[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82824-en":3,"doc-seo-82824-105":29,"detail-sidebar-cat-0-en-105":83},{"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},82824,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Mechanism-Level Routing Failure in LLMs over Lean-verified Algebraic Structures","An empirical study investigates structural routing failure in large language models (LLMs) when proof-mechanism labels must be selected from a fixed closed template set for compact mathematical objects drawn from a FiberRing formalization in Lean 4. A mechanism-level routing ceiling is identified: under blind conditions, template accuracy is limited, while providing a mechanism-bearing Lean verdict/witness cue increases accuracy substantially. The dominant error pattern is a CRT→ring-equivalence misroute, with two cue-resistant boundary failures described and cross-corpus checks reported.","arXiv :2607 .04534v 1 [ cs .CL] 5 Jul 2026  \nMechanism-level routing failure in LLMs  \nover Lean-verified algebraic structures Manuel Israel Cázares 1 Wenlin Zhang2 Haobo Ma3  \n1 Bytepro AI, Mazatlán, México  \n2 National University of Singapore / Omega Institute  \n3 ChronoAI / Omega Institute  \nCorrespondence: [hello@bytepro.ai](hello@bytepro.ai), [e1327962@u.nus.edu](e1327962@u.nus.edu), [aloning@gmail.com](aloning@gmail.com)  \nAbstract  \nWe present an empirical study of structural routing failure in large language models (LLMs) over a formally verified algebraic corpus. The task requires selecting the correct proof-mechanism label—from a fixed closed template set—for compact mathematical objects drawn from the FiberRing formalization in Lean 4 [6], where each evaluation item is anchored to a Lean-verified artifact and assigned a proof-mechanism label from the corresponding named certificate family.  \nOur central finding is a mechanism-level routing ceiling: under blind conditions (no structural cue), gpt-oss-120b achieves 80.3% template accuracy on 22 clean-anchor FiberRing items (n = 66 evaluations; temperature = 0, seed = 0, reasoning = low), while Llama 3.3 70Breaches 68.2% . Exposing a mechanism-bearing Lean verdict / witness cue drawn from the formalization (Condition A2) raises accuracy to 90.9% and 81.8% respectively—a gap of +10 .6 and +13 .6 percentage points that we term cue-induced routing uplift.  \nThe dominant failure mode is a CRT → ring-equivalence misroute: gpt-oss-120b misroutes 7 of 12 CRT items (58 .3%) in the blind condition, and zero in the cue-conditioned condition. We show this error has a direct correlate in the Lean corpus: every CRT row is presented as a ring equivalence at the outer type level, while the finer mechanism is the coprime Fibonacci-modulus factorization followed by ZMod .chineseRemainder. The model identifies the broad algebraic type but fails to resolve the finer proof mechanism without the cue.  \nTwo cue-resistant failure modes persist in both conditions across both models:  \n• stable-value-map → ring-structure  \n• prime-power-fiber-specialization → ring-equivalence  \nThese errors identify two different boundary phenomena in the Lean formalization. The stable-value-map → ring-structure error is a clean formal distinction with strong surfacelanguage overlap: map-preservation rows mention addition, multiplication, zero, one, and ring homomorphism, but their proof role is transport through toZMod, not construction of the ring instance on Xm . The prime-power-fiber-specialization → ring-equivalence error is a true granularity boundary: X4  Z/8Z is broadly a ring equivalence, but the intended mechanism label is the direct prime-power specialization of the general stable-value equivalence, with no nontrivial coprime CRT split.  \nA cross-model dissociation in Llama 3.3 70B is notable: verdict accuracy is identical in both conditions (95 .5%), while template accuracy improves 13.6 pp with the cue. This confirms that truth inference and proof-mechanism classification are separable capacities—consistent with the Lean formalization, where a statement’s truth value and its certificate family are encoded at different layers.  \nA cross-corpus extension (Set B; 6 POM/CollisionKernel items, 19 labels, 72 additional evaluations) provides a small cross-module check: the CRT-granularity compression reappears  \nwith entirely different labels (CRT-budget-bound → CRT is cue-resistant), and an inverse crossmodel dissociation emerges—gpt-oss-120b shows verdict uplift with template accuracy flat at 100.0%, while Llama shows template uplift (+27 .8 pp) with verdict flat—providing additional evidence that truth inference and mechanism classification are separable capacities.  \nThese findings extend the router hypothesis introduced in Cázares (2026) [1] from equational reasoning to formal algebraic structures: LLMs act as structural classifiers routing over broad mathematical types, and fail specifically at the","cbCaitoFG3Ww5e2J","https://ap.wps.com/l/cbCaitoFG3Ww5e2J","pdf",436363,1,11,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"Do cue-resistant failure modes still occur with the cue?\",\"answer\":\"Yes. Two persistent errors are stable-value-map → ring-structure and prime-power-fiber-specialization → ring-equivalence, indicating boundary granularity issues in the Lean formalization.\"}]",1784183216,28,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":27},"mechanism-level-routing-failure-in-llms-over-lean-verified-algebraic-structures","",{"@graph":35,"@context":77},[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/mechanism-level-routing-failure-in-llms-over-lean-verified-algebraic-structures/82824/",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],{"name":72,"@type":73,"acceptedAnswer":74},"Do cue-resistant failure modes still occur with the cue?","Question",{"text":75,"@type":76},"Yes. Two persistent errors are stable-value-map → ring-structure and prime-power-fiber-specialization → ring-equivalence, indicating boundary granularity issues in the Lean formalization.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]