[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86434-en":3,"doc-seo-86434-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86434,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Feedback Coupled Memory Systems in Continuous Time","Feedback-Coupled Memory Systems (FCMS) formalizes closed-loop coordination using four operators, with two—an agent update operator and an environmental update operator—originally left axiomatically unspecified. The work resolves this by defining the agent update via Mechanism-Based Intelligence (MBI) using a decentralized price mechanism and economic incentives, and defining the environment update via the Coupled Memory Graph Process (CMGP), a non-Markovian physical memory substrate. The resulting continuous-time FCMS instance proves Lyapunov global dissipativity under a computable stability threshold, generalizing discrete FCMS and CMGP bifurcation conditions, and validated through numerical simulation and mean-field checks.","arXiv :2607 .097 14v 1 [ cs .AI] 24 Jun 2026  \nFeedback-Coupled Memory Systems in Continuous Time  \nStefano Grassia∗  \na Bangkok University, Phahonyothin Rd, Khlong Nueng, Khlong Luang  \nDistrict, Pathum Thani 12120, Thailand  \nJune 24, 2026  \nAbstract  \nThe Feedback-Coupled Memory Systems (FCMS) architecture formalizes closedloop coordination through four abstract operators, two of which—the agent update operator 􀔕􀖄 and the environmental update operator Ψ—are left axiomatically undefined in the original framework. To address this, 􀔕􀖄 is defined by Mechanism-Based Intelligence (MBI), where agents update locally through a decentralized price mechanism and economic principles, and Ψ is defined by the Coupled Memory Graph Process (CMGP), a non-Markovian framework where the environment is treated as a physical substrate that records and responds to trajectory history coherently without external forcing. The resulting continuous-time FCMS instantiation achieves Lyapunov global dissipativity governed by the computable threshold 4􁅬 2 \u003C 2􁅱􁅷􁅭 2 . This generalizes both the discrete FCMS stability condition 4􁅱􁅬 2 \u003C 􁅭 and CMGP’s physical bifurcation threshold 􁅫 􀕾 = 1/􀔀, confirming that memory dissipation must outpace feedback gain as a universal organizing principle. Numerical simulation with 􀔃 = 2 agents and mean-field validation at 􀔃 = 106 confirm the stability threshold and the self-reinforcing coordination cascade that emerges when it is violated.  \nKeywords: feedback-coupled memory systems, continuous-time coordination, Lyapunov dissipativity, Hopf bifurcation, mechanism design, graph Laplacian dynamics, memory engine, early warning signals  \n1 Introduction  \nDistributed agents interacting through persistent environments must achieve collective order without centralized control, a challenge that lies at the intersection of economics, dynamical systems theory, and multi-agent artificial intelligence [Hayek, 1945] . This work extends FCMS [Grassi, 2026] to continuous time by providing explicit functional forms for its two axiomatically defined operators 􀔕􀖄 and Ψ . FCMS is a closed-loop dynamical architecture formalizing this feedback loop through four operators 􀷠 , Φ , 􀔕􀖄 , Ψ, reviewed in Section 2. FCMS  \n∗ Corresponding author: [stefano.g@bu.ac.th](stefano.g@bu.ac.th)  \nproves that under dissipativity the system admits a bounded forward-invariant region, coordination cannot be reduced to static optimization, and bidirectional coupling is necessary. However, the operators 􀔕􀖄 and Ψ are defined axiomatically in FCMS leaving their continuoustime instantiation an open problem. To close this gap, this paper proposes to instantiate them with two frameworks from the literature: Mechanism-Based Intelligence (MBI)[Grassi, 2025] and the Coupled Memory Graph Process (CMGP) [Sarkar, 2025a,b] . MBI is a mechanism-design framework where the Differentiable Price Mechanism (DPM) computes incentives as a Vickrey–Clarke–Groves (VCG)-equivalent signal [Vickrey, 1961 , Clarke, 1971 , Groves, 1973] guaranteeing Dominant Strategy Incentive Compatibility (DSIC) and convergence [Hurwicz and Reiter, 2006] . MBI’s discrete gradient update x􀖄,􀖏+1 = x􀖄,􀖏 + 􁅱􀓼􀖄,􀖏 , where 􀓼􀖄 = −∇x􀕎 ℒglobal is the DPM incentive signal, is the natural candidate for the continuous-time instantiation of 􀔕􀖄 , becoming the gradient flow 􀖄 = −􁅱∇x􀕎 ℒ􀖄 + 􁅹􀖄 in the continuous limit. CMGP is a physics graph framework where memory-driven feedback generates coherence without external forcing [Sarkar, 2025a,b] . CMGP’s continuous memory field 􁆉􀖏 􀔈(r, 􀔣) = −􁅫􀖎 􀔈 + 􀓶 ∫0􀖏 Θ􀖎 (􀔣 − 􁅽 )􀓼 􁇐 (r − r(􁅽 )) 􀔓􁅽 , governing how a physical substrate records and responds to trajectory history, is the natural candidate for the continuous-time instantiation of Ψ, translated to the discrete multi-agent graph setting as evolving edge weights 􀖄􀖅 = −􁅭􀔦􀖄􀖅 + 􁆁(x􀖄 , x􀖅 )􀶭 􀖄􀖅 . Taken together, the continuous-time FCMS yields a fully specified closed-loop system whose global stability is governed by the computable thresh","cbCaiuhiEtGlcVVH","https://ap.wps.com/l/cbCaiuhiEtGlcVVH","pdf",745407,5,1,20,"English","en",105,"# Introduction\n# Theoretical Background\n## MBI\n## CMGP\n## FCMS","[{\"question\":\"What problem does the paper address in the FCMS framework?\",\"answer\":\"It addresses that two FCMS operators are left axiomatically undefined, so the paper provides explicit continuous-time instantiations for both the agent update and environmental update operators.\"},{\"question\":\"How is the agent update operator defined in the continuous-time model?\",\"answer\":\"The agent update is defined using Mechanism-Based Intelligence via a decentralized Differentiable Price Mechanism, yielding an incentive signal that becomes a gradient-flow form in the continuous limit.\"},{\"question\":\"What determines the system’s stability in continuous time?\",\"answer\":\"Global stability and Lyapunov global dissipativity are governed by a computable threshold condition stating that memory dissipation must outpace feedback gain; this threshold generalizes both discrete FCMS and CMGP bifurcation criteria.\"}]",1784211727,50,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"feedback-coupled-memory-systems-in-continuous-time","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/feedback-coupled-memory-systems-in-continuous-time/86434/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the paper address in the FCMS framework?","Question",{"text":76,"@type":77},"It addresses that two FCMS operators are left axiomatically undefined, so the paper provides explicit continuous-time instantiations for both the agent update and environmental update operators.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the agent update operator defined in the continuous-time model?",{"text":81,"@type":77},"The agent update is defined using Mechanism-Based Intelligence via a decentralized Differentiable Price Mechanism, yielding an incentive signal that becomes a gradient-flow form in the continuous limit.",{"name":83,"@type":74,"acceptedAnswer":84},"What determines the system’s stability in continuous time?",{"text":85,"@type":77},"Global stability and Lyapunov global dissipativity are governed by a computable threshold condition stating that memory dissipation must outpace feedback gain; 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