{ "model": { "rank": 2547, "code": "thing-q134237", "model_id": "vr.tr.square-root", "name": "square root", "purpose": "Let an agent explain square roots and their properties, relay computation methods, extensions to complex numbers and matrices, and history from mathematical references, describe applications, and distinguish the principal square root from other roots, from squaring and from higher nth roots.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q134237/", "tier": 2, "score": 103, "payload": { "layer": "wikidata", "aliases": [ "square root of a matrix", "square root of a 2 by 2 matrix", "theta characteristic" ], "aliasCount": 3, "merged": 3, "knownIn": 103, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q134237", "url": "https://www.wikidata.org/wiki/Q134237", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T13:11:03Z", "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": 4, "engine": "claude" }, "metaModel": { "id": "THING-Q134237", "registryId": "vr.tr.square-root", "name": "square root", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q134237/", "model": { "registry_id": "vr.tr.square-root", "name": "square root", "purpose": "Let an agent explain square roots and their properties, relay computation methods, extensions to complex numbers and matrices, and history from mathematical references, describe applications, and distinguish the principal square root from other roots, from squaring and from higher nth roots.", "definition": "A number that, multiplied by itself, gives a given number, so that every positive real number has a positive principal square root and a negative one, denoted by the radical sign, with extensions to complex numbers, to matrices where a square root of a matrix is a matrix whose square is the original, including explicit formulas for 2 by 2 matrices, and to abstract settings such as theta characteristics in algebraic geometry; square roots appear throughout mathematics, science and computation.", "what_it_is_for": "Not applicable; a mathematical operation.", "affordances": [ "explain the concept", "relay computation and extensions", "describe applications", "distinguish related operations" ], "distinguishing_features": [ "Inverse of squaring", "Two roots for positive numbers", "Irrational values", "Matrix and abstract extensions" ], "appearance": "Not a visible object; the radical symbol.", "visual_identification": [ "Number whose square is a given number", "Principal square root, square root of a matrix, 2 by 2 matrix formula, theta characteristic", "Cube and nth roots are other roots; squaring is the inverse operation" ], "physical_properties": [ { "quantity": "square root of 2", "typical_range": "1.41421356", "unit": "value", "note": "irrational" }, { "quantity": "radical sign introduced", "typical_range": "1525", "unit": "year", "note": "Christoph Rudolff" }, { "quantity": "Newton s method", "typical_range": "converges quadratically", "unit": "note", "note": "" } ], "families_and_kinds": [ "square roots of positive reals", "square roots of negative numbers and complex numbers", "square roots of matrices including the 2 by 2 case", "square roots in modular arithmetic", "theta characteristics as square roots of the canonical bundle", "algorithms for computing square roots" ], "related_models": [ { "relation": "is a kind of", "target": "nth root", "why": "in registry terms" }, { "relation": "is the inverse of", "target": "square", "why": "as an operation" }, { "relation": "is extended to", "target": "matrix", "why": "as matrix square roots" }, { "relation": "is computed by", "target": "Newton s method", "why": "among other algorithms" } ], "identifiers": [], "standards_and_regulation": [ "IEEE 754 specifies correctly rounded square root", "No regulation" ], "failure_modes_and_hazards": [ "Forgetting the negative root or the principal root convention", "Errors with square roots of negatives", "Non-unique matrix square roots" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What a square root is.", "rationale": "Science.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a square root, what is the principal root, and how does it differ from other roots and from squaring?", "kind": "definition" }, { "text": "Is the question about real numbers, complex numbers, matrices or algebraic geometry?", "kind": "boundary" } ] } ] }, { "id": "properties", "name": "Properties", "description": "Properties.", "findings": [ { "id": "properties-finding", "name": "Properties", "description": "Properties.", "questions": [ { "text": "What are the algebraic properties of square roots, and why is the square root of 2 irrational?", "kind": "definition" }, { "text": "Which entry fits irrational number?", "kind": "action" } ] } ] } ] }, { "id": "compute", "name": "Compute", "description": "Computation.", "rationale": "Practice.", "layers": [ { "id": "methods", "name": "Methods", "description": "Methods.", "findings": [ { "id": "methods-finding", "name": "Methods", "description": "Methods.", "questions": [ { "text": "How are square roots computed by hand, by Newton s method and in hardware?", "kind": "action" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "numeric", "name": "Numeric", "description": "Numerical issues.", "findings": [ { "id": "numeric-finding", "name": "Numeric", "description": "Numeric.", "questions": [ { "text": "How do floating-point systems compute square roots, and what precision issues arise?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "extend", "name": "Extend", "description": "Extensions.", "rationale": "Science.", "layers": [ { "id": "complex", "name": "Complex", "description": "Complex numbers.", "findings": [ { "id": "complex-finding", "name": "Complex", "description": "Complex.", "questions": [ { "text": "How are square roots of negative and complex numbers defined, and what is the branch cut?", "kind": "provenance" }, { "text": "Which entry fits complex number?", "kind": "action" } ] } ] }, { "id": "matrices", "name": "Matrices", "description": "Matrices and geometry.", "findings": [ { "id": "matrices-finding", "name": "Matrices", "description": "Matrices.", "questions": [ { "text": "What is a square root of a matrix, how is the 2 by 2 case solved, and what is a theta characteristic?", "kind": "provenance" }, { "text": "Which entry fits square root of a matrix?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Applications and history.", "rationale": "Context.", "layers": [ { "id": "applications", "name": "Applications", "description": "Applications.", "findings": [ { "id": "applications-finding", "name": "Applications", "description": "Applications.", "questions": [ { "text": "Where do square roots appear in geometry, statistics, physics and computing?", "kind": "provenance" }, { "text": "Which entry fits Pythagorean theorem?", "kind": "action" } ] } ] }, { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did Babylonian, Greek, Indian and Renaissance mathematicians treat square roots?", "kind": "provenance" }, { "text": "Which entry fits the history of mathematics?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should square root of a matrix and theta characteristic be separate entries?", "How should mathematical references be linked?", "The registry entry has merged aliases naming an advanced concept in algebraic geometry; should it be split off?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "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 root, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain square roots and their properties, relay computation methods, extensions to complex numbers and matrices, and history from mathematical references, describe applications, and distinguish the principal square root from other roots, from squaring and from higher nth roots.", "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: square root of a matrix, square root of a 2 by 2 matrix, theta characteristic.", "3 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 103 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 root, 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. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "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": [ "square root of a matrix", "square root of a 2 by 2 matrix", "theta characteristic" ] } ], "standing": "Described in 103 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 root from another, and a square root 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 root, 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 root, 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 root 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 root, 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 root, 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 root 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 root 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 root, 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 root, 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 root 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 root from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a square root, 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 root, 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 root 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 root 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 root between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a square root, 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 root, 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 root 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 root, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a square root 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 root, 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 root, 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 root is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What square root is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of square root are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is square root operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of square root and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of square root?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as square root?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a square root 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 root and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a square root, 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 root, 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 root 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 root is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a square root, 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 root, 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 root is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }