[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83880-en":3,"doc-seo-83880-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83880,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Knowledge Knows, Verbalization Tells Disentangling Latent Directions for Mathematical Solvability in LLMs","Although LLMs have improved at mathematical reasoning, identifying whether a problem is solvable is a persistent challenge. Prior work often analyzes internal solvability beliefs, while treating verbalization mainly as behavior, not as a separate internal representation. This paper probes solvability knowledge and verbalization independently in hidden states, showing distinct linearly decodable directions across multiple LLMs. It attributes fabrication chiefly to verbalization changes, and finds that unsolvability prompting and activation steering can mechanistically improve abstention.","Knowledge Knows, Verbalization Tells: Disentangling Latent Directions for  \nMathematical Solvability in LLMs  \nNikolaos Xiros* Maria-Eleni Zoumpoulidi* Georgios Paraskevopoulos  \nInstitute for Language and Speech Processing, Athena Research Center, Greece {n.xiros,m.zoumpoulidi,[g.paraskevopoulos}@athenarc.gr](g.paraskevopoulos}@athenarc.gr)  \narXiv :2607 .050 13v 1 [ cs .CL] 6 Jul 2026  \nAbstract  \nAlthough LLMs have made significant progress in mathematical reasoning, determining whether a mathematical problem is solvable remains a fundamental yet challenging capability. While recent studies have probed internal representations of model solvability beliefs, verbalization has primarily been studied behaviorally rather than as an internal representation, limiting its analysis and manipulation. We address this gap by separately probing representations of solvability knowledge and verbalization, allowing us to disentangle the two within model hidden states. Across multiple LLMs, we show that knowledge and verbalization are encoded as distinct, linearly decodable representationsand that fabrication is primarily associated with changes in verbalization rather than the underlying knowledge. Prompting with unsolvability cues reduces fabrication primarily by shifting verbalization, while activation steering demonstrates that these representations can be mechanistically manipulated to improve model abstention.  \n1 Introduction  \nAssessing the solvability of a problem is a core yet challenging aspect of mathematical reasoning, even for experts, motivating a growing body of research on large language models’(LLMs) ability to recognize unsolvable problems.  \nExisting work spans the development of benchmarks and evaluation of prompting techniques (Xue et al., 2025), as well as mechanistic interpretability approaches that probe the internal and geometric representations of solvability beliefs (Liu et al., 2026 ; Sanyal et al., 2025) . Such analyses suggest that models often possess latent knowledge about a problem’s unsolvability but fail to express it in their responses, while also identifying interventions that can mitigate this behavior (Liu et al.,  \n*Equal contribution. Author order determined by coin flip.  \nFigure 1: Overview of our approach. Given a problem, we probe two complementary directions in LLM representations: knowledge, capturing whether the model internally recognizes the problem as unsolvable, and verbalization, capturing whether it explicitly communicates this judgment. The final generated response typically follows the verbalization direction.  \n2026 ; Sanyal et al., 2025) .  \nIn parallel, research on answerability in broader, primarily non-mathematical domains has evolved along similar methodological lines (Kirichenko et al. (2026), Slobodkin et al. (2023), Lavi et al.(2026)) . These parallels suggest that insights from one setting may naturally transfer to the other. More broadly, prior work on chain-of-thought faithfulness shows that models’ verbalized reasoning often diverges from their underlying computations (Turpin et al., 2023 ; Lanham et al., 2023), fails to acknowledge influential information (Chen et al., 2025 ; Mirtaheri and Belkin, 2026), and can arise from a misalignment between internal knowledge and prediction representations (Park et al., 2026 ; Shen et al., 2026) .  \nAlthough these works advance our understanding of solvability detection and reveal a gap between internal knowledge and verbalized reasoning, they study verbalization only through textual reasoning traces, hindering its analysis and manipulation. We instead disentangle internal representations of solvability knowledge and verbalization, enabling us  \nto study their interaction and influence on model behavior. Inspired by Park et al. (2026), who apply this framework to general multiple-choice QA scenarios, we study the interplay between these two representations in LLMs. Specifically, we probe two complementary directions: knowledge, capt","cbCainFnyfSlCm6u","https://ap.wps.com/l/cbCainFnyfSlCm6u","pdf",1438373,4,1,14,"English","en",105,"# Introduction\n# Related Work\n## LLMs and Solvability-Answerability","[{\"question\":\"Why is solvability detection difficult for LLMs even with advances in mathematical reasoning?\",\"answer\":\"The document frames solvability recognition as a fundamental yet challenging capability; models may fail to recognize unsolvable problems and can instead fabricate solutions.\"},{\"question\":\"How does the paper distinguish solvability knowledge from verbalization?\",\"answer\":\"It separately probes two complementary directions in model hidden states: knowledge capturing internal recognition of unsolvability, and verbalization capturing whether the model explicitly communicates that judgment.\"},{\"question\":\"What interventions reduce fabrication and improve abstention?\",\"answer\":\"Prompting with unsolvability cues reduces fabrication mainly by shifting verbalization, and activation steering shows the representations can be manipulated to improve model abstention.\"}]",1784191194,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"knowledge-knows-verbalization-tells-disentangling-latent-directions-for-mathematical-solvability-in-llms","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/knowledge-knows-verbalization-tells-disentangling-latent-directions-for-mathematical-solvability-in-llms/83880/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","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},"Why is solvability detection difficult for LLMs even with advances in mathematical reasoning?","Question",{"text":75,"@type":76},"The document frames solvability recognition as a fundamental yet challenging capability; models may fail to recognize unsolvable problems and can instead fabricate solutions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper distinguish solvability knowledge from verbalization?",{"text":80,"@type":76},"It separately probes two complementary directions in model hidden states: knowledge capturing internal recognition of unsolvability, and verbalization capturing whether the model explicitly communicates that judgment.",{"name":82,"@type":73,"acceptedAnswer":83},"What interventions reduce fabrication and improve abstention?",{"text":84,"@type":76},"Prompting with unsolvability cues reduces fabrication mainly by shifting verbalization, and activation steering shows the representations can be manipulated to improve model abstention.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]