[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82130-en":3,"doc-seo-82130-105":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},82130,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","A Formalization of the Mean-Field Derivation of the Vlasov Equation","We formalize a mean-field derivation of the nonlinear Vlasov equation in the Lean 4 proof assistant through an “AI-assisted formalization game” directed by a mathematician. The game is won when the Lean development compiles without sorry, and a machine check guarantees that theorems rely only on Lean’s foundational axioms. As a case study, the work proves well-posedness results via Dobrushin’s mean-field route, including existence, uniqueness, stability, and a mean-field limit, and it yields a self-contained reusable layer. Quantitative timing is reported as observations from one run.","arXiv :2607 .08986v 1 [ cs .AI] 9 Jul 2026  \nA Formalization of the Mean-Field Derivation of the Vlasov Equation  \nMathematician in the loop: AI-assisted Lean formalization as a strategy game  \nJoseph K. Miller∗  \nJuly 13, 2026  \nAbstract  \nWe formalize a research result in the Lean 4 proof assistant by having a mathematician direct an AI system, and frame the activity as a formalization game. The objective is to turn a LATEX document into Lean. The game is won when the development compiles, contains no sorry, anda machine check shows the target theorems rest on Lean’s foundational axioms alone. Reuse is a second check, by a definition we introduce: whether the development yields a self-contained layer of general mathematics the wider library could absorb.  \nThe case study is a complete, axiom-clean formalization of well-posedness for the nonlinear Vlasov equation via Dobrushin’s mean-field route — existence, uniqueness, the stability estimate and mean-field limit, and a short-window superposition principle (weak solutions are Lagrangian) . The human’s role here was to direct, not to write proofs: to scope the definitions, steer the decompositions, and triage the library’s gaps; the AI agent executed. The formalization certifies the proof of each statement as written; whether the written statement is the intended theorem stays the mathematician’s judgment. The optimal-transport machinery that fell out of the build (in particular, properties of the Wasserstein-1 metric and the Kantorovich– Rubinstein duality theorem) separates into a self-contained layer (Definition 2.2) that compiles against Mathlib alone: about a sixth of the development (49 of 299 declarations), behind a 22-declaration interface with no reverse dependency. The headline theorems ran in about a week, the full development in about a month. We report the quantitative claims as observations of one game, not as general laws. The game’s rules name no particular system, so the methodological framing is meant to outlast the tools of any one run.  \nContents  \n1 Introduction 2  \n1. 1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  \n1.2 Choosing the target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n1.3 What we formalize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5  \n2 The formalization game 7  \n2.1 Objective, win condition, self-contained layer ...................... 7  \n2.2 The three phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9  \n2.3 The strategy framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11  \n∗ Stanford University, [jkm314@stanford.edu](jkm314@stanford.edu) ; Massachusetts Institute of Technology, [jkm314@mit.edu. The](jkm314@mit.edu. The) development is public. Source: [https://github.com/Hydrodynamical/Vlasov_Meanfield_Formalization](https://github.com/Hydrodynamical/Vlasov_Meanfield_Formalization. Blueprint)[. Blueprint](https://github.com/Hydrodynamical/Vlasov_Meanfield_Formalization. Blueprint) site and  \nAPI documentation: [https://hydrodynamical.github.io/Vlasov_Meanfield_Formalization/](https://hydrodynamical.github.io/Vlasov_Meanfield_Formalization/) .  \n3 The mean-field derivation, formalized 13  \n3. 1 From Newton to an exact weak solution . . . . . . . . . . . . . . . . . . . . . . . . . 13  \n3.2 Wasserstein-1, in two faces ................................. 14  \n3.3 The dynamical core ..................................... 15  \n3.4 The superposition principle, and why C2 ......................... 16  \n3.5 The reusability outcome .................................. 17  \n4 Assessment and outlook 18  \n4.1 The verification discipline ................................. 18  \n4.2 Timeframe and the division of labor . . . . . . . . . . . . . . . . . . . . . . . . . . . 19  \n4.3 What generalizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19  \n4.4 Limita","cbCaii0dpdG5YAy2","https://ap.wps.com/l/cbCaii0dpdG5YAy2","pdf",407598,1,26,"English","en",105,"# Introduction\n## Related work\n## Choosing the target\n## What we formalize\n# The formalization game\n## Objective, win condition, self-contained layer\n## The three phases\n## The strategy framework\n# The mean-field derivation, formalized\n## From Newton to an exact weak solution\n## Wasserstein-1, in two faces\n## The dynamical core\n## The superposition principle\n## The reusability outcome\n# Assessment and outlook\n## The verification discipline\n## Timeframe and the division of labor\n## What generalizes\n## Limitations\n## Outlook","[{\"question\":\"What is the main goal of this document’s Lean formalization work?\",\"answer\":\"To formalize a research result in Lean 4 by converting an existing LATEX statement into verified Lean code, certified by compilation without sorry and reliance only on Lean’s foundational axioms.\"},{\"question\":\"How does the “formalization game” define winning and trustworthiness?\",\"answer\":\"Winning requires the development to compile, contain no sorry, and have the checked theorems depend only on Lean’s foundational axioms. A further reuse check evaluates whether the development forms a self-contained general-mathematics layer absorbable by the wider library.\"},{\"question\":\"What does the case study on the Vlasov equation cover?\",\"answer\":\"A complete axiom-clean formalization of well-posedness for the nonlinear Vlasov equation via Dobrushin’s mean-field route, including existence, uniqueness, a stability estimate, a mean-field limit, and a short-window superposition principle with weak solutions characterized in Lagrangian form.\"}]",1784178365,66,{"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},"a-formalization-of-the-mean-field-derivation-of-the-vlasov-equation","",{"@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/a-formalization-of-the-mean-field-derivation-of-the-vlasov-equation/82130/",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 is the main goal of this document’s Lean formalization work?","Question",{"text":74,"@type":75},"To formalize a research result in Lean 4 by converting an existing LATEX statement into verified Lean code, certified by compilation without sorry and reliance only on Lean’s foundational axioms.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the “formalization game” define winning and trustworthiness?",{"text":79,"@type":75},"Winning requires the development to compile, contain no sorry, and have the checked theorems depend only on Lean’s foundational axioms. A further reuse check evaluates whether the development forms a self-contained general-mathematics layer absorbable by the wider library.",{"name":81,"@type":72,"acceptedAnswer":82},"What does the case study on the Vlasov equation cover?",{"text":83,"@type":75},"A complete axiom-clean formalization of well-posedness for the nonlinear Vlasov equation via Dobrushin’s mean-field route, including existence, uniqueness, a stability estimate, a mean-field limit, and a short-window superposition principle with weak solutions characterized in Lagrangian form.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]