Energy-guided Recursive Model

# Energy-guided Recursive Model
Source: [https://arxiv.org/html/2607.10128](https://arxiv.org/html/2607.10128)
Yifei Zhao1,Ying Tang1,2,3,4

1Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China 2School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China 3Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China, Chengdu 611731, China 4Non\-classical Information Science Basic Discipline Research Center of Sichuan Province, University of Electronic Science and Technology of China, Chengdu 611731, China

###### 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 theEnergy\-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\. WithD=64D=64recurrent steps andK=128K=128candidates, ERM reaches optimal solutions on Sudoku \(98\.97%98\.97\\%\), Pencil Puzzle Bench \(PPBench,88\.04%88\.04\\%\) and Maze \(99\.30%99\.30\\%\), 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\.

## 1Introduction

Test\-time computation is now a central route to stronger reasoning, but its success depends on how extra computation is converted into a final answer\. For autoregressive language models, this conversion is usually explicit: additional budget produces longer chains of thought, multiple sampled rationales, tree\-structured searches, or verifier\-ranked completions\[Weiet al\.,[2022](https://arxiv.org/html/2607.10128#bib.bib28); Wanget al\.,[2023](https://arxiv.org/html/2607.10128#bib.bib29); Yaoet al\.,[2023](https://arxiv.org/html/2607.10128#bib.bib30); Cobbeet al\.,[2021](https://arxiv.org/html/2607.10128#bib.bib31); Lightmanet al\.,[2024](https://arxiv.org/html/2607.10128#bib.bib32); Snellet al\.,[2025](https://arxiv.org/html/2607.10128#bib.bib33)\]\. These methods show that inference\-time work can substitute for part of model\-scale growth, but they also rely on observable text or learned reward models whose scores can be separated from the hidden computation that produced the answer\. Latent iterative reasoners make this problem harder because additional depth or breadth produces hidden trajectories rather than explicit derivations\. The open question is therefore not only how to generate more trajectories, but which trajectory should be trusted when no explicit derivation or external verifier is available\.

Loop models provide the architectural basis for latent test\-time reasoning by reusing a shared learned operator within one forward computation\. This idea appears in early algorithm\-learning systems and adaptive\-depth networks, including Neural GPUs\[Kaiser and Sutskever,[2016](https://arxiv.org/html/2607.10128#bib.bib16)\], adaptive computation time\[Graves,[2016](https://arxiv.org/html/2607.10128#bib.bib17)\], Universal Transformers\[Dehghaniet al\.,[2019](https://arxiv.org/html/2607.10128#bib.bib18)\], implicit models\[Genget al\.,[2021](https://arxiv.org/html/2607.10128#bib.bib20)\]and deep equilibrium models\[Baiet al\.,[2019](https://arxiv.org/html/2607.10128#bib.bib19)\]\. Recent looped and recurrent\-depth transformers\[Giannouet al\.,[2023](https://arxiv.org/html/2607.10128#bib.bib21); Yanget al\.,[2024](https://arxiv.org/html/2607.10128#bib.bib22); Saunshiet al\.,[2025](https://arxiv.org/html/2607.10128#bib.bib23)\]show that repeated shared blocks can act as iterative algorithms, latent thoughts, or compute\-scalable language\-model components\[Geipinget al\.,[2025](https://arxiv.org/html/2607.10128#bib.bib24); Schöneet al\.,[2025](https://arxiv.org/html/2607.10128#bib.bib25); Baeet al\.,[2025](https://arxiv.org/html/2607.10128#bib.bib26); Zhuet al\.,[2025](https://arxiv.org/html/2607.10128#bib.bib27)\]\. Structured reasoning models such as the Hierarchical Reasoning Model \(HRM\)\[Wanget al\.,[2025](https://arxiv.org/html/2607.10128#bib.bib12)\], Tiny Recursive Model \(TRM\)\[Jolicoeur\-Martineau,[2025](https://arxiv.org/html/2607.10128#bib.bib11)\], Equilibrium Reasoners \(EqR\)\[Huanget al\.,[2026](https://arxiv.org/html/2607.10128#bib.bib15)\], and Probabilistic Tiny Recursive Model \(PTRM\)\[Sghaieret al\.,[2026](https://arxiv.org/html/2607.10128#bib.bib10)\]further show that compact recurrent networks can solve Sudoku, Maze, ARC\-style tasks, and related algorithmic benchmarks by repeatedly refining latent states\. A mechanistic analysis of HRM also shows that recursive trajectories can be trapped by spurious fixed points, which makes selection among stochastic candidates part of the reasoning problem rather than a minor implementation detail\[Ren and Liu,[2026](https://arxiv.org/html/2607.10128#bib.bib13)\]\. These works establish recurrent depth and stochastic breadth as useful compute axes, but they leave a persistent selection gap: hidden trajectories are still usually reduced by learned halting heads, token confidence, residuals, or voting rules rather than by an explicit criterion of task compatibility\.

Energy\-based modeling offers a principled language for this missing selection layer because it represents computation as scalar compatibility over configurations\. Classical Hopfield networks\[Hopfield,[1982](https://arxiv.org/html/2607.10128#bib.bib4)\], Boltzmann machines\[Ackleyet al\.,[1985](https://arxiv.org/html/2607.10128#bib.bib5)\], and energy\-based learning\[LeCunet al\.,[2006](https://arxiv.org/html/2607.10128#bib.bib6)\]define inference through low\-energy states, while dense and modern Hopfield networks connect associative retrieval to high\-capacity memories and attention\-like updates\[Krotov and Hopfield,[2016](https://arxiv.org/html/2607.10128#bib.bib34); Ramsaueret al\.,[2021](https://arxiv.org/html/2607.10128#bib.bib8)\]\. The Energy Transformer \(ET\) makes the connection more explicit by designing representation updates that descend an attention\-based energy\[Hooveret al\.,[2023](https://arxiv.org/html/2607.10128#bib.bib9)\]\. However, most of this literature attaches energy to model training, representation dynamics, or memory retrieval, whereas recursive reasoners at test time need an energy over decoded candidate solutions\. This distinction matters because a q\-head, a confidence score, or a majority statistic measures only one projection of candidate quality and may miss global task constraints even when a correct candidate is present in the rollout pool\.

![Refer to caption](https://arxiv.org/html/2607.10128v1/x1.png)Figure 1:Energy\-guided Recursive Model for Latent Iterative Reasoning\.\(a\) A classic TRM\-style recursive reasoner, such as PTRM\[Sghaieret al\.,[2026](https://arxiv.org/html/2607.10128#bib.bib10)\]and EqR\[Huanget al\.,[2026](https://arxiv.org/html/2607.10128#bib.bib15)\], uses depthDDand breadthKKto 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 provides final ranking and supports parallel tempering \(PT\), linking the selection with energy\-guided sampling\. The bottom shows the performance of various models on the three representative reasoning tasks, where ERM reaches the oracle as the best possible accuracy if any generated candidate is correct\.We introduce theEnergy\-guided Recursive Model \(ERM\), which leverages an explicitly constructed Hopfield\-network\-based energy to facilitate the selection of reasoning trajectories\.Figure[1](https://arxiv.org/html/2607.10128#S1.F1)illustrates the main setting: an EqR\-style reasoner first producesKKcandidate outputs by recurrent computation, and ERM ranks those candidates with memories of valid task structures\. The memory bank changes with the task: Sudoku uses row, column, and box permutation memories; Pencil Puzzle Bench \(PPBench\) uses local puzzle\-rule memories plus global distance potentials\[Waugh,[2026](https://arxiv.org/html/2607.10128#bib.bib14)\]; and Maze uses an input\-conditioned shortest\-path memory in the reported global\-memory run\. Across Sudoku, PPBench, and Maze, ERM reaches the shared\-candidate oracle under the sameD=64D=64,K=128K=128rollout budget, and its energy can also guide parallel tempering\. The contribution is therefore both methodological and diagnostic: test\-time compute in recursive reasoners should be evaluated by separating candidate generation from energy\-based selection, and explicit energies can close the selection gap when the correct trajectory is already available\.

## 2Related Work

Latent iterative reasoning models use recurrent computation to solve structured tasks that are difficult for shallow feedforward predictors\. The Hierarchical Reasoning Model \(HRM\) and Tiny Recursive Model \(TRM\) show that compact recurrent architectures can solve Sudoku, Maze, and Abstraction and Reasoning Corpus \(ARC\)\-style tasks with far fewer parameters than large autoregressive models\[Wanget al\.,[2025](https://arxiv.org/html/2607.10128#bib.bib12); Jolicoeur\-Martineau,[2025](https://arxiv.org/html/2607.10128#bib.bib11)\]\. EqR studies reasoning as convergence toward learned attractors, while probabilistic recursive models emphasize stochastic exploration of hidden states\[Huanget al\.,[2026](https://arxiv.org/html/2607.10128#bib.bib15); Sghaieret al\.,[2026](https://arxiv.org/html/2607.10128#bib.bib10)\]\. This line is complementary to test\-time scaling for autoregressive models: rather than sampling many textual derivations, the system samples or iterates hidden configurations and then decodes a structured answer\. We focus on the inference layer that decides which trajectory should be trusted once such a reasoner has produced multiple candidates\.

Energy\-based models and Hopfield networks provide a long\-standing formalism for assigning scalar compatibility to configurations\. Classical Hopfield networks use an energy landscape whose minima correspond to stored memories\[Hopfield,[1982](https://arxiv.org/html/2607.10128#bib.bib4)\]; Boltzmann machines and energy\-based learning generalize this principle to probabilistic and discriminative settings\[Ackleyet al\.,[1985](https://arxiv.org/html/2607.10128#bib.bib5); LeCunet al\.,[2006](https://arxiv.org/html/2607.10128#bib.bib6)\]\. Modern Hopfield networks show that attention\-like retrieval can be interpreted as associative memory over exponentially many patterns\[Ramsaueret al\.,[2021](https://arxiv.org/html/2607.10128#bib.bib8)\], and the Energy Transformer gives a contemporary example of representation updates organized by explicit energy descent\[Hooveret al\.,[2023](https://arxiv.org/html/2607.10128#bib.bib9)\]\. We borrow the memory\-energy interpretation, but our energy is attached to test\-time candidate selection and sampling rather than to the full training objective of a new architecture\.

Verifier and reranking methods are close in spirit but differ in what they optimize\. Confidence selectors prefer candidates with high local probability, q\-head selectors use a learned reliability signal, and majority voting prefers consensus across sampled outputs\. These signals can work well when correctness correlates with local certainty, and PTRM shows that a q\-head can be an effective verifier on several recursive\-reasoning benchmarks\[Sghaieret al\.,[2026](https://arxiv.org/html/2607.10128#bib.bib10)\]\. However, the same paper also exposes verifier headroom on Maze\-like tasks, where pass@KKrises faster than q\-head selection, which is exactly the gap our energy design targets\. ERM turns task constraints into explicit memory sets or memory\-distance potentials, so the selector can measure compatibility with structured rules instead of relying only on model\-internal certainty\.

## 3Method

### 3\.1Problem Setup

We study test\-time reasoning as energy\-based selection over a finite candidate pool\. Letxxdenote an input puzzle, and let a latent iterative reasoner with recurrent depthDDproduceKKcandidate outputs𝒴​\(x\)=\{y1,…,yK\}\\mathcal\{Y\}\(x\)=\\\{y\_\{1\},\\ldots,y\_\{K\}\\\}through independent rollouts, perturbations, or parallel\-tempering chains\. Candidateyky\_\{k\}is a structured discrete assignment over positionsi∈\{1,…,N\}i\\in\\\{1,\\ldots,N\\\}\. A selector assigns each candidate a scalar energyE​\(yk;x\)E\(y\_\{k\};x\), where lower energy means higher compatibility with the task\. The selected answer is

y^=yk^,k^=arg⁡mink∈\{1,…,K\}⁡E​\(yk;x\)\.\\hat\{y\}=y\_\{\\hat\{k\}\},\\qquad\\hat\{k\}=\\arg\\min\_\{k\\in\\\{1,\\ldots,K\\\}\}E\(y\_\{k\};x\)\.\(1\)This equation also defines the common baselines as energies: confidence uses negative mean token confidence, q\-head uses the negative learned halting logit, and majority voting uses a negative agreement score\. The difference is that ERM constructsEEfrom task memories rather than from only model\-internal certainty\.

### 3\.2Energy\-guided Recursive Model

ERM is derived from the Modern Hopfield view of energy\-based memory retrieval\. In a Modern Hopfield network, a query stateqqis compared with a memory setℳ\\mathcal\{M\}, and retrieval lowers an energy whenqqis similar to at least one stored memory\[Ramsaueret al\.,[2021](https://arxiv.org/html/2607.10128#bib.bib8)\]\. A compact form of this retrieval energy is

EMHN​\(q;ℳ\)=12​∥q∥2−τ​log​∑m∈ℳexp⁡\(sim​\(q,m\)τ\),E\_\{\\mathrm\{MHN\}\}\(q;\\mathcal\{M\}\)=\\frac\{1\}\{2\}\\lVert q\\rVert^\{2\}\-\\tau\\log\\sum\_\{m\\in\\mathcal\{M\}\}\\exp\\\!\\left\(\\frac\{\\mathrm\{sim\}\(q,m\)\}\{\\tau\}\\right\),\(2\)wheresim​\(q,m\)\\mathrm\{sim\}\(q,m\)is the similarity between the query and memory,τ\\tauis the retrieval temperature, and the squared\-norm term regularizes the continuous query state\. The part we need for selection is the log\-sum\-exp memory term: it softly retrieves the nearest compatible memory, and it approaches nearest\-memory retrieval asτ\\taudecreases\.

ERM turns the same Hopfield retrieval energy into a task\-structured selector over candidate solutions\. A task is decomposed intoJJfactors, such as Sudoku rows, PPBench clue neighborhoods, or Maze path constraints\. Factorjjhas an input\-conditioned memory setℳj​\(x\)\\mathcal\{M\}\_\{j\}\(x\), and each memorym∈ℳj​\(x\)m\\in\\mathcal\{M\}\_\{j\}\(x\)represents one valid local or global structure\. We replace the Modern Hopfield similarity by the negative memory distance−β​dj​\(yk,m\)\-\\beta d\_\{j\}\(y\_\{k\},m\), sum the retrieval energy over task factors, and add a global violation penaltyG​\(yk,x\)G\(y\_\{k\},x\)for rules that are not easy to enumerate locally:

EERM​\(yk;x\)=μ​G​\(yk,x\)−∑j=1Jτ​log​∑m∈ℳj​\(x\)exp⁡\(−β​dj​\(yk,m\)τ\)E\_\{\\mathrm\{ERM\}\}\(y\_\{k\};x\)=\\mu G\(y\_\{k\},x\)\-\\sum\_\{j=1\}^\{J\}\\tau\\log\\sum\_\{m\\in\\mathcal\{M\}\_\{j\}\(x\)\}\\exp\\\!\\left\(\-\\frac\{\\beta d\_\{j\}\(y\_\{k\},m\)\}\{\\tau\}\\right\)\(3\)Heredj​\(yk,m\)d\_\{j\}\(y\_\{k\},m\)is zero when the candidate matches memorymmon factorjjand grows as the candidate violates that memory,β\\betacontrols how strongly memory distance is penalized,τ\\taukeeps the same role as the Hopfield retrieval temperature,G​\(yk,x\)G\(y\_\{k\},x\)is a global rule\-violation score, andμ\\muis its weight\. Equation[3](https://arxiv.org/html/2607.10128#S3.E3)is therefore the memory\-retrieval part of the Modern Hopfield free\-energy form applied to structured reasoning candidates: the log\-sum\-exp softly retrieves the nearest valid memory for each factor, and the low\-temperature limit becomes a nearest\-memory verifier\. The global term is included because some constraints, such as connectivity, are impractical to enumerate as small local memories but can still be measured as scalar violations\.

This simplified formula includes confidence and hard verification as special cases\. If each factor is one cell and its memories are the possible token labels, the low\-temperature nearest\-memory score reduces to choosing the most confident token\. If a task has one complete memorym⋆​\(x\)m^\{\\star\}\(x\), as in the reported Maze global\-memory run, then ERM reduces to distance from that single memory\. Thus ERM is not a different kind of object from ordinary selection scores; it is a structured way to specify which valid memories the selector should trust\.

### 3\.3Parallel Tempering over Candidate Energies

The same energy can be used for sampling, not only for reranking\. For an energyE​\(y;x\)E\(y;x\)and a temperatureT\>0T\>0, define the tempered distribution

πT​\(y∣x\)∝exp⁡\(−E​\(y;x\)T\),\\pi\_\{T\}\(y\\mid x\)\\propto\\exp\\\!\\left\(\-\\frac\{E\(y;x\)\}\{T\}\\right\),\(4\)where smallTTconcentrates on low\-energy candidates and largeTTallows broader exploration\. Parallel tempering \(PT\) runs replicas at temperaturesT1<⋯<TRT\_\{1\}<\\cdots<T\_\{R\}and periodically proposes swaps between neighboring replicas\. If replicas at temperaturesTρT\_\{\\rho\}andTηT\_\{\\eta\}currently hold candidates with energiesEρE\_\{\\rho\}andEηE\_\{\\eta\}, the swap is accepted with probability

a=min⁡\{1,exp⁡\[\(1Tρ−1Tη\)​\(Eρ−Eη\)\]\}\.a=\\min\\left\\\{1,\\exp\\left\[\\left\(\\frac\{1\}\{T\_\{\\rho\}\}\-\\frac\{1\}\{T\_\{\\eta\}\}\\right\)\(E\_\{\\rho\}\-E\_\{\\eta\}\)\\right\]\\right\\\}\.\(5\)PT is useful when the energy does more than rank an existing pool: it can change which states are visited by letting hot replicas explore and cold replicas exploit\. In our experiments, we report both the PT\-selected accuracy and the PT oracle accuracy, because their difference separates sampling quality from final energy ranking\.

### 3\.4Task Instantiations

Sudoku instantiates ERM with row, column, and box permutation memories\. For a candidateyky\_\{k\}, letLk​\[i,d\]L\_\{k\}\[i,d\]be the log\-probability that celliihas digitd∈\{1,…,9\}d\\in\\\{1,\\ldots,9\\\}\. Each unituuis a row, column, or3×33\\times 3box, and its memory setℳu\\mathcal\{M\}\_\{u\}contains the9\!9\!permutation matrices that assign each digit exactly once\. The unit score is

Su​\(k\)=τ​log​∑σ∈S9exp⁡\(1τ​∑r=19Lk​\[ir,σ​\(r\)\]\),S\_\{u\}\(k\)=\\tau\\log\\sum\_\{\\sigma\\in S\_\{9\}\}\\exp\\\!\\left\(\\frac\{1\}\{\\tau\}\\sum\_\{r=1\}^\{9\}L\_\{k\}\[i\_\{r\},\\sigma\(r\)\]\\right\),\(6\)whereS9S\_\{9\}is the set of all digit permutations,σ\\sigmais one such permutation, and\(i1,…,i9\)\(i\_\{1\},\\ldots,i\_\{9\}\)are the cells in unituu\. The sum is evaluated exactly as a log\-permanent by dynamic programming rather than by enumerating all permutations\. Clue cells are clamped to their input digits, so the energy measures compatibility with valid completions of the given puzzle\.

PPBench uses the same ERM principle with puzzle\-type\-specific memories and global distance potentials\. For enumerable local rules, a factor uses legal binary or multi\-class patterns as memories and scores a candidate by soft retrieval over normalized Hamming distance\. Lightup uses numbered\-clue exact\-count memories and line\-of\-sight at\-most\-one\-bulb memories; Tapa uses cyclic run\-pattern memories around clues; Heyawake uses room\-count and adjacency memories; Nurikabe uses no\-2×22\\times 2\-sea memories and island/sea potentials; and the Sudoku subset uses permutation memories\. Connectivity and coverage rules enter through the global violation termG​\(yk,x\)G\(y\_\{k\},x\)in Equation[3](https://arxiv.org/html/2607.10128#S3.E3), because enumerating every globally valid board would be intractable\. In the reported PPBench run, we set the model\-confidence mixing weight toγ=0\\gamma=0, use memory\-distance retrieval withβ=128\\beta=128, and use global distance weight6464, so the score is a pure memory\-distance energy rather than a confidence fallback\.

Maze illustrates why local and global memories must be distinguished\. A local path\-degree ERM can define valid memories around each cell: start and goal cells should have path degree one, while interior path cells should have degree two\. This is a legitimate local Hopfield factor memory, but it cannot rule out disconnected cycles or locally valid path fragments\. The reported oracle\-aligned Maze run therefore uses a global input\-conditioned shortest\-path memory\. For each input mazexx, breadth\-first search constructs the unique shortest pathmBFS​\(x\)m\_\{\\mathrm\{BFS\}\}\(x\)from start to goal, and the score is

SMaze​\(yk;x\)=β​sim​\(yk,mBFS​\(x\)\),EMaze​\(yk;x\)=−SMaze​\(yk;x\),S\_\{\\mathrm\{Maze\}\}\(y\_\{k\};x\)=\\beta\\,\\mathrm\{sim\}\(y\_\{k\},m\_\{\\mathrm\{BFS\}\}\(x\)\),\\qquad E\_\{\\mathrm\{Maze\}\}\(y\_\{k\};x\)=\-S\_\{\\mathrm\{Maze\}\}\(y\_\{k\};x\),\(7\)wheresim\\mathrm\{sim\}is the fraction of matching tokens andβ=64\\beta=64\. This is the single\-memory, low\-temperature limit of Equation[3](https://arxiv.org/html/2607.10128#S3.E3); it should be interpreted as a strong rule\-derived global verifier energy, not as a weak local constraint\. The broader Maze ERM family is local plus global, but the reported shared\-candidate result sets the local weight to zero so the global memory is decisive\.

### 3\.5Design Recipe for New Tasks

The energy view gives a reusable recipe for tasks beyond the experiments in this paper\. First, define the candidate variables and the valid token or object states\. Second, identify small factors whose legal assignments can be enumerated as memories, such as permutations, adjacency patterns, count constraints, local transitions, or example\-conditioned transformations\. Third, add global distance potentials only for rules that cannot be represented faithfully by local memories, such as connectivity, reachability, coverage, or uniqueness\. Fourth, set confidence mixing to zero unless it is deliberately being evaluated as a separate hybrid, because otherwise model likelihood can lift a confident but invalid candidate above a valid one\. Finally, report the oracle, the selector\-oracle gap, and PT oracle versus PT\-selected accuracy; these diagnostics show whether the bottleneck is candidate generation, energy design, or final selection\.

## 4Experiments

### 4\.1Experimental Setup

We evaluate whether explicit energies improve selection among EqR rollouts\. The main setting uses recurrent depthD=64D=64andK=128K=128candidates unless otherwise stated\. The task suite contains Sudoku\-Lite, the five\-type PPBench validation split derived from Pencil Puzzle Bench\[Waugh,[2026](https://arxiv.org/html/2607.10128#bib.bib14)\], and the official EqR Maze\-Unique setting\. We report exact accuracy, which requires the entire structured answer to match the target, and token accuracy, which measures average per\-position correctness\. Exact accuracy is the primary metric because all three tasks contain global constraints where a small number of token errors can invalidate a solution\.

We compare ERM with four non\-oracle selectors\. The one\-rollout baseline uses a single EqR prediction\. Whole\-sequence majority chooses the most frequent complete answer among theKKcandidates when applicable, while cell\-majority voting is additionally reported for PPBench and Maze diagnostics\. The q\-head selector chooses the candidate with the largest learned EqR halting score, and confidence chooses the candidate with the largest mean valid\-token confidence\. The candidate oracle reports whether any of theKKcandidates is exactly correct, so it is not a deployable method but an upper bound on selection from the same pool\. Because Maze exposes a particularly sharp distinction between local confidence and global path compatibility, Figure[2](https://arxiv.org/html/2607.10128#S4.F2)also shows how the selector energy and exact accuracy evolve over recurrent depth\.

![Refer to caption](https://arxiv.org/html/2607.10128v1/x2.png)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 fromD=1D=1toD=64D=64withK=128K=128Maze rollouts\. Confidence uses negative mean valid\-token maximum log\-probability, while q\-head and confidence quickly become saturated selector scores\. ERM uses the global BFS shortest\-path memory from Equation[7](https://arxiv.org/html/2607.10128#S3.E7); its energy changes monotonically without sudden jumps, behaving as a true compatibility energy over Maze candidates\.Table 1:Shared\-Candidate Selection atD=64D=64,K=128K=128\.All methods select from the same candidate pool for each task\. Base denotes the one\-rollout baseline, Conf\. denotes confidence selection, the oracle is the best possible exact accuracy inside that pool, and the ERM gap is oracle minus ERM\.
### 4\.2Structured Energies Approach the Shared\-Candidate Oracle

The shared\-candidate results show that the largest immediate gain comes from using task memories to rank existing trajectories\. On EqR Sudoku, the baseline solves1829/20481829/2048puzzles, whileK=128K=128candidate generation raises the oracle to2024/2048=98\.83%2024/2048=98\.83\\%\. ERM selects2024/2048=98\.83%2024/2048=98\.83\\%, three examples below the oracle and five examples above confidence at2019/2048=98\.58%2019/2048=98\.58\\%\. Thus most remaining Sudoku errors are candidate\-generation failures, while the small residual selector gap is exactly the kind of ranking error tested by the PT experiment below\.

PPBench is the strongest evidence that structured energy improves selection beyond confidence and q\-head\. The baseline solves206/301206/301held\-out puzzles, whole\-sequence majority solves238/301238/301, confidence solves243/301243/301, and q\-head solves251/301251/301\. ERM solves265/301265/301, exactly matching the candidate oracle and improving over q\-head by 14 puzzles\. The improvement follows the method design: PPBench contains heterogeneous local and global rules, so a selector that measures distance to legal memories can reject confident candidates with rule violations that a generic confidence score does not see\.

Maze confirms that energy design must match the true constraint level of the task\. The one\-rollout official Maze checkpoint obtains89\.10%89\.10\\%exact accuracy on the 30\-by\-30 unique\-path test set, whileK=128K=128candidates contain a correct solution for99\.30%99\.30\\%of examples\. Whole\-sequence majority and confidence reach97\.80%97\.80\\%, q\-head reaches97\.30%97\.30\\%, and the global\-memory ERM reaches the99\.30%99\.30\\%oracle\. This result is not evidence that local path\-degree memories are sufficient; rather, it shows that when the input\-conditioned shortest\-path memory is provided as a zero\-temperature Hopfield memory, the correct candidate can be selected whenever it appears in the EqR rollout pool\.

![Refer to caption](https://arxiv.org/html/2607.10128v1/x3.png)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 accuracy for the same energy on a full0–100%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 or improves the corresponding original selector, and ERM remains the strongest selected result because its structured energy stays aligned with the candidate oracle\.Table 2:Parallel Tempering with Different Energies\.Original is the accuracy of the same selector on the shared EqR candidate pool, selected accuracy is the final answer chosen after PT sampling, and oracle accuracy is pass@KKwithin the PT\-generated candidates\.
### 4\.3Energy\-Guided Sampling with Parallel Tempering

Parallel tempering tests whether an energy can guide exploration, not just choose among a fixed set of candidates\. Table[2](https://arxiv.org/html/2607.10128#S4.T2)therefore reports the original shared\-pool selector, the PT\-selected answer, and the PT oracle for the same energy\. On Sudoku, PT\-ERM reaches2027/2048=98\.97%2027/2048=98\.97\\%exact accuracy, three more exact solves than original ERM at2024/2048=98\.83%2024/2048=98\.83\\%and seven more than PT\-confidence or PT\-q\-head\. Its selected accuracy equals its own oracle, which means ERM did not lose correct candidates at the final selection step\. Because PT uses the same permutation\-memory energy for sampling and final scoring, the added evaluation cost is mainly inexpensive scalar energy evaluations and replica\-swap decisions rather than a separate learned verifier\.

PPBench shows a different but informative PT pattern\. PT\-confidence improves over its original shared\-pool selector from80\.73%80\.73\\%to81\.06%81\.06\\%, q\-head remains at83\.39%83\.39\\%, and PT\-ERM obtains the strongest selected result,265/301=88\.04%265/301=88\.04\\%, matching both its original ERM accuracy and its PT oracle\. This closes the final\-selection gap for the completed PT candidate pool and indicates that remaining PPBench failures are candidate\-generation failures rather than energy\-ranking failures\. The diagnostic remains useful for future design: when selected accuracy is below PT oracle, the next improvement should target energy ranking; when PT oracle is low, the next improvement should target candidate generation\.

Maze PT is now complete under the same officialD=64D=64,K=128K=128setting\. Compared with the original selectors, PT\-confidence increases from97\.80%97\.80\\%to979/1000=97\.90%979/1000=97\.90\\%, and PT\-q\-head increases from97\.30%97\.30\\%to974/1000=97\.40%974/1000=97\.40\\%\. PT\-ERM selects993/1000=99\.30%993/1000=99\.30\\%and matches its own PT oracle, indicating that the global shortest\-path memory does not lose correct Maze candidates at the final selection step\. This equals the shared\-candidate ERM result of993/1000=99\.30%993/1000=99\.30\\%, so the Maze PT result should be read as oracle\-matching ranking with the same exact ceiling as the shared\-candidate global\-memory run\.

### 4\.4Training\-Side Energy as Motivation

We also evaluated an EqR\+Energy\-Transformer Sudoku variant as a training\-side motivation for the energy view\. WithD=64D=64,K=16K=16, and a checkpoint at step 75000, the one\-rollout baseline obtains65\.97%65\.97\\%exact accuracy on 2048 Sudoku\-Lite examples\. ET\-energy selection and confidence\-guided inference both reach76\.86%76\.86\\%, majority voting gives65\.28%65\.28\\%, and PT\-ERM gives75\.93%75\.93\\%\. These results show that explicit training\-time energy can be meaningful, but this branch is not the main empirical focus because it is more computationally expensive and more sensitive to the base checkpoint than inference\-time ERM\.

## 5Discussion

The central empirical lesson is that test\-time breadth should be evaluated through an energy\-and\-oracle decomposition\. Across the completed shared\-candidate experiments, EqR rollouts often contain substantially better answers than the baseline prediction, but ordinary selectors do not always recover them\. ERM closes or nearly closes this selector\-oracle gap by replacing generic confidence with memories that encode the task’s valid structures\. This finding is strongest on PPBench, where heterogeneous rules make confidence less reliable, and more diagnostic on Sudoku, where confidence is already near the oracle but ERM provides an explicit energy interpretation and PT closes the remaining three\-example gap\.

The method depends on energy design choices that should be reported as part of the scientific result rather than hidden as implementation details\. In Sudoku, the important choice is to use row, column, and box permutation memories and to clamp clue cells\. In PPBench, the important choices areβ=128\\beta=128,γ=0\\gamma=0, and global distance weight6464, because earlier confidence mixing could raise invalid but high\-likelihood candidates\. In Maze, the important choice is even more consequential: the reported ERM uses a global BFS shortest\-path memory with local weight zero, so it should be described as a strong single\-memory Hopfield limit rather than as a purely local path constraint\. These details make the energy view reproducible and prevent overclaiming about what each experiment demonstrates\.

The main limitation is that the present energies are task\-structured rather than universally learned\. This is appropriate for testing the energy perspective, because Sudoku, PPBench, and Maze all have crisp constraint structures that can be written as memories or distances\. However, it also means that the current Maze global energy is close to a rule\-derived verifier, and the PPBench global potentials require puzzle\-specific engineering\. A stronger future version would learn parts of the memory set or distance potential while preserving the oracle\-gap diagnostics used here\. Such a version would keep the advantage of energy\-based selection while reducing manual task design\.

The broader implication is that energy functions provide a shared interface between attractor dynamics, verifiers, and test\-time compute\. A latent reasoner supplies candidate configurations; an energy states what compatibility means; and selection or PT turns the energy into an inference procedure\. This interface is useful even when the energy is simple, because it tells researchers where to look next: generate better candidates when the oracle is low, design better memories when the selector misses the oracle, and improve sampling when PT selected accuracy matches PT oracle but both remain below the shared\-candidate ceiling\. In this sense, energy is not only a model component but a methodology for making reasoning\-time computation measurable and improvable\.

## 6Conclusion

We presented an energy view of test\-time scaling for latent iterative reasoners\. In this view, a recurrent rollout is treated as a candidate configuration, and the selector that chooses among rollouts is treated as an energy over that configuration\. ERM instantiates this energy with Hopfield task memories and global distance potentials\. Across Sudoku, PPBench, and Maze, ERM reaches or nearly reaches the shared\-candidate oracle under theD=64D=64,K=128K=128setting, while PT\-ERM gives the strongest completed energy\-guided sampling results on the same task suite\. These results show that the main bottleneck in latent reasoning is often not only candidate generation, but also the scalar principle used to decide which candidate should be trusted\.

The results also clarify the assumptions under which the present method is most informative\. Our strongest energies use task structure: Sudoku uses permutation memories, PPBench uses puzzle\-specific local and global rule memories, and Maze uses an input\-conditioned BFS shortest\-path memory\. This design makes the experiments reproducible and diagnostic, but it also means that the current energies should be understood as structured task verifiers rather than as universal learned energies\. A natural next step is therefore to learn parts of the memory sets, distance potentials, or energy weights while preserving the oracle\-gap diagnostics used here\. More broadly, energy\-based inference provides a common interface between attractor dynamics, verifiers, and test\-time compute, turning additional reasoning budget into a measurable selection problem rather than an undirected increase in breadth\.

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## Appendix ANotation Table

Table 3:Notation Used in the Main Text\.The table is breakable so every symbol remains visible in the appendix rather than being hidden inside an oversized float\.SymbolMeaningxxInput puzzle or reasoning instance\.qqQuery state in the Modern Hopfield retrieval energy\.∥q∥2\\lVert q\\rVert^\{2\}Squared Euclidean norm of the Modern Hopfield query state\.DDRecurrent reasoning depth used by the base EqR model\.KKNumber of candidate rollouts or sampled trajectories\.𝒴​\(x\)\\mathcal\{Y\}\(x\)Candidate pool produced for inputxx\.yyGeneric candidate output variable used when defining a distribution\.yky\_\{k\}Thekk\-th candidate structured output\.kkCandidate index inside the pool𝒴​\(x\)\\mathcal\{Y\}\(x\)\.k^\\hat\{k\},y^\\hat\{y\}Selected candidate index and selected structured answer\.NNNumber of output positions or cells\.E​\(yk;x\)E\(y\_\{k\};x\)Energy assigned to candidateyky\_\{k\}for inputxx; lower is better\.EρE\_\{\\rho\},EηE\_\{\\eta\}Energies of two PT replicas proposed for swapping\.EMHNE\_\{\\mathrm\{MHN\}\}Modern Hopfield retrieval energy in Equation[2](https://arxiv.org/html/2607.10128#S3.E2)\.EERME\_\{\\mathrm\{ERM\}\}ERM memory\-distance energy in Equation[3](https://arxiv.org/html/2607.10128#S3.E3)\.JJNumber of task factors used by ERM\.jjFactor index, such as a Sudoku row or a PPBench clue neighborhood\.ℳ\\mathcal\{M\}Generic memory set used in the Modern Hopfield retrieval energy\.ℳj​\(x\)\\mathcal\{M\}\_\{j\}\(x\)Input\-conditioned memory set for factorjj\.mmOne stored memory pattern in eitherℳ\\mathcal\{M\}orℳj​\(x\)\\mathcal\{M\}\_\{j\}\(x\)\.m⋆​\(x\)m^\{\\star\}\(x\)A single complete task memory used in the hard\-verification special case\.dj​\(yk,m\)d\_\{j\}\(y\_\{k\},m\)Distance between candidateyky\_\{k\}and memorymmon factorjj\.τ\\tauSoft\-retrieval temperature for the memory log\-sum\-exp\.G​\(yk,x\)G\(y\_\{k\},x\)Global rule\-violation score for constraints not enumerated locally\.μ\\muWeight of the global rule\-violation term\.TTSampling temperature in the tempered distribution\.πT​\(y∣x\)\\pi\_\{T\}\(y\\mid x\)Tempered distribution over candidate outputs at temperatureTT\.T1,…,TRT\_\{1\},\\ldots,T\_\{R\}Parallel\-tempering replica temperatures\.TρT\_\{\\rho\},TηT\_\{\\eta\}Temperatures of two PT replicas proposed for swapping\.RRNumber of PT replicas\.ρ\\rho,η\\etaReplica indices used in the PT swap rule\.aaSwap acceptance probability between two PT replicas\.Lk​\[i,d\]L\_\{k\}\[i,d\]Log\-probability that candidatekkassigns digit or tokenddto positionii\.ii,ddOutput position index and digit/token index\.uuSudoku structural unit, such as a row, column, or box\.ℳu\\mathcal\{M\}\_\{u\}Memory set for Sudoku unituu\.Su​\(k\)S\_\{u\}\(k\)Sudoku ERM score for unituuand candidatekk\.rrPosition index inside a Sudoku unit\.iri\_\{r\}Therr\-th cell in a Sudoku unit\.σ\\sigmaA permutation assigning digits to positions inside a Sudoku unit\.S9S\_\{9\}Set of all permutations of the nine Sudoku digits\.mBFS​\(x\)m\_\{\\mathrm\{BFS\}\}\(x\)BFS\-derived shortest\-path memory for a unique\-path maze input\.sim\\mathrm\{sim\}Similarity function used in Modern Hopfield retrieval and in the Maze global memory\.SMazeS\_\{\\mathrm\{Maze\}\}Maze global\-memory score in Equation[7](https://arxiv.org/html/2607.10128#S3.E7)\.EMazeE\_\{\\mathrm\{Maze\}\}Maze global\-memory energy, defined as−SMaze\-S\_\{\\mathrm\{Maze\}\}\.β\\betaScale parameter for memory similarity or distance\.γ\\gammaOptional model\-confidence mixing weight; set to zero in main ERM runs\.
## Appendix BAdditional Experimental Details

### B\.1EqR\+ET Sudoku Motivation

![Refer to caption](https://arxiv.org/html/2607.10128v1/x4.png)Figure 4:Training with an Energy Transformer Architecture\.This appendix experiment uses the EqR\+ET Sudoku checkpoint withD=64D=64,K=16K=16, and PT temperatures from0\.0010\.001to1\.01\.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 as motivation for the energy view rather than as a main empirical claim\. The architectural ET score and ordinary confidence both reach1574/2048=76\.86%1574/2048=76\.86\\%, showing that an explicit energy\-like signal can be useful even when it is attached to a weaker training\-side checkpoint\. However, PT\-confidence reaches only1555/2048=75\.93%1555/2048=75\.93\\%, which suggests that poor base\-model quality can make PT ineffective: if the sampled states are low quality or the confidence landscape is poorly aligned with exact correctness, temperature mixing may explore more states without improving final selection\. This failure mode motivates the main paper’s inference\-time ERM setting, where task memories, selector\-oracle diagnostics, and PT\-selected versus PT\-oracle accuracy separate energy design from checkpoint quality\.

Table 4:EqR\+ET Sudoku Appendix Result\.PT\-confidence uses confidence as the PT energy; its small drop relative to direct confidence selection indicates that a weak checkpoint can limit energy\-guided sampling\.
### B\.2Task\-Specific ERM Examples

Sudoku ERM can be interpreted as exact soft retrieval over all valid memories for each structural unit\. The row, column, and box memory sets are not learned from labels at evaluation time; they are the task’s known local validity rules\. The log\-permanent computation evaluates Equation[6](https://arxiv.org/html/2607.10128#S3.E6)exactly for each unit and each candidate, which avoids replacing the structured memory with an ad hoc violation count\.

PPBench ERM uses a mixed local\-global decomposition because the benchmark combines several puzzle families\. Local factors cover small enumerable rules such as clue counts, forbidden adjacencies, no\-2×22\\times 2patterns, and run\-length memories\. Global distance potentials cover constraints that are difficult to enumerate as local memories, including connectivity and coverage\. The reported configuration usesγ=0\\gamma=0so that the energy is not secretly using model confidence as a fallback selector\.

Maze ERM should be read as a demonstration of the boundary between local consistency and global task memory\. The local path\-degree energy is a valid Hopfield factor memory, but it is insufficient for oracle alignment because disconnected cycles can satisfy many local degree rules\. The global BFS memory supplies the unique shortest\-path solution implied by the input maze, and the reported ERM selects the candidate closest to that memory\. This makes the experiment valuable as a candidate\-selection diagnostic, while also making it a strong verifier\-style energy rather than a weak local selector\.

## Appendix CFigure Source Data

The plotting scripts export the numerical source data used in the figures as CSV files\. The Maze depth\-curve figure uses per\-depth selector energies, selected exact accuracies, and oracle diagnostics for q\-head, confidence, and ERM on the official Maze\-Unique candidate pool\. The PT figure uses completed Sudoku, PPBench, and Maze PT results, the shared\-selector source table records the exact accuracies used in Table[1](https://arxiv.org/html/2607.10128#S4.T1), and the EqR\+ET appendix figure records the five Sudoku\-Lite exact accuracies shown in Figure[4](https://arxiv.org/html/2607.10128#A2.F4)\. These files are included to make figure values auditable against the corresponding experiment logs and result JSON files\.

## Appendix DSelf\-Review Checklist

The current draft makes three supported claims\. First, shared\-candidate ERM reaches the oracle on PPBench and Maze and is within three Sudoku examples of the shared\-candidate oracle; this is supported by Table[1](https://arxiv.org/html/2607.10128#S4.T1)\. Second, ERM is a Hopfield energy; this is supported by Equation[3](https://arxiv.org/html/2607.10128#S3.E3)and by the task\-specific memory definitions\. Third, PT\-ERM can guide sampling in addition to reranking and reaches the Sudoku PT oracle; this is supported by the completed Sudoku, PPBench, and Maze PT results in Table[2](https://arxiv.org/html/2607.10128#S4.T2)\.