{ "model": { "rank": 4865, "code": "thing-q12510", "model_id": "vr.tr.torus", "name": "torus", "purpose": "Let an agent explain the torus, relay geometry, topology, higher-dimensional and algebraic forms and applications from mathematics and physics sources, describe the named forms and flag the loosely filed alias, and distinguish the torus from the sphere, annulus, solid torus, Klein bottle and toroidal inductors as objects.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q12510/", "tier": 2, "score": 72, "payload": { "layer": "wikidata", "aliases": [ "tie ring", "maximal torus", "Clifford torus", "supertoroid" ], "aliasCount": 4, "merged": 4, "knownIn": 72, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.torus", "source": { "dataset": "wikidata", "item": "Q12510", "url": "https://www.wikidata.org/wiki/Q12510", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T23:23:27Z", "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-Q12510", "registryId": "vr.tr.torus", "name": "torus", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q12510/", "model": { "registry_id": "vr.tr.torus", "name": "torus", "purpose": "Let an agent explain the torus, relay geometry, topology, higher-dimensional and algebraic forms and applications from mathematics and physics sources, describe the named forms and flag the loosely filed alias, and distinguish the torus from the sphere, annulus, solid torus, Klein bottle and toroidal inductors as objects.", "definition": "A surface of revolution generated by revolving a circle in three-dimensional space about an axis in its plane that does not intersect the circle, forming a doughnut shape, and in topology any space homeomorphic to the product of two circles, a closed orientable surface of genus one; related forms the registry aliases name include the Clifford torus, a flat torus embedded in four-dimensional space, maximal tori, abelian subgroups central to Lie group theory, and supertoroids, generalised torus shapes used in computer graphics; the alias tie ring may refer to a ring-shaped mechanical or fastening part and appears loosely filed. Tori appear in physics as the shape of tokamak plasma chambers.", "what_it_is_for": "Describing ring-shaped surfaces and spaces.", "affordances": [ "explain geometry and topology", "relay higher-dimensional and algebraic forms", "describe named forms", "relay applications" ], "distinguishing_features": [ "Genus one", "Product of circles", "Flat metrics possible", "Many applications" ], "appearance": "A doughnut or ring shape.", "visual_identification": [ "Doughnut-shaped surface", "Tie ring, maximal torus, Clifford torus, supertoroid, ring torus", "Spheres have genus zero; annuli are flat rings; solid tori include the interior; Klein bottles are non-orientable; toroidal inductors are components" ], "physical_properties": [ { "quantity": "genus", "typical_range": "1", "unit": "count", "note": "" }, { "quantity": "Euler characteristic", "typical_range": "0", "unit": "count", "note": "" }, { "quantity": "surface area", "typical_range": "4 pi squared R r", "unit": "formula", "note": "ring torus" } ], "families_and_kinds": [ "ring, horn and spindle tori", "flat tori such as the Clifford torus", "n-dimensional tori", "maximal tori in Lie groups", "supertoroids" ], "related_models": [ { "relation": "is a kind of", "target": "closed manifold", "why": "in registry terms" }, { "relation": "is the product of", "target": "circle", "why": "two circles" }, { "relation": "includes", "target": "Clifford torus", "why": "" }, { "relation": "is contrasted with", "target": "Klein bottle", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "No regulation" ], "failure_modes_and_hazards": [ "Confusing torus and solid torus", "Loosely filed mechanical alias", "Confusing embedded and abstract tori" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What a torus is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a torus, and how does it differ from a sphere, annulus, solid torus, Klein bottle and toroidal components?", "kind": "definition" }, { "text": "Is the question about geometry, topology, Lie groups, graphics or physics?", "kind": "boundary" } ] } ] }, { "id": "forms", "name": "Forms", "description": "Named forms.", "findings": [ { "id": "forms-finding", "name": "Forms", "description": "Forms.", "questions": [ { "text": "What are Clifford tori, maximal tori and supertoroids, and what might tie ring refer to?", "kind": "definition" }, { "text": "Which entry fits the specific form?", "kind": "action" } ] } ] } ] }, { "id": "geometry", "name": "Geometry", "description": "Geometry and topology.", "rationale": "Science.", "layers": [ { "id": "surface", "name": "Surface", "description": "Surface of revolution.", "findings": [ { "id": "surface-finding", "name": "Surface", "description": "Surface.", "questions": [ { "text": "How are area and volume of a ring torus calculated?", "kind": "measurement" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "topology", "name": "Topology", "description": "Topology.", "findings": [ { "id": "topology-finding", "name": "Topology", "description": "Topology.", "questions": [ { "text": "Why is the torus a genus-one surface, and how is it built from a square?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "advanced", "name": "Advanced", "description": "Higher forms.", "rationale": "Science.", "layers": [ { "id": "clifford", "name": "Clifford torus", "description": "Clifford torus.", "findings": [ { "id": "clifford-finding", "name": "Clifford torus", "description": "Clifford torus.", "questions": [ { "text": "How does the Clifford torus sit flat in four dimensions?", "kind": "provenance" }, { "text": "Which entry fits Clifford torus?", "kind": "action" } ] } ] }, { "id": "lie", "name": "Lie groups", "description": "Maximal tori.", "findings": [ { "id": "lie-finding", "name": "Lie groups", "description": "Lie groups.", "questions": [ { "text": "What role do maximal tori play in the theory of compact Lie groups?", "kind": "provenance" }, { "text": "Which entry fits maximal torus?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Applications.", "rationale": "Context.", "layers": [ { "id": "tokamak", "name": "Tokamak", "description": "Fusion devices.", "findings": [ { "id": "tokamak-finding", "name": "Tokamak", "description": "Tokamak.", "questions": [ { "text": "Why are tokamak plasma chambers toroidal?", "kind": "provenance" }, { "text": "Which entry fits tokamak?", "kind": "action" } ] } ] }, { "id": "games", "name": "Games", "description": "Toroidal worlds.", "findings": [ { "id": "games-finding", "name": "Games", "description": "Games.", "questions": [ { "text": "Why do wrap-around video game maps form a torus?", "kind": "provenance" }, { "text": "Which entry fits wraparound (video games)?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "The registry alias tie ring should be reviewed and moved to mechanical entries if appropriate", "Should maximal torus be a separate entry?", "How should topology references 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 torus, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain the torus, relay geometry, topology, higher-dimensional and algebraic forms and applications from mathematics and physics sources, describe the named forms and flag the loosely filed alias, and distinguish the torus from the sphere, annulus, solid torus, Klein bottle and toroidal inductors as objects.", "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: tie ring, maximal torus, Clifford torus, supertoroid.", "4 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 72 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 torus, 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": [ "tie ring", "maximal torus", "Clifford torus", "supertoroid" ] } ], "standing": "Described in 72 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 torus from another, and a torus 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 torus, 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 torus, 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 torus 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 torus, 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 torus, 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 torus 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 torus 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 torus, 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 torus, 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 torus 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 torus from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a torus, 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 torus, 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 torus 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 torus 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 torus between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a torus, 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 torus, 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 torus 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 torus, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a torus 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 torus, 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 torus, 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 torus 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 torus 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 torus 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 torus 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 torus 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 torus?", "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 torus?", "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 torus 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 torus and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a torus, 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 torus, 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 torus 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 torus is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a torus, 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 torus, 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 torus 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 } } } }