[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-328148-105":3,"doc-detail-328148-en":85,"detail-sidebar-cat-0-en-105":103},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":84},105,"en","the-rank-one-corner-how-much-value-equivalence-does-a-task-need-from-a-world-model","The Rank-One Corner - How Much Value Equivalence Does a Task Need from a World Model?","","A learned world model is commonly evaluated by reconstruction fidelity or reward prediction, but tasks rely only on a small subset of predictive coordinates we call closure. The study shows that the fraction of closure represented by a latent is governed by the dimensionality of the objective used for training, not by model capacity or observations. Using a DreamerV3 setup with known ground-truth closure, scalar value equivalence installs only a 1D projection, recovering R2 from 0.10 to 0.76 when the full objective is used. Sweeping objective dimensionality from one to four installs matching closure directions, and the same rank pattern appears in the model’s own value head, with capacity-matched checks excluding alternatives. A companion closed-loop task marks the regime boundary: reconstruction alone yields task structure, and the scalar objective suffices when cheaper signals cannot recover it, making value equivalence dimensional rather than all-or-nothing.",{"@graph":14,"@context":77},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/the-rank-one-corner-how-much-value-equivalence-does-a-task-need-from-a-world-model/328148/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/the-rank-one-corner-how-much-value-equivalence-does-a-task-need-from-a-world-model/328148.png","ImageObject",300,407,{"name":42,"@type":43},"Theodora","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-04","2026-09-21",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69,73],{"name":60,"@type":61,"acceptedAnswer":62},"What does the paper mean by a world model’s “closure”?","Question",{"text":63,"@type":64},"Closure is the small set of predictive coordinates that a task’s queries depend on and that evolve together under dynamics. It is narrower than reconstructing all observations or predicting reward.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What determines how much closure a latent represents?",{"text":68,"@type":64},"The dimensionality of the training objective determines the represented fraction of closure. The paper finds this is not driven by model capacity or by the observations used for training.",{"name":70,"@type":61,"acceptedAnswer":71},"How does changing value equivalence from scalar to full objectives affect closure recovery?",{"text":72,"@type":64},"Using an aligned scalar (single-reward) objective recovers only a small fraction of closure (R2 ≈ 0.10), while using the full multi-dimensional objective recovers much more (R2 ≈ 0.76). The recoverable structure increases as objective dimensionality grows.",{"name":74,"@type":61,"acceptedAnswer":75},"Does the “rank-one corner” behavior depend on how the prediction head is implemented?",{"text":76,"@type":64},"No. The paper reports the same staircase-shaped dependence on objective dimensionality when training with an auxiliary linear probe versus using the model’s own value head, indicating the effect is dimensional rather than an artifact of head form.","https://schema.org",{"og:url":32,"og:type":79,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":81,"canonical":32},"index,follow",{"doc_id":83,"site_id":7},328148,1790827763,{"code":4,"msg":86,"data":87},"success",{"doc_id":83,"user_id":88,"nickname":42,"user_avatar":89,"doc_module":4,"category_id":90,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":91,"file_id":92,"file_url":93,"file_type":94,"file_size":95,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":96,"language":97,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":98,"faqs":99,"seo_title":100,"seo_description":12,"update_tm":101,"read_time":102,"preview_page_count":4},687197207919,"https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"arXiv :2607 .06640v 1 [ cs .LG] 7 Jul 2026  \nThe Rank-One Corner:  \nHow Much Value Equivalence Does a Task Need from  \na World Model?  \nDonna Vakalis  \nMila – Quebec AI Institute  \nUniversité de Montréal  \nMontréal, Québec  \n[donna.vakalis@olympian.org](donna.vakalis@olympian.org)  \nAbstract  \nA learned world model is usually judged by how faithfully it reconstructs its observations or predicts reward, as though quality were something the model simply has or lacks. But what a task actually needs from a model is narrower:  \nthe few predictive coordinates its queries depend on, which we call the closure.  \nWe show that how much of that closure a latent comes to represent is set not by the model’s capacity or its observations but by the dimensionality of the objective it is trained against, and we measure this directly on a DreamerV3 stack in a controlled environment whose ground-truth closure is known. An aligned scalar value signal—the objective at the heart of value equivalence—installs only a onedimensional projection of a closure that needs several dimensions: read through a single linear probe, the recoverable structure rises from R2 = 0 .10 to 0.76 as the scalar is replaced by the full objective. Sweeping the objective’s dimensionality from one to four installs exactly that many predictive directions through an auxiliary regression head, and the same staircase appears—at attenuated magnitude but the same rank—through the model’s own value head, so the dissociation is dimensional rather than an artifact of head form. Capacity-matched comparisons and in-situ pressure checks rule out the obvious alternatives. The law governs a regime, and we measure its boundary: on a companion closed-loop task whose structure is directly observable frame by frame, reconstruction installs that structure and the scalar objective suffices—the objective decides what a latent represents precisely where cheaper training signals cannot already recover it. Value equivalence is therefore not all-or-nothing but dimensional: the familiar single-reward objective is its rank-one corner, and a model installs as much of a task’s structure as the objective it is asked to predict.  \n1 Introduction  \nA learned world model is typically trained to reconstruct what it observes or to predict the reward it receives, and its quality is then reported as a single number: a reconstruction error, or a return under planning. This treats fidelity as something a model either has or lacks. But a model that reconstructsits inputs perfectly can still be useless for control, and a model that predicts reward well can fail to support the longer-horizon reasoning a task demands. What a world model should capture is not its observations in general but the structure its downstream queries actually depend on—the small set of collective coordinates that carry those queries and evolve together under the dynamics. We call this set the queries’ closure, and we take model quality to be how much of the closure a trained latent comes to represent. Ask where the Moon will be at midnight: the closure is six slowly drifting orbital elements, no matter how many pixels of night sky we record.  \nPreprint.  \nValue equivalence is dimensional  \nsingle-reward value equivalence (d = 1) is the insufficient rank-1 corner of a graded law  \na  \nc losure insta l led R 2 (L ← z)  \n0.8  \n0.6  \n0.4  \n0.2  \n0.0  \nsame latent, same linear probe  \n\n| 0.765\u003Cbr>\u003Cbr> sufficiency floor |  |\n| --- | --- |\n|  | the rank-1\u003Cbr>corner\u003Cbr>0.10\u003Cbr>|\n\nsingle-reward VE (d = 1)  \nfull objective (d = k)  \nb  \ninstalled c losure rank  \n4  \n3  \n2  \n1  \ninstalled rank = objective dimensionality  \n1 2 3 4 objective dimensionality d  \nFigure 1: Value equivalence is dimensional. Left: read through one linear probe, an aligned scalar (single-reward) objective installs a fraction 0.10 of the query closure, while the full objective installs 0.76 in the same configuration. Right: sweeping the objective’s target dimensionality installs","cbCaipBqOu3qh8qY","https://ap.wps.com/l/cbCaipBqOu3qh8qY","pdf",718516,22,"English","# Introduction\n## Value equivalence is dimensional","[{\"question\":\"What does the paper mean by a world model’s “closure”?\",\"answer\":\"Closure is the small set of predictive coordinates that a task’s queries depend on and that evolve together under dynamics. It is narrower than reconstructing all observations or predicting reward.\"},{\"question\":\"What determines how much closure a latent represents?\",\"answer\":\"The dimensionality of the training objective determines the represented fraction of closure. The paper finds this is not driven by model capacity or by the observations used for training.\"},{\"question\":\"How does changing value equivalence from scalar to full objectives affect closure recovery?\",\"answer\":\"Using an aligned scalar (single-reward) objective recovers only a small fraction of closure (R2 ≈ 0.10), while using the full multi-dimensional objective recovers much more (R2 ≈ 0.76). The recoverable structure increases as objective dimensionality grows.\"},{\"question\":\"Does the “rank-one corner” behavior depend on how the prediction head is implemented?\",\"answer\":\"No. The paper reports the same staircase-shaped dependence on objective dimensionality when training with an auxiliary linear probe versus using the model’s own value head, indicating the effect is dimensional rather than an artifact of head form.\"}]","The Rank-One Corner - How Much Value Equivalence Does a Task Need from a World Model? | PDF",1789978743,55,{"code":4,"msg":86,"data":104},[105,109,113,117,122,127,131,134,139,142,146],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":114,"show_sort_weight":115,"slug":116},"Exam",70,"exam",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":119,"show_sort_weight":120,"slug":121},5,"Comic",60,"comic",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":125,"slug":126},6,"Technology",50,"technology",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":129,"slug":130},"Healthcare",40,"healthcare",{"id":90,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":132,"slug":133},30,"research-report",{"id":135,"doc_module":4,"doc_module_name":25,"category_name":136,"show_sort_weight":137,"slug":138},9,"Religion & Spirituality",20,"religion-spirituality",{"id":137,"doc_module":4,"doc_module_name":25,"category_name":140,"show_sort_weight":137,"slug":141},"World Cup","world-cup",{"id":143,"doc_module":4,"doc_module_name":25,"category_name":144,"show_sort_weight":143,"slug":145},10,"Lifestyle","lifestyle",{"id":147,"doc_module":4,"doc_module_name":25,"category_name":148,"show_sort_weight":118,"slug":149},19,"General","general"]