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arxiv: 2605.21488 · v1 · pith:TLPOOVCSnew · submitted 2026-05-20 · 💻 cs.LG

Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning

Benhao Huang , Zhengyang Geng , Zico Kolter This is my paper

Pith reviewed 2026-05-21 04:53 UTC · model grok-4.3

classification 💻 cs.LG
keywords equilibrium reasonersattractorslatent dynamical systemstest-time scalingiterative reasoningfixed pointssudoku
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The pith

Neural networks learn task-conditioned attractors in latent space so that iterative updates at test time converge to valid solutions and scale reasoning performance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that generalizable reasoning emerges when models are trained to form latent dynamical systems whose stable fixed points align with correct task solutions. If this holds, then simply running more internal iterations at test time lets the network refine its output adaptively, using greater depth for harder cases without external verifiers or hand-crafted rules. Experiments on Sudoku-Extreme demonstrate the effect: feedforward baselines reach only 2.6 percent accuracy, yet the same architecture unrolled to the equivalent of 40,000 layers exceeds 99 percent when convergence to solution attractors is strong. A sympathetic reader would care because the account supplies a concrete mechanism for why test-time scaling works and shows that performance gains track directly with attractor alignment rather than with raw iteration count alone.

Core claim

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. By unrolling up to the equivalent of 40,000 layers, scalable latent reasoning boosts the 2.

What carries the argument

Equilibrium Reasoners (EqR), models whose latent updates implement task-conditioned dynamical systems that converge to stable fixed points at valid solutions.

If this is right

  • Accuracy on hard instances rises in direct proportion to how closely latent trajectories approach the solution attractor.
  • Simpler problems converge in few steps while difficult ones require and benefit from thousands of additional iterations.
  • Compute can be allocated adaptively by monitoring convergence speed during test-time execution.
  • No external verifiers or task-specific priors are required for the scaling effect to appear.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same attractor-training recipe could be applied to other structured reasoning domains such as theorem proving or planning to produce analogous test-time scaling.
  • The perspective offers a possible dynamical explanation for why chain-of-thought or scratchpad methods improve performance in language models.
  • One could deliberately perturb training to misalign attractors and test whether reasoning accuracy collapses while feedforward accuracy remains intact.

Load-bearing premise

Generalizable reasoning requires learning task-conditioned attractors whose stable fixed points are exactly the valid solutions rather than arising from some other internal process.

What would settle it

Finding high accuracy on Sudoku-Extreme while the iterated latent states fail to converge to the known solution configurations, or observing that accuracy improvements decouple from measured convergence strength.

Figures

Figures reproduced from arXiv: 2605.21488 by Benhao Huang, Zhengyang Geng, Zico Kolter.

Figure 1
Figure 1. Figure 1: Scalable test-time compute emerges from convergence to neural attractors. We evaluate Equilibrium Reasoners (EqRs) with varying inference budgets and plot exact accuracy against the fixed-point residual ∥fθ(z; x) − z∥ (where z denotes latent states; lower indicates better convergence). Point color encodes the number of iterations on a log scale. Remarkably, while training is capped at 16 iterations, the le… view at source ↗
Figure 2
Figure 2. Figure 2: Landscape alignment enables scalable and general￾izable reasoning. The lower surface depicts the task-metric land￾scape over solutions, while the upper surface depicts the model’s learned internal landscape over latent states. We conceptualize EqR as learning a task-conditioned fixed-point system z ⋆ = fθ(z ⋆ ; x), whose attractors form an internal landscape in latent space. Train￾ing shapes this internal … view at source ↗
Figure 3
Figure 3. Figure 3: Pareto heatmaps illustrating two-axis test-time scaling with iterations ("depth", D), and stochastic trajectories from multiple initializations ("breadth", B). Left: fixed-point residual r = ∥fθ(z; x) − z∥ (power-law normalized), where x stands for model input and z for latent states. Right: prediction error (lower is better, log-scaled). Breadth scaling becomes effective only beyond a minimum number of mo… view at source ↗
Figure 4
Figure 4. Figure 4: Controlled construction path. We build a controlled path from a feedforward model to a scalable iterative reasoner by progressively adding five ingredients: (1) Weight-tied parameters across iteration steps; (2) Gradient truncation with detached carry to stabilize optimization through long trajectories and reduce cost; (3) Segmented online training to shape intermediate solver states through interleaved pa… view at source ↗
Figure 5
Figure 5. Figure 5: Datasets and Evaluations. We evaluate our models and baselines on Sudoku-Extreme, a challenging 9 × 9 Sudoku benchmark for long-horizon constraint satisfaction, and on Maze-Unique, a uniquely solvable variant of Maze-hard-1k designed to remove ambiguity caused by multiple shortest paths. Additional details are provided in Appendix C. This attractor view keeps the core convergence claim with￾out requiring c… view at source ↗
Figure 6
Figure 6. Figure 6: Four attractor landscape modes visualized by residual and task error. We run 512 random initializations over 256 Sudoku-Extreme examples, project trajectories to 2D with PCA, and color by sequence-level error: (a) no correct attractor; (b) correct and spurious attractors; (c) a narrow correct basin; (d) a broad, mostly unique correct attractor. a trajectory after it reaches a useful basin, while breadth in… view at source ↗
Figure 7
Figure 7. Figure 7: Test-time depth-scaling efficiency on Sudoku-Lite. At a matched exact-accuracy target of 92.99%, EqR requires 3.76× fewer NFEs than the baseline, and EqR+ACT improves this to 11.34× fewer NFEs. The gain is therefore not explained by in￾creased test-time compute alone: the proposed training interven￾tions make the same accuracy reachable with substantially less inference compute. 0 32 # Restarts (k) 0.75 0.… view at source ↗
Figure 8
Figure 8. Figure 8: Breadth-scaling efficiency of aggregation rules on Sudoku. Expected accuracy versus the number of restarts for majority vote and Top-1 Converged across TRM, TRM+RI, and EqR (TRM+RI+NI). After applying RI, especially RI+NI, Top-1 Converged becomes more compute-efficient than majority vote and reaches a higher accuracy ceiling. Convergence becomes a reliable selection signal after land￾scape shaping. Majorit… view at source ↗
Figure 9
Figure 9. Figure 9: Iterative depth improves generalization under a matched layer-evaluation budget. On Sudoku, increasing the iteration steps changes both optimization and generalization: the shallow iterative model reaches higher training accuracy than the feedforward depth baseline at its best evaluation checkpoint, while deeper iterative models trade slightly lower training accuracy for substantially higher evaluation acc… view at source ↗
Figure 10
Figure 10. Figure 10: Gradient flow in segmented online training. Left: a segment loss supervises the current solver state while backpropagating only through the immediately preceding local computation graph, marked by the red dashed region. Right: over a long trajectory, SOT changes the optimizer time scale: each segment performs a local backward pass and an immediate parameter update, then the next segment continues from the… view at source ↗
Figure 11
Figure 11. Figure 11: Residual diagnostic for late anchors. At the terminal-loss detached-carry checkpoint, the rollout residual ∥zt − zt−1∥ drops sharply in the early iterations and then changes only slightly over the late trajectory. The shifted-log scale keeps this late plateau visible: the change from iterations 8 to 12 is small compared with the early 2 to 8 drop, supporting the use of late anchors that supervise states a… view at source ↗
Figure 12
Figure 12. Figure 12: Ablation of ACT halting mechanisms. We compare several ACT variants (including oracle halting based on ground-truth correctness) and report both task performance and the resulting average number of iteration steps at 50k training steps; oracle halting tends to overfit and collapses toward near-single-step behavior. Backward and parameter-update costs cℓ One local loss/head backward. cB Segment parameter-b… view at source ↗
Figure 13
Figure 13. Figure 13: An example reasoning trajectory (length 32) produced by TRM on a Sudoku-Extreme puzzle. Correct predictions are highlighted in green. The figure suggests that the model’s reasoning is not strictly sequential in the way an algorithmic solver would be. Instead, it exhibits “erase then retry” behavior, with partial revisions that overwrite earlier choices. We mark one representative location with a red circl… view at source ↗
Figure 14
Figure 14. Figure 14: Maze benchmark examples. Maze-1k contains inputs with multiple valid shortest paths, while Maze-Unique restricts the benchmark to uniquely solvable instances. 0 33 66 99 132 165 198 231 Rating 0 20000 40000 60000 80000 100000 Count variance=997.12 Train Ratings mean=29.87 0 19 38 57 76 95 114 133 152 Rating 0 50 100 150 200 250 300 Count variance=464.04 Lite-Eval Ratings mean=20.92 0 49 98 147 196 245 294… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Equilibrium Reasoners (EqR), a framework for scalable latent reasoning in which neural networks learn task-conditioned attractors whose stable fixed points are hypothesized to correspond to valid solutions. The approach scales test-time compute along depth (by increasing the number of iterations) and breadth (by aggregating multiple stochastic trajectories), with the claim that performance gains are tightly coupled to stronger convergence toward solution-aligned attractors. On Sudoku-Extreme, unrolling dynamics equivalent to 40,000 layers raises accuracy from 2.6% (feedforward baseline) to over 99%.

Significance. If the attractor mechanism can be shown to operate independently of label-based supervision, the work would supply a dynamical-systems account of why iterative latent models generalize and scale, together with a practical method for adaptive test-time compute allocation. The reported scaling results on a hard constraint-satisfaction task would be a notable empirical contribution to the literature on test-time reasoning.

major comments (2)
  1. [Abstract / Hypothesis] Abstract and opening hypothesis paragraph: the claim that fixed points 'correspond to valid solutions' and that convergence is 'tightly coupled' to accuracy risks circularity if solution-alignment is measured by proximity to the same ground-truth labels used to compute the 99% accuracy figure. The manuscript must demonstrate that the learned fixed points satisfy the underlying task constraints (Sudoku rules, etc.) on held-out instances even when label information is withheld from the convergence metric.
  2. [Empirical Results] Empirical section reporting the Sudoku-Extreme results: the abstract states large gains and a correlation between convergence and accuracy, yet supplies no dataset splits, error bars, ablation controls, or explicit separation between the attractor fixed-point property and the supervised accuracy signal. Without these, the support for the mechanistic attractor hypothesis cannot be evaluated.
minor comments (1)
  1. [Methods / Scaling Description] Clarify the precise mapping from iteration count to 'equivalent of 40,000 layers' and whether this equivalence is architectural or merely computational.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments highlight important points for clarifying the independence of the attractor mechanism from label supervision and for improving the empirical reporting. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract / Hypothesis] Abstract and opening hypothesis paragraph: the claim that fixed points 'correspond to valid solutions' and that convergence is 'tightly coupled' to accuracy risks circularity if solution-alignment is measured by proximity to the same ground-truth labels used to compute the 99% accuracy figure. The manuscript must demonstrate that the learned fixed points satisfy the underlying task constraints (Sudoku rules, etc.) on held-out instances even when label information is withheld from the convergence metric.

    Authors: We agree that explicit separation is necessary to substantiate the mechanistic claim. Convergence in EqR is defined via the fixed-point residual of the latent dynamics (||Δx|| < ε), which does not reference ground-truth labels during iteration. In the revision we will add a dedicated analysis on held-out Sudoku-Extreme instances showing that trajectories reaching this label-free convergence criterion decode to states satisfying all Sudoku constraints (unique rows, columns, and blocks) at rates that track the reported accuracy gains. This will demonstrate that the attractor property operates independently of the supervised accuracy metric. revision: yes

  2. Referee: [Empirical Results] Empirical section reporting the Sudoku-Extreme results: the abstract states large gains and a correlation between convergence and accuracy, yet supplies no dataset splits, error bars, ablation controls, or explicit separation between the attractor fixed-point property and the supervised accuracy signal. Without these, the support for the mechanistic attractor hypothesis cannot be evaluated.

    Authors: We acknowledge that these details were insufficiently reported. The revised manuscript will specify the exact train/validation/test splits for Sudoku-Extreme, report standard error bars across multiple random seeds, and include targeted ablations that compare convergent EqR trajectories against non-convergent iterative baselines while holding supervision fixed. These additions will make the empirical support for the attractor hypothesis directly evaluable. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper hypothesizes that generalizable reasoning arises from task-conditioned attractors whose fixed points correspond to valid solutions, then reports that test-time scaling gains are empirically coupled to convergence toward solution-aligned attractors, with accuracy improvements demonstrated on Sudoku-Extreme via unrolling iterations. No equations, fitting procedures, or derivations are shown that reduce any claimed prediction or result to its inputs by construction. There are no self-citations, uniqueness theorems, or ansatzes invoked in the provided text that would create load-bearing circularity. The central claims remain independent of the accuracy metric and are presented as an empirical observation rather than a definitional equivalence, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review is limited to the abstract; the ledger therefore records only the core hypothesis and postulated mechanism explicitly stated there.

axioms (1)
  • domain assumption Stable fixed points of the learned latent dynamics correspond to valid task solutions
    Invoked in the first paragraph as the basis for the entire approach.
invented entities (1)
  • task-conditioned attractors no independent evidence
    purpose: Stable fixed points in latent space that encode correct solutions for a given task
    Newly hypothesized mechanism introduced to explain generalization and scalable reasoning

pith-pipeline@v0.9.0 · 5743 in / 1326 out tokens · 39381 ms · 2026-05-21T04:53:29.026844+00:00 · methodology

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Reference graph

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