[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83434-en":3,"doc-seo-83434-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},83434,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","A Mathematical Theory of Value: A Synthesis on Goal-Directed Agency Under Resource Constraints","Proposes value as a lawful structural quantity created, destroyed, and exchanged by goal-directed agents, in the same formal category as information once stripped of moral, price, and psychological semantics. Using Shannon’s method, defines value as conversion rate from physical resources to goal-progress relative to an agent-fixed goal frame, yielding a logarithmic law via scale invariance. Establishes a coding bound ∆G ≤ I(X;Y) and an exact decomposition into potential minus dissipation to quantify misalignment and waste.","arXiv :2606 . 12502v2 [physics .soc-ph] 3 Jul 2026  \nA Mathematical Theory of Value  \na synthesis on goal-directed agency under resource constraints  \nCheng Qian  \n2026  \nAbstract  \nWe propose that value—the quantity that goal-directed agents create, destroy, and exchange— is a lawful structural quantity, in the same category as information, once stripped of its semantic clothing (morality, price, psychology) . Following the method of Shannon (1948), we make one ruthless abstraction: value is the rate at which an agent converts a physical resource into goal-progress, relative to a frame fixed by the agent’s goal. A scale-invariance axiom forces a logarithmic measure of value, V = Pi ki ln ei , via a Cauchy functional equation; the compounding dynamics of a reinvested resource force the same form independently via the ergodicity argument of Peters (2019) . The two routes are kin rather than independent—the scale-invariance axiom is the static shadow of multiplicative compounding—so their agreement is a consistency check on the form, not an independent over-determination. From the compounding dynamics we also derive a coding theorem of value: the rate at which an agent can create value through a perception channel Y of the world X is bounded by the mutual information, ∆G ≤ I (X;Y ), achieved by Bayes-proportional allocation; and realized value decomposes exactly as available potential minus dissipation, G = D (q ∥r) − D(q ∥p), identifying misalignment with measurable waste. For populations we show value is frame-relative while price is frame-independent, that a fleet which pools its resource and fuses its perception is one agent and so inherits the capacity ceiling, Gfleet ≤ I (X;Y1:m) ≤ H (X) (a corollary; an earlier sum-form claim was wrong and is corrected in v5), and that the fleet’s operating point is a Kelly portfolio over agents selected by an emergent price. A dynamical layer gives the equations of motion and an is/ought asymmetry—beliefs have a target the world supplies, goals do not—from which alignment emerges as a control-stability condition with a closed-form residual misalignment. We then test the single-frame laws on live language models in a pre-registered scale-up across three task domains and a ten-model, five-family ladder (0 .5B–8B): perception mutual information tracks realized capability rather than parameter count (Spearman ρ = 0 .977 pooled over 30 model×domain points), out-of-sample ∆G tracks I (X;Y ) (slope CI excludes 0), and over-confidence is measurable dissipation in every domain—the two single-frame laws generalize. A further pre-registered test shows the out-of-sample bridge ∆G ∼ I (X;Y ) is shape-invariant: pooled across four qualitatively different task shapes—classification, reasoning (GSM8K), sequential decision, and code (MBPP), all with discrete gold so I is computed, not estimated (n = 42)—the slope is 0 .953, statistically indistinguishable from the classification-only value, promoting the bridge from a demonstration toward a law. A fleet-pricing experiment is reported scoped and primary-metric-first: valuepricing recovers cost-aware routing from first principles—it ties good hand-tuned routing under a token budget, beats a cost-blind router under a compute budget, and does not outperform a cost-aware engineer; its contribution is principled measurement, not outperformance. The paper’s stated continuation gate has since been run (pre-registered, on a frontier-model population): the coupled capacity-region prediction—the growth-gap law, coalition submodularity with a designed XOR synergy control, the joint ceiling, and Kelly selection—is confirmed within its frozen bands on real agents, the first real-agent confirmation of a prediction no component theory makes separately; the mean-field residual-scaling law ∥Vg∥/γ, by contrast, found no domain —  \ncapable populations hold no goal dispersion (V → 0)—and is retired to its mathematical scope.  \nThe laws hold in the smooth, concave (diminishin","cbCaikQPCfjmITV6","https://ap.wps.com/l/cbCaikQPCfjmITV6","pdf",390485,1,20,"English","en",105,"# Introduction\n# Statics: the measure, the limit, the Second Law\n## The measure: a logarithmic law of value\n## The capacity theorem: value is bounded by information\n## The Second Law of Value\n# Multi-agent: price and the fleet\n## Cross-frame value and the frame-independence of price\n## The fleet capacity region\n# Dynamics: motion and alignment\n## The equations of motion and the is/ought asymmetry\n## Alignment as a stability condition\n# Evidence\n## Synthetic validation\n## Real agents\n## Generalization: a pre-registered scale-up","[{\"question\":\"How does the paper define value for goal-directed agents under resource constraints?\",\"answer\":\"Value is defined as the rate at which an agent converts a physical resource into goal-progress relative to a frame fixed by its goal. This structural definition is treated as analogous to information once semantic dressing is removed.\"},{\"question\":\"What are the main theoretical results connecting value and information?\",\"answer\":\"A coding theorem bounds realized value increase by mutual information, with ∆G ≤ I(X;Y) achieved by Bayes-proportional allocation. Value also decomposes exactly into potential minus dissipation, identifying misalignment with measurable waste.\"},{\"question\":\"How does the paper test and validate the proposed laws?\",\"answer\":\"It reports pre-registered experiments on live language models, including a scale-up across multiple domains and model sizes, and an out-of-sample bridge test across qualitatively different task shapes. Results show that mutual information tracks realized capability and that dissipation corresponds to over-confidence, supporting generalization.\"}]",1784187693,50,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"a-mathematical-theory-of-value-a-synthesis-on-goal-directed-agency-under-resource-constraints","",{"@graph":35,"@context":84},[36,53,67],{"@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/a-mathematical-theory-of-value-a-synthesis-on-goal-directed-agency-under-resource-constraints/83434/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"How does the paper define value for goal-directed agents under resource constraints?","Question",{"text":74,"@type":75},"Value is defined as the rate at which an agent converts a physical resource into goal-progress relative to a frame fixed by its goal. This structural definition is treated as analogous to information once semantic dressing is removed.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What are the main theoretical results connecting value and information?",{"text":79,"@type":75},"A coding theorem bounds realized value increase by mutual information, with ∆G ≤ I(X;Y) achieved by Bayes-proportional allocation. Value also decomposes exactly into potential minus dissipation, identifying misalignment with measurable waste.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the paper test and validate the proposed laws?",{"text":83,"@type":75},"It reports pre-registered experiments on live language models, including a scale-up across multiple domains and model sizes, and an out-of-sample bridge test across qualitatively different task shapes. Results show that mutual information tracks realized capability and that dissipation corresponds to over-confidence, supporting generalization.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,113,118,121,125,128,132],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":28,"slug":112},6,"Technology","technology",{"id":114,"doc_module":4,"doc_module_name":45,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":119,"slug":120},30,"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":21,"slug":124},9,"Religion & Spirituality","religion-spirituality",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":126,"show_sort_weight":21,"slug":127},"World Cup","world-cup",{"id":129,"doc_module":4,"doc_module_name":45,"category_name":130,"show_sort_weight":129,"slug":131},10,"Lifestyle","lifestyle",{"id":133,"doc_module":4,"doc_module_name":45,"category_name":134,"show_sort_weight":105,"slug":135},19,"General","general"]