{ "model": { "rank": 18198, "code": "thing-q867707", "model_id": "vr.tr.special-function", "name": "special function", "purpose": "Let an agent explain special functions as a category, identify major families and their origins, relay reference sources and computational methods, and distinguish special functions from elementary functions.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q867707/", "tier": 2, "score": 30, "payload": { "layer": "wikidata", "aliases": [ "spherical Hankel function", "associated Legendre polynomials", "modified Bessel function", "Scorer's function", "trigonometric integral", "elliptic integral", "theta function", "log sine integral", "Legendre function", "Wigner 3-j symbol", "Hankel function", "Weierstrass functions", "Whittaker function", "Airy function", "Chebyshev polynomial", "spin-weighted spherical harmonics", "spinor spherical harmonics", "vector spherical harmonics", "Clebsch–Gordan coefficient" ], "aliasCount": 19, "merged": 19, "knownIn": 30, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q867707", "url": "https://www.wikidata.org/wiki/Q867707", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T01:51:10Z", "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": 3, "engine": "claude" }, "metaModel": { "id": "THING-Q867707", "registryId": "vr.tr.special-function", "name": "special function", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q867707/", "model": { "registry_id": "vr.tr.special-function", "name": "special function", "purpose": "Let an agent explain special functions as a category, identify major families and their origins, relay reference sources and computational methods, and distinguish special functions from elementary functions.", "definition": "A mathematical function with an established name and notation because of its importance in analysis, physics and applied mathematics, beyond the elementary functions, such as the gamma and beta functions, Bessel and modified Bessel functions, spherical Hankel functions, Legendre and associated Legendre polynomials, elliptic integrals, the error function, trigonometric integrals, Airy and Scorer functions and hypergeometric functions; special functions arise as solutions of differential equations and integrals and are tabulated in standard references.", "what_it_is_for": "Solving equations and integrals in science and engineering.", "affordances": [ "explain the category", "identify major families", "relay references and computation", "distinguish from elementary functions" ], "distinguishing_features": [ "Named and tabulated", "Differential equation origins", "Physical applications", "Extensive identities" ], "appearance": "Not a visible object; named functions with graphs and tables.", "visual_identification": [ "Named non-elementary functions", "Gamma, Bessel, Legendre, elliptic, error, Airy, hypergeometric", "Elementary functions are built from polynomials, exponentials and trigonometry" ], "physical_properties": [], "families_and_kinds": [ "gamma, beta and related functions", "Bessel, modified Bessel and Hankel functions including spherical forms", "orthogonal polynomials such as Legendre and associated Legendre", "elliptic integrals and functions", "error function and trigonometric integrals", "Airy and Scorer functions", "hypergeometric functions", "zeta and related functions" ], "related_models": [ { "relation": "is a kind of", "target": "function", "why": "in registry terms" }, { "relation": "is contrasted with", "target": "elementary function", "why": "the simpler class" }, { "relation": "arises from", "target": "differential equation", "why": "in physics" }, { "relation": "is catalogued in", "target": "NIST Digital Library of Mathematical Functions", "why": "the standard reference" } ], "identifiers": [ { "scheme": "DLMF", "value_or_pattern": "chapters", "note": "NIST reference" } ], "standards_and_regulation": [ "No regulation; notation conventions from the DLMF and ISO 80000-2" ], "failure_modes_and_hazards": [ "Notation and normalisation inconsistencies", "Numerical instability in computation", "Confusing families" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What special functions are.", "rationale": "Mathematics.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition and scope.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What makes a function special, and how does the category relate to elementary functions?", "kind": "definition" }, { "text": "Is the question about special functions in general or a specific family?", "kind": "boundary" } ] } ] }, { "id": "families", "name": "Families", "description": "Major families.", "findings": [ { "id": "families-finding", "name": "Families", "description": "Families.", "questions": [ { "text": "What are the major families, and where does each arise?", "kind": "definition" }, { "text": "Which entry fits the specific function?", "kind": "action" } ] } ] } ] }, { "id": "use", "name": "Use", "description": "Applications.", "rationale": "Application.", "layers": [ { "id": "physics", "name": "Physics", "description": "Physics and engineering.", "findings": [ { "id": "physics-finding", "name": "Physics", "description": "Physics.", "questions": [ { "text": "How do Bessel, Legendre, Airy and hypergeometric functions arise in physics and engineering problems?", "kind": "provenance" }, { "text": "Which entry fits the specific application?", "kind": "action" } ] } ] }, { "id": "statistics", "name": "Statistics", "description": "Statistics and probability.", "findings": [ { "id": "statistics-finding", "name": "Statistics", "description": "Statistics.", "questions": [ { "text": "How do the gamma, beta and error functions appear in statistics?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] } ] }, { "id": "compute", "name": "Compute", "description": "Computation and references.", "rationale": "Practice.", "layers": [ { "id": "computation", "name": "Computation", "description": "Numerical computation.", "findings": [ { "id": "computation-finding", "name": "Computation", "description": "Computation.", "questions": [ { "text": "How are special functions computed reliably in software libraries?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] }, { "id": "references", "name": "References", "description": "Reference works.", "findings": [ { "id": "references-finding", "name": "References", "description": "References.", "questions": [ { "text": "What are the standard references, from Abramowitz and Stegun to the DLMF?", "kind": "provenance" }, { "text": "Which entry fits the DLMF?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "History and theory.", "rationale": "Context.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did special functions develop from Euler and Bessel to the twentieth century?", "kind": "provenance" }, { "text": "Which entry fits the history of analysis?", "kind": "action" } ] } ] }, { "id": "theory", "name": "Theory", "description": "Unifying theory.", "findings": [ { "id": "theory-finding", "name": "Theory", "description": "Theory.", "questions": [ { "text": "How do hypergeometric functions and group theory unify many special functions?", "kind": "provenance" }, { "text": "Which entry fits hypergeometric function?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should each family be a separate entry?", "How should the DLMF be linked?", "The registry entry has merged aliases naming specific functions; should they be split off?" ], "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 special function, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain special functions as a category, identify major families and their origins, relay reference sources and computational methods, and distinguish special functions from elementary functions.", "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: spherical Hankel function, associated Legendre polynomials, modified Bessel function, Scorer's function, trigonometric integral, elliptic integral, theta function, log sine integral, Legendre function, Wigner 3-j symbol, Hankel function, Weierstrass functions.", "19 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 30 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 special function, 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": [ "spherical Hankel function", "associated Legendre polynomials", "modified Bessel function", "Scorer's function", "trigonometric integral", "elliptic integral", "theta function", "log sine integral", "Legendre function", "Wigner 3-j symbol", "Hankel function", "Weierstrass functions" ] } ], "standing": "Described in 30 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 19 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 special function from another, and a special function 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 special function, 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 special function, 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 special function 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 special function, 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 special function, 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 special function 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 special function 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 special function, 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 special function, 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 special function 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 special function from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a special function, 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 special function, 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 special function 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 special function 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 special function between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a special function, 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 special function, 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 special function 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 special function, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a special function 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 special function, 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 special function, 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 special function 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 special function 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 special function 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 special function 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 special function 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 special function?", "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 special function?", "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 special function 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 special function and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a special function, 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 special function, 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 special function 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 special function is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a special function, 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 special function, 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 special function 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 } } } }