{ "model": { "rank": 7143, "code": "thing-q26401", "model_id": "vr.tr.vertex", "name": "vertex", "purpose": "Let an agent explain vertices across geometry, graph theory and graphics, relay definitions, counts such as Euler s formula and uses from mathematics and computing sources, describe the Voronoi pole alias, and distinguish vertices from edges, faces, nodes in networks generally, apexes and the astrological or anatomical senses of vertex.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q26401/", "tier": 2, "score": 57, "payload": { "layer": "wikidata", "aliases": [ "Voronoi pole" ], "aliasCount": 1, "merged": 1, "knownIn": 57, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.vertex", "source": { "dataset": "wikidata", "item": "Q26401", "url": "https://www.wikidata.org/wiki/Q26401", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T22:45:28Z", "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-Q26401", "registryId": "vr.tr.vertex", "name": "vertex", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q26401/", "model": { "registry_id": "vr.tr.vertex", "name": "vertex", "purpose": "Let an agent explain vertices across geometry, graph theory and graphics, relay definitions, counts such as Euler s formula and uses from mathematics and computing sources, describe the Voronoi pole alias, and distinguish vertices from edges, faces, nodes in networks generally, apexes and the astrological or anatomical senses of vertex.", "definition": "A point where two or more lines, edges or curves meet, as in the corners of polygons and polyhedra, the extreme points of polytopes and simplices, the nodes of graphs, where vertices are joined by edges, and the vertex of a parabola or other conic, its turning point; in computational geometry a Voronoi pole is a Voronoi vertex farthest from a sample point, used to approximate the medial axis in surface reconstruction. In computer graphics vertices carry positions and attributes such as colour and normals for rendering meshes.", "what_it_is_for": "Naming meeting points in geometry and graphs.", "affordances": [ "explain vertices across fields", "relay Euler s formula and counts", "describe the Voronoi pole", "distinguish other senses" ], "distinguishing_features": [ "Meeting point", "Degree or valence", "Cross-disciplinary", "Data carrier in graphics" ], "appearance": "Not a physical object; drawn as corners of shapes or dots in graphs.", "visual_identification": [ "Meeting point of edges or curves", "Voronoi pole, corner, node, extreme point", "Edges join vertices; faces are bounded regions; network nodes may carry data; apex is a top vertex; anatomical vertex is the top of the head" ], "physical_properties": [ { "quantity": "Euler s polyhedron formula", "typical_range": "V - E + F = 2", "unit": "note", "note": "convex polyhedra" }, { "quantity": "cube vertices", "typical_range": "8", "unit": "count", "note": "" }, { "quantity": "Voronoi poles in surface reconstruction", "typical_range": "Amenta and Bern, 1998-1999", "unit": "note", "note": "power crust and crust algorithms" } ], "families_and_kinds": [ "polygon and polyhedron vertices", "graph vertices", "conic vertices", "Voronoi vertices and poles", "mesh vertices" ], "related_models": [ { "relation": "is a kind of", "target": "point", "why": "in registry terms" }, { "relation": "is joined by", "target": "edge (geometry)", "why": "" }, { "relation": "appears in", "target": "Euler characteristic", "why": "" }, { "relation": "is contrasted with", "target": "face (geometry)", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "No regulation; graphics API specifications define vertex formats" ], "failure_modes_and_hazards": [ "Confusing vertex and edge counts", "Mixing mathematical and anatomical senses", "Treating Voronoi poles as generic vertices" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What a vertex is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a vertex, and how does it differ from edges, faces, network nodes, apexes and anatomical senses?", "kind": "definition" }, { "text": "Is the question about geometry, graphs, graphics or another field?", "kind": "boundary" } ] } ] }, { "id": "pole", "name": "Voronoi pole", "description": "Voronoi pole.", "findings": [ { "id": "pole-finding", "name": "Voronoi pole", "description": "Voronoi pole.", "questions": [ { "text": "What is a Voronoi pole, and how is it used in surface reconstruction?", "kind": "definition" }, { "text": "Which entry fits Voronoi diagram?", "kind": "action" } ] } ] } ] }, { "id": "geometry", "name": "Geometry", "description": "Geometry.", "rationale": "Science.", "layers": [ { "id": "polytopes", "name": "Polytopes", "description": "Polytopes.", "findings": [ { "id": "polytopes-finding", "name": "Polytopes", "description": "Polytopes.", "questions": [ { "text": "How are vertices defined for polytopes and simplices?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "euler", "name": "Euler", "description": "Euler s formula.", "findings": [ { "id": "euler-finding", "name": "Euler", "description": "Euler.", "questions": [ { "text": "How does Euler s formula relate vertices, edges and faces?", "kind": "measurement" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "graphs", "name": "Graphs", "description": "Graph theory.", "rationale": "Science.", "layers": [ { "id": "degree", "name": "Degree", "description": "Vertex degree.", "findings": [ { "id": "degree-finding", "name": "Degree", "description": "Degree.", "questions": [ { "text": "What is vertex degree, and what does the handshake lemma say?", "kind": "provenance" }, { "text": "Which entry fits degree (graph theory)?", "kind": "action" } ] } ] }, { "id": "colouring", "name": "Colouring", "description": "Vertex colouring.", "findings": [ { "id": "colouring-finding", "name": "Colouring", "description": "Colouring.", "questions": [ { "text": "What is vertex colouring, and what does the four colour theorem state?", "kind": "provenance" }, { "text": "Which entry fits four color theorem?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Graphics.", "rationale": "Context.", "layers": [ { "id": "graphics", "name": "Graphics", "description": "Vertices in rendering.", "findings": [ { "id": "graphics-finding", "name": "Graphics", "description": "Graphics.", "questions": [ { "text": "How do vertex shaders process mesh vertices?", "kind": "provenance" }, { "text": "Which entry fits vertex shader?", "kind": "action" } ] } ] }, { "id": "conics", "name": "Conics", "description": "Parabola vertex.", "findings": [ { "id": "conics-finding", "name": "Conics", "description": "Conics.", "questions": [ { "text": "How is the vertex of a parabola found?", "kind": "measurement" }, { "text": "Which entry fits parabola?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should Voronoi pole be a separate entry?", "Should graph vertex be disambiguated from polytope vertex?", "How should mathematics 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 vertex, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain vertices across geometry, graph theory and graphics, relay definitions, counts such as Euler s formula and uses from mathematics and computing sources, describe the Voronoi pole alias, and distinguish vertices from edges, faces, nodes in networks generally, apexes and the astrological or anatomical senses of vertex.", "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: Voronoi pole.", "1 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 57 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 vertex, 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": [ "Voronoi pole" ] } ], "standing": "Described in 57 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 1 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 vertex from another, and a vertex 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 vertex, 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 vertex, 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 vertex 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 vertex, 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 vertex, 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 vertex 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 vertex 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 vertex, 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 vertex, 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 vertex 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 vertex from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a vertex, 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 vertex, 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 vertex 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 vertex 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 vertex between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a vertex, 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 vertex, 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 vertex 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 vertex, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a vertex 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 vertex, 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 vertex, 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 vertex 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 vertex 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 vertex 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 vertex 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 vertex 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 vertex?", "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 vertex?", "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 vertex 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 vertex and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a vertex, 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 vertex, 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 vertex 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 vertex is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a vertex, 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 vertex, 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 vertex 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 } } } }