{ "model": { "rank": 5051, "code": "thing-q6520159", "model_id": "vr.tr.fourier-transform", "name": "Fourier transform", "purpose": "Let an agent explain the Fourier transform and its variants, relay definitions, properties and algorithms from mathematical references, describe applications, and distinguish continuous, discrete and time-frequency forms.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q6520159/", "tier": 2, "score": 70, "payload": { "layer": "wikidata", "aliases": [ "Constant Q transform", "sliding DFT", "DFT matrix", "discrete Fourier transform", "short-time Fourier transform", "Multiresolution Fourier transform", "Fourier transform on groups", "Friedel's law", "Gabor transform" ], "aliasCount": 9, "merged": 10, "knownIn": 70, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q6520159", "url": "https://www.wikidata.org/wiki/Q6520159", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T02:44:35Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 3, "engine": "claude" }, "metaModel": { "id": "THING-Q6520159", "registryId": "vr.tr.fourier-transform", "name": "Fourier transform", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q6520159/", "model": { "registry_id": "vr.tr.fourier-transform", "name": "Fourier transform", "purpose": "Let an agent explain the Fourier transform and its variants, relay definitions, properties and algorithms from mathematical references, describe applications, and distinguish continuous, discrete and time-frequency forms.", "definition": "An integral transform that decomposes a function of time or space into the frequencies that make it up, expressing a signal as a sum or integral of sinusoids, with the continuous Fourier transform for functions on the real line, the discrete Fourier transform for finite sequences computed efficiently by the fast Fourier transform and represented by the DFT matrix, the short-time and sliding DFT for time-varying spectra, and variants such as the constant Q and multiresolution transforms; the Fourier transform underlies signal processing, physics, imaging and data compression.", "what_it_is_for": "Not applicable; a mathematical operation.", "affordances": [ "explain the transform and variants", "relay properties and algorithms", "describe applications", "distinguish forms" ], "distinguishing_features": [ "Frequency decomposition", "Linearity and invertibility", "Convolution theorem", "Fast algorithms" ], "appearance": "Not a visible object; spectra and formulas.", "visual_identification": [ "Decomposition of signals into frequencies", "Continuous FT, DFT and FFT, DFT matrix, STFT, sliding DFT, constant Q, multiresolution", "Laplace and wavelet transforms are related but distinct; Fourier series apply to periodic functions" ], "physical_properties": [ { "quantity": "FFT complexity", "typical_range": "O(n log n)", "unit": "operations", "note": "Cooley-Tukey 1965" }, { "quantity": "Fourier memoir", "typical_range": "1822", "unit": "year", "note": "Theorie analytique de la chaleur" } ], "families_and_kinds": [ "continuous Fourier transform", "Fourier series", "discrete-time Fourier transform", "discrete Fourier transform, DFT matrix and fast Fourier transform", "short-time Fourier transform and sliding DFT", "constant Q transform and multiresolution Fourier transform", "fractional and quantum Fourier transforms" ], "related_models": [ { "relation": "is a kind of", "target": "integral transform", "why": "in registry terms" }, { "relation": "is computed by", "target": "fast Fourier transform", "why": "for discrete data" }, { "relation": "is related to", "target": "Laplace transform", "why": "as a special case on the imaginary axis" }, { "relation": "is applied in", "target": "signal processing", "why": "and many fields" } ], "identifiers": [], "standards_and_regulation": [ "No regulation; standard definitions with sign and normalisation conventions varying by field" ], "failure_modes_and_hazards": [ "Convention mismatches in sign and scaling", "Aliasing and leakage in discrete transforms", "Confusing transform variants" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What the Fourier transform is.", "rationale": "Mathematics.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is the Fourier transform, and how does it represent a function in terms of frequencies?", "kind": "definition" }, { "text": "Is the question about the continuous transform, the DFT, a time-frequency variant or a related transform?", "kind": "boundary" } ] } ] }, { "id": "variants", "name": "Variants", "description": "Variants.", "findings": [ { "id": "variants-finding", "name": "Variants", "description": "Variants.", "questions": [ { "text": "How do the DFT, FFT, STFT, sliding DFT, constant Q and multiresolution transforms relate?", "kind": "definition" }, { "text": "Which entry fits the specific variant?", "kind": "action" } ] } ] } ] }, { "id": "properties", "name": "Properties", "description": "Properties and computation.", "rationale": "Mathematics.", "layers": [ { "id": "properties", "name": "Properties", "description": "Properties.", "findings": [ { "id": "properties-finding", "name": "Properties", "description": "Properties.", "questions": [ { "text": "What properties such as linearity, shift, convolution and Parseval theorems hold?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "algorithms", "name": "Algorithms", "description": "Algorithms.", "findings": [ { "id": "algorithms-finding", "name": "Algorithms", "description": "Algorithms.", "questions": [ { "text": "How do FFT algorithms work, and what conventions and pitfalls such as aliasing matter?", "kind": "provenance" }, { "text": "Which entry fits fast Fourier transform?", "kind": "action" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Application.", "layers": [ { "id": "signals", "name": "Signals", "description": "Signal processing.", "findings": [ { "id": "signals-finding", "name": "Signals", "description": "Signals.", "questions": [ { "text": "How is the transform used in audio, communications, filtering and compression?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] }, { "id": "science", "name": "Science", "description": "Science and imaging.", "findings": [ { "id": "science-finding", "name": "Science", "description": "Science.", "questions": [ { "text": "How does the transform appear in physics, crystallography, MRI and spectroscopy?", "kind": "provenance" }, { "text": "Which entry fits the specific application?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "History and teaching.", "rationale": "Context.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did Fourier analysis develop from Fourier heat work to the FFT?", "kind": "provenance" }, { "text": "Which entry fits the history of Fourier analysis?", "kind": "action" } ] } ] }, { "id": "teaching", "name": "Teaching", "description": "Teaching.", "findings": [ { "id": "teaching-finding", "name": "Teaching", "description": "Teaching.", "questions": [ { "text": "How is the Fourier transform taught, and what intuitions help?", "kind": "provenance" }, { "text": "Which entry fits mathematics education?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should the DFT and STFT be separate primary entries?", "How should mathematical references be linked?", "The registry entry has merged aliases naming variants and a matrix; 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 Fourier transform, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain the Fourier transform and its variants, relay definitions, properties and algorithms from mathematical references, describe applications, and distinguish continuous, discrete and time-frequency forms.", "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: Constant Q transform, sliding DFT, DFT matrix, discrete Fourier transform, short-time Fourier transform, Multiresolution Fourier transform, Fourier transform on groups, Friedel's law, Gabor transform.", "10 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 70 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 Fourier transform, 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": [ "Constant Q transform", "sliding DFT", "DFT matrix", "discrete Fourier transform", "short-time Fourier transform", "Multiresolution Fourier transform", "Fourier transform on groups", "Friedel's law", "Gabor transform" ] } ], "standing": "Described in 70 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 10 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 Fourier transform from another, and a Fourier transform 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 Fourier transform, 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 Fourier transform, 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 Fourier transform 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 Fourier transform, 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 Fourier transform, 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 Fourier transform 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 Fourier transform 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 Fourier transform, 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 Fourier transform, 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 Fourier transform 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 Fourier transform from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a Fourier transform, 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 Fourier transform, 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 Fourier transform 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 Fourier transform 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 Fourier transform between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a Fourier transform, 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 Fourier transform, 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 Fourier transform 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 Fourier transform, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a Fourier transform 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 Fourier transform, 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 Fourier transform, 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 Fourier transform 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 Fourier transform 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 Fourier transform 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 Fourier transform 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 Fourier transform 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 Fourier transform?", "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 Fourier transform?", "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 Fourier transform 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 Fourier transform and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a Fourier transform, 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 Fourier transform, 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 Fourier transform 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 Fourier transform is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a Fourier transform, 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 Fourier transform, 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 Fourier transform 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 } } } }