{ "model": { "rank": 7824, "code": "thing-q381040", "model_id": "vr.tr.complex-conjugate", "name": "complex conjugate", "purpose": "Let an agent explain the complex conjugate and its properties, show how it is used in computation and theory, and support learning of complex numbers.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q381040/", "tier": 2, "score": 53, "payload": { "layer": "wikidata", "aliases": [], "aliasCount": 0, "merged": 0, "knownIn": 53, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.complex-conjugate", "source": { "dataset": "wikidata", "item": "Q381040", "url": "https://www.wikidata.org/wiki/Q381040", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T21:25:36Z", "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-Q381040", "registryId": "vr.tr.complex-conjugate", "name": "complex conjugate", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q381040/", "model": { "registry_id": "vr.tr.complex-conjugate", "name": "complex conjugate", "purpose": "Let an agent explain the complex conjugate and its properties, show how it is used in computation and theory, and support learning of complex numbers.", "definition": "For a complex number a plus b i, the complex number a minus b i obtained by reversing the sign of the imaginary part, denoted with a bar or an asterisk, so that a number multiplied by its conjugate gives the square of its modulus; complex conjugation is used to divide complex numbers, find moduli, describe roots of real polynomials, and in physics and signal processing.", "what_it_is_for": "Operations with complex numbers.", "affordances": [ "define and compute the conjugate", "explain its properties", "show uses in division, moduli and polynomials", "describe uses in physics and engineering" ], "distinguishing_features": [ "Involution", "Preserves modulus", "Respects addition and multiplication", "Geometrically a reflection" ], "appearance": "Not physical; a reflection across the real axis in the complex plane.", "visual_identification": [ "Sign of imaginary part reversed", "Reflection in the real axis", "The negative of a complex number reverses both parts" ], "physical_properties": [], "families_and_kinds": [ "conjugate of a complex number", "conjugate roots of real polynomials", "conjugate transpose of matrices", "conjugation in quaternions and other algebras" ], "related_models": [ { "relation": "is a kind of", "target": "complex number", "why": "category" }, { "relation": "is related to", "target": "modulus", "why": "via product" }, { "relation": "is related to", "target": "complex plane", "why": "geometric view" }, { "relation": "is related to", "target": "polynomial", "why": "conjugate root theorem" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical notation conventions" ], "failure_modes_and_hazards": [ "Confusing conjugate with negative or reciprocal", "Notation clashes between bar and asterisk", "Forgetting conjugation in inner products" ], "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 the complex conjugate of a given number, and how is it written?", "kind": "definition" }, { "text": "How is it seen geometrically in the complex plane?", "kind": "definition" } ] } ] }, { "id": "properties", "name": "Properties", "description": "Properties.", "findings": [ { "id": "properties-finding", "name": "Properties", "description": "Properties.", "questions": [ { "text": "Which properties does conjugation have with respect to sums, products, moduli and inverses?", "kind": "definition" }, { "text": "Why is conjugation an involution?", "kind": "definition" } ] } ] } ] }, { "id": "compute", "name": "Compute", "description": "Using the conjugate.", "rationale": "Calculation.", "layers": [ { "id": "division", "name": "Division", "description": "Division and moduli.", "findings": [ { "id": "division-finding", "name": "Division", "description": "Division.", "questions": [ { "text": "How is the conjugate used to divide complex numbers and compute moduli?", "kind": "action" }, { "text": "How is a given expression simplified using conjugates?", "kind": "action" } ] } ] }, { "id": "roots", "name": "Roots", "description": "Conjugate roots.", "findings": [ { "id": "roots-finding", "name": "Roots", "description": "Roots.", "questions": [ { "text": "Why do non-real roots of real polynomials occur in conjugate pairs?", "kind": "definition" }, { "text": "How is this used to factor polynomials?", "kind": "action" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Science and engineering.", "layers": [ { "id": "physics", "name": "Physics", "description": "Quantum mechanics and waves.", "findings": [ { "id": "physics-finding", "name": "Physics", "description": "Physics.", "questions": [ { "text": "How is complex conjugation used in quantum mechanics, wave analysis and inner products?", "kind": "definition" }, { "text": "What is the conjugate transpose of a matrix?", "kind": "definition" } ] } ] }, { "id": "signals", "name": "Signals", "description": "Signal processing.", "findings": [ { "id": "signals-finding", "name": "Signals", "description": "Signals.", "questions": [ { "text": "How does conjugation appear in Fourier transforms and matched filters?", "kind": "definition" }, { "text": "How do software libraries implement it?", "kind": "provenance" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Teaching.", "rationale": "Education.", "layers": [ { "id": "teach", "name": "Teach", "description": "Teaching conjugates.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can the complex conjugate be taught with geometry and examples?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] }, { "id": "extend", "name": "Extend", "description": "Generalisations.", "findings": [ { "id": "extend-finding", "name": "Extend", "description": "Generalisations.", "questions": [ { "text": "How does conjugation generalise to quaternions, field extensions and algebras?", "kind": "definition" }, { "text": "Which entry fits those topics?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should complex numbers be a single entry?", "How should notation conventions be linked?", "How should applications 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 complex conjugate, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain the complex conjugate and its properties, show how it is used in computation and theory, and support learning of complex numbers.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Described in 53 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 complex conjugate, 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": [], "standing": "Described in 53 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one complex conjugate from another, and a complex conjugate 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 complex conjugate, 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 complex conjugate, 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 complex conjugate 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 complex conjugate, 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 complex conjugate, 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 complex conjugate 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 complex conjugate 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 complex conjugate, 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 complex conjugate, 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 complex conjugate 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 complex conjugate from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a complex conjugate, 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 complex conjugate, 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 complex conjugate 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 complex conjugate 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 complex conjugate between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a complex conjugate, 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 complex conjugate, 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 complex conjugate 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 complex conjugate, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a complex conjugate 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 complex conjugate, 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 complex conjugate, 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 complex conjugate 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 complex conjugate 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 complex conjugate 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 complex conjugate 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 complex conjugate 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 complex conjugate and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a complex conjugate, 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 complex conjugate, 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 complex conjugate 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 complex conjugate is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a complex conjugate, 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 complex conjugate, 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 complex conjugate 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 } } } }