[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86198-en":3,"doc-seo-86198-105":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},86198,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Efficient Test-Time Optimization for Multi-Agent Proof Autoformalization","Full-proof autoformalization connects extensive mathematical proofs in natural language with formally verified reasoning, raising the ceiling for trustworthy machine-assisted theorem proving. Unlike statement-level formalization, proof autoformalization is a long-horizon coordination problem across many claims and dependencies, where prior inference-time repair is costly or unguided. TOMAP introduces a Decomposer-Formalizer-Prover pipeline with verification-guided test-time optimization and semantic proof rubrics, focusing compute on refining decomposition via an evolving-prompt loop driven by verification progress. Experiments on PROOFFLOWBENCH show 19.0% improvement in correctness and faithfulness with lower test-time cost, with most gains achieved in early decomposition iterations for budget selection.","arXiv :2607 . 1 1307v 1 [ cs .AI] 13 Jul 2026  \nEFFICIENT TEST-TIME OPTIMIZATION FOR MULTI-AGENT PROOF  \nAUTOFORMALIZATION  \nA PREPRINT  \nTian-Shuo Liu1,2,*, Shiyuan Zhang1,*, Zijie Geng3, Haoyu Liu1, Runjie Xu1, Pengyuan Wang1, Lei Yuan1, Yang Yu1,2,⋄  \n1 National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China  \n& School of Artificial Intelligence, Nanjing University, Nanjing, China  \n2Polixir Technologies, Nanjing, China  \n3 University of Science and Technology of China, Hefei, China  \n*  \nEqual contribution  \n⋄ [Corresponding: yuy@nju.edu.cn](Corresponding: yuy@nju.edu.cn)  \nABSTRACT  \nFull-proof autoformalization bridges extensive mathematical proofs in natural language with formally validated reasoning, offering a pathway to elevate the ceiling of verifiable mathematical reasoning. Unlike statement-level formalization, proof autoformalization is a long-horizon challenge requiring coordination of claims, contexts, and dependencies across many proof steps, yet has only recently come under focused study. Current approaches either rely on costly model training or apply excessive, unguided repair at inference time. To this end, we introduce TOMAP, a multi-agent framework that structures proof autoformalization as a Decomposer-Formalizer-Prover pipeline with efficient test-time optimization guided by formal verification and semantic rubrics for proof quality. Rather than distributing test-time compute across all agents, we perform bottleneck analysis and identify the Decomposer as the critical bottleneck: the quality of its atomic, self-contained proof units directly determines whether downstream agents can successfully formalize and prove each step. TOMAP therefore treats the Formalizer and Prover as downstream executors and efficiently focuses test-time compute on Decomposer refinement. This refinement follows a loop inspired by GEPA, evolving prompts over candidate decompositions and using formal verification progress together with semantic proof rubrics to define a Pareto frontier that guides the next decomposition update. Experiments on PROOFFLOWBENCH show that TOMAP improves over the best previous method by 19.0% when evaluated by both syntactic correctness and semantic faithfulness, while requiring lower test-time cost. Scaling analysis shows that most gains emerge within a few iterations of decomposition evolution, guiding test-time budget selection.  \n1 Introduction  \nLarge language models have made rapid progress in mathematical reasoning [Wang et al., 2026a, Lewkowycz et al., 2022, Azerbayev et al., 2023] . Among these advances, mathematical proof remains a particularly stringent target, as plausible solutions do not guarantee that intermediate claims, dependencies, and side conditions are logically valid. This has motivated a growing body of work on combining informal reasoning with formal verification, where proof assistants provide machine-checkable guarantees [Dekoninck et al., 2025, Jiang et al., 2022, Ren et al., 2025, Chen et al., 2025] . Autoformalization, translating informal mathematics into formal languages, is the central interface in this effort [Wu et al., 2022] . Early work focused on statement-level formalization. More recently, full-proof autoformalization has gained increasing attention, as translating complete proofs more effectively bridges informal and formal mathematics by leveraging the abundance of natural-language proofs while retaining the rigorous verifiability of formal systems.  \nThis shift, however, fundamentally changes the autoformalization problem. A theorem statement is a localized translation target [Wu et al., 2022], whereas a full proof is a long-horizon construction of intermediate claims, local assumptions, and dependencies that must compose into a globally checkable whole [Jiang et al., 2022] . Different stages of this translation demand distinct LLM capabilities acquired during post-training: natural-language reasoning for decomposition, forma","cbCaikGtZsllAF0N","https://ap.wps.com/l/cbCaikGtZsllAF0N","pdf",738589,1,30,"English","en",105,"# Introduction\n## Problem of full-proof autoformalization\n## Agentic Decomposer-Formalizer-Prover pipeline\n## Efficient test-time optimization via bottleneck analysis","[{\"question\":\"What is the main goal of TOMAP in proof autoformalization?\",\"answer\":\"TOMAP aims to improve full-proof autoformalization by structuring the process into specialized agents and applying efficient test-time optimization guided by formal verification and semantic rubrics.\"},{\"question\":\"Why is test-time optimization challenging in full-proof settings?\",\"answer\":\"Verifier feedback typically arrives only after a long generation chain, rarely indicating which earlier decision caused failure or what precise change should be made next under a limited budget.\"},{\"question\":\"How does TOMAP decide where to spend test-time compute?\",\"answer\":\"TOMAP performs bottleneck analysis and identifies the Decomposer as the critical stage, because the quality of atomic, self-contained proof units and their dependencies largely determines whether downstream formalization and proving succeed.\"}]",1784209337,76,{"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},"efficient-test-time-optimization-for-multi-agent-proof-autoformalization","",{"@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/efficient-test-time-optimization-for-multi-agent-proof-autoformalization/86198/",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 TOMAP in proof autoformalization?","Question",{"text":74,"@type":75},"TOMAP aims to improve full-proof autoformalization by structuring the process into specialized agents and applying efficient test-time optimization guided by formal verification and semantic rubrics.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why is test-time optimization challenging in full-proof settings?",{"text":79,"@type":75},"Verifier feedback typically arrives only after a long generation chain, rarely indicating which earlier decision caused failure or what precise change should be made next under a limited budget.",{"name":81,"@type":72,"acceptedAnswer":82},"How does TOMAP decide where to spend test-time compute?",{"text":83,"@type":75},"TOMAP performs bottleneck analysis and identifies the Decomposer as the critical stage, because the quality of atomic, self-contained proof units and their dependencies largely determines whether downstream formalization and proving succeed.","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},[92,96,100,104,109,114,119,121,126,129,133],{"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":21,"slug":120},"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":124,"slug":125},9,"Religion & Spirituality",20,"religion-spirituality",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":124,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]