{ "model": { "rank": 8242, "code": "thing-q245102", "model_id": "vr.tr.triangular-number", "name": "triangular number", "purpose": "Let an agent explain triangular numbers by formula, properties, proofs and applications for learners and puzzle solvers.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q245102/", "tier": 2, "score": 52, "payload": { "layer": "wikidata", "aliases": [ "pentagonal triangular number" ], "aliasCount": 1, "merged": 1, "knownIn": 52, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q245102", "url": "https://www.wikidata.org/wiki/Q245102", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T15:11:43Z", "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": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q245102", "registryId": "vr.tr.triangular-number", "name": "triangular number", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q245102/", "model": { "registry_id": "vr.tr.triangular-number", "name": "triangular number", "purpose": "Let an agent explain triangular numbers by formula, properties, proofs and applications for learners and puzzle solvers.", "definition": "A number that counts the objects in an equilateral triangular arrangement, given by n(n+1)/2 for the nth triangular number: 1, 3, 6, 10, 15 and so on; triangular numbers appear in combinatorics as the number of pairs among n+1 items, and some are also square or pentagonal.", "what_it_is_for": "Number theory, combinatorics and recreational mathematics.", "affordances": [ "compute the nth triangular number", "test whether a number is triangular", "explain the handshake and pairing connection", "relate to square and other figurate numbers" ], "distinguishing_features": [ "Figurate number", "Sum of first n integers", "Counts pairs", "Simple test for membership" ], "appearance": "Not physical; dots arranged in triangles and sequences of numbers.", "visual_identification": [ "Triangular dot patterns", "Formula n(n+1)/2", "Square numbers form square arrays" ], "physical_properties": [], "families_and_kinds": [ "triangular numbers", "square triangular numbers", "pentagonal triangular numbers", "tetrahedral numbers (3D analogue)" ], "related_models": [ { "relation": "is a kind of", "target": "polygonal number", "why": "category" }, { "relation": "is related to", "target": "square-free integer", "why": "number theory" }, { "relation": "is related to", "target": "Gaussian integer", "why": "number theory" }, { "relation": "is listed in", "target": "OEIS", "why": "reference" } ], "identifiers": [ { "scheme": "OEIS", "value_or_pattern": "A000217", "note": "triangular numbers" } ], "standards_and_regulation": [ "No specific regulation" ], "failure_modes_and_hazards": [ "Off-by-one errors in indexing", "Confusing with tetrahedral numbers" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "compute", "name": "Compute", "description": "Formula.", "rationale": "The formula is simple.", "layers": [ { "id": "nth", "name": "Nth", "description": "Computing values.", "findings": [ { "id": "nth-finding", "name": "Nth", "description": "Computing.", "questions": [ { "text": "What is the nth triangular number for this n?", "kind": "measurement" }, { "text": "How is the formula derived?", "kind": "definition" } ] } ] }, { "id": "test", "name": "Test", "description": "Membership test.", "findings": [ { "id": "test-finding", "name": "Test", "description": "Testing.", "questions": [ { "text": "Is this number triangular?", "kind": "boundary" }, { "text": "Which test using 8n+1 applies?", "kind": "definition" } ] } ] } ] }, { "id": "properties", "name": "Properties", "description": "Facts.", "rationale": "Properties link to other numbers.", "layers": [ { "id": "pairs", "name": "Pairs", "description": "Counting pairs.", "findings": [ { "id": "pairs-finding", "name": "Pairs", "description": "Counting pairs.", "questions": [ { "text": "Why does the number of handshakes among n people equal a triangular number?", "kind": "definition" }, { "text": "How does it relate to binomial coefficients?", "kind": "definition" } ] } ] }, { "id": "squares", "name": "Squares", "description": "Square triangular numbers.", "findings": [ { "id": "squares-finding", "name": "Squares", "description": "Square triangular numbers.", "questions": [ { "text": "Which numbers are both triangular and square?", "kind": "provenance" }, { "text": "How are they found?", "kind": "definition" } ] } ] } ] }, { "id": "proofs", "name": "Proofs", "description": "Reasoning.", "rationale": "Proofs teach.", "layers": [ { "id": "visual", "name": "Visual", "description": "Visual proofs.", "findings": [ { "id": "visual-finding", "name": "Visual", "description": "Visual proofs.", "questions": [ { "text": "How does a picture show that two triangular numbers form a rectangle?", "kind": "action" }, { "text": "Which other visual proofs exist?", "kind": "provenance" } ] } ] }, { "id": "induction", "name": "Induction", "description": "Formal proof.", "findings": [ { "id": "induction-finding", "name": "Induction", "description": "Induction.", "questions": [ { "text": "How is the formula proved by induction?", "kind": "definition" }, { "text": "What is the Gauss anecdote, according to historians?", "kind": "provenance" } ] } ] } ] }, { "id": "learning", "name": "Learning", "description": "Teaching.", "rationale": "Patterns engage learners.", "layers": [ { "id": "teach", "name": "Teach", "description": "Classroom use.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "Which activities introduce triangular numbers to learners?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] }, { "id": "puzzles", "name": "Puzzles", "description": "Recreational maths.", "findings": [ { "id": "puzzles-finding", "name": "Puzzles", "description": "Puzzles.", "questions": [ { "text": "Which puzzles use triangular numbers, such as bowling pin arrangements?", "kind": "provenance" }, { "text": "How can they be solved?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should figurate number families be separate entries?", "How should OEIS be linked?", "How should puzzles be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "unit, quantity or measure", "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 triangular number, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain triangular numbers by formula, properties, proofs and applications for learners and puzzle solvers.", "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: pentagonal triangular number.", "1 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 triangular number, 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: a unit is a convention. What is seen is an instrument reading in it." ], "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": [ "pentagonal triangular number" ] } ], "standing": "Described in 52 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 1 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 triangular number from another, and a triangular number 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 triangular number, 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 triangular number, 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 triangular number 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 triangular number, 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 triangular number, 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 triangular number 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 triangular number 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 triangular number, 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 triangular number, 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 triangular number 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 triangular number from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a triangular number, 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 triangular number, 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 triangular number 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 triangular number 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 triangular number between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a triangular number, 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 triangular number, 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 triangular number 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 triangular number, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a triangular number 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 triangular number, 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 triangular number, 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 triangular number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definition-and-realisation", "name": "Definition, dimension and realisation", "description": "What triangular number is defined as and how it is realised in practice.", "rationale": "A unit is not the quantity it measures, and its definition has a history that older data still depends on.", "layers": [ { "id": "definition", "name": "Definition and dimension", "description": "The defining relation and the quantity it measures.", "findings": [ { "id": "definition-record", "name": "Definition, body and date", "description": "What defines it now, and what defined it before.", "questions": [ { "id": "definition-record-q01", "text": "How is triangular number defined, by which body, and since when?", "kind": "definition" }, { "id": "definition-record-q02", "text": "What was the previous definition, and what data still depends on it?", "kind": "provenance" } ] } ] }, { "id": "conversion", "name": "Conversion and compatibility", "description": "Relations to other units, exact or approximate.", "findings": [ { "id": "conversion-record", "name": "Conversions and their exactness", "description": "Factors, and whether each is exact by definition.", "questions": [ { "id": "conversion-record-q01", "text": "What conversions relate triangular number to other units, and which are exact by definition?", "kind": "measurement" }, { "id": "conversion-record-q02", "text": "Where does rounding a conversion change a decision, and how should that be recorded?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a triangular number 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 triangular number and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a triangular number, 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 triangular number, 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 triangular number 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 triangular number is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a triangular number, 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 triangular number, 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 triangular number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 4, "layers": 8, "findings": 10, "questions": 28 } } } }