[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-328134-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-328134-en":130},{"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":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","cross-trajectory-chimera-interventions-reveal-dissociable-roles-of-weight-magnitude-and-direction-in-grokking","Cross-Trajectory Chimera Interventions Reveal Dissociable Roles of Weight Magnitude and Direction in Grokking","","Which properties of a partially trained neural network transfer causally to a separately trained network, and which do not, is addressed through cross-trajectory chimera interventions. Two networks trained from different seeds are combined by decomposing weights into norm (magnitude) and direction, then recombining magnitude from one run with direction from the other. Across modular-arithmetic grokking tasks, direction transfer drives donor-specific circuit identity while norm mainly affects grokking delay. Interpolating directions along geodesics yields a threshold-like identity switch predicted by recipient norm class, localized via an adaptive bisection measurement tool.",{"@graph":14,"@context":73},[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/cross-trajectory-chimera-interventions-reveal-dissociable-roles-of-weight-magnitude-and-direction-in-grokking/328134/",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/cross-trajectory-chimera-interventions-reveal-dissociable-roles-of-weight-magnitude-and-direction-in-grokking/328134.png","ImageObject",300,407,{"name":42,"@type":43},"Theodora","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-16","2026-09-21",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",8,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What question do cross-trajectory chimera interventions answer that within-trajectory interventions cannot?","Question",{"text":63,"@type":64},"They test causal portability: whether a property from a partially trained checkpoint can determine outcomes when transplanted into a differently trained run, not just within a single run.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How is the chimera constructed from two independently trained networks?",{"text":68,"@type":64},"Each weight vector is decomposed into magnitude (norm) and a unit direction; training then continues after recombining the norm from one run with the direction from the other.",{"name":70,"@type":61,"acceptedAnswer":71},"What do the results show about the roles of weight direction versus weight norm in grokking?",{"text":72,"@type":64},"Direction transfers a donor-specific component tied to final circuit spectral identity, while norm carries only a modest, spatially distributed effect on grokking delay without measurable identity information.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},328134,1790567316,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":97,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":55,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":112,"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","arXiv :2607 .06628v 1 [ cs .LG] 7 Jul 2026  \nCross-Trajectory Chimera Interventions Reveal Dissociable Roles of Weight Magnitude and Direction in Grokking  \nTruong Xuan Khanh  \nH&K Research Studio, Clevix LLC, Hanoi, Vietnam  \n[khanh@clevix. vn](khanh@clevix. vn)  \nJuly 9, 2026  \nAbstract  \nWhich properties of a partially trained network are causally portable to a different, independently trained network, and which are not? Interventions on a single training trajectory cannot answer this:  \nthey can show that a property is necessary for an outcome within one run, but not whether it is portable across runs. We introduce cross-trajectory chimera interventions: given two networks trained from different random seeds on the same task, we decompose each weight vector into a magnitude (norm) and a direction (unit vector), recombine the magnitude of one run with the direction of another to forma chimera, and continue training. Across two modular-arithmetic tasks that exhibit grokking, we find a clean dissociation. The direction carries a transferable component associated with the final circuit’s spectral identity: implanting a donor’s direction while keeping the recipient’s norm drives the continued run toward the donor’s eventual circuit in 40/40 independent recombinations, and this transfer is specific to the donor’s content rather than to the mere angular displacement (an angle-matched random control produces no such shift) . Interpolating the direction along the geodesic between two runs reveals that this transfer is threshold-like rather than a continuous blend. Most notably, cross-trajectory intervention reveals that transferability itself depends on the recipient’s dynamical state: the interpolation threshold at which identity flips from recipient-like to donor-like is predicted by the recipient’s weight norm, separating perfectly by norm class across all 20 pairs and both tasks (joint exact permutation probability 1.9×10−4) . We localize the threshold to a resolution of ±1/64 using an adaptive bisection procedure that we contribute as a reusable measurement tool for stability-under-protocol interventions. By contrast, the norm carries only a modest, spatially distributed effect on grokking delay and no measurable identity information. Together these results map a concrete geometric division of labour: direction indexes which solution a trajectory approaches, while norm governs how susceptible that identity is to being overwritten.  \n1 Introduction  \nGrokking—the delayed onset of generalization long after a network has fit its training data—has become a controlled setting for asking how neural networks transition from memorization to structured solutions [Power et al., 2022, Nanda et al., 2023] . A large body of recent work intervenes on a single training run to probe this transition: freezing, rescaling, or projecting weights along a trajectory and observing the effect on delay or on the emerging circuit [Nanda et al., 2023, Varma et al., 2023] . Such within-trajectory interventions, however, cannot answer a distinct question: of everything a partially trained checkpoint contains, which parts are portable—able to determine an outcome when transplanted into a different, independently trained run? Within-trajectory interventions can establish that a property is necessary for an outcome inside one run; they cannot establish whether that property is portable—able to produce the outcome when placed in a different run’s context. Distinguishing these two questions is our organizing concern throughout the paper: we do not ask why direction encodes circuit identity, but whether it is causally portable, and if so, under what conditions.  \nWe study this question through a decomposition of the weight vector into a radial part (its norm) and an angular part (its unit direction), θ = r · u. This decomposition itself is not new: the interplay of weight normand direction in grokking has been analyzed within single trajectories [Liu et al., 20","cbCaimlLTo7N7Ji1","https://ap.wps.com/l/cbCaimlLTo7N7Ji1","pdf",455250,12,"English","# Abstract\n# 1 Introduction\n## Problem framing: portability vs within-trajectory necessity\n## Weight decomposition into radial and angular parts\n## Cross-trajectory causal probe via chimera recombination\n## Key findings: direction identity vs norm delay; threshold-like transfer","[{\"question\":\"What question do cross-trajectory chimera interventions answer that within-trajectory interventions cannot?\",\"answer\":\"They test causal portability: whether a property from a partially trained checkpoint can determine outcomes when transplanted into a differently trained run, not just within a single run.\"},{\"question\":\"How is the chimera constructed from two independently trained networks?\",\"answer\":\"Each weight vector is decomposed into magnitude (norm) and a unit direction; training then continues after recombining the norm from one run with the direction from the other.\"},{\"question\":\"What do the results show about the roles of weight direction versus weight norm in grokking?\",\"answer\":\"Direction transfers a donor-specific component tied to final circuit spectral identity, while norm carries only a modest, spatially distributed effect on grokking delay without measurable identity information.\"}]","Cross-Trajectory Chimera Interventions Reveal Dissociable Roles of Weight Magnitude and Direction in Grokking | PDF",1789978722]