{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T18:08:38Z",
    "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",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-06T12:00:00Z",
      "ingestedAt": "2026-10-06T07:37:03+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "revisions": [
      {
        "at": "2026-10-06T22:51:00+00:00",
        "engine": "codex",
        "fields": [
          "definition",
          "visual_identification"
        ],
        "answering_review": "03bfae3eec646a58a819ba703b5a89dd9a28ba9b6cbec4c0801990ee852e0d02"
      }
    ],
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-07T10:10:30+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q3924032",
    "registryId": "vr.tr.mathematical-property",
    "name": "mathematical property",
    "version": "0.2.0-wave.4",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QLT"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q3924032/",
  "model": {
    "registry_id": "vr.tr.mathematical-property",
    "name": "mathematical property",
    "purpose": "Let an agent explain mathematical properties as predicates, relay examples across analysis, topology and geometry from mathematics references, describe the properties the registry aliases name, and distinguish properties from structures, invariants, relations of higher arity and physical properties.",
    "definition": "A property that a mathematical object may have, expressed as a predicate or unary relation over a class of objects, such as continuity, differentiability, convexity or square-integrability of functions, geometric continuity of curves, stability properties of systems, and productive properties preserved under products in topology; the registry treats such properties as functions from objects to truth values.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the concept",
      "relay examples",
      "describe named properties",
      "distinguish related concepts"
    ],
    "distinguishing_features": [
      "Predicate",
      "Defines subclasses",
      "Preservation questions",
      "Across all fields"
    ],
    "appearance": "Not a visible object; a predicate.",
    "visual_identification": [
      "A mathematical property is a predicate that mathematical objects may satisfy.",
      "Examples include stability, square-integrability, productive properties and geometric continuity, while degrees of freedom is a count rather than a truth-valued property.",
      "Structures add operations, invariants are preserved quantities, relations may relate several objects, and physical properties are measured."
    ],
    "physical_properties": [
      {
        "quantity": "formal sense",
        "typical_range": "unary predicate or subset",
        "unit": "note",
        "note": ""
      },
      {
        "quantity": "square-integrable",
        "typical_range": "finite integral of the square",
        "unit": "note",
        "note": "L2"
      },
      {
        "quantity": "geometric continuity",
        "typical_range": "G0, G1, G2 levels",
        "unit": "note",
        "note": "curves"
      }
    ],
    "families_and_kinds": [
      "analytic properties such as continuity and square-integrability",
      "topological properties including productive properties",
      "geometric properties such as geometric continuity",
      "dynamical properties such as stability",
      "counting notions such as degrees of freedom",
      "unary relations in logic"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "property",
        "why": "in registry terms"
      },
      {
        "relation": "is formalised as",
        "target": "unary relation",
        "why": ""
      },
      {
        "relation": "is contrasted with",
        "target": "mathematical structure",
        "why": ""
      },
      {
        "relation": "is exemplified by",
        "target": "continuous function",
        "why": "having continuity"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "No regulation"
    ],
    "failure_modes_and_hazards": [
      "Confusing properties with structures",
      "Registry aliases mixing properties and counts",
      "Vagueness of the general notion"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Explain mathematical properties such as continuity or convexity, citing mathematical sources.",
        "State precisely the class of objects a property is defined on."
      ],
      "must_not": [
        "Claim a function or system has a property without proof or a cited source.",
        "Confuse a property with a structure or with a count such as degrees of freedom.",
        "Present a stability claim as certified when it has not been checked."
      ],
      "requires_human": [
        "Relying on a claimed property in a safety-critical design."
      ]
    },
    "ethics": {
      "considerations": [
        "Engineers rely on properties such as stability; unchecked claims can cause failures.",
        "Mathematical errors in teaching propagate to students."
      ],
      "affected_parties": [
        "Students",
        "Engineers relying on proofs"
      ]
    },
    "owners": {
      "steward": "Nobody owns mathematical concepts; they are held in common.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q3924032: mathematical property",
      "url": "https://www.wikidata.org/wiki/Q3924032",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Property (mathematics)",
      "url": "https://en.wikipedia.org/wiki/Property_%28mathematics%29",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "understand",
        "name": "Understand",
        "description": "What a mathematical property is.",
        "rationale": "Science.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a mathematical property, and how does it differ from a structure, invariant, relation and physical property?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about logic, a field or a specific property?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "examples",
            "name": "Examples",
            "description": "Named properties.",
            "findings": [
              {
                "id": "examples-finding",
                "name": "Examples",
                "description": "Examples.",
                "questions": [
                  {
                    "text": "What are stability, square-integrability, productive properties, geometric continuity, degrees of freedom and unary relations?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits the specific property?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "logic",
        "name": "Logic",
        "description": "Logic.",
        "rationale": "Science.",
        "layers": [
          {
            "id": "predicates",
            "name": "Predicates",
            "description": "Predicates.",
            "findings": [
              {
                "id": "predicates-finding",
                "name": "Predicates",
                "description": "Predicates.",
                "questions": [
                  {
                    "text": "How are properties formalised as predicates and sets?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "preservation",
            "name": "Preservation",
            "description": "Preservation.",
            "findings": [
              {
                "id": "preservation-finding",
                "name": "Preservation",
                "description": "Preservation.",
                "questions": [
                  {
                    "text": "What does it mean for a property to be preserved under products, maps or limits?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "fields",
        "name": "Fields",
        "description": "Fields.",
        "rationale": "Science.",
        "layers": [
          {
            "id": "analysis",
            "name": "Analysis",
            "description": "Analysis.",
            "findings": [
              {
                "id": "analysis-finding",
                "name": "Analysis",
                "description": "Analysis.",
                "questions": [
                  {
                    "text": "What are key properties of functions in analysis?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits square-integrable function?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "geometry",
            "name": "Geometry",
            "description": "Geometry and dynamics.",
            "findings": [
              {
                "id": "geometry-finding",
                "name": "Geometry",
                "description": "Geometry.",
                "questions": [
                  {
                    "text": "How are geometric continuity and stability defined?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits stability theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "Philosophy.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "philosophy",
            "name": "Philosophy",
            "description": "Philosophy.",
            "findings": [
              {
                "id": "philosophy-finding",
                "name": "Philosophy",
                "description": "Philosophy.",
                "questions": [
                  {
                    "text": "How do philosophers of mathematics treat properties?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits philosophy of mathematics?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "freedom",
            "name": "Freedom",
            "description": "Degrees of freedom.",
            "findings": [
              {
                "id": "freedom-finding",
                "name": "Freedom",
                "description": "Freedom.",
                "questions": [
                  {
                    "text": "What are degrees of freedom in mechanics and statistics?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits degrees of freedom?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should the named properties be separate primary entries?",
    "How should mathematics references be linked?",
    "The registry alias degree of freedom is a count rather than a property; should it be split off?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}