{ "model": { "rank": 6138, "code": "thing-q190549", "model_id": "vr.tr.functional-analysis", "name": "functional analysis", "purpose": "Enable an AI agent to recognise a functional-analysis problem, record the hypotheses that govern its mathematical objects, and judge which deductions or operations are justified.", "family": "Thing Registry", "category": "Information and virtual systems", "status": "research-draft", "kind": "thing", "plane": "INF", "domain": "INF.KNW", "industry": "", "version": "", "url": "/models/thing/q190549/", "tier": 2, "score": 62, "payload": { "layer": "wikidata", "aliases": [ "operator theory", "Compression as functional analysis", "nonlinear functional analysis", "uniform algebra" ], "aliasCount": 4, "merged": 4, "knownIn": 62, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q190549", "url": "https://www.wikidata.org/wiki/Q190549", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-07T11:44:15Z", "providers": [ "Codex", "Grok" ], "missingProviders": [], "pass": 2, "cost": { "grok": { "usd": 0.00342176, "turns": 1, "seconds": 44, "error": null }, "codex": { "seconds": 74, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.3\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a07bad-c961-78e1-af54-28bb8d34f293\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: functional analysis\nRegistry id: vr.tr.functional-analysis\nPlane / domain: INF / INF.KNW\nRegistry definition: (none recorded)\nNames folded into this " } } }, "metaModel": { "id": "THING-Q190549", "registryId": "vr.tr.functional-analysis", "name": "functional analysis", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "INF.KNW" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q190549/", "model": { "registry_id": "vr.tr.functional-analysis", "name": "functional analysis", "purpose": "Enable an AI agent to recognise a functional-analysis problem, record the hypotheses that govern its mathematical objects, and judge which deductions or operations are justified.", "definition": "Functional analysis is the branch of mathematics that studies vector spaces of functions together with linear operators on those spaces, typically in the setting of Banach and Hilbert spaces, in order to treat infinite-dimensional problems arising from analysis, differential equations, and operator theory.", "scope_statement": "This model provisionally treats functional analysis as the mathematical study of topological vector spaces and operators, owning the relationships among space structure, convergence, duality and operator behaviour that determine whether an argument applies.", "in_scope": [ "Vector spaces equipped with norms, inner products or compatible vector-space topologies", "Completeness, convergence and compactness conditions relevant to arguments on these spaces", "Continuous linear functionals, dual spaces and duality relationships", "Linear operators, their domains and their continuity, closure and spectral properties", "The hypotheses, conclusions and applicability boundaries of functional-analytic results" ], "out_of_scope": [ "Functional decomposition of products, organisations or engineering systems", "Behavioural functional analysis of antecedents and consequences", "Finite-dimensional matrix computations that do not require functional-analytic structure", "Standalone construction of measures and integrals", "Domain-specific differential equations, physical systems and optimisation objectives", "Software implementation and benchmarking of numerical solvers" ], "distinguishing_features": [ "The entry must concern mathematical spaces and maps; describing what a system does indicates a different meaning of functional analysis.", "An argument depends on a specified topology, norm or inner product, rather than vector-space algebra alone.", "Questions about limits, continuity or compactness identify the topology in which those properties are asserted.", "Operator claims identify the source space, target space and domain, rather than treating every operator as an everywhere-defined matrix.", "A theorem application can be checked against structural hypotheses such as completeness or continuity, rather than accepted from a subject label." ], "characteristics": [ { "name": "Scalar field", "kind": "category", "unit_or_values": "real; complex; unspecified", "why_it_matters": "The scalar field affects duality conventions and the interpretation of spectral claims." }, { "name": "Space structure", "kind": "category", "unit_or_values": "topological vector space; locally convex space; normed space; Banach space; inner-product space; Hilbert space; other explicitly defined structure", "why_it_matters": "Records which structures and associated hypotheses an argument may use." }, { "name": "Completeness status", "kind": "state", "unit_or_values": "established complete; established incomplete; unresolved, relative to the specified structure", "why_it_matters": "Prevents use of conclusions requiring completeness before that requirement is established." }, { "name": "Convergence structure", "kind": "category", "unit_or_values": "norm; weak; weak-star; strong operator; weak operator; other explicitly specified topology", "why_it_matters": "A convergence statement is meaningful only with its underlying space and topology." }, { "name": "Operator domain relationship", "kind": "relation", "unit_or_values": "D(T) contained in X, with T mapping D(T) into Y; record whether D(T) equals X and whether it is dense in X", "why_it_matters": "Controls which inputs and operator constructions are admissible." }, { "name": "Operator norm", "kind": "measurement", "unit_or_values": "nonnegative real value or certified bound for a bounded linear operator, relative to named source and target norms", "why_it_matters": "Supports quantitative continuity and perturbation estimates." }, { "name": "Hypothesis verification status", "kind": "state", "unit_or_values": "proved; assumed; disproved; unresolved, for each required hypothesis", "why_it_matters": "Separates conditional deductions from established conclusions." } ], "affordances": [ "Classify a mathematical setting by the space structures and operator properties it actually establishes.", "Check whether a functional-analytic theorem applies and identify missing hypotheses.", "Compare convergence or compactness claims while preserving their topology and quantifiers.", "Determine whether restriction, extension, completion, adjoint formation or inversion is justified.", "Identify a counterexample or failed hypothesis that blocks a proposed deduction.", "Transfer a result through a verified embedding or isomorphism while recording which structures are preserved." ] }, "sources": [ { "id": "iso-80000-2", "title": "ISO 80000-2:2019 Quantities and units - Part 2: Mathematics", "url": "https://www.iso.org/standard/64973.html", "what_it_supports": "ISO vocabulary and notation for mathematical objects used in functional analysis (spaces, operators, norms).", "url_status": "blocked" }, { "id": "ams-msc2020", "title": "Mathematics Subject Classification 2020, class 46 - Functional analysis", "url": "https://msc2020.org/", "what_it_supports": "Authoritative taxonomic identifier and kind-structure for the field as used by zbMATH, MathSciNet, and journals.", "url_status": "live" }, { "id": "wikidata-q190549", "title": "functional analysis (Q190549)", "url": "https://www.wikidata.org/wiki/Q190549", "what_it_supports": "Wikidata item, aliases, and same-as links that identify the mathematical sense of the term.", "url_status": "live" }, { "id": "encyclopedia-of-mathematics", "title": "Functional analysis", "url": "https://encyclopediaofmath.org/wiki/Functional_analysis", "what_it_supports": "Specialist definition, historical scope, and neighbouring fields as used in the mathematical literature.", "url_status": "live" } ], "structure": { "bundles": [ { "id": "spaces-and-structural-hypotheses", "name": "Spaces and structural hypotheses", "description": "Records the mathematical spaces and the structures on which functional-analytic reasoning depends.", "rationale": "Space names alone do not establish the hypotheses needed for an argument.", "layers": [ { "id": "space-identification", "name": "Space identification", "description": "Identifies elements, scalar field and the relationships that define the space.", "findings": [ { "id": "elements-and-equality", "name": "Elements and equality", "description": "Establishes what counts as an element, including whether elements are functions or equivalence classes.", "questions": [ { "text": "What are the elements, scalar field and equality relation of each space under consideration?", "kind": "definition", "id": "elements-and-equality-q01" }, { "text": "If elements are equivalence classes of functions, which proposed operations are independent of the representative?", "kind": "boundary", "id": "elements-and-equality-q02" } ] } ] }, { "id": "topology-and-completion", "name": "Topology and completion", "description": "Records the chosen topology and evidence for structural properties such as completeness.", "findings": [ { "id": "structure-certification", "name": "Structure certification", "description": "Distinguishes a declared norm, inner product or topology from properties actually established for it.", "questions": [ { "text": "Which norm, inner product, seminorm family or topology is fixed, and which structural properties have been proved?", "kind": "definition", "id": "structure-certification-q01" }, { "text": "If completeness is required but unavailable, what completion is intended and which maps extend to it?", "kind": "action", "id": "structure-certification-q02" } ] } ] } ] }, { "id": "convergence-and-compactness", "name": "Convergence and compactness", "description": "Records how limits and compactness are interpreted and used in the chosen spaces.", "rationale": "Changing topology or silently replacing nets with sequences can change the validity of an argument.", "layers": [ { "id": "convergence-claims", "name": "Convergence claims", "description": "Makes the topology, limiting object and quantifiers of a convergence claim explicit.", "findings": [ { "id": "limit-mode-and-witness", "name": "Limit mode and witness", "description": "Captures what converges, in which sense, and how convergence is established.", "questions": [ { "text": "Is the claim about a sequence, net or filter, and in which explicitly named topology does it converge?", "kind": "definition", "id": "limit-mode-and-witness-q01" }, { "text": "What estimate or test against functionals or vectors establishes the claimed convergence?", "kind": "measurement", "id": "limit-mode-and-witness-q02" } ] } ] }, { "id": "compactness-mechanisms", "name": "Compactness mechanisms", "description": "Identifies the exact compactness property used to obtain limits or convergent subobjects.", "findings": [ { "id": "compactness-scope", "name": "Compactness scope", "description": "Separates compactness, relative compactness and sequential compactness in their stated topology.", "questions": [ { "text": "Which set is compact, relatively compact or sequentially compact, and in which topology?", "kind": "definition", "id": "compactness-scope-q01" }, { "text": "What verified hypothesis licenses the particular subsequence or subnet extraction used in the argument?", "kind": "boundary", "id": "compactness-scope-q02" } ] } ] } ] }, { "id": "duality-and-representation", "name": "Duality and representation", "description": "Records continuous functionals, dual pairings and representations that convert abstract statements into usable tests.", "rationale": "Dual-space identifications and representation formulas require explicit conventions and hypotheses.", "layers": [ { "id": "dual-pairings", "name": "Dual pairings", "description": "Defines which functionals belong to the dual and how they act on the original space.", "findings": [ { "id": "continuous-dual-identification", "name": "Continuous dual identification", "description": "Specifies the continuous dual and distinguishes it from the algebraic dual.", "questions": [ { "text": "Does the stated dual mean continuous linear functionals for the chosen topology, and what pairing convention is used?", "kind": "definition", "id": "continuous-dual-identification-q01" }, { "text": "What proof or source supports any identification of this dual with a concrete function or sequence space?", "kind": "provenance", "id": "continuous-dual-identification-q02" } ] } ] }, { "id": "bidual-and-functional-constructions", "name": "Bidual and functional constructions", "description": "Tracks passage to the bidual and the hypotheses for extending or representing functionals.", "findings": [ { "id": "canonical-map-and-extensions", "name": "Canonical map and extensions", "description": "Records which duality constructions are available and which preserve the required properties.", "questions": [ { "text": "What is the canonical map into the bidual, and is its injectivity, isometry or surjectivity established in this setting?", "kind": "boundary", "id": "canonical-map-and-extensions-q01" }, { "text": "Which theorem permits the proposed extension or representation of a functional, and what does it establish about norm preservation or uniqueness?", "kind": "action", "id": "canonical-map-and-extensions-q02" } ] } ] } ] }, { "id": "operators-and-spectral-questions", "name": "Operators and spectral questions", "description": "Records operator domains, regularity and the conditions under which inverse or spectral reasoning is meaningful.", "rationale": "An operator formula without its domain and ambient spaces is insufficient to judge permissible operations.", "layers": [ { "id": "operator-definition-and-regularity", "name": "Operator definition and regularity", "description": "Identifies the map and independently records boundedness, density and graph properties.", "findings": [ { "id": "domain-and-graph-properties", "name": "Domain and graph properties", "description": "Captures the operator's domain and evidence for properties needed by subsequent constructions.", "questions": [ { "text": "What are the source space, target space, domain and action of the operator, and is linearity established?", "kind": "definition", "id": "domain-and-graph-properties-q01" }, { "text": "Which of boundedness, dense definition, closedness and closability are established, and by what estimate or argument?", "kind": "provenance", "id": "domain-and-graph-properties-q02" } ] } ] }, { "id": "inversion-adjoints-and-spectrum", "name": "Inversion, adjoints and spectrum", "description": "Checks the prerequisites for derived operators and spectral conclusions.", "findings": [ { "id": "derived-operator-admissibility", "name": "Derived operator admissibility", "description": "Makes inverse, adjoint and resolvent constructions conditional on their actual definitions and hypotheses.", "questions": [ { "text": "For the proposed inverse or adjoint, what definition is used, what is its domain, and which existence conditions are verified?", "kind": "action", "id": "derived-operator-admissibility-q01" }, { "text": "For a spectral claim, what scalar field and operator setting are fixed, and what evidence determines whether T minus lambda times the identity has the required bounded inverse?", "kind": "boundary", "id": "derived-operator-admissibility-q02" } ] } ] } ] }, { "id": "theorem-applicability-and-obstructions", "name": "Theorem applicability and obstructions", "description": "Connects proposed deductions to exact theorem statements and records why an argument succeeds or fails.", "rationale": "An agent must distinguish a familiar theorem name from a justified application to the present spaces and operators.", "layers": [ { "id": "hypothesis-matching", "name": "Hypothesis matching", "description": "Matches each required theorem hypothesis to evidence in the current setting.", "findings": [ { "id": "theorem-application-record", "name": "Theorem application record", "description": "Records the exact result being invoked and the status of every hypothesis.", "questions": [ { "text": "Which exact sourced version of the proposed result, such as uniform boundedness, open mapping or closed graph, is being invoked?", "kind": "provenance", "id": "theorem-application-record-q01" }, { "text": "Which required hypotheses are proved, assumed, disproved or unresolved for the spaces, operator family and topologies at hand?", "kind": "boundary", "id": "theorem-application-record-q02" } ] } ] }, { "id": "counterexamples-and-repair", "name": "Counterexamples and repair", "description": "Records obstructions and justified changes that could make an intended deduction available.", "findings": [ { "id": "failed-inference-and-repair", "name": "Failed inference and repair", "description": "Distinguishes an inapplicable theorem from a false conclusion and identifies precise repair options.", "questions": [ { "text": "Does a verified counterexample refute the desired conclusion, or is only the proposed theorem application blocked?", "kind": "boundary", "id": "failed-inference-and-repair-q01" }, { "text": "Would restricting the domain, strengthening a hypothesis or changing the topology justify a revised conclusion, and what must be proved again?", "kind": "action", "id": "failed-inference-and-repair-q02" } ] } ] } ] }, { "id": "evidence-and-external-alignment", "name": "Evidence and external alignment", "description": "What the world already says about this thing, gathered so the model can be checked against it.", "rationale": "A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.", "layers": [ { "id": "reported-evidence", "name": "Reported evidence", "description": "Findings from the breadth pass, kept separate from the structural claims.", "findings": [ { "id": "evidence-kinds", "name": "Kinds and varieties", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "real and complex Banach-space theory", "Hilbert-space and inner-product operator theory", "spectral theory of linear operators", "distribution and Sobolev-space analysis", "nonlinear functional analysis", "applied functional analysis for PDEs and integral equations", "operator algebras and C*-algebraic analysis" ], "questions": [ { "id": "evidence-kinds-q01", "text": "Which of these kinds and varieties hold for the sense of functional analysis this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-identifiers", "name": "Identifiers and schemes", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ { "scheme": "Wikidata", "value_or_pattern": "Q190549", "note": "Mathematical sense: study of function spaces and operators." }, { "scheme": "MSC 2020", "value_or_pattern": "46-XX", "note": "Primary subject class Functional analysis; notable subclasses 46Bxx (normed linear spaces and Banach spaces), 46Cxx (inner product spaces and Hilbert spaces), 46Exx (linear function spaces and their duals), 46Fxx (distributions, generalized functions, distribution spaces), 46Hxx (topological algebras, normed rings and algebras, Banach algebras), 46Jxx (commutative Banach algebras and commutative topological algebras), 46Lxx (selfadjoint operator algebras, C*- and von Neumann algebras), 46Nxx (miscellaneous applications of functional analysis)." }, { "scheme": "MSC 2020 (history)", "value_or_pattern": "46-03", "note": "History of functional analysis." }, { "scheme": "ISO 80000-2", "value_or_pattern": "clause on vector spaces, norms, inner products, and operators", "note": "Does not assign a single code to the field; supplies the quantity and notation layer used when the subject is written down." } ], "questions": [ { "id": "evidence-identifiers-q01", "text": "Which of these identifiers and schemes hold for the sense of functional analysis this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-standards-and-regulation", "name": "Standards and regulation", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "ISO 80000-2:2019 Quantities and units - Part 2: Mathematics (International Organization for Standardization): notation and names for spaces, operators, norms, and inner products.", "MSC 2020 (zbMATH Open / Mathematical Reviews): classification scheme used to index the literature of the field.", "No statute uniquely governs the mathematical discipline; professional publication and teaching follow journal, society, and national curriculum rules rather than a legal code." ], "questions": [ { "id": "evidence-standards-and-regulation-q01", "text": "Which of these standards and regulation hold for the sense of functional analysis this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-real-world-use", "name": "Real-world use", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "Core graduate course and research field in pure mathematics departments.", "Working language for existence, uniqueness, and stability theorems for linear and nonlinear PDEs.", "Spectral and operator methods in quantum mechanics, control theory, and signal processing.", "Theoretical backbone of finite-element, spectral, and other Galerkin numerical methods.", "Classification and search key (MSC 46-XX) in MathSciNet, zbMATH, and arXiv math.FA." ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of functional analysis this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-measurements", "name": "Typical measurements", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ { "quantity": "dimension of the underlying space", "typical_range": "infinite (typically separable infinite-dimensional); finite-dimensional cases are the linear-algebra degeneration", "unit": "dimension (dimensionless)" }, { "quantity": "operator norm", "typical_range": "0 to unbounded; bounded operators have finite norm", "unit": "depends on the pair of norms on domain and codomain" }, { "quantity": "spectrum of a bounded operator", "typical_range": "nonempty compact subset of the complex plane for Banach-algebra elements; for self-adjoint Hilbert-space operators, a nonempty compact subset of the reals", "unit": "same as the scalar field (usually complex numbers, dimensionless in abstract setting)" } ], "questions": [ { "id": "evidence-measurements-q01", "text": "Which of these typical measurements hold for the sense of functional analysis this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-failure-modes-and-hazards", "name": "Failure modes and hazards", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "Unbounded or discontinuous operators: algebraic inverses that are not continuous, so formal solution formulae are unstable.", "Loss of compactness: bounded sequences need not have convergent subsequences, so existence proofs that work in finite dimensions fail.", "Spectral pathology: residual or continuous spectrum with no eigenvalues, so \"diagonalization\" heuristics fail.", "Choice of topology: weak, weak*, and strong topologies give different duals, compactness, and convergence, so a proof in the wrong topology is invalid.", "Confusion of senses of the name (see neighbours) produces wrongly scoped models, catalogues, or teaching modules." ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of functional analysis this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-regional-variation", "name": "Regional variation", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ "English \"functional analysis\" and French \"analyse fonctionnelle\" are the mathematical field; German \"Funktionalanalysis\" likewise.", "In systems engineering, software requirements, and some ISO/IEC systems-and-software documents, \"functional analysis\" means decomposition of functions of a system, not the mathematical field.", "In behaviour analysis and clinical psychology, \"functional analysis\" means experimental identification of the contingencies that maintain a behaviour (Iwata et al. and later ABA usage).", "Russian-language literature historically emphasized operator theory and \"функциональный анализ\" as a pillar of the Moscow and Leningrad schools, with a stronger operator-algebra and PDE accent than some Western undergraduate sequences." ], "questions": [ { "id": "evidence-regional-variation-q01", "text": "Which of these regional variation hold for the sense of functional analysis this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-neighbours", "name": "Neighbouring kinds and how to tell them apart", "description": "Reported by the breadth pass; each item needs checking against its source before it becomes normative.", "evidence": [ { "name": "linear algebra", "difference": "Linear algebra treats finite-dimensional vector spaces; functional analysis is the infinite-dimensional, topological, and operator-theoretic continuation. Finite dimension is the special case in which all norms are equivalent and all linear maps are continuous." }, { "name": "real and complex analysis", "difference": "Classical analysis studies functions, limits, integrals, and series on R^n or C; functional analysis studies spaces of such functions and operators between them." }, { "name": "operator theory", "difference": "Operator theory is the study of linear (and some nonlinear) operators; it is a core chapter of functional analysis rather than a synonym for the whole field, which also includes the geometry of Banach spaces, distributions, and topological vector spaces." }, { "name": "systems-engineering functional analysis", "difference": "That practice decomposes system functions, interfaces, and requirements (e.g. INCOSE/ISO/IEC/IEEE 15288 processes). It does not study Banach or Hilbert spaces. Test: if the objects are requirements, functions, and interfaces of a system, it is not this field." }, { "name": "behavioural functional analysis (ABA)", "difference": "That method identifies antecedents and consequences of behaviour. Test: if the data are observed behaviours and contingencies, it is not this field." }, { "name": "functional programming / functional design", "difference": "Programming and design senses concern pure functions or functional decomposition of software, not topological vector spaces." } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of functional analysis this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Does this registry entry intend mathematical functional analysis, or another established use of the same name?", "Which existing Vercy models already own mathematical spaces, operators or theorem applications, and where should this model reference those models instead of duplicating them?", "Should the initial scope cover general topological vector spaces or concentrate on normed, Banach and Hilbert spaces?", "How much specialised coverage is required for nonlinear functional analysis, operator algebras and distribution spaces?", "Which authoritative references and theorem variants should establish the first researched set of hypothesis checks and counterexamples?" ], "statistics": { "bundles": 6, "layers": 11, "findings": 18, "questions": 28 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "discipline or field of knowledge", "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 analysis, record what state it is in, and decide what may be done with it.", "whatItIs": "Enable an AI agent to recognise a functional-analysis problem, record the hypotheses that govern its mathematical objects, and judge which deductions or operations are justified.", "characteristics": { "substance": "information", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "read and interpreted" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: operator theory, Compression as functional analysis, nonlinear functional analysis, uniform algebra.", "4 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 62 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 analysis, and on what evidence are they told apart?" ], "whatItIsMadeOf": "content that has to be carried by something else", "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": [ "read it and act on what it says" ], "howItIsRecognised": [], "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": [ "operator theory", "Compression as functional analysis", "nonlinear functional analysis", "uniform algebra" ] } ], "standing": "Described in 62 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 4 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 analysis from another, and a functional analysis 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 analysis, 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 analysis, 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 analysis 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 analysis, 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 analysis, 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 analysis 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 analysis 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 analysis, 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 analysis, 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 analysis 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 analysis from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a functional analysis, 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 analysis, 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 analysis 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 analysis 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 analysis between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a functional analysis, 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 analysis, 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 analysis 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 analysis, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a functional analysis 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 analysis, 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 analysis, 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 analysis is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "content-and-access", "name": "Content, encoding and access", "description": "What a functional analysis says, how it is encoded and who may read it.", "rationale": "An informational thing carries content that can be copied, versioned and withheld, none of which its physical carrier explains.", "layers": [ { "id": "content-and-encoding", "name": "Content and encoding", "description": "The content itself, its format and its language.", "findings": [ { "id": "content-and-format", "name": "Content, format and language", "description": "What a functional analysis contains and in what form it is held.", "questions": [ { "id": "content-and-format-q01", "text": "What identifies and describes the content of a functional analysis, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "content-and-format-q02", "text": "Who or what asserted this about the content of a functional analysis, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "content-and-format-q03", "text": "What may an agent decide or do once the content of a functional analysis is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "access-and-rights", "name": "Access, rights and retention", "description": "Who may read, copy or change it, and for how long it is kept.", "findings": [ { "id": "access-rules", "name": "Access rules and retention", "description": "The permissions attached to a functional analysis and the period it survives.", "questions": [ { "id": "access-rules-q01", "text": "What identifies and describes access to a functional analysis, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "access-rules-q02", "text": "Who or what asserted this about access to a functional analysis, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "access-rules-q03", "text": "What may an agent decide or do once access to a functional analysis is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "subject-and-method", "name": "Subject matter and method", "description": "What functional analysis studies and how it establishes anything.", "rationale": "A field is identified by its questions and its methods, and those are what distinguish it from the fields next to it.", "layers": [ { "id": "subject", "name": "Subject matter", "description": "The questions it takes as its own.", "findings": [ { "id": "subject-record", "name": "Questions and objects of study", "description": "What it is about, stated so a neighbouring field can be told apart.", "questions": [ { "id": "subject-record-q01", "text": "What questions and objects does functional analysis take as its own?", "kind": "definition" }, { "id": "subject-record-q02", "text": "Which questions does it share with a neighbouring field, and who claims them?", "kind": "boundary" } ] } ] }, { "id": "method", "name": "Method and evidence", "description": "How claims are established and what counts as evidence.", "findings": [ { "id": "method-record", "name": "Methods and standards of evidence", "description": "The methods used and what they are taken to establish.", "questions": [ { "id": "method-record-q01", "text": "By what methods does functional analysis establish claims, and what counts as sufficient evidence?", "kind": "provenance" }, { "id": "method-record-q02", "text": "What kind of claim can this field not settle, and where should an agent look instead?", "kind": "action" } ] } ] } ] }, { "id": "institutions-and-classification", "name": "Institutions and classification", "description": "Where functional analysis is practised and how catalogues place it.", "rationale": "The institutional footprint is the evidence that a field exists as a field rather than as a topic.", "layers": [ { "id": "institutions", "name": "Institutional markers", "description": "Societies, journals, degrees and departments.", "findings": [ { "id": "institution-record", "name": "Bodies and venues", "description": "Where the field organises itself.", "questions": [ { "id": "institution-record-q01", "text": "Which societies, journals or degrees mark functional analysis as an established field?", "kind": "provenance" }, { "id": "institution-record-q02", "text": "What would show that it is emerging or dissolving rather than established?", "kind": "boundary" } ] } ] }, { "id": "classification", "name": "Classification schemes", "description": "Where library and research classifications put it.", "findings": [ { "id": "classification-record", "name": "Codes and their disagreements", "description": "The classification codes that place it, and where they differ.", "questions": [ { "id": "classification-record-q01", "text": "Which classification schemes place functional analysis, under what codes?", "kind": "definition" }, { "id": "classification-record-q02", "text": "Where do those schemes disagree about its scope?", "kind": "boundary" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a functional analysis 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 analysis and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a functional analysis, 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 analysis, 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 analysis 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 analysis is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a functional analysis, 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 analysis, 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 analysis is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 6, "layers": 12, "findings": 14, "questions": 38 } } } }