{ "model": { "rank": 8173, "code": "thing-q191290", "model_id": "vr.tr.ordered-pair", "name": "ordered pair", "purpose": "Let an agent explain ordered pairs and their set-theoretic definition, show their uses across mathematics and computing, and distinguish them from unordered pairs and other structures.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q191290/", "tier": 2, "score": 52, "payload": { "layer": "wikidata", "aliases": [ "key pair", "Si yuan shu", "directed edge", "key–value pair", "ordered pair of points", "directed hypergraph", "hyperarc", "(B, N) pair", "irregular pair", "complex128", "integer couple", "tuple of integers", "t-structure", "topological pair", "adjoint functor", "fundamental pair of periods", "Gelfand pair", "betrothed numbers", "signed set", "CW-pair", "Galois connection" ], "aliasCount": 21, "merged": 21, "knownIn": 52, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q191290", "url": "https://www.wikidata.org/wiki/Q191290", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T21:19:23Z", "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": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q191290", "registryId": "vr.tr.ordered-pair", "name": "ordered pair", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q191290/", "model": { "registry_id": "vr.tr.ordered-pair", "name": "ordered pair", "purpose": "Let an agent explain ordered pairs and their set-theoretic definition, show their uses across mathematics and computing, and distinguish them from unordered pairs and other structures.", "definition": "A pair of objects in which the order matters, written (a, b), so that (a, b) differs from (b, a) unless a equals b, the basic construction behind coordinates, functions as sets of pairs, directed edges in graphs, key-value pairs in data structures and cryptographic key pairs; the ordered pair is defined in set theory and is a two-element tuple.", "what_it_is_for": "Representing two things in order.", "affordances": [ "explain the definition and notation", "show uses in coordinates, relations and graphs", "explain key-value and key pairs in computing", "distinguish from unordered pairs" ], "distinguishing_features": [ "Order matters", "Two components", "Set-theoretically definable", "Foundational" ], "appearance": "Not physical; written as (a, b).", "visual_identification": [ "Two elements with order", "(a, b) differs from (b, a)", "An unordered pair or set {a, b} ignores order" ], "physical_properties": [], "families_and_kinds": [ "ordered pairs of numbers as coordinates", "ordered pairs in relations and functions", "directed edges in graphs", "key-value pairs in computing", "cryptographic key pairs" ], "related_models": [ { "relation": "is a kind of", "target": "tuple", "why": "category" }, { "relation": "is related to", "target": "Cartesian product", "why": "sets of pairs" }, { "relation": "is related to", "target": "function", "why": "defined as sets of pairs" }, { "relation": "is related to", "target": "graph theory", "why": "directed edges" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical notation conventions", "Data format standards for key-value structures" ], "failure_modes_and_hazards": [ "Confusing ordered and unordered pairs", "Confusing (a, b) with an open interval", "Mixing mathematical and computing senses" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "What it is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition and notation.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is an ordered pair, and how is it defined in set theory, for example by the Kuratowski definition?", "kind": "definition" }, { "text": "When are two ordered pairs equal?", "kind": "definition" } ] } ] }, { "id": "contrast", "name": "Contrast", "description": "Ordered versus unordered.", "findings": [ { "id": "contrast-finding", "name": "Contrast", "description": "Contrast.", "questions": [ { "text": "How does an ordered pair differ from a set of two elements or an interval?", "kind": "definition" }, { "text": "How does notation disambiguate?", "kind": "definition" } ] } ] } ] }, { "id": "maths", "name": "Maths", "description": "Mathematical uses.", "rationale": "Applications.", "layers": [ { "id": "coordinates", "name": "Coordinates", "description": "Coordinates and products.", "findings": [ { "id": "coordinates-finding", "name": "Coordinates", "description": "Coordinates.", "questions": [ { "text": "How do ordered pairs represent points in a plane and elements of a Cartesian product?", "kind": "definition" }, { "text": "How are relations and functions built from them?", "kind": "definition" } ] } ] }, { "id": "graphs", "name": "Graphs", "description": "Directed edges.", "findings": [ { "id": "graphs-finding", "name": "Graphs", "description": "Graphs.", "questions": [ { "text": "How are directed edges and directed hypergraphs represented with ordered pairs and tuples?", "kind": "definition" }, { "text": "How does this extend to n-tuples?", "kind": "definition" } ] } ] } ] }, { "id": "computing", "name": "Computing", "description": "Computing uses.", "rationale": "Data.", "layers": [ { "id": "keyvalue", "name": "Key-value", "description": "Key-value pairs.", "findings": [ { "id": "keyvalue-finding", "name": "Key-value", "description": "Key-value pairs.", "questions": [ { "text": "What is a key-value pair, and how is it used in maps, dictionaries and configuration?", "kind": "definition" }, { "text": "How does order matter in pairs versus in maps?", "kind": "definition" } ] } ] }, { "id": "keypair", "name": "Key pair", "description": "Cryptographic key pairs.", "findings": [ { "id": "keypair-finding", "name": "Key pair", "description": "Key pairs.", "questions": [ { "text": "What is a cryptographic key pair, and how do public and private keys relate?", "kind": "definition" }, { "text": "Is the mathematical or the cryptographic sense meant?", "kind": "boundary" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Teaching and history.", "rationale": "Education.", "layers": [ { "id": "teach", "name": "Teach", "description": "Teaching ordered pairs.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can ordered pairs be taught, from coordinates to functions?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] }, { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How was the ordered pair formalised, and what were earlier uses such as the Chinese four unknowns method?", "kind": "provenance" }, { "text": "Which definitions have been proposed?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should tuples be a single entry?", "How should notation ambiguities be handled?", "How should computing senses be linked?" ], "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 ordered pair, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain ordered pairs and their set-theoretic definition, show their uses across mathematics and computing, and distinguish them from unordered pairs and other structures.", "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: key pair, Si yuan shu, directed edge, key–value pair, ordered pair of points, directed hypergraph, hyperarc, (B, N) pair, irregular pair, complex128, integer couple, tuple of integers.", "21 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 52 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 ordered pair, 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": [ "key pair", "Si yuan shu", "directed edge", "key–value pair", "ordered pair of points", "directed hypergraph", "hyperarc", "(B, N) pair", "irregular pair", "complex128", "integer couple", "tuple of integers" ] } ], "standing": "Described in 52 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 21 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 ordered pair from another, and a ordered pair 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 ordered pair, 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 ordered pair, 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 ordered pair 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 ordered pair, 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 ordered pair, 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 ordered pair 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 ordered pair 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 ordered pair, 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 ordered pair, 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 ordered pair 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 ordered pair from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a ordered pair, 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 ordered pair, 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 ordered pair 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 ordered pair 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 ordered pair between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a ordered pair, 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 ordered pair, 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 ordered pair 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 ordered pair, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a ordered pair 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 ordered pair, 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 ordered pair, 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 ordered pair 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 ordered pair 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 ordered pair 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 ordered pair 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 ordered pair 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 ordered pair?", "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 ordered pair?", "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 ordered pair 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 ordered pair and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a ordered pair, 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 ordered pair, 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 ordered pair 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 ordered pair is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a ordered pair, 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 ordered pair, 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 ordered pair 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 } } } }