[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84326-en":3,"doc-seo-84326-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"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},84326,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","A First-Principles Theory of Slow Thinking and Active Perception","A first-principles framework is proposed to formalize thinking and perception through mathematical modeling. The approach lifts and projects probability distributions across observable and latent spaces to approximate complex data using simple function families such as neural networks. An “active lifting” theory is introduced via latent sequence sampling and an intrinsic objective that reduces uncertainty at maximum rate. The work yields a design space for slow-thinking language models, with static subspaces, hierarchical upgrade paths, internal-time inference, and training objectives resembling minimum-length coding and language invention. It also outlines techniques for representation learning across modalities and potential mitigation of policy collapse.","arXiv :2607 .08 196v 1 [ cs .AI] 9 Jul 2026  \nA First-Principles Theory of Slow Thinking and Active Perception  \nHongkang Yang 1,2 , Zhi-Qin John Xu3 , Feiyu Xiong 1,2 , and Weinan E4,2,B  \n1 MemTensor (Shanghai) Technology Co. , Ltd.  \n2 Institute for Advanced Algorithms Research, Shanghai  \n3 Shanghai JiaoTong University  \n4 Peking University  \nAbstract  \nAs part of a series on first-principles modeling of cognitive functions, this paper attempts to provide a mathematical formulation of thinking and perception. It formally derives slow thinking or more generally, active perception, and encompasses the design, training and inference of slow thinking large language models. Our starting point is the lifting and projection of probability distributions on the observable and latent spaces, with the objective of representing complex data distributions by simple function families such as neural networks. A theory called “active lifting”is proposed, based on the sampling of latent sequences and an intrinsic drive to reduce uncertainty with maximum rate. It derives a large design space, containing the slow thinking models ina subspace that we call the static theory. These models are positioned on the representation hierarchy and sampler hierarchy induced by the static theory, and can be upgraded by climbing the two hierarchies. Active lifting further derives an inference process with an internal timeaxis, and a training objective that resembles minimum-length coding as well as the invention of languages. Thus, it characterizes the agency of perception, including the emergence of the slow thinking formats. Technical by-products of this theory include a three-stage pathway for improving slow thinking models, a unified approach to constructing encoders and generative models for all data modalities, a priori formation of human-like visual representations, and a possible solution to policy collapse.  \nCognitive  \nPhenomena  \nMaroscopic Mechanisms  \nMicroscopic Mechanism  \nPerceptual  \nInference  \nMemory  \nSystem I & II  \nEpisodic vs. semantic  \nSelective memory, future thinking  \nMemory indexing, Memory hierarchy, Memory replay  \nVisual concepts & schema  \nInformationSeeking Behavior  \nExploration,  \nSelf-reflection,  \nProve & conjecture,  \nInvention & Use  \nRepresentation  \nInvention of math, formal systems, and scientific methods  \nInvention of games  \nThinking  \nof Language  \nComposition & relations,  \nImpossible Triangle of Read-Write Operations  \nAction CoarseGraining under Speed-Accuracy Tradeoff  \nActive Lifting  \nInformation Gain  \nthrough Micro-environment  \nTrajectory  \nUnified Theory (To be discussed)  \nFigure 0.1: The modeling hierarchy of cognitive functions. The coverage of this paper is marked in red. The other topics are left for our paper series and this figure is subject to constant update. Examples of cognitive phenomena (in gray) are included to illustrate their diversity.  \nB Corresponding author: [weinan@math.pku.edu.cn](weinan@math.pku.edu.cn)  \nExplanatory sampler  \nPredictive sampler  \nIdentity sampler  \nFast thinking models  \nForgetful latent  \nStatic lifting of simple projection  \nActive lifting  \nFigure 0.2: Road-map for improving slow thinking models, as a by-product of our theory. This figure summarizes the improvements proposed in Sections 5 .5-5.7 and 6 . The two axes are the representation hierarchy (illustrated in Figures 2.1 and 5.3) and the sampler hierarchy (Figure 3.1) .  \nContents  \n1 Introduction 4  \n1.1 Motivating Examples ...................................... 6  \n1.2 Overview ............................................ 9  \n2 Separation of Approximation Ability 11  \n2.1 From Learning Ability to Approximation Ability ...................... 11  \n2.2 Sequence Data ......................................... 13  \n2.3 Probability Space ........................................ 14  \n2.4 Parametrization ......................................... 15  \n2.5 Separation, Part I ...................................","cbCaierl3U63nLXF","https://ap.wps.com/l/cbCaierl3U63nLXF","pdf",8290342,1,118,"English","en",105,"# Introduction\n# Separation of Approximation Ability\n# Latent Sampling and Optimal Samplers\n# Unified Objective\n# Derivation and Improvement of Existing Models","[{\"question\":\"What core idea does the paper use to model thinking and perception?\",\"answer\":\"It lifts and projects probability distributions between observable and latent spaces to represent complex data with simple function families such as neural networks.\"},{\"question\":\"What is “active lifting” in this theory?\",\"answer\":\"“Active lifting” is a proposed mechanism based on sampling latent sequences while pursuing an intrinsic drive to reduce uncertainty at the maximum rate.\"},{\"question\":\"How does the paper connect the theory to slow-thinking language models?\",\"answer\":\"It derives a design space that includes slow-thinking models inside a static theory subspace, then uses representation and sampler hierarchies to upgrade models, along with an internal-time inference process and a training objective related to minimum-length coding and language invention.\"}]",1784194833,297,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-first-principles-theory-of-slow-thinking-and-active-perception","",{"@graph":35,"@context":85},[36,53,68],{"@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/a-first-principles-theory-of-slow-thinking-and-active-perception/84326/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What core idea does the paper use to model thinking and perception?","Question",{"text":75,"@type":76},"It lifts and projects probability distributions between observable and latent spaces to represent complex data with simple function families such as neural networks.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is “active lifting” in this theory?",{"text":80,"@type":76},"“Active lifting” is a proposed mechanism based on sampling latent sequences while pursuing an intrinsic drive to reduce uncertainty at the maximum rate.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper connect the theory to slow-thinking language models?",{"text":84,"@type":76},"It derives a design space that includes slow-thinking models inside a static theory subspace, then uses representation and sampler hierarchies to upgrade models, along with an internal-time inference process and a training objective related to minimum-length coding and language 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