[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86498-en":3,"doc-seo-86498-105":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},86498,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Laguerre Geometry for Interpreting Large Language Models","Existing hypotheses treat an LLM concept as a point, a linear direction, or a Gaussian cluster, leaving the emergence mechanism and measurement unclear. This work characterizes concept geometry using Laguerre Geometry, defining a concept as a Laguerre-Voronoi cell (or union of cells) to strictly define, measure, and separate concepts. Laguerre weights reveal inclusion and hierarchy. The geometry is integrated into transformer dynamics via a coupled piecewise-linear decomposition that enables Geometric Lens for exact concept reading and Laguerre Autoencoder for visual reasoning.","arXiv :2607 . 10578v 1 [ cs .AI] 12 Jul 2026  \nLaguerre Geometry for Interpreting Large Language Models  \nA PREPRINT  \nChunwei Ma* 1 and Russell D. Wolfinger†1  \n1JMP Statistical Discovery, Cary, NC, USA  \nJuly 14, 2026  \nABSTRACT  \nExisting hypotheses represent a concept in an LLM as a single point, a linear direction, or a Gaussian cluster, yet it remains unclear how and why such structures emerge. Here, we show that concept geometry can be precisely characterized via Laguerre Geometry, in which a concept is defined asa region—a Laguerre-Voronoi cell or a union of cells—allowing us to strictly define, measure, and separate concepts. Building on this formulation, we show that finer-grained concept structures, such as inclusion and hierarchy, are naturally revealed by the Laguerre weights. We then push this geometry inside the transformer. Decomposing each layer into piecewise-linear operators, we show that a token’s hidden trajectory is governed by two coupled mechanisms: a static tree of self-contained piecewise-linear flow, and a dynamic transport that hops the trajectory across trees when cross-token attention fires. This decomposition yields Geometric Lens, a training-free, hyperparameter-free method for reading out the exact concept a hidden vector encodes at any layer. We also develop Laguerre Autoencoder, a 2D visualizer that renders both the decision geometry and a model’s full reasoning trajectory in one view. Finally, we move beyond explanatory geometry toward actionable interpretability, showing that Geometric Lens recovers the correct factual token when a model is prompted with in-context interference. The code is available on GitHub♯ .  \n1 Introduction  \nLarge Language Models (LLMs) have achieved remarkable breakthroughs not only in general question-answering and conversation, but also in coding [Li et al., 2022, Chen et al., 2021], mathematical reasoning [Romera-Paredeset al., 2024], multimodal (image, audio, and video) generation [Wu et al., 2023], and scientific discovery [Ghareebet al., 2026, Aygün et al., 2026] . Yet, although every atomic internal computation within an LLM is ontologically well understood, the reason these computations, when composed, give rise to a high degree of intelligence remains largely epistemologically opaque—warranting deeper scientific inquiry.  \nA typical LLM is built on decoder-only transformers and trained autoregressively, with the model optimized to correctly predict the next token given a preceding piece of text. How this simple training objective gives rise to emergent capabilities such as comprehension and reasoning remains one of the central mysteries of LLMs. In moving from next-token probability distributions to emergent capabilities, the first natural question is how concepts are organized within an LLM’s internal representations.  \nTo answer this question, an intensively studied idea is the linear representation hypothesis (LRH) [Mikolov et al., 2013a, Elhage et al., 2022, Nanda et al., 2023, Gurnee et al., 2024, Park et al., 2023]—the informal hypothesis that semantic concepts are represented linearly in the representation spaces of LLMs. LRH has been widely substantiated across various contexts, including linear probes [Hewitt & Manning, 2019, Tenney et al., 2019], vector arithmetic [Mikolovet al., 2013b, Park et al., 2023, Zou et al., 2023], intervention and activation steering [Li et al., 2023, Turner et al., 2023],  \n∗[chunwei.ma@jmp.com](chunwei.ma@jmp.com)  \n†[russ.wolfinger@jmp.com](russ.wolfinger@jmp.com)  \n♯[https://github.com/horsepurve/Geometric-Lens](https://github.com/horsepurve/Geometric-Lens)  \n(A) Oriented spheres in Laguerre Geometry (B) Hyponyms/hypernyms in Phi-2 (C) An example  \nfound by Laguerre Geometry Laguerre-Voronoi Diagram  \nsubsumption internal tangent  \nexternal tangent disjoint  \n( Laguerre we ights)  \n0.650  \n0.625  \n0.600  \n0.575  \n0.550  \n0.525  \n0.500  \n_Pets  \n_Animals  \n__bbereaestsds _mammalian  \n_leash  \n_barking_canine  \n_puppie","cbCaitPlGebRUxtH","https://ap.wps.com/l/cbCaitPlGebRUxtH","pdf",1232705,3,1,26,"English","en",105,"# Introduction\n## Linear Representation Hypothesis Branches\n## Remaining Open Questions","[{\"question\":\"How does the paper define a concept using Laguerre Geometry?\",\"answer\":\"A concept is defined as a region in representation space, specifically a Laguerre-Voronoi cell or a union of such cells, enabling strict definition, measurement, and separation of concepts.\"},{\"question\":\"What additional concept structures are revealed by Laguerre weights?\",\"answer\":\"Finer-grained concept structures such as inclusion and hierarchy emerge naturally from the Laguerre weights.\"},{\"question\":\"How does Geometric Lens read out a concept encoded in a hidden vector?\",\"answer\":\"By decomposing transformer layers into piecewise-linear operators, Geometric Lens provides a training-free, hyperparameter-free method that reads out the exact concept a hidden vector encodes at any layer, and can recover the correct factual token under in-context interference.\"}]",1784212205,66,{"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},"laguerre-geometry-for-interpreting-large-language-models","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/laguerre-geometry-for-interpreting-large-language-models/86498/",4,{"url":51,"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","2026-07-28","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},"How does the paper define a concept using Laguerre Geometry?","Question",{"text":75,"@type":76},"A concept is defined as a region in representation space, specifically a Laguerre-Voronoi cell or a union of such cells, enabling strict definition, measurement, and separation of concepts.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What additional concept structures are revealed by Laguerre weights?",{"text":80,"@type":76},"Finer-grained concept structures such as inclusion and hierarchy emerge naturally from the Laguerre weights.",{"name":82,"@type":73,"acceptedAnswer":83},"How does Geometric Lens read out a concept encoded in a hidden vector?",{"text":84,"@type":76},"By decomposing transformer layers into piecewise-linear operators, Geometric Lens provides a training-free, hyperparameter-free method that reads out the exact concept a hidden vector encodes at any layer, and can recover the correct factual token under in-context 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