{ "model": { "rank": 15793, "code": "thing-q579600", "model_id": "vr.tr.functional", "name": "functional", "purpose": "Let an agent explain functionals and their kinds, relay their role in functional analysis, the calculus of variations and physics from mathematics references, describe the constructions the registry aliases name, and distinguish functionals from functions, operators and higher-order functions in programming.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q579600/", "tier": 2, "score": 33, "payload": { "layer": "wikidata", "aliases": [ "modulus of continuity", "integral functional", "blossom", "Energy functional", "entanglement witness" ], "aliasCount": 5, "merged": 5, "knownIn": 33, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q579600", "url": "https://www.wikidata.org/wiki/Q579600", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T18:08:38Z", "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-Q579600", "registryId": "vr.tr.functional", "name": "functional", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q579600/", "model": { "registry_id": "vr.tr.functional", "name": "functional", "purpose": "Let an agent explain functionals and their kinds, relay their role in functional analysis, the calculus of variations and physics from mathematics references, describe the constructions the registry aliases name, and distinguish functionals from functions, operators and higher-order functions in programming.", "definition": "In mathematics, a map from a space of functions or vectors to its field of scalars, as in linear functionals of functional analysis, integral functionals such as the action and energy functionals of physics that are minimised in the calculus of variations, and the modulus of continuity as a functional measuring uniform continuity; related constructions include the blossom, a symmetric multi-affine map associated with a polynomial in spline theory, and entanglement witnesses, functionals used to detect quantum entanglement.", "what_it_is_for": "Not applicable; a mathematical object.", "affordances": [ "explain the concept", "relay roles in analysis and physics", "describe named constructions", "distinguish related concepts" ], "distinguishing_features": [ "Scalar output", "Function inputs", "Linear and nonlinear", "Variational use" ], "appearance": "Not a visible object; a mapping.", "visual_identification": [ "Map from functions or vectors to scalars", "Integral functional, energy functional, modulus of continuity, blossom, entanglement witness", "Functions map numbers; operators map functions to functions; higher-order functions are programming constructs" ], "physical_properties": [ { "quantity": "Riesz representation theorem", "typical_range": "1907-1909", "unit": "note", "note": "linear functionals on Hilbert spaces" }, { "quantity": "calculus of variations", "typical_range": "17th-18th centuries", "unit": "note", "note": "Euler, Lagrange" }, { "quantity": "dual space", "typical_range": "set of continuous linear functionals", "unit": "note", "note": "" } ], "families_and_kinds": [ "linear functionals and dual spaces", "integral functionals", "energy and action functionals", "functionals measuring regularity such as the modulus of continuity", "blossoms in spline theory", "entanglement witnesses in quantum information" ], "related_models": [ { "relation": "is a kind of", "target": "function", "why": "in registry terms" }, { "relation": "is studied in", "target": "functional analysis", "why": "" }, { "relation": "is minimised in", "target": "calculus of variations", "why": "" }, { "relation": "is contrasted with", "target": "operator", "why": "which returns functions" } ], "identifiers": [], "standards_and_regulation": [ "No regulation" ], "failure_modes_and_hazards": [ "Confusing functionals with operators", "Registry aliases naming loosely related constructions", "Assuming linearity" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What a functional is.", "rationale": "Science.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a functional, and how does it differ from a function, operator and programming higher-order function?", "kind": "definition" }, { "text": "Is the question about analysis, physics, splines or quantum information?", "kind": "boundary" } ] } ] }, { "id": "examples", "name": "Examples", "description": "Named constructions.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Examples.", "questions": [ { "text": "What are integral and energy functionals, the modulus of continuity, blossoms and entanglement witnesses?", "kind": "definition" }, { "text": "Which entry fits the specific construction?", "kind": "action" } ] } ] } ] }, { "id": "analysis", "name": "Analysis", "description": "Functional analysis.", "rationale": "Science.", "layers": [ { "id": "linear", "name": "Linear", "description": "Linear functionals.", "findings": [ { "id": "linear-finding", "name": "Linear", "description": "Linear.", "questions": [ { "text": "What are linear functionals, dual spaces and the Riesz and Hahn-Banach theorems?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "variations", "name": "Variations", "description": "Calculus of variations.", "findings": [ { "id": "variations-finding", "name": "Variations", "description": "Variations.", "questions": [ { "text": "How are functionals minimised with Euler-Lagrange equations?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "applications", "name": "Applications", "description": "Applications.", "rationale": "Application.", "layers": [ { "id": "physics", "name": "Physics", "description": "Physics.", "findings": [ { "id": "physics-finding", "name": "Physics", "description": "Physics.", "questions": [ { "text": "How do action and energy functionals and density functional theory use functionals?", "kind": "provenance" }, { "text": "Which entry fits density functional theory?", "kind": "action" } ] } ] }, { "id": "quantum", "name": "Quantum", "description": "Quantum information.", "findings": [ { "id": "quantum-finding", "name": "Quantum", "description": "Quantum.", "questions": [ { "text": "How do entanglement witnesses detect entanglement?", "kind": "provenance" }, { "text": "Which entry fits entanglement witness?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Splines and history.", "rationale": "Context.", "layers": [ { "id": "splines", "name": "Splines", "description": "Blossoms.", "findings": [ { "id": "splines-finding", "name": "Splines", "description": "Splines.", "questions": [ { "text": "What is the blossoming principle in spline theory?", "kind": "provenance" }, { "text": "Which entry fits blossom (functional)?", "kind": "action" } ] } ] }, { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did Volterra, Hadamard and others develop functionals?", "kind": "provenance" }, { "text": "Which entry fits the history of functional analysis?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should blossom and entanglement witness be separate primary entries?", "How should mathematics references be linked?", "The registry entry has merged aliases naming constructions from different fields; 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 functional, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain functionals and their kinds, relay their role in functional analysis, the calculus of variations and physics from mathematics references, describe the constructions the registry aliases name, and distinguish functionals from functions, operators and higher-order functions in programming.", "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: modulus of continuity, integral functional, blossom, Energy functional, entanglement witness.", "5 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 33 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 functional, 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": [ "modulus of continuity", "integral functional", "blossom", "Energy functional", "entanglement witness" ] } ], "standing": "Described in 33 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 5 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 functional from another, and a functional 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 functional, 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 functional, 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 functional 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 functional, 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 functional, 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 functional 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 functional 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 functional, 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 functional, 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 functional 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 functional from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a functional, 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 functional, 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 functional 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 functional 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 functional between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a functional, 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 functional, 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 functional 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 functional, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a functional 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 functional, 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 functional, 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 functional 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 functional 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 functional 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 functional 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 functional 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 functional?", "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 functional?", "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 functional 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 functional and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a functional, 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 functional, 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 functional 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 functional is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a functional, 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 functional, 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 functional 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 } } } }