{ "model": { "rank": 9471, "code": "thing-q214881", "model_id": "vr.tr.curvature", "name": "curvature", "purpose": "Let an agent explain curvature and its kinds, compute curvature for curves and surfaces, relate curvature to physics and engineering applications, and support learning.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.STA", "industry": "", "version": "", "url": "/models/thing/q214881/", "tier": 2, "score": 47, "payload": { "layer": "wikidata", "aliases": [ "optical corrections", "positive curvature", "scalar curvature", "space curvature", "mean curvature", "normal curvature", "geodesic curvature", "principal curvature", "torsion of a curve", "constant curvature", "spacetime curvature", "meridional curvature" ], "aliasCount": 12, "merged": 12, "knownIn": 47, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.curvature", "source": { "dataset": "wikidata", "item": "Q214881", "url": "https://www.wikidata.org/wiki/Q214881", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T00:11:08Z", "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-Q214881", "registryId": "vr.tr.curvature", "name": "curvature", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.STA" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q214881/", "model": { "registry_id": "vr.tr.curvature", "name": "curvature", "purpose": "Let an agent explain curvature and its kinds, compute curvature for curves and surfaces, relate curvature to physics and engineering applications, and support learning.", "definition": "A measure of how much a curve, surface or space deviates from being straight or flat, expressed as a reciprocal length for curves and in forms such as Gaussian, mean, normal, scalar and sectional curvature for surfaces and manifolds, positive or negative in sign; curvature is central to differential geometry, describes lenses and optical corrections, roads and rails, and in general relativity the curvature of spacetime by mass and energy.", "what_it_is_for": "Deviation from straightness or flatness.", "affordances": [ "explain kinds of curvature", "compute curvature", "relate to physics and engineering", "support learning" ], "distinguishing_features": [ "Reciprocal length", "Signed", "Intrinsic and extrinsic forms", "Broad application" ], "appearance": "Not physical; a quantity describing bending.", "visual_identification": [ "Reciprocal of radius for curves", "Gaussian, mean, scalar for surfaces and spaces", "Slope measures steepness, not bending" ], "physical_properties": [], "families_and_kinds": [ "curvature of plane and space curves", "Gaussian and mean curvature of surfaces", "sectional, Ricci and scalar curvature of manifolds", "curvature in optics", "spacetime curvature" ], "related_models": [ { "relation": "is a kind of", "target": "physical quantity", "why": "category" }, { "relation": "is a kind of", "target": "geometric measure", "why": "category" }, { "relation": "is related to", "target": "trapezoid", "why": "another geometric entry" }, { "relation": "is related to", "target": "space", "why": "curved space" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical definitions and sign conventions", "Road and rail curvature design standards", "Optical surface specifications" ], "failure_modes_and_hazards": [ "Sign convention confusion", "Confusing intrinsic and extrinsic curvature", "Unsafe road or rail curvature" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What curvature is.", "rationale": "Mathematics.", "layers": [ { "id": "curves", "name": "Curves", "description": "Curvature of curves.", "findings": [ { "id": "curves-finding", "name": "Curves", "description": "Curves.", "questions": [ { "text": "How is the curvature of a curve defined and computed, and what is the radius of curvature?", "kind": "definition" }, { "text": "What is torsion for space curves?", "kind": "definition" } ] } ] }, { "id": "surfaces", "name": "Surfaces", "description": "Surfaces and manifolds.", "findings": [ { "id": "surfaces-finding", "name": "Surfaces", "description": "Surfaces.", "questions": [ { "text": "What are Gaussian, mean, normal, sectional and scalar curvature, and which are intrinsic?", "kind": "definition" }, { "text": "Which entry fits a specific curvature?", "kind": "action" } ] } ] } ] }, { "id": "compute", "name": "Compute", "description": "Calculations.", "rationale": "Practice.", "layers": [ { "id": "calculate", "name": "Calculate", "description": "Computing curvature.", "findings": [ { "id": "calculate-finding", "name": "Calculate", "description": "Computing.", "questions": [ { "text": "How is the curvature of this curve or surface computed, and which sign convention applies?", "kind": "action" }, { "text": "Which tools compute curvature numerically?", "kind": "provenance" } ] } ] }, { "id": "interpret", "name": "Interpret", "description": "Interpreting curvature.", "findings": [ { "id": "interpret-finding", "name": "Interpret", "description": "Interpretation.", "questions": [ { "text": "What do positive, negative and zero curvature mean geometrically?", "kind": "definition" }, { "text": "How does the Gauss-Bonnet theorem relate curvature to topology?", "kind": "definition" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Context.", "layers": [ { "id": "physics", "name": "Physics", "description": "Curvature in physics.", "findings": [ { "id": "physics-finding", "name": "Physics", "description": "Physics.", "questions": [ { "text": "How does general relativity describe gravity as spacetime curvature?", "kind": "provenance" }, { "text": "Which entry fits general relativity?", "kind": "action" } ] } ] }, { "id": "engineering", "name": "Engineering", "description": "Engineering and optics.", "findings": [ { "id": "engineering-finding", "name": "Engineering", "description": "Engineering.", "questions": [ { "text": "How is curvature used in road and rail design, lens optics and optical corrections, and manufacturing?", "kind": "provenance" }, { "text": "Which standards apply?", "kind": "provenance" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "History and teaching.", "rationale": "Education.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did the concept of curvature develop from Newton and Euler to Gauss and Riemann?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "teach", "name": "Teach", "description": "Teaching.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can curvature be taught with visual and hands-on examples?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should each curvature type be a separate entry?", "How should engineering standards be linked?", "How should physics resources be linked?" ], "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 curvature, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain curvature and its kinds, compute curvature for curves and surfaces, relate curvature to physics and engineering applications, and support learning.", "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: optical corrections, positive curvature, scalar curvature, space curvature, mean curvature, normal curvature, geodesic curvature, principal curvature, torsion of a curve, constant curvature, spacetime curvature, meridional curvature.", "12 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 47 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 curvature, 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": [ "optical corrections", "positive curvature", "scalar curvature", "space curvature", "mean curvature", "normal curvature", "geodesic curvature", "principal curvature", "torsion of a curve", "constant curvature", "spacetime curvature", "meridional curvature" ] } ], "standing": "Described in 47 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 12 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 curvature from another, and a curvature 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 curvature, 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 curvature, 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 curvature 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 curvature, 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 curvature, 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 curvature 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 curvature 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 curvature, 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 curvature, 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 curvature 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 curvature from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a curvature, 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 curvature, 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 curvature 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 curvature 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 curvature between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a curvature, 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 curvature, 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 curvature 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 curvature, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a curvature 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 curvature, 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 curvature, 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 curvature 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 curvature 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 curvature 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 curvature 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 curvature 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 curvature?", "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 curvature?", "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 curvature 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 curvature and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a curvature, 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 curvature, 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 curvature 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 curvature is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a curvature, 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 curvature, 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 curvature 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 } } } }