{ "model": { "rank": 18202, "code": "thing-q50706", "model_id": "vr.tr.square-free-integer", "name": "square-free integer", "purpose": "Let an agent explain square-free integers by definition, tests, properties and links to number theory functions.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q50706/", "tier": 2, "score": 30, "payload": { "layer": "wikidata", "aliases": [ "Heegner number", "Zeisel number" ], "aliasCount": 2, "merged": 2, "knownIn": 30, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q50706", "url": "https://www.wikidata.org/wiki/Q50706", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T12:19:31Z", "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-Q50706", "registryId": "vr.tr.square-free-integer", "name": "square-free integer", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q50706/", "model": { "registry_id": "vr.tr.square-free-integer", "name": "square-free integer", "purpose": "Let an agent explain square-free integers by definition, tests, properties and links to number theory functions.", "definition": "A positive integer that is not divisible by any perfect square other than 1, so that in its prime factorisation no prime appears more than once; examples are 1, 2, 3, 5, 6, 7, 10, and the natural density of square-free integers is 6/pi^2, about 0.6079.", "what_it_is_for": "Number theory and algebra.", "affordances": [ "test whether a number is square-free", "explain the Mobius function connection", "compute the density of square-free numbers", "find sequences in OEIS" ], "distinguishing_features": [ "No square divisor above 1", "Distinct prime factors", "Density 6/pi^2", "Linked to the Mobius function" ], "appearance": "Not physical; integers and their factorisations.", "visual_identification": [ "No repeated prime factor", "Mobius function nonzero", "A square number is the opposite extreme" ], "physical_properties": [ { "quantity": "natural density", "typical_range": "6/pi^2, about 0.6079", "unit": "", "note": "share of integers that are square-free" } ], "families_and_kinds": [ "square-free integers", "primes (a subset)", "sphenic numbers", "square-free elements in rings" ], "related_models": [ { "relation": "is a kind of", "target": "positive integer", "why": "category" }, { "relation": "is characterised by", "target": "Mobius function", "why": "function" }, { "relation": "is related to", "target": "prime factorisation", "why": "concept" }, { "relation": "is listed in", "target": "OEIS", "why": "reference" } ], "identifiers": [ { "scheme": "OEIS", "value_or_pattern": "A005117", "note": "sequence of square-free numbers" } ], "standards_and_regulation": [ "No specific regulation" ], "failure_modes_and_hazards": [ "Forgetting that 1 is square-free", "Slow tests for very large numbers" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "definition", "name": "Definition", "description": "What it means.", "rationale": "Definition is simple.", "layers": [ { "id": "test", "name": "Test", "description": "Checking.", "findings": [ { "id": "test-finding", "name": "Test", "description": "Testing.", "questions": [ { "text": "Is this number square-free?", "kind": "boundary" }, { "text": "What is its prime factorisation?", "kind": "measurement" } ] } ] }, { "id": "examples", "name": "Examples", "description": "Examples.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Examples.", "questions": [ { "text": "Which numbers below a given bound are square-free?", "kind": "measurement" }, { "text": "Why is 1 included?", "kind": "definition" } ] } ] } ] }, { "id": "properties", "name": "Properties", "description": "Theory.", "rationale": "Properties link to functions.", "layers": [ { "id": "mobius", "name": "Mobius", "description": "Mobius function.", "findings": [ { "id": "mobius-finding", "name": "Mobius", "description": "Mobius function.", "questions": [ { "text": "How does the Mobius function detect square-free numbers?", "kind": "definition" }, { "text": "What is its value here?", "kind": "measurement" } ] } ] }, { "id": "density", "name": "Density", "description": "Proportion.", "findings": [ { "id": "density-finding", "name": "Density", "description": "Density.", "questions": [ { "text": "Why is the density of square-free numbers 6/pi^2?", "kind": "definition" }, { "text": "How is it related to the zeta function?", "kind": "definition" } ] } ] } ] }, { "id": "computation", "name": "Computation", "description": "Algorithms.", "rationale": "Large numbers are harder.", "layers": [ { "id": "algorithm", "name": "Algorithm", "description": "Tests.", "findings": [ { "id": "algorithm-finding", "name": "Algorithm", "description": "Algorithms.", "questions": [ { "text": "How can square-freeness be tested efficiently?", "kind": "definition" }, { "text": "Is it as hard as factoring?", "kind": "boundary" } ] } ] }, { "id": "sequence", "name": "Sequence", "description": "OEIS.", "findings": [ { "id": "sequence-finding", "name": "Sequence", "description": "Sequence data.", "questions": [ { "text": "What does the OEIS entry for square-free numbers contain?", "kind": "provenance" }, { "text": "Which related sequences exist?", "kind": "provenance" } ] } ] } ] }, { "id": "generalisation", "name": "Generalisation", "description": "Beyond integers.", "rationale": "Concept extends.", "layers": [ { "id": "rings", "name": "Rings", "description": "Algebra.", "findings": [ { "id": "rings-finding", "name": "Rings", "description": "Square-free elements.", "questions": [ { "text": "What does square-free mean for polynomials or ring elements?", "kind": "definition" }, { "text": "Which textbooks treat it?", "kind": "provenance" } ] } ] }, { "id": "teaching", "name": "Teaching", "description": "Explaining.", "findings": [ { "id": "teaching-finding", "name": "Teaching", "description": "Teaching.", "questions": [ { "text": "How can square-free numbers be explained with examples?", "kind": "action" }, { "text": "Which exercises help?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should related number classes be separate entries?", "How should OEIS be linked?", "How should algebraic generalisations 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-free integer, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain square-free integers by definition, tests, properties and links to number theory functions.", "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: Heegner number, Zeisel number.", "2 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 30 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-free integer, 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": [ "Heegner number", "Zeisel number" ] } ], "standing": "Described in 30 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 2 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-free integer from another, and a square-free integer 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-free integer, 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-free integer, 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-free integer 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-free integer, 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-free integer, 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-free integer 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-free integer 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-free integer, 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-free integer, 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-free integer 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-free integer from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a square-free integer, 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-free integer, 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-free integer 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-free integer 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-free integer between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a square-free integer, 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-free integer, 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-free integer 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-free integer, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a square-free integer 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-free integer, 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-free integer, 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-free integer 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-free integer 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-free integer 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-free integer 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-free integer 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-free integer and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a square-free integer, 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-free integer, 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-free integer 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-free integer is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a square-free integer, 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-free integer, 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-free integer 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 } } } }