[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86129-en":3,"doc-seo-86129-105":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},86129,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Energy Calculus: A Compositional Algebra of Energy in Computational Systems","Energy is a binding constraint for AI scaling but lacks a formal calculus comparable to computation, communication, and learning. Existing systems optimize specific granularities, making cross-system reasoning about total energy difficult because energy is treated as a monolithic scalar measured after the fact. Energy calculus introduces energy elements and energy signatures, composable via sequential and parallel operators over DAG executions, grounded in seven axioms with uncertainty propagation and a reduction theorem to context-independence.","arXiv :2607 . 11087v1 [ cs .DC] 13 Jul 2026  \nThe ML. ENERGY Initiative  \nEnergy Calculus  \nA Compositional Algebra of Energy in Computational Systems  \nMosharaf Chowdhury∗ , Jae-Won Chung∗ , Jeff J. Ma, Nishil Talati, Ruofan Wu  \nEnergy is a binding constraint for AI scaling, yet it lacks the formal treatment that computation, communication, and learning have long enjoyed. Recent systems demonstrate large energy savings, but each targets a specific granularity and structure; one cannot combine frequency scaling from one system with critical-path analysis from another and reason about their joint effect on total energy. Energy remains a monolithic scalar that is measured after the fact and optimized with point solutions that do not generalize.  \nWe propose energy calculus, a compositional algebra that treats energy as a first-class primitive. It buildson energy elements, units of computation whose energy we can reliably measure, each carrying an energy signature that comprises its time, its static and dynamic energy, the hardware operating point and execution context under which we measured it, and the associated measurement uncertainty. Three operators (sequential, same-device parallel, and cross-device parallel) compose signatures along the same structure as the computation itself, covering arbitrary DAG-structured executions. The algebra rests on seven axioms that capture how hardware consumes energy, and it exhibits two properties distinctive to energy among computing resources: sequential composition commutes only when elements are mutually contextinsensitive, and sequential composition does not distribute over parallel composition. We also present a Reduction Theorem that recovers simple context-independent algebra whenever interactions fall below measurement uncertainty, so practitioners pay for context dependence only where the physics demands it. Uncertainty propagates through every composition, so each prediction carries an error bound. Finally, we show that the same operators extend from energy totals to time–energy Pareto frontiers, so reasoning about tradeoffs composes with the same algebra.  \nAuthors listed alphabetically. ∗ Equal contribution. Correspondence: [calculus@ml.energy](calculus@ml.energy)  \nContents  \n1 Introduction 4  \n2 Background 5  \n2.1 Energy in Computational Systems ................................... 5  \n2.2 Computation as Structured Composition ............................... 6  \n2.3 Measurement Uncertainty ........................................ 6  \n2.4 Composing the Energy of a Transformer Attention Layer ....................... 6  \n3 Energy Calculus: Foundations 7  \n3.1 Energy Elements and Context Variables ................................ 8  \n3.2 Energy Signatures ............................................ 8  \n3.3 Composing Energy Elements ...................................... 10  \n3.4 Interaction Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12  \n4 Reduction to Context-Independence 13  \n4.1 Context-Insensitivity ........................................... 13  \n4.2 The Reduction Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15  \n4.3 The Context-Independent Form .................................... 16  \n4.4 The Interaction Graph .......................................... 16  \n4. 5 Axioms and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17  \n5 Properties of Energy Calculus 18  \n5.1 Algebraic Properties ........................................... 18  \n5.2 Uncertainty and Measurement ..................................... 21  \n6 Application: Operating Frontiers Under Context-Independence 22  \n6.1 Operating Set and Frontier ....................................... 23  \n6.2 Frontier Composition .......................................... 23  \n7 Discussion 25  \n7. 1 What the Framework Covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ","cbCailw83wcqk5g0","https://ap.wps.com/l/cbCailw83wcqk5g0","pdf",1166840,4,1,40,"English","en",105,"# Introduction\n# Background\n## Energy in Computational Systems\n## Computation as Structured Composition\n## Measurement Uncertainty\n## Composing the Energy of a Transformer Attention Layer\n# Energy Calculus: Foundations\n## Energy Elements and Context Variables\n## Energy Signatures\n## Composing Energy Elements\n## Interaction Mechanisms\n# Reduction to Context-Independence\n## Context-Insensitivity\n## The Reduction Theorem\n## The Context-Independent Form\n## The Interaction Graph\n## Axioms and Notation\n# Properties of Energy Calculus\n## Algebraic Properties\n## Uncertainty and Measurement\n# Application: Operating Frontiers Under Context-Independence\n## Operating Set and Frontier\n## Frontier Composition\n# Discussion\n## What the Framework Covers\n## What the Framework Does Not Cover\n## Open Problems\n# Related Work\n# Conclusion","[{\"question\":\"Why is existing AI energy optimization hard to combine across systems?\",\"answer\":\"Each system typically targets a specific granularity and structure, so frequency scaling from one and critical-path analysis from another cannot be combined reliably to reason about total energy. Energy is often treated as a monolithic scalar measured after the fact, with point optimizations that do not generalize.\"},{\"question\":\"What is the core idea of energy calculus?\",\"answer\":\"Energy calculus treats energy as a first-class primitive using energy elements and energy signatures. Each element records time, static and dynamic energy, the hardware operating point and execution context, and the associated measurement uncertainty, which can be composed using sequential and parallel operators over DAG-structured executions.\"},{\"question\":\"How does energy calculus handle context dependence and uncertainty?\",\"answer\":\"The framework is based on seven axioms describing how hardware consumes energy and distinguishes properties unique to energy, including non-commutativity under context dependence and lack of distributivity over parallel composition. Uncertainty propagates through every composition, and a Reduction Theorem recovers simpler context-independent algebra when interactions fall below measurement uncertainty.\"}]",1784208728,101,{"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},"energy-calculus-a-compositional-algebra-of-energy-in-computational-systems","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/energy-calculus-a-compositional-algebra-of-energy-in-computational-systems/86129/",{"url":52,"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","2026-07-27","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},"Why is existing AI energy optimization hard to combine across systems?","Question",{"text":75,"@type":76},"Each system typically targets a specific granularity and structure, so frequency scaling from one and critical-path analysis from another cannot be combined reliably to reason about total energy. Energy is often treated as a monolithic scalar measured after the fact, with point optimizations that do not generalize.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the core idea of energy calculus?",{"text":80,"@type":76},"Energy calculus treats energy as a first-class primitive using energy elements and energy signatures. Each element records time, static and dynamic energy, the hardware operating point and execution context, and the associated measurement uncertainty, which can be composed using sequential and parallel operators over DAG-structured executions.",{"name":82,"@type":73,"acceptedAnswer":83},"How does energy calculus handle context dependence and uncertainty?",{"text":84,"@type":76},"The framework is based on seven axioms describing how hardware consumes energy and distinguishes properties unique to energy, including non-commutativity under context dependence and lack of distributivity over parallel composition. Uncertainty propagates through every composition, and a Reduction Theorem recovers simpler context-independent algebra when interactions fall below measurement uncertainty.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":22,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]