{ "model": { "rank": 5653, "code": "thing-q168817", "model_id": "vr.tr.algebraic-number", "name": "algebraic number", "purpose": "Let an agent handle algebraic numbers by definition, minimal polynomial, degree and related structures in number theory.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q168817/", "tier": 2, "score": 65, "payload": { "layer": "wikidata", "aliases": [ "ideal number", "square root of natural numbers", "algebraic integer", "Perron number" ], "aliasCount": 4, "merged": 4, "knownIn": 65, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.algebraic-number", "source": { "dataset": "wikidata", "item": "Q168817", "url": "https://www.wikidata.org/wiki/Q168817", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T09:42:29Z", "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-Q168817", "registryId": "vr.tr.algebraic-number", "name": "algebraic number", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q168817/", "model": { "registry_id": "vr.tr.algebraic-number", "name": "algebraic number", "purpose": "Let an agent handle algebraic numbers by definition, minimal polynomial, degree and related structures in number theory.", "definition": "A complex number that is a root of a non-zero polynomial with rational (equivalently integer) coefficients, such as square roots of integers and roots of unity; numbers that are not algebraic, like pi and e, are transcendental.", "what_it_is_for": "Number theory, algebra and exact computation.", "affordances": [ "find the minimal polynomial of a number", "determine degree", "distinguish algebraic and transcendental numbers", "work with algebraic integers" ], "distinguishing_features": [ "Countable set", "Closed under arithmetic operations", "Includes all rationals", "Algebraic integers form a ring" ], "appearance": "Numbers expressed with radicals or as polynomial roots.", "visual_identification": [ "Roots of integer polynomials", "Degree equals minimal polynomial degree", "Pi and e are transcendental" ], "physical_properties": [], "families_and_kinds": [ "rational numbers", "quadratic irrationals", "algebraic integers", "roots of unity", "Perron and Pisot numbers" ], "related_models": [ { "relation": "is a kind of", "target": "complex number", "why": "category" }, { "relation": "is contrasted with", "target": "transcendental number", "why": "complement" }, { "relation": "is studied in", "target": "algebraic number theory", "why": "field" }, { "relation": "forms", "target": "number field", "why": "structure" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 notation" ], "failure_modes_and_hazards": [ "Assuming irrational means transcendental", "Numerical approximation errors" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "definition", "name": "Definition", "description": "Is it algebraic.", "rationale": "Polynomials decide.", "layers": [ { "id": "test", "name": "Test", "description": "Polynomial.", "findings": [ { "id": "test-finding", "name": "Test", "description": "Algebraicity.", "questions": [ { "text": "Is this number a root of a polynomial with integer coefficients?", "kind": "boundary" }, { "text": "Which polynomial?", "kind": "definition" } ] } ] }, { "id": "transcendental", "name": "Transcendental", "description": "Contrast.", "findings": [ { "id": "transcendental-finding", "name": "Transcendental", "description": "Transcendence.", "questions": [ { "text": "Is the number known to be transcendental?", "kind": "provenance" }, { "text": "Or is its status open?", "kind": "boundary" } ] } ] } ] }, { "id": "structure", "name": "Structure", "description": "Degree.", "rationale": "Degree measures complexity.", "layers": [ { "id": "minimal", "name": "Minimal polynomial", "description": "Degree.", "findings": [ { "id": "minimal-finding", "name": "Minimal polynomial", "description": "Minimal polynomial.", "questions": [ { "text": "What is its minimal polynomial and degree?", "kind": "measurement" }, { "text": "What are its conjugates?", "kind": "definition" } ] } ] }, { "id": "integers", "name": "Algebraic integers", "description": "Monic polynomials.", "findings": [ { "id": "integers-finding", "name": "Algebraic integers", "description": "Algebraic integers.", "questions": [ { "text": "Is it an algebraic integer?", "kind": "boundary" }, { "text": "In which ring?", "kind": "definition" } ] } ] } ] }, { "id": "computation", "name": "Computation", "description": "Exact arithmetic.", "rationale": "Computers can handle them exactly.", "layers": [ { "id": "exact", "name": "Exact", "description": "Computer algebra.", "findings": [ { "id": "exact-finding", "name": "Exact", "description": "Exact computation.", "questions": [ { "text": "How can it be represented exactly in software?", "kind": "action" }, { "text": "Which system supports it?", "kind": "provenance" } ] } ] }, { "id": "approximation", "name": "Approximation", "description": "Numerics.", "findings": [ { "id": "approximation-finding", "name": "Approximation", "description": "Approximation.", "questions": [ { "text": "How accurate is the numerical approximation?", "kind": "measurement" }, { "text": "Could rounding mislead?", "kind": "boundary" } ] } ] } ] }, { "id": "theory", "name": "Theory", "description": "Number fields.", "rationale": "Algebraic numbers build fields.", "layers": [ { "id": "fields", "name": "Fields", "description": "Number fields.", "findings": [ { "id": "fields-finding", "name": "Fields", "description": "Number fields.", "questions": [ { "text": "Which number field does it generate?", "kind": "definition" }, { "text": "What is its degree?", "kind": "measurement" } ] } ] }, { "id": "history", "name": "History", "description": "Development.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did the theory develop, including ideal numbers?", "kind": "provenance" }, { "text": "Which mathematicians contributed?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should algebraic integers be a separate entry?", "How should computer algebra be linked?", "How should number fields 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 algebraic number, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent handle algebraic numbers by definition, minimal polynomial, degree and related structures in number theory.", "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: ideal number, square root of natural numbers, algebraic integer, Perron number.", "4 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 65 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 algebraic 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": [ "ideal number", "square root of natural numbers", "algebraic integer", "Perron number" ] } ], "standing": "Described in 65 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 4 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 algebraic number from another, and a algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic number from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic number between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a algebraic 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 algebraic 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 algebraic 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 algebraic number, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic 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 algebraic number and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a algebraic 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 algebraic 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 algebraic 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 algebraic number is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a algebraic 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 algebraic 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 algebraic 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 } } } }