Pith. sign in

REVIEW 4 major objections 7 minor 37 references

Explicit energy over valid task structures can select the correct answer from recursive-reasoning rollouts, matching the best possible pool accuracy on Sudoku, Pencil Puzzle Bench, and Maze.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 07:20 UTC pith:A6MYKUSA

load-bearing objection A clean demonstration that explicit task-structure energies can replace learned selectors for recursive reasoners and match the oracle on three benchmarks, though the ranking property of the hand-built potentials is not stress-tested. the 4 major comments →

arxiv 2607.10128 v2 pith:A6MYKUSA submitted 2026-07-11 cs.LG stat.ML

Energy-guided Recursive Model

Yifei Zhao , Ying Tang This is my paper
classification cs.LG stat.ML
keywords energy-based inferencerecursive reasoninglatent reasoningtest-time scalingHopfield memoryparallel temperingSudokupuzzle solving
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that the selection step in latent recursive reasoning should be a principled energy over task-valid structures, not a learned confidence head or heuristic voting. It introduces the Energy-guided Recursive Model (ERM), which scores each candidate rollout by its distance to enumerable valid local patterns plus a global rule-violation penalty. With 64 recurrent steps and 128 candidates, ERM reaches the shared-candidate oracle on all three benchmarks: 98.97% on Sudoku, 88.04% on Pencil Puzzle Bench, and 99.30% on Maze, without adding any learned parameters. The central message is that when a correct candidate already exists in the rollout pool, an energy built from the task's own rules can find it, and the same energy can guide parallel-tempering sampling.

Core claim

The paper claims that test-time scaling in latent iterative reasoners fails mainly at selection, not generation. In a candidate pool produced by a recursive reasoner, the correct answer is often present but ordinary selectors miss it; ERM instead evaluates every decoded candidate with a Hopfield-type retrieval energy defined by memories of valid structures. For Sudoku, the memories are row, column, and box permutations; for Pencil Puzzle Bench, the memories are puzzle-rule patterns plus global distance potentials for connectivity and coverage; for Maze, the memories encode valid simple paths and distance is computed through format, connectivity, degree, cycle, and length penalties. In the ma

What carries the argument

The load-bearing object is the ERM energy in Equation 3: E_ERM(y_k;x) = mu G(y_k,x) - sum_j tau log sum_{m in M_j(x)} exp(-beta d_j(y_k,m)/tau). It is a retrieval energy: each factor j has an enumerable memory set M_j(x) of valid local patterns, d_j is a distance from the candidate to a memory, and the log-sum-exp behaves like nearest-memory retrieval at small temperature; G(y_k,x) is an implicit global memory distance for constraints that cannot be listed locally, such as connectivity and coverage. The energy replaces the learned q-head or majority vote as the selector, and the same scalar can define a tempered distribution for parallel tempering.

Load-bearing premise

The hand-constructed energy functions, especially the global connectivity and coverage penalties, are faithful enough to rank every valid candidate above every invalid one on the test distribution; if any approximate penalty misranks a pair, ERM would miss the oracle.

What would settle it

Construct or find a test instance where the candidate pool contains a correct solution but ERM's energy selects an incorrect candidate, for example by adversarially adding a disconnected candidate with lower global penalty on Pencil Puzzle Bench, or by measuring whether the Maze connectivity penalty ever scores a broken path below a valid one. A systematic sweep over corrupted candidates would reveal such a false ranking.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • If ERM's central claim holds, test-time compute in recursive reasoners should be measured as a two-part problem: candidate generation and energy-based selection, with the oracle gap telling which part to improve.
  • The energy can also serve as an early-stopping signal: on Maze, stopping when the ERM energy stabilizes uses only 7.60 average recurrent steps and retains 99.60% exact accuracy, suggesting that full depth is often unnecessary.
  • Parallel tempering with the ERM energy can improve sampling, not just reranking: on Sudoku, PT-ERM reaches 98.97% and matches its own oracle, showing no correct candidates are lost at the final selection step.
  • The oracle-matching results imply that remaining errors on these benchmarks are candidate-generation failures, not selection failures: if the correct answer appears in the pool, ERM finds it.
  • The method's design recipe — list local valid factors, add global distance penalties only for non-local rules, and report oracle-vs-selected gaps — generalizes as a diagnostic for other benchmark tasks.

Where Pith is reading between the lines

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

  • The same energy-selection recipe could transfer to autoregressive language models: for tasks with checkable constraints, a rule-distance verifier could replace a learned reward model when ranking sampled chains, though the paper does not test this.
  • Because ERM adds no learned parameters, its oracle-matching performance marks an upper bound on what selection from a fixed candidate pool can achieve; further gains on these tasks would have to come from more diverse or better-trained generation.
  • The energy's value as a stopping diagnostic suggests a testable extension: learn part of the memory set or distance potential on one task and transfer it to a sibling task, using the oracle-gap metrics as an automatic guide.
  • A natural stress test would be to corrupt the global potentials adversarially and measure when ERM drops below the oracle; the paper reports no such failure analysis, so the robustness boundary of the hand-built energies remains open.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The paper proposes Energy-guided Recursive Model (ERM), a test-time selection mechanism for latent iterative reasoners. ERM replaces learned q-heads or majority voting with an explicit Hopfield-style energy that scores K candidate rollouts by their distance to hand-constructed task memories, optionally with a global penalty term for non-local constraints. The energy is also used to drive parallel tempering. On Sudoku, PPBench, and Maze at D=64, K=128, the paper reports that ERM exactly matches the shared-candidate oracle (98.83%, 88.04%, 99.30% respectively), and that PT-ERM reaches 98.97% on Sudoku. The authors interpret this as evidence that, given an energy faithful to task structure, the bottleneck in latent reasoning is candidate generation rather than selection.

Significance. If the claims hold, the paper makes a useful methodological contribution: it separates candidate generation from candidate selection, introduces an oracle-gap diagnostic, and shows that an explicit verifier-style energy can match the oracle on three structured benchmarks with no learned selector parameters. The paper is honest about limitations, states the setting of key hyperparameters, and provides inference pseudocode and source-data availability. The central scientific claim is not circular: the energy is derived from known task rules rather than fitted to labels. However, the paper's broader significance is limited by the task-specific, hand-engineered nature of the energies, and the load-bearing claim that the energy ranks every valid candidate above every invalid candidate is not established beyond the finite test pools. The lack of a precise definition of the global potentials, the unexplained discrepancy between the D=64 and D=16 Maze results, and the absence of a direct PTRM model comparison are substantive gaps that should be addressed.

major comments (4)
  1. [§3.4, Eqs. (7)-(8), Appendix B.2] The central claim that ERM selects the correct candidate whenever one exists requires G(y_k,x)—the global penalty for PPBench and Maze—to rank every valid candidate below every invalid candidate. The paper describes G only verbally as a distance to 'connectivity and coverage' (PPBench) or 'format, connectivity, degree, cycle, and length penalties' (Maze), and Appendix B.2 explicitly calls these potentials 'puzzle-specific engineering' and 'close to a structured verifier.' No formula, algorithm, or proof is given that G is zero iff the candidate is valid and positive otherwise. Oracle-matching on the K=128 test pools is compatible with false positives/negatives on near-miss invalid candidates that did not appear in those pools. Please provide the exact definition of G, a margin/failure analysis, and preferably an adversarial or near-miss evaluation.
  2. [§4.2, Table 1 and Appendix B.3, Table 5] There is an unexplained inconsistency in Maze results. Table 1 reports D=64, K=128 oracle and ERM both at 99.30% (993/1000 in the text). Appendix B.3, Table 5 reports, on 'the same 1000-example test split,' D=16, K=128 oracle and ERM both at 99.90% (999/1000). A shorter recurrent depth yielding a strictly higher oracle is surprising and needs explanation: are these different candidate-generation runs, different random seeds, or different splits? If stochastic, report variance or seeds. As written, the discrepancy undermines the reproducibility of the headline D=64 result and the claimed monotone energy diagnostic in Figure 2.
  3. [Abstract; §4.1, Table 1] The abstract claims ERM 'improves upon recent PTRM and EqR,' but the paper does not provide a direct model-to-model comparison with PTRM. Table 1 compares selectors on the same EqR-generated candidate pool only; no PTRM candidate pools or PTRM-trained checkpoints are evaluated. The PTRM bars in Figure 1 are not described with their settings in the text or tables. Either add a direct comparison under matched D, K, and training setup, or narrow the abstract claim to 'improves over q-head/confidence/majority selectors on EqR rollouts.'
  4. [§3.4, §5, Eqs. (7)-(8)] The energy has several hand-set hyperparameters—β=128, global weight 64, Maze local weight 0.25, retrieval temperature τ, and confidence mixing γ=0—with no sensitivity analysis. The paper itself notes that earlier confidence-mixing settings 'could raise invalid but high-likelihood candidates,' indicating that the ranking is sensitive to these weights. Because the central claim is that the energy is a faithful selector, a sensitivity study (e.g., varying each weight by factors of 2–10 and reporting exact accuracy/oracle gap) is necessary to demonstrate that the result is not a fragile tuning artifact. An analytical argument that the exact-verifier term dominates for the chosen weights would also suffice.
minor comments (7)
  1. [Abstract] The phrase 'optimal solutions' should be replaced by 'exact solutions' to avoid implying global optimality beyond the benchmark's exact-match criterion.
  2. [Eq. (3) and Eq. (6)] Equation (3) includes a distance scale β in the exponent, but Eq. (6) for Sudoku uses log-probabilities directly without β. State explicitly how β is absorbed in the Sudoku instantiation.
  3. [§3.4] The statement that the Sudoku inner sum 'is evaluated exactly by a log-permanent dynamic program' needs a reference or pseudocode; as written, it is not reproducible for a reader unfamiliar with permanent computations.
  4. [Appendix B.3] The early-stopping rule ('relative change in ERM energy below 0.01 for two consecutive checks, after at least four recurrent steps') should specify the check interval and how the final candidate is chosen after stopping.
  5. [Figure 1] The figure contains labels such as 'Deepseek-V4' and 'PTRM (q-head)' that are not explained in the caption or text. Either describe these baselines fully or remove them from the figure.
  6. [Table 6] The 'Equivalent layers' multipliers (42 for Sudoku/PPBench, 15 for Maze) are presented without a source or derivation; cite the TRM/EqR accounting or explain the conversion.
  7. [Notation, Table 3] The symbol γ is listed in the notation table and mentioned in the text as a confidence mixing weight, but it does not appear in Eq. (3) or Eq. (7). Clarify where γ would enter if nonzero.

Circularity Check

0 steps flagged

No significant circularity: ERM's energy is a hand-constructed task-rule verifier, not a fitted predictor; oracle-matching is a validation of the selector rather than a hidden reuse of its inputs.

full rationale

The paper's derivation chain is self-contained rather than circular. The ERM energy is explicitly constructed from known task rules: Sudoku uses row/column/box permutation memories, PPBench uses enumerable legal local patterns plus global connectivity/coverage potentials, and Maze uses a rule-defined simple-path memory set with format, connectivity, degree, cycle, and length penalties. These are not learned from labels and are not fitted to the reported test-set outcomes; Equation (3) defines a general energy and Equations (6)-(8) are specializations. Because the energy is built from validity criteria, oracle-matching is a consequence of construction when the energy is exact, but the paper openly acknowledges that the PPBench global potentials 'require puzzle-specific engineering' and that the Maze energy is 'close to a structured verifier' (Section 5; Appendix B.2). Thus the zero-gap results are presented as evidence that the hand-built potentials rank correctly on the test distribution, not as a prediction derived from a fitted input. The remaining concern—that approximate global potentials could mis-rank near-miss invalid candidates—is a robustness/correctness risk, not circularity. There are no load-bearing self-citations and no imported uniqueness claims; references to EqR, PTRM, Hopfield networks, and related works are external baselines or background. The central contribution, replacing a learned q-head or majority vote with an explicit task-energy selector, is a legitimate and independently evaluable design, even if its oracle-matching behavior is partly by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 3 axioms · 0 invented entities

The method uses no new physical entities; it relies on hand-set hyperparameters and standard domain assumptions about the tasks and the generator. The energy itself is a constructed scoring function, not a new conserved quantity or force.

free parameters (5)
  • β (PPBench distance scale) = 128
    Set to 128 in Equation 7; hand-chosen, no sensitivity analysis.
  • global weight (PPBench) = 64
    Weight on G(y_k,x) in Equation 7; hand-chosen.
  • Maze local memory weight = 0.25
    Weight on local Hopfield term in Equation 8; hand-chosen.
  • τ (retrieval temperature) = not specified
    Used in Equations 3, 6, 7, 8; value not reported, but likely fixed.
  • γ (confidence mixing) = 0
    Set to zero in main ERM runs to avoid confidence fallback.
axioms (3)
  • standard math Modern Hopfield retrieval energy (Equation 2) is accepted as a standard energy-based memory model.
    Borrowed from Ramsauer et al. [28]; used to motivate Equation 3.
  • domain assumption The task rules of Sudoku, PPBench, and Maze are known and can be expressed as enumerable memories or distance potentials.
    Used throughout Section 3.4 to instantiate the energy.
  • domain assumption The EqR generator (fixed) produces a candidate pool where the oracle accuracy is high at D=64, K=128.
    The selection method's benefit depends on the pool containing correct answers; oracle values are 98.83%, 88.04%, 99.30%.

pith-pipeline@v1.3.0-alltime-deepseek · 13937 in / 9755 out tokens · 103600 ms · 2026-08-02T07:20:56.534449+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Energy-guided Recursive Model." pith.science (2026). https://pith.science/paper/A6MYKUSA

@misc{pith2026260710128,
  author       = {Pith},
  title        = {Pith review of: Energy-guided Recursive Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A6MYKUSA}},
  note         = {Machine review of arXiv:2607.10128}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Recursive reasoning models address structured problems by repeatedly updating latent states of small neural networks. However, their test-time scaling lacks a principled inference mechanism: increasing depth or stochastic breadth generates more trajectories without a clear criterion for selection, and existing methods predominantly rely on additional q-heads or heuristic voting. Here, we develop the Energy-guided Recursive Model (ERM), which introduces an intrinsic selection principle based on explicit Hopfield energies. ERM leverages Hopfield-type memories of valid local or global structures to define the selector over candidate trajectories. The resulting energy seamlessly integrates with energy-based techniques such as parallel tempering to enhance sampling efficiency and ranking. With $D=64$ recurrent steps and $K=128$ candidates, ERM reaches optimal solutions on Sudoku ($98.97\%$), Pencil Puzzle Bench (PPBench, $88.04\%$) and Maze ($99.30\%$), improving upon recent Probabilistic Tiny Recursive Model and Equilibrium Reasoners. These results suggest that incorporating explicit energy functions into recursive reasoning offers a principled path toward more effective inference.

Figures

Figures reproduced from arXiv: 2607.10128 by Yifei Zhao, Ying Tang.

Figure 1
Figure 1. Figure 1: Energy-guided Recursive Model for Latent Iterative Reasoning. (a) A classic TRM￾style recursive reasoner, such as PTRM [22] and EqR [21], uses depth D and breadth K to generate candidate trajectories, then selects by q-head or majority voting. (b) ERM replaces this selector by constructing a Hopfield-network-based energy over task-structured memories and chooses the lowest-energy candidate. (c) The energy … view at source ↗
Figure 1
Figure 1. Figure 1: Energy-guided Recursive Model for Latent Iterative Reasoning. (a) TRM-style recur￾sive reasoners, such as PTRM [1] and EqR [2], use depth D and breadth K to generate candidate trajectories, then select by q-head or majority voting. (b) ERM replaces this selector with a Hopfield￾network-based energy over task-structured memories and chooses the lowest-energy candidate. (c) The energy ranks candidates and su… view at source ↗
Figure 2
Figure 2. Figure 2: Maze Selector Dynamics over Recurrent Depth. The solid curve in each panel shows the mean selected energy, and the dashed gray curve uses the right axis to show exact accuracy as recurrent depth increases from D = 1 to D = 64 with K = 128 Maze rollouts. Confidence uses negative mean valid-token maximum log-probability, while q-head and confidence quickly become saturated selector scores. ERM uses the globa… view at source ↗
Figure 3
Figure 3. Figure 3: Parallel Tempering Compared with Original Selection. Hollow bars show the original shared-pool selector accuracy, and solid bars show the parallel-tempering (PT)-selected exact accu￾racy for the same energy on a full 0–100% vertical scale. PT reuses each scalar energy for sampling and final scoring, adding only lightweight energy evaluations and swap decisions. Across Sudoku, PPBench, and Maze, PT matches … view at source ↗
Figure 4
Figure 4. Figure 4: Training with an Energy Transformer Architecture. This appendix experiment uses the EqR+ET Sudoku checkpoint with D = 64, K = 16, and PT temperatures from 0.001 to 1.0. ET energy and confidence both improve over the one-rollout baseline, serving as motivation for the main inference-time ERM design, whereas PT-confidence falls slightly below direct confidence selection. showing that an explicit energy-like … view at source ↗
Figure 4
Figure 4. Figure 4: Training with an Energy Transformer Architecture. This appendix experiment uses the EqR+ET Sudoku checkpoint with D = 64, K = 16, and PT temperatures from 0.001 to 1.0. ET energy and confidence both improve over the one-rollout baseline, serving as motivation for the main inference-time ERM design, whereas PT-confidence falls slightly below direct confidence selection. The EqR+ET Sudoku result is included … view at source ↗

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 9 linked inside Pith

  1. [1]

    Probabilistic tiny recursive model.arXiv preprint arXiv:2605.19943, 2026

    Amin Sghaier, Ali Parviz, and Alexia Jolicoeur-Martineau. Probabilistic tiny recursive model.arXiv preprint arXiv:2605.19943, 2026. URLhttps://arxiv.org/abs/2605. 19943

    Pith/arXiv arXiv 2026

  2. [2]

    Equilibrium reasoners: Learning attrac- tors enables scalable reasoning

    Benhao Huang, Zhengyang Geng, and Zico Kolter. Equilibrium reasoners: Learning attrac- tors enables scalable reasoning. InProc. Int. Conf. Mach. Learn., 2026. URLhttps: //openreview.net/forum?id=lh95PnOlpM

    2026

  3. [3]

    Chi, Quoc V

    Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed H. Chi, Quoc V . Le, and Denny Zhou. Chain-of-thought prompting elicits reasoning in large language models. InAdv. Neural Inf. Process. Syst., volume 35, pages 24824–24837,

  4. [4]

    Le, Ed H

    Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc V . Le, Ed H. Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. Self-consistency improves chain of thought rea- soning in language models. InProc. Int. Conf. Learn. Represent., 2023. URLhttps: //openreview.net/forum?id=1PL1NIMMrw

    2023

  5. [5]

    Griffiths, Yuan Cao, and Karthik Narasimhan

    Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Thomas L. Griffiths, Yuan Cao, and Karthik Narasimhan. Tree of thoughts: Deliberate problem solving with large language models. InAdv. Neural Inf. Process. Syst., volume 36,

  6. [7]

    Let’s verify step by step

    Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harrison Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. Let’s verify step by step. In Proc. Int. Conf. Learn. Represent., 2024. URLhttps://openreview.net/forum? id=v8L0pN6EOi

    2024

  7. [8]

    Scaling LLM test-time compute optimally can be more effective than scaling model parameters

    Charlie Snell, Jaehoon Lee, Kelvin Xu, and Aviral Kumar. Scaling LLM test-time compute optimally can be more effective than scaling model parameters. InProc. Int. Conf. Learn. Represent., 2025. URLhttps://openreview.net/forum?id=4FWAwZtd2n

    2025

  8. [9]

    Neural GPUs learn algorithms

    Łukasz Kaiser and Ilya Sutskever. Neural GPUs learn algorithms. InProc. Int. Conf. Learn. Represent., 2016. URLhttps://arxiv.org/abs/1511.08228

    Pith/arXiv arXiv 2016

  9. [10]

    Adaptive computation time for recurrent neural networks.arXiv preprint arXiv:1603.08983, 2016

    Alex Graves. Adaptive computation time for recurrent neural networks.arXiv preprint arXiv:1603.08983, 2016. URLhttps://arxiv.org/abs/1603.08983

    Pith/arXiv arXiv 2016

  10. [11]

    Universal transformers

    Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Łukasz Kaiser. Universal transformers. InProc. Int. Conf. Learn. Represent., 2019. URLhttps:// openreview.net/forum?id=HyzdRiR9Y7

    2019

  11. [12]

    On training implicit models

    Zhengyang Geng, Xin-Yu Zhang, Shaojie Bai, Yisen Wang, and Zhouchen Lin. On training implicit models. InAdv. Neural Inf. Process. Syst., volume 34, pages 24247– 24260, 2021. URLhttps://proceedings.neurips.cc/paper/2021/hash/ cb8da6767461f2812ae4290eac7cbc42-Abstract.html

    2021

  12. [13]

    Zico Kolter, and Vladlen Koltun

    Shaojie Bai, J. Zico Kolter, and Vladlen Koltun. Deep equilibrium models. InAdv. Neu- ral Inf. Process. Syst., volume 32, 2019. URLhttps://papers.nips.cc/paper/ 8358-deep-equilibrium-models

    2019

  13. [14]

    Lee, and Dimitris Papailiopoulos

    Angeliki Giannou, Shashank Rajput, Jy-yong Sohn, Kangwook Lee, Jason D. Lee, and Dimitris Papailiopoulos. Looped transformers as programmable computers. InProc. Int. Conf. Mach. Learn., 2023. URLhttps://proceedings.mlr.press/v202/ giannou23a.html

    2023

  14. [15]

    Looped transformers are better at learning learning algorithms

    Liu Yang, Kangwook Lee, Robert Nowak, and Dimitris Papailiopoulos. Looped transformers are better at learning learning algorithms. InProc. Int. Conf. Learn. Represent., 2024. URL https://openreview.net/forum?id=HHbRxoDTxE

    2024

  15. [16]

    Nikunj Saunshi, Nishanth Dikkala, Zhiyuan Li, Sanjiv Kumar, and Sashank J. Reddi. Rea- soning with latent thoughts: On the power of looped transformers. InProc. Int. Conf. Learn. Represent., 2025. URLhttps://openreview.net/forum?id=din0lGfZFd

    2025

  16. [17]

    Bar- toldson, Bhavya Kailkhura, Abhinav Bhatele, and Tom Goldstein

    Jonas Geiping, Sean McLeish, Neel Jain, John Kirchenbauer, Siddharth Singh, Brian R. Bar- toldson, Bhavya Kailkhura, Abhinav Bhatele, and Tom Goldstein. Scaling up test-time com- pute with latent reasoning: A recurrent depth approach. InAdv. Neural Inf. Process. Syst.,

  17. [18]

    Implicit language models are rnns: Balancing parallelization and expressivity

    Mark Sch ¨one, Babak Rahmani, Heiner Kremer, Fabian Falck, Hitesh Ballani, and Jannes Gladrow. Implicit language models are rnns: Balancing parallelization and expressivity. In Proc. Int. Conf. Mach. Learn., 2025. URLhttps://openreview.net/forum?id= 5EbiopWH6e

    2025

  18. [19]

    Mixture-of-recursions: Learning dynamic recursive depths for adaptive token-level computation

    Sangmin Bae, Yujin Kim, Reza Bayat, Sungnyun Kim, Jiyoun Ha, Tal Schuster, Adam Fisch, Hrayr Harutyunyan, Ziwei Ji, Aaron Courville, and Se-Young Yun. Mixture-of-recursions: Learning dynamic recursive depths for adaptive token-level computation. InAdv. Neural Inf. Process. Syst., 2025. URLhttps://openreview.net/forum?id=QuqsEIVWIG

    2025

  19. [20]

    Scaling latent reasoning via looped language mod- els.arXiv preprint arXiv:2510.25741, 2025

    Rui-Jie Zhu, Zixuan Wang, Kai Hua, Tianyu Zhang, Ziniu Li, Haoran Que, Boyi Wei, Zixin Wen, Fan Yin, He Xing, Lu Li, Jiajun Shi, Kaijing Ma, Taylor Kergan, Andrew Smith, Xingwei Qu, Mude Hui, Bohong Wu, Qiyang Min, Hongzhi Huang, Xun Zhou, Wei Ye, Jiaheng Liu, Jian Yang, Yunfeng Shi, Chenghua Lin, Enduo Zhao, Tianle Cai, Ge Zhang, Wenhao Huang, Yoshua Ben...

    Pith/arXiv arXiv 2025

  20. [21]

    Hierarchical reasoning model.arXiv preprint arXiv:2506.21734, 2025

    Guan Wang, Jin Li, Yuhao Sun, Xing Chen, Changling Liu, Yue Wu, Meng Lu, Sen Song, and Yasin Abbasi-Yadkori. Hierarchical reasoning model.arXiv preprint arXiv:2506.21734, 2025. URLhttps://arxiv.org/abs/2506.21734

    Pith/arXiv arXiv 2025

  21. [22]

    Less is more: Recursive reasoning with tiny networks.arXiv preprint arXiv:2510.04871, 2025

    Alexia Jolicoeur-Martineau. Less is more: Recursive reasoning with tiny networks.arXiv preprint arXiv:2510.04871, 2025. URLhttps://arxiv.org/abs/2510.04871

    Pith/arXiv arXiv 2025

  22. [23]

    Are your reasoning models reasoning or guessing? a mechanistic analysis of hierarchical reasoning models.arXiv preprint arXiv:2601.10679, 2026

    Zirui Ren and Ziming Liu. Are your reasoning models reasoning or guessing? a mechanistic analysis of hierarchical reasoning models.arXiv preprint arXiv:2601.10679, 2026. doi: 10. 48550/arXiv.2601.10679. URLhttps://arxiv.org/abs/2601.10679

    doi:10.48550/arxiv.2601.10679 2026

  23. [24]

    Hopfield

    John J. Hopfield. Neural networks and physical systems with emergent collective compu- tational abilities.Proc. Natl. Acad. Sci. U.S.A., 79(8):2554–2558, 1982. URLhttps: //doi.org/10.1073/pnas.79.8.2554

    doi:10.1073/pnas.79.8.2554 1982

  24. [25]

    Ackley, Geoffrey E

    David H. Ackley, Geoffrey E. Hinton, and Terrence J. Sejnowski. A learning algorithm for boltzmann machines.Cogn. Sci., 9(1):147–169, 1985. URLhttps://doi.org/10. 1016/S0364-0213(85)80012-4

    1985

  25. [26]

    A tutorial on energy-based learning

    Yann LeCun, Sumit Chopra, Raia Hadsell, Marc’Aurelio Ranzato, and Fu Jie Huang. A tutorial on energy-based learning. InPredicting Structured Data, pages 191–246. MIT Press, 2006. URLhttp://yann.lecun.com/exdb/publis/pdf/lecun-06.pdf

    2006

  26. [27]

    Hopfield

    Dmitry Krotov and John J. Hopfield. Dense associative memory for pattern recognition. InAdv. Neural Inf. Process. Syst., volume 29, 2016. URLhttps://proceedings.neurips. cc/paper/2016/hash/eaae339c4d89fc102edd9dbdb6a28915-Abstract. html

    2016

  27. [28]

    Hopfield networks is all you need

    Hubert Ramsauer, Bernhard Sch ¨afl, Johannes Lehner, Philipp Seidl, Michael Widrich, Thomas Adler, Lukas Gruber, Markus Holzleitner, Milena Pavlovi´c, Geir Kjetil Sandve, Victor Greiff, David Kreil, Michael Kopp, G ¨unter Klambauer, Johannes Brandstetter, and Sepp Hochreiter. Hopfield networks is all you need. InProc. Int. Conf. Learn. Represent., 2021. U...

    2021

  28. [29]

    Zaki, and Dmitry Krotov

    Benjamin Hoover, Yuchen Liang, Bao Pham, Rameswar Panda, Hendrik Strobelt, Duen Horng Chau, Mohammed J. Zaki, and Dmitry Krotov. Energy trans- former. InAdv. Neural Inf. Process. Syst., volume 36, pages 27532–27559,

  29. [30]

    Pencil puzzle bench: A benchmark for multi-step verifiable reasoning.arXiv preprint arXiv:2603.02119, 2026

    Justin Waugh. Pencil puzzle bench: A benchmark for multi-step verifiable reasoning.arXiv preprint arXiv:2603.02119, 2026. URLhttps://arxiv.org/abs/2603.02119

    arXiv 2026

  30. [31]

    Solve the loop: Attractor models for language and reasoning.arXiv preprint arXiv:2605.12466, 2026

    Jacob Fein-Ashley and Paria Rashidinejad. Solve the loop: Attractor models for language and reasoning.arXiv preprint arXiv:2605.12466, 2026. doi: 10.48550/arXiv.2605.12466. URL https://doi.org/10.48550/ARXIV.2605.12466. 11

    Pith/arXiv arXiv doi:10.48550/arxiv.2605.12466 2026

  31. [32]

    Konstantin Rusch, and Antonio Orvieto

    Sajad Movahedi, Vera Milovanovi´c, Shlomo Libo Feigin, Alexander Theus, Thomas Hofmann, Valentina Boeva, T. Konstantin Rusch, and Antonio Orvieto. Fixed-point reasoners: Stable and adaptive deep looped transformers.arXiv preprint arXiv:2606.18206, 2026. doi: 10.48550/ arXiv.2606.18206. URLhttps://arxiv.org/abs/2606.18206

    Pith/arXiv arXiv doi:10.48550/arxiv.2606.18206 2026

  32. [33]

    Learning iterative reasoning through energy minimization

    Yilun Du, Shuang Li, Joshua Tenenbaum, and Igor Mordatch. Learning iterative reasoning through energy minimization. InProc. Int. Conf. Mach. Learn., volume 162, pages 5570– 5582, 2022. URLhttps://proceedings.mlr.press/v162/du22d.html. Code: https://github.com/yilundu/irem_code_release. 12 A NOTATIONTABLE Table 3:Notation Used in the Main Text.The table is...

    2022

  33. [34]

    URLhttps://proceedings.neurips.cc/paper_files/paper/2023/ hash/57a9b97477b67936298489e3c1417b0a-Abstract-Conference.html

    2023

  34. [2021]

    URLhttps://arxiv.org/abs/2110.14168

    Pith/arXiv arXiv

  35. [2022]

    URLhttps://proceedings.neurips.cc/paper_files/paper/2022/ hash/9d5609613524ecf4f15af0f7b31abca4-Abstract-Conference.html. 9

    2022

  36. [2023]

    URLhttps://proceedings.neurips.cc/paper_files/paper/2023/ hash/271db9922b8d1f4dd7aaef84ed5ac703-Abstract-Conference.html

    2023

  37. [2025]

    URLhttps://openreview.net/forum?id=S3GhJooWIC. 10