{ "model": { "rank": 14903, "code": "thing-q621461", "model_id": "vr.tr.symmetric-relation", "name": "symmetric relation", "purpose": "Let an agent define symmetric relations, give examples and counterexamples, explain related properties and the symmetric closure, and separate the mathematical concept from unrelated aliases in the registry entry.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q621461/", "tier": 2, "score": 35, "payload": { "layer": "wikidata", "aliases": [ "genetic relationship", "arene substitution pattern", "empty relation", "symmetric closure", "transversality", "comparability" ], "aliasCount": 6, "merged": 6, "knownIn": 35, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q621461", "url": "https://www.wikidata.org/wiki/Q621461", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T22:30:46Z", "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-Q621461", "registryId": "vr.tr.symmetric-relation", "name": "symmetric relation", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q621461/", "model": { "registry_id": "vr.tr.symmetric-relation", "name": "symmetric relation", "purpose": "Let an agent define symmetric relations, give examples and counterexamples, explain related properties and the symmetric closure, and separate the mathematical concept from unrelated aliases in the registry entry.", "definition": "A binary relation on a set in which whenever a is related to b, b is related to a, such as equality, being a sibling of, or being connected to; symmetric relations include equivalence relations and compatibility relations, contrast with antisymmetric and asymmetric relations such as ordering, and any relation can be extended to its symmetric closure by adding the reverse pairs.", "what_it_is_for": "A basic property of relations in mathematics and logic.", "affordances": [ "define and test symmetry", "give examples and counterexamples", "explain related properties and closure", "separate unrelated aliases" ], "distinguishing_features": [ "Symmetry property", "Undirected representation", "Part of equivalence relations", "Symmetric closure" ], "appearance": "Not a visible object; an undirected graph or a symmetric matrix when drawn.", "visual_identification": [ "If a R b then b R a", "Undirected graph or symmetric adjacency matrix", "Antisymmetric relations such as less than or equal are different; reflexive and transitive are separate properties" ], "physical_properties": [], "families_and_kinds": [ "equivalence relations", "compatibility and comparability relations", "the empty relation, symmetric vacuously", "symmetric closures of relations", "symmetric relations in graphs and matrices", "relations in specific domains such as transversality in geometry or genetic relationship, listed as aliases" ], "related_models": [ { "relation": "is a kind of", "target": "endorelation", "why": "in registry terms, a relation on one set" }, { "relation": "is a kind of", "target": "binary relation", "why": "with the symmetry property" }, { "relation": "is part of", "target": "equivalence relation", "why": "together with reflexivity and transitivity" }, { "relation": "is contrasted with", "target": "antisymmetric relation", "why": "the opposite property" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 for mathematical notation" ], "failure_modes_and_hazards": [ "Confusing symmetric with reflexive or transitive", "Assuming symmetry in directed data", "Sense confusion with unrelated aliases" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "What a symmetric relation is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition and examples.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a symmetric relation, and what are examples and counterexamples?", "kind": "definition" }, { "text": "Is the relation symmetric, antisymmetric, asymmetric or none?", "kind": "boundary" } ] } ] }, { "id": "properties", "name": "Properties", "description": "Related properties.", "findings": [ { "id": "properties-finding", "name": "Properties", "description": "Properties.", "questions": [ { "text": "How does symmetry combine with reflexivity and transitivity in equivalence and compatibility relations?", "kind": "definition" }, { "text": "Which entry fits equivalence relation?", "kind": "action" } ] } ] } ] }, { "id": "compute", "name": "Compute", "description": "Testing and closure.", "rationale": "Computation.", "layers": [ { "id": "test", "name": "Test", "description": "Testing symmetry.", "findings": [ { "id": "test-finding", "name": "Test", "description": "Test.", "questions": [ { "text": "How is symmetry tested for a given relation, matrix or graph?", "kind": "action" }, { "text": "Is the relation in question symmetric?", "kind": "measurement" } ] } ] }, { "id": "closure", "name": "Closure", "description": "Symmetric closure.", "findings": [ { "id": "closure-finding", "name": "Closure", "description": "Closure.", "questions": [ { "text": "How is the symmetric closure constructed, and what are its properties?", "kind": "action" }, { "text": "Which entry fits closure of a relation?", "kind": "action" } ] } ] } ] }, { "id": "represent", "name": "Represent", "description": "Representations.", "rationale": "Theory.", "layers": [ { "id": "graphs", "name": "Graphs", "description": "Graphs and matrices.", "findings": [ { "id": "graphs-finding", "name": "Graphs", "description": "Graphs.", "questions": [ { "text": "How do symmetric relations correspond to undirected graphs and symmetric matrices?", "kind": "provenance" }, { "text": "Which entry fits undirected graph?", "kind": "action" } ] } ] }, { "id": "counting", "name": "Counting", "description": "Counting relations.", "findings": [ { "id": "counting-finding", "name": "Counting", "description": "Counting.", "questions": [ { "text": "How many symmetric relations exist on a finite set?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Uses and teaching.", "rationale": "Context.", "layers": [ { "id": "uses", "name": "Uses", "description": "Uses.", "findings": [ { "id": "uses-finding", "name": "Uses", "description": "Uses.", "questions": [ { "text": "Where do symmetric relations appear in mathematics, computing and the sciences?", "kind": "provenance" }, { "text": "Which entry fits the specific application?", "kind": "action" } ] } ] }, { "id": "teach", "name": "Teach", "description": "Teaching.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How are relation properties taught, and which are confused?", "kind": "provenance" }, { "text": "Which entry fits discrete mathematics?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should symmetric closure be a separate entry?", "How should logic references be linked?", "The registry entry has merged aliases for unrelated concepts such as arene substitution patterns; should they be removed?" ], "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 symmetric relation, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent define symmetric relations, give examples and counterexamples, explain related properties and the symmetric closure, and separate the mathematical concept from unrelated aliases in the registry entry.", "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: genetic relationship, arene substitution pattern, empty relation, symmetric closure, transversality, comparability.", "6 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 35 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 symmetric relation, 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": [ "genetic relationship", "arene substitution pattern", "empty relation", "symmetric closure", "transversality", "comparability" ] } ], "standing": "Described in 35 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 6 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 symmetric relation from another, and a symmetric relation 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 symmetric relation, 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 symmetric relation, 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 symmetric relation 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 symmetric relation, 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 symmetric relation, 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 symmetric relation 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 symmetric relation 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 symmetric relation, 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 symmetric relation, 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 symmetric relation 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 symmetric relation from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a symmetric relation, 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 symmetric relation, 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 symmetric relation 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 symmetric relation 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 symmetric relation between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a symmetric relation, 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 symmetric relation, 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 symmetric relation 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 symmetric relation, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a symmetric relation 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 symmetric relation, 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 symmetric relation, 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 symmetric relation 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 symmetric relation 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 symmetric relation 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 symmetric relation 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 symmetric relation 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 symmetric relation?", "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 symmetric relation?", "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 symmetric relation 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 symmetric relation and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a symmetric relation, 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 symmetric relation, 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 symmetric relation 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 symmetric relation is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a symmetric relation, 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 symmetric relation, 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 symmetric relation 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 } } } }