{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T02:31:45Z",
    "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": 3,
    "engine": "claude",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-06T10:00:00Z",
      "ingestedAt": "2026-10-06T07:05:51+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:52:02+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q752723",
    "registryId": "vr.tr.natural-density",
    "name": "natural density",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q752723/",
  "model": {
    "registry_id": "vr.tr.natural-density",
    "name": "natural density",
    "purpose": "Let an agent explain natural density, relay its definition, properties and examples from mathematical references, describe related densities, and distinguish it from probability and other density notions.",
    "definition": "A measure in number theory of how large a subset of the natural numbers is, defined as the limit, if it exists, of the proportion of numbers up to n that belong to the set as n grows, with upper and lower densities when the limit does not exist; even numbers have density one half, primes have density zero, and natural density underlies results such as Szemeredi theorem and the study of arithmetic progressions.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the definition",
      "relay examples and properties",
      "describe related densities",
      "distinguish from other notions"
    ],
    "distinguishing_features": [
      "Limit definition",
      "Finite additivity only",
      "Upper and lower variants",
      "Combinatorial applications"
    ],
    "appearance": "Not a visible object; a limit of ratios.",
    "visual_identification": [
      "Asymptotic proportion of a subset of the naturals",
      "Natural, upper, lower, logarithmic and Schnirelmann densities",
      "Probability measures are countably additive; natural density is not"
    ],
    "physical_properties": [
      {
        "quantity": "density of even numbers",
        "typical_range": "1/2",
        "unit": "value",
        "note": ""
      },
      {
        "quantity": "density of primes",
        "typical_range": "0",
        "unit": "value",
        "note": ""
      },
      {
        "quantity": "density of squarefree numbers",
        "typical_range": "6/pi^2",
        "unit": "value",
        "note": "about 0.608"
      }
    ],
    "families_and_kinds": [
      "natural or asymptotic density",
      "upper and lower asymptotic density",
      "logarithmic density",
      "Schnirelmann density",
      "Banach density"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "number of entities",
        "why": "in registry terms, as a limiting proportion"
      },
      {
        "relation": "is used in",
        "target": "additive combinatorics",
        "why": "such as Szemeredi theorem"
      },
      {
        "relation": "is related to",
        "target": "prime number theorem",
        "why": "for the density of primes"
      },
      {
        "relation": "is contrasted with",
        "target": "probability measure",
        "why": "which is countably additive"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "No regulation; standard mathematical definitions"
    ],
    "failure_modes_and_hazards": [
      "Assuming the limit always exists",
      "Treating density as a probability",
      "Confusing density variants"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Explain natural density and compute it for examples such as even numbers or primes.",
        "Distinguish natural density from upper and lower density and from other notions such as logarithmic density."
      ],
      "must_not": [
        "State that a set has a natural density when the limit has not been shown to exist.",
        "Treat density zero as meaning the set is finite or empty.",
        "Present a conjectured density as a proven theorem."
      ],
      "requires_human": [
        "Publishing a claimed new result about densities under a person's name."
      ]
    },
    "ethics": {
      "considerations": [
        "Misstated results mislead students and researchers who build on them.",
        "Credit for results should go to those who proved them."
      ],
      "affected_parties": [
        "Students and learners",
        "Researchers in number theory"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; the mathematical community maintains it.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q752723: natural density",
      "url": "https://www.wikidata.org/wiki/Q752723",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Natural density",
      "url": "https://en.wikipedia.org/wiki/Natural_density",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "understand",
        "name": "Understand",
        "description": "What natural density is.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "How is natural density defined, and when does it exist?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about natural density or a probability or other density notion?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "examples",
            "name": "Examples",
            "description": "Examples.",
            "findings": [
              {
                "id": "examples-finding",
                "name": "Examples",
                "description": "Examples.",
                "questions": [
                  {
                    "text": "What are the densities of even numbers, primes, squarefree numbers and other classic sets?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Which entry fits the specific set?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "properties",
        "name": "Properties",
        "description": "Properties.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "properties",
            "name": "Properties",
            "description": "Properties.",
            "findings": [
              {
                "id": "properties-finding",
                "name": "Properties",
                "description": "Properties.",
                "questions": [
                  {
                    "text": "What properties does natural density have, and why is it only finitely additive?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "variants",
            "name": "Variants",
            "description": "Variants.",
            "findings": [
              {
                "id": "variants-finding",
                "name": "Variants",
                "description": "Variants.",
                "questions": [
                  {
                    "text": "How do upper, lower, logarithmic, Schnirelmann and Banach densities differ?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits the specific variant?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "apply",
        "name": "Apply",
        "description": "Applications.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "combinatorics",
            "name": "Combinatorics",
            "description": "Additive combinatorics.",
            "findings": [
              {
                "id": "combinatorics-finding",
                "name": "Combinatorics",
                "description": "Combinatorics.",
                "questions": [
                  {
                    "text": "How does density appear in Szemeredi theorem and Erdos conjectures on arithmetic progressions?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "analytic",
            "name": "Analytic",
            "description": "Analytic number theory.",
            "findings": [
              {
                "id": "analytic-finding",
                "name": "Analytic",
                "description": "Analytic.",
                "questions": [
                  {
                    "text": "How does the prime number theorem give the density of primes and related results?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits prime number theorem?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and teaching.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did the concept develop in the work of Schnirelmann, Erdos and others?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "teaching",
            "name": "Teaching",
            "description": "Teaching.",
            "findings": [
              {
                "id": "teaching-finding",
                "name": "Teaching",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How is natural density taught, and what intuitions help?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits mathematics education?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should Schnirelmann density be a separate entry?",
    "How should mathematical references be linked?",
    "How should applications be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}