[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85822-en":3,"doc-seo-85822-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85822,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Energy-Guided Recursive Model","Recursive reasoning models solve structured tasks by repeatedly updating latent states in compact neural networks, yet their test-time scaling lacks a principled inference mechanism. As depth or stochastic breadth increases, more hidden trajectories are generated without a clear selection criterion, and prior approaches often depend on extra q-heads or heuristic voting. The Energy-guided Recursive Model (ERM) introduces an intrinsic selector using explicit Hopfield energies derived from task-structured memories. ERM’s energy also supports energy-based sampling such as parallel tempering to improve efficiency and ranking, achieving strong results on Sudoku, PPBench, and Maze.","arXiv :2607 . 10 128v 1 [ cs .LG] 11 Jul 2026  \nENERGY-GUIDED RECURSIVE MODEL  \nYifei Zhao 1 , ∗ Ying Tang 1 ,2 ,3 ,4†  \n1Institute of Fundamental and Frontier Sciences,  \nUniversity of Electronic Science and Technology of China, Chengdu 611731, China  \n2 School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China  \n3 Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China, Chengdu 611731, China  \n4Non-classical Information Science Basic Discipline Research Center of Sichuan Province,  \nUniversity of Electronic Science and Technology of China, Chengdu 611731, China  \nABSTRACT  \nRecursive reasoning models address structured problems by repeatedly updating  \nlatent states of small neural networks. However, their test-time scaling lacks a  \nprincipled inference mechanism: increasing depth or stochastic breadth generates  \nmore trajectories without a clear criterion for selection, and existing methods pre  \ndominantly rely on additional q-heads or heuristic voting. Here, we develop the  \nEnergy-guided Recursive Model (ERM), which introduces an intrinsic selec  \ntion principle based on explicit Hopfield energies. ERM leverages Hopfield-type  \nmemories of valid local or global structures to define the selector over candidate  \ntrajectories. The resulting energy seamlessly integrates with energy-based tech  \nniques such as parallel tempering to enhance sampling efficiency and ranking.  \nWith D = 64 recurrent steps and K = 128 candidates, ERM reaches optimal so  \nlutions on Sudoku (98 .97%), Pencil Puzzle Bench (PPBench, 88.04%) and Maze  \n(99 .30%), improving upon recent Probabilistic Tiny Recursive Model and Equilib  \nrium Reasoners. These results suggest that incorporating explicit energy functions  \ninto recursive reasoning offers a principled path toward more effective inference.  \n1 INTRODUCTION  \nTest-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 [1, 2, 3, 4, 5, 6] . 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.  \nLoop 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 [7], adaptive computation time [8], Universal Transformers [9], implicit models [10] and deep equilibrium models [11] . Recent looped andrecurrent-depth transformers [12, 13, 14] show that repeated shared blocks can act as iterative algorithms, latent thoughts, or compute-scalable language-model components [15, 16, 17, 18] . Structured reasoning models such as the Hierarchical Reasoning Model (HRM) [19], Tiny Recursive Model (TRM) [20], Equilibrium Reasoners (EqR) [21], and Probabilistic Tiny Recursive Model (PTRM) [22] further show that compact recurrent networks can solve Sudoku, Maze, ARC-style  \n∗ Correspondence authors: [202521210324@std.uestc.edu.cn](202521210324@std.uestc.edu.cn)  \n†Correspondence authors: [jamestang23@gmail.com](jamestang23@gmail.com)  \n|  |  | Sudoku (Oracle=98 .97)\u003Cbr> 97.90 98.54 98.97 \u003Cbr>89.31\u003Cbr>\u003Cbr","cbCaivSgfm2kM49d","https://ap.wps.com/l/cbCaivSgfm2kM49d","pdf",1066466,1,14,"English","en",105,"# Abstract\n# Introduction\n## Test-time computation and trajectory selection\n## Loop models for latent reasoning\n## Structured recursive reasoners and the selection gap\n# Figure 1: Energy-guided recursive model for latent iterative reasoning","[{\"question\":\"What problem does the Energy-guided Recursive Model (ERM) address?\",\"answer\":\"ERM addresses the lack of a principled test-time selection mechanism in recursive reasoning, where increasing depth or breadth generates many hidden trajectories without a clear criterion for choosing which one to trust.\"},{\"question\":\"How does ERM select among candidate trajectories?\",\"answer\":\"ERM replaces q-head or majority voting selection with an intrinsic selector built from explicit Hopfield energies computed over task-structured memories, choosing the lowest-energy candidate.\"},{\"question\":\"Which tasks and metrics does ERM report improvements on?\",\"answer\":\"ERM reports strong performance on Sudoku (98.97%), PPBench (88.04%), and Maze (99.30%), improving over recent Probabilistic Tiny Recursive Model and Equilibrium Reasoners.\"}]",1784206468,35,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"energy-guided-recursive-model","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/energy-guided-recursive-model/85822/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the Energy-guided Recursive Model (ERM) address?","Question",{"text":74,"@type":75},"ERM addresses the lack of a principled test-time selection mechanism in recursive reasoning, where increasing depth or breadth generates many hidden trajectories without a clear criterion for choosing which one to trust.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does ERM select among candidate trajectories?",{"text":79,"@type":75},"ERM replaces q-head or majority voting selection with an intrinsic selector built from explicit Hopfield energies computed over task-structured memories, choosing the lowest-energy candidate.",{"name":81,"@type":72,"acceptedAnswer":82},"Which tasks and metrics does ERM report improvements on?",{"text":83,"@type":75},"ERM reports strong performance on Sudoku (98.97%), PPBench (88.04%), and Maze (99.30%), improving over recent Probabilistic Tiny Recursive Model and Equilibrium 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