{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.convergence","code":"thing-q1287023","url":"https://ver.cy/models/thing/q1287023/","name":"convergence","alternateNames":["weak convergence","normal convergence","compact convergence","convergence of measures","Delta-convergence","Epi-convergence","Almost convergent sequence","Flat convergence","uniform absolute-convergence","Gromov–Hausdorff convergence","weak convergence of measures"],"kind":"thing","status":"published","version":"0.2.0-wave.4","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"In mathematics, the property of a sequence, series, function or process approaching a limit, with many notions the registry aliases name including weak convergence in normed spaces and of probability measures, convergence of measures more generally, normal convergence of function series, compact convergence or uniform convergence on compact sets, Delta-convergence in metric spaces and epi-convergence used in optimisation and variational analysis; other important modes include pointwise, uniform, almost sure, in probability and in distribution convergence, and the relationships between modes, such as uniform convergence implying pointwise convergence but not conversely, are central to analysis and probability.","purpose":"Let an agent explain convergence, relay major modes and their relationships from analysis and probability sources, describe the named notions, and distinguish mathematical convergence from divergence, limits as values, Cauchy sequences as a criterion and convergence in other senses such as technological or economic convergence.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Many distinct modes","Implications between modes","Topology-dependent","Foundation of analysis"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What convergence is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is convergence in mathematics, and how does it differ from divergence, limits, Cauchy sequences and non-mathematical convergence?","kind":"definition"},{"text":"Is the question about a specific mode of convergence or homework-level series tests?","kind":"boundary"}]}]},{"id":"notions","name":"Notions","description":"Named notions.","findings":[{"id":"notions-finding","name":"Notions","description":"Notions.","questions":[{"text":"What are weak, normal, compact, Delta and epi-convergence and convergence of measures?","kind":"definition"},{"text":"Which entry fits the specific notion?","kind":"action"}]}]}]},{"id":"analysis","name":"Analysis","description":"Analysis.","layers":[{"id":"uniform","name":"Uniform","description":"Pointwise and uniform.","findings":[{"id":"uniform-finding","name":"Uniform","description":"Uniform.","questions":[{"text":"How do pointwise and uniform convergence differ, and why does it matter for continuity?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"series","name":"Series","description":"Series tests.","findings":[{"id":"series-finding","name":"Series","description":"Series.","questions":[{"text":"What tests determine whether a series converges?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"probability","name":"Probability","description":"Probability.","layers":[{"id":"modes","name":"Modes","description":"Probabilistic modes.","findings":[{"id":"modes-finding","name":"Modes","description":"Modes.","questions":[{"text":"How do almost sure, in probability and in distribution convergence relate?","kind":"provenance"},{"text":"Which entry fits convergence of random variables?","kind":"action"}]}]},{"id":"weak","name":"Weak convergence","description":"Weak convergence of measures.","findings":[{"id":"weak-finding","name":"Weak convergence","description":"Weak convergence.","questions":[{"text":"What is weak convergence of probability measures?","kind":"provenance"},{"text":"Is the information accurate?","kind":"boundary"}]}]}]},{"id":"context","name":"Context","description":"History.","layers":[{"id":"history","name":"History","description":"Rigour in analysis.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Cauchy and Weierstrass make convergence rigorous, as historians describe?","kind":"provenance"},{"text":"Is the presentation neutral and attributed?","kind":"boundary"}]}]},{"id":"optimisation","name":"Optimisation","description":"Epi-convergence.","findings":[{"id":"optimisation-finding","name":"Optimisation","description":"Optimisation.","questions":[{"text":"Why is epi-convergence used for minimisation problems?","kind":"provenance"},{"text":"Which entry fits variational analysis?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Explain notions of convergence and the differences between them with correct definitions.","Check whether a stated convergence result has the hypotheses it needs.","Flag when a numerical method's apparent convergence has not been proven."],"mustNot":["Conflate modes of convergence, such as pointwise and uniform, in a proof or explanation.","Swap limits, sums or integrals without stating the justification.","Present numerical evidence as proof that a series or algorithm converges.","Attribute a theorem to the wrong person or misstate its conditions."],"requiresHuman":["Relying on a convergence claim for a safety-critical calculation or engineering result."]},"ethics":{"considerations":["Errors about convergence in engineering and finance can lead to unsafe or costly decisions.","Teaching wrong definitions misleads students who trust the source."],"affectedParties":["Students and learners","Engineers relying on numerical results","Researchers"]},"owners":{"steward":"Nobody owns the concept; it is held in common by mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q3924032","type":"parent","note":"registry parent class"},{"target":"mathematical property","type":"related","note":"in registry terms"},{"target":"limit of a sequence","type":"related"},{"target":"uniform convergence","type":"related"},{"target":"divergent series","type":"related"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.convergence","Wikidata: Q1287023 (https://www.wikidata.org/wiki/Q1287023)"]},"properties":{"applicability":"not-applicable","items":["Cauchy s Cours d analyse: 1821 year - rigorous limits","uniform convergence concept: 1840s note - Seidel, Stokes, Weierstrass","registry parent: mathematical property note"]},"recognition":{"applicability":"optional","items":["Approach to a limit","Weak convergence, normal convergence, compact convergence, convergence of measures, Delta-convergence, epi-convergence","Divergence is failure to converge; limits are the values approached; Cauchy sequences give a criterion; other convergence senses are non-mathematical","Not a physical object; expressed in definitions, theorems and graphs approaching a value."]},"capabilities":{"applicability":"required","items":["explain the concept","relay modes of convergence","describe named notions","relay relationships between modes"]},"hazards":{"applicability":"required","items":["Confusing modes of convergence","Swapping limits without justification","Confusing mathematical and everyday senses"]},"interfaces":{"applicability":"required","items":["Not applicable"]},"context":{"applicability":"required","items":["Describing approach to limits in mathematics.","convergence of sequences and series","convergence of functions","convergence in probability theory","convergence of measures","variational convergence"]}},"sources":[{"title":"Wikidata item Q1287023: convergence","url":"https://www.wikidata.org/wiki/Q1287023","note":"identity and sense of the item"},{"title":"Wikipedia: Convergence (mathematics)","url":"https://en.wikipedia.org/wiki/Convergence_%28mathematics%29","note":"general description of the item"}],"openQuestions":["Should each mode be a separate entry?","Should epi-convergence be moved to optimisation entries?","How should mathematical reference works be linked?"],"resources":{"spec":"/models/things/publications/thing-q1287023/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1287023/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-14T01:02:47Z","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 2026-10-06T19:51:54+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}}