variant in Appendix A.4 . Empirically, mild damping ( λ = 0.05 \lambda=0.05 ) combined with small path noise ( β = 0.01 \beta=0.01 ) performs best among the tested variants. At test time, one can increase β \beta under breadth scaling to foster exploration, analogous to temperature scaling.

## 6 Experiments

We organize the experiments around two questions. First, we ask what ingredients turn a feedforward model into a strong iterative model. Second, based on the iterative backbone, we test whether landscape-shaping interventions improve accuracy and make depth and breadth scaling reliable.

#### Task representation.

Each puzzle is serialized into a token sequence. The input sequence encodes the unsolved puzzle, and the target sequence encodes its solution, as illustrated in Fig. 5 . The sequence length is fixed during inference for a given task, since each task uses a fixed grid size, but it differs across tasks, e.g., Sudoku ( 9 × 9 ) (9\times 9) versus Maze ( 30 × 30 ) (30\times 30) . See more details in Appendix C .

#### Evaluation metrics.

By default, i.e., without breadth scaling, we report exact accuracy , which equals 1 only if all tokens are correct and 0 otherwise. Under breadth scaling with B B independent restarts, we consider three evaluation metrics in this paper: (1) Averaged exact accuracy , the mean exact-match accuracy over the B B restarts ( B = 1 B{=}1 reduces to the standard single run); (2) Top-1 convergence accuracy , which selects the restart with the smallest average residual over the final few iterations ( L = 3 L{=}3 ) and checks whether its prediction is correct; and (3) Majority-vote accuracy , which predicts by majority vote across restarts. Formal definitions are given in Appendix D.3 .

#### Baselines.

For Sec. 6.1 , we begin with a feedforward baseline and then introduce iterative components incrementally. For Sec. 6.2 , we also report HRM ( Wang et al., 2025 ) and TRM ( Jolicoeur-Martineau, 2025 ) baselines without our training interventions or inference extrapolation. For the TRM comparison, the backbone is matched at the block level: we use TRM’s task-specific settings for Sudoku and Maze, and each block has a task-dependent token mixer followed by an MLP. The mixer is an MLP-mixer on Sudoku and self-attention on Maze. This makes the comparison about the training and inference changes on a comparable iterative backbone, instead of a new block architecture. The feedforward baselines use 42 blocks on Sudoku and 15 blocks on Maze; the corresponding weight-tied variants use 2 blocks with 21 iterations and 1 block with 15 iterations, respectively. See Appendix D.1 for full details.

### 6.1 From Feedforward Models to Iterative Models

We study the gain from turning a feedforward model into an iterative one in this section. Tab. 2 starts from a vanilla feedforward predictor and adds weight-tying, long unrolls, hierarchy, and ACT.

Tab. 2 shows a monotonic construction path. Compared with the feedforward baseline under the same budgets, weight-tied models show significant improvement (from 2.6% to 32.6%); when we further scale up the depth with the supervision and update schedule needed to train long unrolls, the performance improves to 74.7% 3 3 3 Scaling the depth of weight-tied models requires a well-designed supervision and optimization schedule. We include the details in Appendix A.2 . . Hierarchical recurrence provides a smaller additional gain in this setting, ACT training allows difficulty-aware allocation of training-time compute per sample, and further improves inference performance to 84.8%. The ACT row has lower training accuracy because the model learns to stop early to prevent overfitting on easier examples while allocating more computation to learn harder ones, which gives the best evaluation accuracy in the construction path. Full ablations, update-schedule variants, ACT diagnostics, FLOPs accounting, and supporting figures are in Appendix A.2 .

#### Summary and takeaway.

Overall, the evidence supports a specific training principle: Weight-tied models create iterative capacity, but realizing that capacity at larger depth requires well-designed training strategies to shape the dynamics so that late-iteration states stay aligned with the task objective under finite stability and memory constraints.

### 6.2 From Iterative Models to Equilibrium Reasoners

Training with the proposed interventions (Sec. 5 ) improves both accuracy and the reliability of test-time scaling, consistent with shaping a more favorable attractor landscape. Across tasks, we observe three consistent effects: (i) higher baseline accuracy without additional test-time compute, (ii) a higher scaling ceiling as inference uses more depth, with further gains when depth is combined with breadth, and (iii) stronger empirical alignment between residual convergence and task correctness, which makes convergence-based selection effective under breadth scaling.

In Tab. 3 , baseline denotes the final TRM-style iterative model from Tab. 2 , before applying randomized state initialization or noise injection. We compare our methods against baselines on Sudoku and Maze in Tab. 3 and Tab. 4 .

Landscape shaping improves accuracy and stability without extra inference compute.

At the training-time compute budget, models trained with randomized state initialization and noise injection consistently outperform the same backbone trained without these interventions. In Tab. 3 , the full RI+NI training improves Sudoku from 84.8 84.8 to 86.4 86.4 and Maze from 44.9 44.9 to 82.2 82.2 , with RI alone already raising Maze accuracy to 68.6 68.6 . These gains hold on both training and evaluation splits, suggesting that correct attractors become reachable for a larger fraction of inputs within the same compute budget (Sec. 4.2 ). The same interventions also improve inference stability, as reflected by stronger path independence (Tab. 12 , Appendix A.4 ).

Shaped landscapes raise the test-time scaling ceiling.

After applying the landscape-shaping interventions, increasing depth first improves within-trajectory refinement, and combining the deeper rollout with breadth scaling further improves coverage across restarts. Tab. 4 makes this effect explicit: at B = 1 B{=}1 , increasing the depth from D = 16 D{=}16 to D = 64 D{=}64 raises EqR from 86.4 86.4 to 93.0 93.0 on Sudoku and from 82.2 82.2 to 88.9 88.9 on Maze. Combining this deeper rollout with breadth scaling ( B = 128 B{=}128 ) further increases accuracy to 99.8 99.8 on Sudoku and 93.0 93.0 on Maze. This pattern suggests that landscape shaping supports both forms of test-time scaling: deeper rollouts improve a single trajectory, while broader restarts cover more attractor basins. This is consistent with the landscape modes in Fig. 6 : interventions primarily mitigate modes (b) and (c) by enlarging correct basins and reducing spurious trapping.

Convergence becomes a reliable selection signal after landscape shaping.

Majority voting and convergence-based selection are both breadth-scaling strategies, but they exploit different signals. Majority voting aggregates decoded outputs, whereas convergence-based selection picks the run with the strongest convergence signal, e.g., the smallest average residual over the final few iterations. After learning a favorable landscape, convergence aligns better with task correctness, so selecting the Top-1 Converged run becomes a reliable and compute-efficient selection rule. We visualize this trend on Sudoku in Fig. 8 : as breadth increases, Top-1 Converged achieves comparable or higher expected accuracy than majority vote for the same number of restarts, with our training interventions.

This selection rule is not universally valid: for the baseline TRM, residual reduction can indicate convergence to spurious attractors, so Top-1 Converged can underperform majority voting. It becomes reliable only after shaping the landscape so residual convergence tracks task error. This distinction is important: convergence is useful here not as a task-agnostic certificate, but as a learned proxy whose reliability depends on the attractor landscape.

#### Summary.

Overall, the results suggest that the proposed interventions improve baseline accuracy and stability, raise the test-time scaling ceiling across depth and breadth, and strengthen convergence–correctness alignment. As a result, convergence becomes a more reliable selection signal under breadth scaling, offering a more efficient alternative to majority voting once the landscape is well shaped. Additional experiments in Appendix A.5 show that the same recipe also improves Mini-ARC performance and transfers to a Transformer backbone.

### 6.3 Adaptive Computation for Budget-Elastic Inference