{ "model": { "rank": 6927, "code": "thing-q50705", "model_id": "vr.tr.square-number", "name": "square number", "purpose": "Let an agent handle square numbers by definition, properties, tests and related number families.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q50705/", "tier": 2, "score": 58, "payload": { "layer": "wikidata", "aliases": [ "centered octagonal number", "square triangular number", "pentagonal square number" ], "aliasCount": 3, "merged": 3, "knownIn": 58, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q50705", "url": "https://www.wikidata.org/wiki/Q50705", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T10:36:56Z", "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-Q50705", "registryId": "vr.tr.square-number", "name": "square number", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q50705/", "model": { "registry_id": "vr.tr.square-number", "name": "square number", "purpose": "Let an agent handle square numbers by definition, properties, tests and related number families.", "definition": "An integer that is the square of an integer, such as 0, 1, 4, 9, 16 and 25; also a polygonal number that can be arranged as a square array of dots.", "what_it_is_for": "Arithmetic, number theory and geometry.", "affordances": [ "test whether a number is a perfect square", "list square numbers", "explain properties such as last digits", "relate to Pythagorean triples" ], "distinguishing_features": [ "n squared", "Sum of first n odd numbers", "Restricted last digits", "Basis of many identities" ], "appearance": "Numbers shown as square dot arrays.", "visual_identification": [ "n x n dot patterns", "Perfect squares such as 144", "Square roots undo squaring" ], "physical_properties": [], "families_and_kinds": [ "perfect squares", "square triangular numbers", "centred square numbers", "squares in Pythagorean triples" ], "related_models": [ { "relation": "is a kind of", "target": "polygonal number and perfect power", "why": "category" }, { "relation": "is related to", "target": "square root", "why": "inverse" }, { "relation": "appears in", "target": "Pythagorean theorem", "why": "geometry" }, { "relation": "is studied in", "target": "number theory", "why": "field" } ], "identifiers": [ { "scheme": "OEIS", "value_or_pattern": "A000290", "note": "integer sequence" } ], "standards_and_regulation": [], "failure_modes_and_hazards": [ "Floating point errors in square tests", "Confusing square and square root" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "test", "name": "Test", "description": "Is it square.", "rationale": "Tests identify squares.", "layers": [ { "id": "check", "name": "Check", "description": "Perfect square.", "findings": [ { "id": "check-finding", "name": "Check", "description": "Square test.", "questions": [ { "text": "Is this number a perfect square?", "kind": "boundary" }, { "text": "What is its integer square root?", "kind": "measurement" } ] } ] }, { "id": "quick", "name": "Quick tests", "description": "Last digits.", "findings": [ { "id": "quick-finding", "name": "Quick tests", "description": "Quick tests.", "questions": [ { "text": "Which last digits rule out a square?", "kind": "definition" }, { "text": "How reliable is the test?", "kind": "boundary" } ] } ] } ] }, { "id": "properties", "name": "Properties", "description": "Patterns.", "rationale": "Squares have patterns.", "layers": [ { "id": "odd-sums", "name": "Odd sums", "description": "Sum of odds.", "findings": [ { "id": "odd-sums-finding", "name": "Odd sums", "description": "Sums of odd numbers.", "questions": [ { "text": "Why is n squared the sum of the first n odd numbers?", "kind": "definition" }, { "text": "How can it be shown visually?", "kind": "action" } ] } ] }, { "id": "differences", "name": "Differences", "description": "Gaps.", "findings": [ { "id": "differences-finding", "name": "Differences", "description": "Differences.", "questions": [ { "text": "What is the difference between consecutive squares?", "kind": "measurement" }, { "text": "Why?", "kind": "definition" } ] } ] } ] }, { "id": "related", "name": "Related", "description": "Families.", "rationale": "Squares link to other ideas.", "layers": [ { "id": "pythagorean", "name": "Pythagorean", "description": "Triples.", "findings": [ { "id": "pythagorean-finding", "name": "Pythagorean", "description": "Pythagorean triples.", "questions": [ { "text": "Which Pythagorean triples involve these squares?", "kind": "definition" }, { "text": "How are they generated?", "kind": "action" } ] } ] }, { "id": "special", "name": "Special", "description": "Square triangular.", "findings": [ { "id": "special-finding", "name": "Special", "description": "Special squares.", "questions": [ { "text": "Is this number also triangular or pentagonal?", "kind": "boundary" }, { "text": "Which sequence lists such numbers?", "kind": "provenance" } ] } ] } ] }, { "id": "computing", "name": "Computing", "description": "Code.", "rationale": "Code must be exact.", "layers": [ { "id": "exact", "name": "Exact", "description": "Integer arithmetic.", "findings": [ { "id": "exact-finding", "name": "Exact", "description": "Exact computation.", "questions": [ { "text": "How can a perfect square be tested exactly in code?", "kind": "action" }, { "text": "Why avoid floating point?", "kind": "boundary" } ] } ] }, { "id": "teaching", "name": "Teaching", "description": "Learning.", "findings": [ { "id": "teaching-finding", "name": "Teaching", "description": "Teaching.", "questions": [ { "text": "How can squares be taught with arrays?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "boundary" } ] } ] } ] } ] }, "openQuestions": [ "Should special square families be separate entries?", "How should OEIS sequences be linked?", "How should geometric squares 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 square number, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent handle square numbers by definition, properties, tests and related number families.", "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: centered octagonal number, square triangular number, pentagonal square number.", "3 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 58 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 square 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": [ "centered octagonal number", "square triangular number", "pentagonal square number" ] } ], "standing": "Described in 58 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 3 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 square number from another, and a square 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 square 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 square 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 square 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 square 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 square 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 square 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 square 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 square 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 square 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 square 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 square number from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a square 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 square 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 square 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 square 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 square number between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a square 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 square 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 square 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 square number, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a square 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 square 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 square 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 square 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 square 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 square 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 square 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 square 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 square number and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a square 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 square 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 square 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 square number is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a square 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 square 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 square 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 } } } }