{ "model": { "rank": 2461, "code": "thing-q80091", "model_id": "vr.tr.integral", "name": "integral", "purpose": "Let an agent explain integrals and integration techniques, compute and check integrals, interpret them in applications, and explain generalisations.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q80091/", "tier": 2, "score": 105, "payload": { "layer": "wikidata", "aliases": [ "optical path length", "absement", "air mass", "tabular integration", "Harish-Chandra integral", "Cauchy principal value", "Birkhoff integral", "Complete Fermi–Dirac integral", "definite integral", "Pettis integral", "complex integral", "partial fractions in integration", "Pearcey integral", "parametric integral", "Monte Carlo integration", "Riemann–Liouville integral", "Barnes integral", "improper integral", "line integral", "Berezin integral", "nonelementary integral", "correlation integral", "Kolmogorov integral", "surface integral", "Khinchin integral", "Selberg integral", "Shell integration", "Henstock–Kurzweil integral", "indefinite integral", "multiple integral", "Bochner integral", "area under the curve", "integral of revolution", "Frullani integral", "line integral of a scalar field", "line integral of a vector field", "contour integral", "curve integral of the second kind", "curve integral of the first kind", "surface integral of a scalar field" ], "aliasCount": 43, "merged": 45, "knownIn": 105, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q80091", "url": "https://www.wikidata.org/wiki/Q80091", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T19:59:16Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q80091", "registryId": "vr.tr.integral", "name": "integral", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q80091/", "model": { "registry_id": "vr.tr.integral", "name": "integral", "purpose": "Let an agent explain integrals and integration techniques, compute and check integrals, interpret them in applications, and explain generalisations.", "definition": "A fundamental concept of calculus assigning a number to a function over an interval or region, interpreted as area, accumulated change or total quantity, with definite and indefinite forms, and generalisations such as line, surface, multiple and Lebesgue integrals and principal values; integrals appear across science as accumulated quantities like optical path length and absement.", "what_it_is_for": "Calculus, analysis and quantitative science.", "affordances": [ "compute definite and indefinite integrals", "explain integration techniques", "interpret integrals in applications", "explain generalisations and numerical methods" ], "distinguishing_features": [ "Accumulation operation", "Inverse of differentiation", "Definite and indefinite", "Many generalisations" ], "appearance": "Not physical; the integral sign and formulas.", "visual_identification": [ "Integral sign with limits", "Area under a curve interpretation", "A derivative measures rate, an integral accumulates" ], "physical_properties": [], "families_and_kinds": [ "definite and indefinite integrals", "multiple, line and surface integrals", "Lebesgue and other measure-theoretic integrals", "improper integrals and principal values", "numerical integration" ], "related_models": [ { "relation": "is a kind of", "target": "unary operation", "why": "category" }, { "relation": "is a kind of", "target": "linear map", "why": "category" }, { "relation": "is studied in", "target": "mathematical analysis", "why": "field" }, { "relation": "is related to", "target": "limit", "why": "defined via limits" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 notation" ], "failure_modes_and_hazards": [ "Sign and bound errors", "Misapplying techniques", "Divergent integrals treated as finite" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "compute", "name": "Compute", "description": "Evaluating integrals.", "rationale": "Techniques.", "layers": [ { "id": "techniques", "name": "Techniques", "description": "Methods.", "findings": [ { "id": "techniques-finding", "name": "Techniques", "description": "Methods.", "questions": [ { "text": "Which technique, such as substitution, parts or partial fractions, evaluates this integral?", "kind": "action" }, { "text": "What is the result, and can it be checked by differentiation?", "kind": "measurement" } ] } ] }, { "id": "numerical", "name": "Numerical", "description": "Numerical integration.", "findings": [ { "id": "numerical-finding", "name": "Numerical", "description": "Numerical integration.", "questions": [ { "text": "How can this integral be approximated numerically, and with what error?", "kind": "action" }, { "text": "When is numerical integration necessary?", "kind": "boundary" } ] } ] } ] }, { "id": "meaning", "name": "Meaning", "description": "Interpretation.", "rationale": "Accumulation.", "layers": [ { "id": "interpret", "name": "Interpret", "description": "What it represents.", "findings": [ { "id": "interpret-finding", "name": "Interpret", "description": "Interpretation.", "questions": [ { "text": "What does this integral represent physically or geometrically?", "kind": "definition" }, { "text": "What are the units of the result?", "kind": "measurement" } ] } ] }, { "id": "fundamental", "name": "Fundamental", "description": "Fundamental theorem.", "findings": [ { "id": "fundamental-finding", "name": "Fundamental", "description": "Fundamental theorem.", "questions": [ { "text": "How does the fundamental theorem of calculus connect integrals and derivatives?", "kind": "definition" }, { "text": "When does it apply?", "kind": "boundary" } ] } ] } ] }, { "id": "advanced", "name": "Advanced", "description": "Generalisations.", "rationale": "Beyond Riemann.", "layers": [ { "id": "types", "name": "Types", "description": "Kinds of integral.", "findings": [ { "id": "types-finding", "name": "Types", "description": "Types.", "questions": [ { "text": "How do line, surface, multiple and Lebesgue integrals generalise the basic integral?", "kind": "definition" }, { "text": "When is each used?", "kind": "provenance" } ] } ] }, { "id": "improper", "name": "Improper", "description": "Improper integrals.", "findings": [ { "id": "improper-finding", "name": "Improper", "description": "Improper integrals.", "questions": [ { "text": "Does this improper integral converge, and what is a principal value?", "kind": "boundary" }, { "text": "Which tests apply?", "kind": "definition" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Teaching.", "rationale": "Intuition and rigour.", "layers": [ { "id": "teach", "name": "Teach", "description": "Teaching integrals.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can integration be taught with intuition and worked examples?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] }, { "id": "applications", "name": "Applications", "description": "Uses in science.", "findings": [ { "id": "applications-finding", "name": "Applications", "description": "Applications.", "questions": [ { "text": "How are integrals used for quantities such as work, probability and optical path length?", "kind": "provenance" }, { "text": "Which examples help?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should each integral type be a separate entry?", "How should computer algebra tools be linked?", "How should applications be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a integral, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain integrals and integration techniques, compute and check integrals, interpret them in applications, and explain generalisations.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: optical path length, absement, air mass, tabular integration, Harish-Chandra integral, Cauchy principal value, Birkhoff integral, Complete Fermi–Dirac integral, definite integral, Pettis integral, complex integral, partial fractions in integration.", "45 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 105 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a integral, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "optical path length", "absement", "air mass", "tabular integration", "Harish-Chandra integral", "Cauchy principal value", "Birkhoff integral", "Complete Fermi–Dirac integral", "definite integral", "Pettis integral", "complex integral", "partial fractions in integration" ] } ], "standing": "Described in 105 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 45 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one integral from another, and a integral from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a integral, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a integral, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a integral is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a integral, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a integral, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a integral is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a integral sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a integral, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a integral, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a integral is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a integral from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a integral, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a integral, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a integral is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a integral passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a integral between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a integral, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a integral, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a integral is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a integral, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a integral is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a integral, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a integral, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a integral is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What integral is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of integral are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is integral operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of integral and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of integral?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as integral?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a integral came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a integral and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a integral, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a integral, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a integral is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a integral is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a integral, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a integral, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a integral is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }