{ "model": { "rank": 17273, "code": "thing-q865746", "model_id": "vr.tr.metric-function", "name": "metric function", "purpose": "Let an agent explain metric functions, relay axioms, examples and uses from mathematics sources, describe the named examples and distinguish the metric tensor sense, flag the misfiled alias, and distinguish metrics from norms, pseudometrics, similarity measures and performance metrics in business or software.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q865746/", "tier": 2, "score": 31, "payload": { "layer": "wikidata", "aliases": [ "Kerr metric", "spacetime metric", "Wikidata:Statistics", "Minkowski distance", "graph distance", "Fubini–Study metric", "Lévy metric", "product metric", "Wasserstein metric", "Variation of information", "string metric", "phylogenetic distance", "graph diameter", "graph radius", "degree of kinship", "graph eccentricity", "Earth mover's distance", "inclusion measure" ], "aliasCount": 18, "merged": 18, "knownIn": 31, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q865746", "url": "https://www.wikidata.org/wiki/Q865746", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-14T01:27:58Z", "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-Q865746", "registryId": "vr.tr.metric-function", "name": "metric function", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q865746/", "model": { "registry_id": "vr.tr.metric-function", "name": "metric function", "purpose": "Let an agent explain metric functions, relay axioms, examples and uses from mathematics sources, describe the named examples and distinguish the metric tensor sense, flag the misfiled alias, and distinguish metrics from norms, pseudometrics, similarity measures and performance metrics in business or software.", "definition": "In mathematics, a function that defines a distance between each pair of elements of a set, satisfying non-negativity with zero only for identical points, symmetry and the triangle inequality, making the set a metric space; examples the registry aliases name include Minkowski distance, the Lp family that includes Euclidean and Manhattan distance, and graph distance, the length of the shortest path between vertices, while the Kerr metric, spacetime metrics and the Fubini-Study metric are metric tensors in differential geometry and general relativity, a related but different sense in which a metric measures lengths infinitesimally and may not be positive definite, and Wikidata:Statistics is a project page and appears misfiled. Metrics are used in analysis, topology, machine learning and data science.", "what_it_is_for": "Measuring distance in mathematical spaces.", "affordances": [ "explain axioms", "relay examples", "distinguish metric tensors", "flag misfiled aliases" ], "distinguishing_features": [ "Triangle inequality", "Symmetry", "Defines topology", "Many concrete examples" ], "appearance": "Not a physical object; defined by formulas and axioms.", "visual_identification": [ "Distance function satisfying metric axioms", "Kerr metric, spacetime metric, Wikidata:Statistics, Minkowski distance, graph distance, Fubini-Study metric", "Norms measure vector size; pseudometrics allow zero distance between distinct points; similarity measures may not satisfy axioms; performance metrics are indicators" ], "physical_properties": [ { "quantity": "metric space concept", "typical_range": "1906", "unit": "year", "note": "Maurice Frechet" }, { "quantity": "axioms", "typical_range": "identity, symmetry, triangle inequality", "unit": "list", "note": "plus non-negativity" }, { "quantity": "Kerr metric", "typical_range": "1963", "unit": "year", "note": "Roy Kerr" } ], "families_and_kinds": [ "Lp and Minkowski distances", "graph and path metrics", "edit and string metrics", "Riemannian and pseudo-Riemannian metric tensors" ], "related_models": [ { "relation": "is a kind of", "target": "function", "why": "in registry terms" }, { "relation": "defines", "target": "metric space", "why": "" }, { "relation": "is contrasted with", "target": "metric tensor", "why": "related sense" }, { "relation": "is contrasted with", "target": "norm (mathematics)", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "Not applicable" ], "failure_modes_and_hazards": [ "Confusing metric functions and metric tensors", "Using non-metric similarities where axioms are assumed", "Misfiled project page alias" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What a metric is.", "rationale": "Science.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a metric function, and how does it differ from norms, pseudometrics, similarity measures and performance metrics?", "kind": "definition" }, { "text": "Is the question about distance functions or metric tensors in geometry and physics?", "kind": "boundary" } ] } ] }, { "id": "examples", "name": "Examples", "description": "Named examples.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Examples.", "questions": [ { "text": "What are Minkowski distance and graph distance, and how are the Kerr, spacetime and Fubini-Study metrics different?", "kind": "definition" }, { "text": "Which entry fits the specific metric?", "kind": "action" } ] } ] } ] }, { "id": "axioms", "name": "Axioms", "description": "Axioms.", "rationale": "Science.", "layers": [ { "id": "triangle", "name": "Triangle inequality", "description": "Triangle inequality.", "findings": [ { "id": "triangle-finding", "name": "Triangle inequality", "description": "Triangle inequality.", "questions": [ { "text": "Why is the triangle inequality essential?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "spaces", "name": "Metric spaces", "description": "Metric spaces.", "findings": [ { "id": "spaces-finding", "name": "Metric spaces", "description": "Metric spaces.", "questions": [ { "text": "How do metrics define open sets and convergence?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "geometry", "name": "Geometry", "description": "Metric tensors.", "rationale": "Science.", "layers": [ { "id": "tensors", "name": "Tensors", "description": "Metric tensors.", "findings": [ { "id": "tensors-finding", "name": "Tensors", "description": "Tensors.", "questions": [ { "text": "How does a metric tensor define lengths and angles on manifolds?", "kind": "provenance" }, { "text": "Which entry fits metric tensor?", "kind": "action" } ] } ] }, { "id": "relativity", "name": "Relativity", "description": "Spacetime metrics.", "findings": [ { "id": "relativity-finding", "name": "Relativity", "description": "Relativity.", "questions": [ { "text": "What does the Kerr metric describe for rotating black holes?", "kind": "provenance" }, { "text": "Is the information accurate?", "kind": "boundary" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Applications.", "rationale": "Context.", "layers": [ { "id": "ml", "name": "Machine learning", "description": "Distance-based methods.", "findings": [ { "id": "ml-finding", "name": "Machine learning", "description": "Machine learning.", "questions": [ { "text": "How are metrics used in nearest-neighbour methods and clustering?", "kind": "provenance" }, { "text": "Which entry fits k-nearest neighbors algorithm?", "kind": "action" } ] } ] }, { "id": "graphs", "name": "Graphs", "description": "Graph distance.", "findings": [ { "id": "graphs-finding", "name": "Graphs", "description": "Graphs.", "questions": [ { "text": "How is shortest-path distance computed in graphs?", "kind": "measurement" }, { "text": "Is the information current?", "kind": "boundary" } ] } ] } ] } ] }, "openQuestions": [ "The registry alias Wikidata:Statistics should be removed", "The registry metric tensor aliases should be moved to differential geometry entries", "Should graph distance be a separate entry?" ], "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 metric function, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain metric functions, relay axioms, examples and uses from mathematics sources, describe the named examples and distinguish the metric tensor sense, flag the misfiled alias, and distinguish metrics from norms, pseudometrics, similarity measures and performance metrics in business or software.", "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: Kerr metric, spacetime metric, Wikidata:Statistics, Minkowski distance, graph distance, Fubini–Study metric, Lévy metric, product metric, Wasserstein metric, Variation of information, string metric, phylogenetic distance.", "18 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 31 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 metric function, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "Kerr metric", "spacetime metric", "Wikidata:Statistics", "Minkowski distance", "graph distance", "Fubini–Study metric", "Lévy metric", "product metric", "Wasserstein metric", "Variation of information", "string metric", "phylogenetic distance" ] } ], "standing": "Described in 31 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 18 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 metric function from another, and a metric function from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a metric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a metric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a metric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a metric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a metric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a metric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a metric function sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a metric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a metric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a metric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a metric function from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a metric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a metric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a metric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a metric function passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a metric function between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a metric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a metric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a metric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a metric function, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a metric function is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a metric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a metric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a metric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What metric function is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of metric function are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is metric function operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of metric function and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of metric function?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as metric function?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a metric function came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a metric function and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a metric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a metric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a metric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a metric function is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a metric function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a metric function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a metric function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }