{ "model": { "rank": 12018, "code": "thing-q215084", "model_id": "vr.tr.analytic-function", "name": "analytic function", "purpose": "Let an agent explain analytic functions, relay the power-series definition, real and complex cases, properties and examples from mathematics sources, describe the named examples and note which are not analytic, and distinguish analytic functions from merely smooth functions, continuous functions, holomorphic functions as the complex case and polynomials.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q215084/", "tier": 2, "score": 40, "payload": { "layer": "wikidata", "aliases": [ "Fresnel integral", "step function", "pluriharmonic function", "algebraic function", "boxcar function", "real analytic function", "Softplus", "Blaschke product", "Nash functions", "transcendental function", "hypertranscendental function" ], "aliasCount": 11, "merged": 11, "knownIn": 40, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q215084", "url": "https://www.wikidata.org/wiki/Q215084", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-14T02:27:14Z", "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": 4, "engine": "claude" }, "metaModel": { "id": "THING-Q215084", "registryId": "vr.tr.analytic-function", "name": "analytic function", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q215084/", "model": { "registry_id": "vr.tr.analytic-function", "name": "analytic function", "purpose": "Let an agent explain analytic functions, relay the power-series definition, real and complex cases, properties and examples from mathematics sources, describe the named examples and note which are not analytic, and distinguish analytic functions from merely smooth functions, continuous functions, holomorphic functions as the complex case and polynomials.", "definition": "In mathematics, an analytic function is one that is locally given by a convergent power series, equal to its own Taylor series near each point, a strong smoothness property; real analytic functions and complex analytic or holomorphic functions have powerful properties, and the registry aliases name related and contrasting examples such as algebraic functions, pluriharmonic functions, and functions that are notably not analytic everywhere, such as the step function, the boxcar function and the Fresnel integral. Analytic functions are central to complex analysis, differential equations and physics.", "what_it_is_for": "Functions locally equal to their power series.", "affordances": [ "explain the definition", "relay real and complex cases", "describe named examples", "distinguish related classes" ], "distinguishing_features": [ "Power-series representation", "Stronger than smooth", "Real and complex cases", "Rigid behaviour" ], "appearance": "Not a physical object; functions and their graphs and series.", "visual_identification": [ "Function locally equal to its power series", "Fresnel integral, step function, pluriharmonic function, algebraic function, boxcar function, real analytic function", "Smooth functions have all derivatives but need not be analytic; continuous functions need not be differentiable; holomorphic functions are complex analytic; polynomials are a simple case" ], "physical_properties": [ { "quantity": "definition", "typical_range": "locally a convergent power series", "unit": "note", "note": "" }, { "quantity": "smooth but not analytic", "typical_range": "exist", "unit": "note", "note": "example bump functions" }, { "quantity": "registry parent", "typical_range": "smooth function", "unit": "note", "note": "" } ], "families_and_kinds": [ "real analytic functions", "complex analytic or holomorphic functions", "entire functions", "algebraic functions" ], "related_models": [ { "relation": "is a kind of", "target": "smooth function", "why": "in registry terms" }, { "relation": "is contrasted with", "target": "smooth function", "why": "not all smooth are analytic" }, { "relation": "includes", "target": "holomorphic function", "why": "complex case" }, { "relation": "is contrasted with", "target": "continuous function", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "Not applicable" ], "failure_modes_and_hazards": [ "Assuming all smooth functions are analytic", "Confusing real and complex analyticity", "Treating step functions as analytic" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What an analytic function is.", "rationale": "Science.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is an analytic function, and how does it differ from merely smooth functions, continuous functions, holomorphic functions and polynomials?", "kind": "definition" }, { "text": "Is the question about real or complex analysis?", "kind": "boundary" } ] } ] }, { "id": "examples", "name": "Examples", "description": "Named examples.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Examples.", "questions": [ { "text": "Which of algebraic functions, the Fresnel integral, step and boxcar functions are analytic, and which are not?", "kind": "definition" }, { "text": "Which entry fits the specific function?", "kind": "action" } ] } ] } ] }, { "id": "theory", "name": "Theory", "description": "Theory.", "rationale": "Science.", "layers": [ { "id": "powerseries", "name": "Power series", "description": "Power series.", "findings": [ { "id": "powerseries-finding", "name": "Power series", "description": "Power series.", "questions": [ { "text": "How does the power-series condition define analyticity?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "smoothvsanalytic", "name": "Smooth versus analytic", "description": "Smooth versus analytic.", "findings": [ { "id": "smoothvsanalytic-finding", "name": "Smooth versus analytic", "description": "Smooth versus analytic.", "questions": [ { "text": "Why are some smooth functions not analytic?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "complex", "name": "Complex", "description": "Complex analysis.", "rationale": "Science.", "layers": [ { "id": "holomorphic", "name": "Holomorphic", "description": "Holomorphic functions.", "findings": [ { "id": "holomorphic-finding", "name": "Holomorphic", "description": "Holomorphic.", "questions": [ { "text": "Why are complex differentiable functions automatically analytic?", "kind": "provenance" }, { "text": "Which entry fits holomorphic function?", "kind": "action" } ] } ] }, { "id": "continuation", "name": "Continuation", "description": "Analytic continuation.", "findings": [ { "id": "continuation-finding", "name": "Continuation", "description": "Continuation.", "questions": [ { "text": "What is analytic continuation?", "kind": "provenance" }, { "text": "Is the information accurate?", "kind": "boundary" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Applications.", "rationale": "Science.", "layers": [ { "id": "applications", "name": "Applications", "description": "Applications.", "findings": [ { "id": "applications-finding", "name": "Applications", "description": "Applications.", "questions": [ { "text": "How are analytic functions used in physics and differential equations?", "kind": "provenance" }, { "text": "Is the presentation neutral and attributed?", "kind": "boundary" } ] } ] }, { "id": "nonanalytic", "name": "Non-analytic", "description": "Non-analytic functions.", "findings": [ { "id": "nonanalytic-finding", "name": "Non-analytic", "description": "Non-analytic.", "questions": [ { "text": "Why are step and boxcar functions not analytic?", "kind": "provenance" }, { "text": "Which entry fits step function?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should holomorphic functions be a separate entry?", "How should non-analytic examples be documented?", "How should analytic continuation 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 analytic function, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain analytic functions, relay the power-series definition, real and complex cases, properties and examples from mathematics sources, describe the named examples and note which are not analytic, and distinguish analytic functions from merely smooth functions, continuous functions, holomorphic functions as the complex case and polynomials.", "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: Fresnel integral, step function, pluriharmonic function, algebraic function, boxcar function, real analytic function, Softplus, Blaschke product, Nash functions, transcendental function, hypertranscendental function.", "11 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 40 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 analytic 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": [ "Fresnel integral", "step function", "pluriharmonic function", "algebraic function", "boxcar function", "real analytic function", "Softplus", "Blaschke product", "Nash functions", "transcendental function", "hypertranscendental function" ] } ], "standing": "Described in 40 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 11 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 analytic function from another, and a analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic function from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a analytic 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 analytic 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 analytic 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 analytic 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 analytic function between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a analytic 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 analytic 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 analytic 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 analytic function, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic 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 analytic function and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a analytic 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 analytic 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 analytic 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 analytic function is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a analytic 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 analytic 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 analytic 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 } } } }