{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.smooth-function","code":"thing-q868473","url":"https://ver.cy/models/thing/q868473/","name":"smooth function","alternateNames":["Schwartz function","local diffeomorphism","Lie group action","bump function","flat function","harmonic map","holonomic function","non-analytic smooth function","smooth immersion","smooth submersion","harmonic morphism"],"kind":"thing","status":"published","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A function that has continuous derivatives of all orders, also called infinitely differentiable or of class C infinity, the standard regularity assumption in differential geometry and analysis, with examples such as polynomials, exponentials and bump functions that are smooth yet vanish outside a bounded set, Schwartz functions that decay rapidly with all derivatives, and flat functions with all derivatives zero at a point; smooth maps between manifolds, local diffeomorphisms, Lie group actions and harmonic maps are built on this notion.","purpose":"Let an agent define smooth functions and related classes, relay their properties and examples from analysis and geometry references, explain their role in manifolds and physics, and distinguish smooth from continuous, differentiable and analytic functions.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Derivatives of all orders","Not necessarily analytic","Partitions of unity","Foundation of manifolds"],"structure":{"bundles":[{"id":"define","name":"Define","description":"Definition.","layers":[{"id":"definition","name":"Definition","description":"Definition and examples.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a smooth function, and what are standard examples?","kind":"definition"},{"text":"Is the function smooth, merely differentiable, continuous or analytic?","kind":"boundary"}]}]},{"id":"classes","name":"Classes","description":"Special classes.","findings":[{"id":"classes-finding","name":"Classes","description":"Classes.","questions":[{"text":"What are bump functions, Schwartz functions and flat functions, and why do they matter?","kind":"definition"},{"text":"Which entry fits the specific class?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Properties and theory.","layers":[{"id":"properties","name":"Properties","description":"Properties.","findings":[{"id":"properties-finding","name":"Properties","description":"Properties.","questions":[{"text":"What properties do smooth functions have, including Taylor expansions and why they need not be analytic?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"geometry","name":"Geometry","description":"Manifolds and maps.","findings":[{"id":"geometry-finding","name":"Geometry","description":"Geometry.","questions":[{"text":"How do smooth maps, diffeomorphisms and Lie group actions build differential geometry?","kind":"provenance"},{"text":"Which entry fits differentiable manifold?","kind":"action"}]}]}]},{"id":"apply","name":"Apply","description":"Applications.","layers":[{"id":"physics","name":"Physics","description":"Physics and PDE.","findings":[{"id":"physics-finding","name":"Physics","description":"Physics.","questions":[{"text":"How are smooth functions used in physics, distributions and partial differential equations, including harmonic maps?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]},{"id":"computation","name":"Computation","description":"Computation and approximation.","findings":[{"id":"computation-finding","name":"Computation","description":"Computation.","questions":[{"text":"How are smooth functions approximated and used in numerical methods and machine learning?","kind":"provenance"},{"text":"Which entry fits smoothing?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and terminology.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did the concept of smoothness and the C^k hierarchy develop?","kind":"provenance"},{"text":"Which entry fits the history of analysis?","kind":"action"}]}]},{"id":"terminology","name":"Terminology","description":"Terminology.","findings":[{"id":"terminology-finding","name":"Terminology","description":"Terminology.","questions":[{"text":"How does the word smooth vary in meaning across mathematical fields?","kind":"provenance"},{"text":"Which entry fits smoothness?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Define, test and use smoothness conditions in analysis and numerical work.","Explain where smoothness assumptions hold and where they fail in a model."],"mustNot":["Assume a function is smooth in a proof or model without checking it.","Hide that a numerical method's error bound depends on an unverified smoothness assumption.","Present a result from a smooth model as valid for data with shocks or kinks."],"requiresHuman":["Relying on a smooth model for decisions with safety or financial consequences."]},"ethics":{"considerations":["Smoothness assumptions in engineering and finance models fail precisely in crises.","Clear statements of assumptions let others check and reuse work."],"affectedParties":["Engineers and modellers","People affected by model-based decisions","Students"]},"owners":{"steward":"Nobody owns the concept; it belongs to shared mathematical knowledge.","roles":[],"masterSystems":[]},"relations":[{"target":"Q11855366","type":"parent","note":"registry parent class"},{"target":"Q140645","type":"parent","note":"registry parent class"},{"target":"Q91933113","type":"parent","note":"registry parent class"},{"target":"continuously differentiable function","type":"related","note":"in registry terms"},{"target":"k times continuously differentiable function","type":"related","note":"in registry terms, for all k"},{"target":"scalar function of scalar variable","type":"related","note":"in registry terms, in the one-variable case"},{"target":"analytic function","type":"related","note":"which equals its Taylor series"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.smooth-function","Wikidata: Q868473 (https://www.wikidata.org/wiki/Q868473)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Infinitely differentiable function","Bump, Schwartz, flat functions, smooth maps","Continuous functions may lack derivatives; analytic functions are smoother still; C^k functions have only k derivatives","A graph without corners or cusps for functions of one variable."]},"capabilities":{"applicability":"required","items":["define and give examples","relay properties","explain role in geometry","distinguish from related classes"]},"hazards":{"applicability":"required","items":["Confusing smooth with analytic","Assuming smoothness where only continuity holds","Conflating distinct special classes"]},"interfaces":{"applicability":"required","items":["No regulation; mathematical conventions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical class of functions.","smooth functions on the real line and Euclidean space","bump functions and partitions of unity","Schwartz functions","flat functions","smooth maps between manifolds and local diffeomorphisms","smooth Lie group actions","harmonic maps as smooth maps satisfying an equation"]}},"sources":[{"title":"Wikidata item Q868473: smooth function","url":"https://www.wikidata.org/wiki/Q868473","note":"identity and sense of the item"},{"title":"Wikipedia: Smoothness","url":"https://en.wikipedia.org/wiki/Smoothness","note":"general description of the item"}],"openQuestions":["Should bump function and Schwartz space be separate entries?","How should analysis references be linked?","The registry entry has merged aliases naming unrelated geometric objects; should they be split off?"],"resources":{"spec":"/models/things/publications/thing-q868473/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q868473/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T01:37:50Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:52:02+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}