[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84103-en":3,"doc-seo-84103-105":29,"detail-sidebar-cat-0-en-105":95},{"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},84103,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Axioms for Physical Reasoning: Codifying the Seiberg–Witten Solution in Lean","Interactive theorem proving is used to verify non-rigorous theoretical physics arguments by replacing long, trusted reasoning with a short, explicit list of named physical postulates that a machine can check. The approach is applied to the Seiberg–Witten solution of N=2 SU(2) super-Yang–Mills (genus one), formalized in Lean 4, while higher-genus generalizations are prepared as an axiomatized skeleton. The paper details proved results, stated assumptions, and their validation via external review and an independent numerical oracle, proposing this workflow as a robust standard for auditing AI-generated physics derivations.","arXiv :2607 .06379v 1 [hep-th] 7 Jul 2026  \nAxioms for physical reasoning:  \ncodifying the Seiberg–Witten solution in Lean ∗  \nMichael R. Douglas †1  \n1 Center Of Mathematical Sciences And Applications, Harvard University, Cambridge, MA 02138 USA  \nJuly 2026  \nAbstract  \nMathematicians have embraced interactive theorem provers with growing enthusiasm—building large shared libraries and machine-checking a string of landmark results. Theoretical physics is different: most of its results are not theorems but justified by arguments the community trusts without a rigorous proof. For many—the one we treat here among them—no rigorous proof is within reach. For 4d Yang–Mills theory, deriving exact rigorous results from first principles would first require constructing the interacting theory nonperturbatively, which is a sizable piece of one of the Clay Millennium prize problems.  \nWe argue here that an interactive theorem prover can be used to verify some non-rigorous physics arguments. The method is to postulate a short list of explicit, named physical postulates, which imply the physical results by virtue of a machine-checkable proof. The trust that remains then rests on that short, inspectable list, and the prover can report, for any downstream result, exactly which assumptions it used. We carry this out for the Seiberg–Witten solution of N = 2 SU(2) super-Yang–Mills—the genus-one case—formalized in Lean 4; the higher-genus SU (N) generalization is developed in the same repository asan axiomatized skeleton and left to future work. We describe what is proved, what is assumed, how the assumptions are checked—external review and an independent numerical oracle—and why this discipline is a sound standard for validating AI-generated results in theoretical physics. What we offer is a discipline, reviewable on its own terms: a reader may take the Seiberg–Witten mathematics on trust and still assess the formalization method.  \nContents  \n1 Introduction 2  \n2 Two kinds of formalization 3  \n3 The Seiberg–Witten solution in two paragraphs 5  \n4 The formalization 5  \n4.1 The dictionary: physical concepts as mathematical objects . . . . . . . . . . . . . . . . . . . . 5  \n4.2 What is assumed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5  \n4.3 What is proved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n4.4 What formalizing forces into the open ............................... 9  \n4.5 A worked check: Argyres–Douglas points from matter, axiom-free ................ 9  \n5 Checking the axioms: validation by independent agreement 10  \n6 Remaining mathematical debt, and future work 11  \n∗Working paper, draft. Companion to the formal development at mrdouglasny/seiberg-witten and to the ICML 2026 talk Validation of AI-generated results in theoretical physics.  \n†[mdouglas@cmsa.fas.harvard.edu](mdouglas@cmsa.fas.harvard.edu)  \n7 Prospects: deriving the physical assumptions 12  \n8 Conclusion 12  \nA The physics hypotheses (H0–H7)—predicates, not axioms 14  \nB The residual mathematical axioms and the headline theorems 16  \n1 Introduction  \nMost of what theoretical physics asserts is not rigorously proved. Across the field, the results we build on are typically not theorems but conclusions of arguments—chains of physical reasoning trusted because they cohere internally, because independent routes to the same quantity agree, and ultimately because they account for experiment. A result can be as certain as anything in science and still have no rigorous proof from first principles; certainty in physics rests on a different footing than in mathematics.  \nIn quantum field theory the gap is very wide. There the deepest results—the spectrum of a gauge theory, its phase structure, an exact low-energy effective action—follow from reasoning based on principles like symmetry, locality, dualities and consistency. The Seiberg–Witten solution [21, 22], our subject below, is a paradigmat","cbCaiqVNFDrYPiUL","https://ap.wps.com/l/cbCaiqVNFDrYPiUL","pdf",622164,1,18,"English","en",105,"# Introduction\n## Two kinds of formalization\n## The Seiberg–Witten solution in two paragraphs\n## The formalization\n### The dictionary: physical concepts as mathematical objects\n### What is assumed\n### What is proved\n### What formalizing forces into the open\n### A worked check: Argyres–Douglas points from matter, axiom-free\n## Checking the axioms: validation by independent agreement\n## Remaining mathematical debt, and future work\n## Prospects: deriving the physical assumptions\n## Conclusion\n# Appendices\n## The physics hypotheses (H0–H7)\n## Residual mathematical axioms and the headline theorems","[{\"question\":\"What problem does the paper address in theoretical physics verification?\",\"answer\":\"Most theoretical physics claims rely on trusted argumentation rather than rigorous proof. The paper targets the risk that AI-generated derivations may include hallucinated assumptions or subtle omissions that are hard to audit by reading entire proofs.\"},{\"question\":\"How does the proposed method make physics arguments machine-auditable?\",\"answer\":\"It replaces a long reasoning chain with a short list of explicit, named physical postulates. A prover checks a machine-checkable proof, and downstream results can report exactly which assumptions were used.\"},{\"question\":\"What exactly is formalized, and in which proof assistant?\",\"answer\":\"The paper formalizes the genus-one case of the Seiberg–Witten solution for N=2 SU(2) super-Yang–Mills in Lean 4. A higher-genus SU(N) generalization is developed as an axiomatized skeleton in the same repository and deferred for future work.\"},{\"question\":\"How are the physical assumptions validated according to the paper?\",\"answer\":\"Validation combines external review with an independent numerical oracle. This cross-checking supports the remaining trust placed only on the short, inspectable physical postulate list.\"}]",1784192822,45,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":27},"axioms-for-physical-reasoning-codifying-the-seibergwitten-solution-in-lean","",{"@graph":35,"@context":89},[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/axioms-for-physical-reasoning-codifying-the-seibergwitten-solution-in-lean/84103/",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",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in theoretical physics verification?","Question",{"text":75,"@type":76},"Most theoretical physics claims rely on trusted argumentation rather than rigorous proof. The paper targets the risk that AI-generated derivations may include hallucinated assumptions or subtle omissions that are hard to audit by reading entire proofs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method make physics arguments machine-auditable?",{"text":80,"@type":76},"It replaces a long reasoning chain with a short list of explicit, named physical postulates. A prover checks a machine-checkable proof, and downstream results can report exactly which assumptions were used.",{"name":82,"@type":73,"acceptedAnswer":83},"What exactly is formalized, and in which proof assistant?",{"text":84,"@type":76},"The paper formalizes the genus-one case of the Seiberg–Witten solution for N=2 SU(2) super-Yang–Mills in Lean 4. A higher-genus SU(N) generalization is developed as an axiomatized skeleton in the same repository and deferred for future work.",{"name":86,"@type":73,"acceptedAnswer":87},"How are the physical assumptions validated according to the paper?",{"text":88,"@type":76},"Validation combines external review with an independent numerical oracle. This cross-checking supports the remaining trust placed only on the short, inspectable physical postulate list.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},"Exam",70,"exam",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},5,"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},6,"Technology",50,"technology",{"id":120,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":122,"slug":123},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":125,"slug":126},30,"research-report",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":130,"slug":131},9,"Religion & Spirituality",20,"religion-spirituality",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":130,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":45,"category_name":141,"show_sort_weight":110,"slug":142},19,"General","general"]