{
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    "publishableCanonical": false,
    "generatedAt": "2026-09-13T03:18:30Z",
    "providers": [
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    "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": {
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      "engine": "claude",
      "producedAt": "2026-10-06T10:00:00Z",
      "ingestedAt": "2026-10-06T07:08:33+00:00",
      "fields": [
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        "ethics",
        "owners"
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  "metaModel": {
    "id": "THING-Q2482753",
    "registryId": "vr.tr.schnirelmann-density",
    "name": "Schnirelmann density",
    "version": "0.2.0-wave.3",
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    "domain": [
      "XCT.QTY"
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  "canonicalUrl": "https://ver.cy/models/thing/q2482753/",
  "model": {
    "registry_id": "vr.tr.schnirelmann-density",
    "name": "Schnirelmann density",
    "purpose": "Let an agent explain Schnirelmann density, relay its definition, properties and use in Schnirelmann theorem from mathematical references, and distinguish it from natural and other densities.",
    "definition": "A measure in additive number theory of how dense a set of natural numbers is, defined by Schnirelmann in 1930 as the infimum over positive integers n of the number of set members from 1 through n divided by n, so that any set missing 1 has density zero; for sets A and B of nonnegative integers both containing zero, with densities alpha and beta calculated without counting zero, their sumset has density at least alpha + beta - alpha * beta; the density underlies Schnirelmann's theorem that every integer greater than one is a sum of a bounded number of primes and is distinct from natural density.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the definition",
      "relay properties and theorems",
      "describe applications",
      "distinguish from other densities"
    ],
    "distinguishing_features": [
      "Infimum definition",
      "Sensitivity to small numbers",
      "Sumset inequality",
      "Prime sum application"
    ],
    "appearance": "Not a visible object; an infimum of ratios.",
    "visual_identification": [
      "Infimum density of a set of naturals",
      "Schnirelmann density and Schnirelmann theorem on sums of primes",
      "Natural density is a limit; upper and lower densities are limsup and liminf"
    ],
    "physical_properties": [
      {
        "quantity": "defined",
        "typical_range": "1930",
        "unit": "year",
        "note": "Lev Schnirelmann"
      },
      {
        "quantity": "Schnirelmann constant",
        "typical_range": "at most 7",
        "unit": "value",
        "note": "Ramare 1995; 3 with Helfgott ternary result"
      }
    ],
    "families_and_kinds": [
      "Schnirelmann density of sets",
      "Mann theorem strengthening the sumset inequality",
      "Schnirelmann theorem and constant",
      "related densities: natural, lower, upper, Banach"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "number of entities",
        "why": "in registry terms, as a limiting proportion"
      },
      {
        "relation": "is contrasted with",
        "target": "natural density",
        "why": "which is a limit"
      },
      {
        "relation": "underlies",
        "target": "Schnirelmann theorem",
        "why": "on sums of primes"
      },
      {
        "relation": "is sharpened by",
        "target": "Mann theorem",
        "why": "on sumset densities"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "No regulation; standard mathematical definitions"
    ],
    "failure_modes_and_hazards": [
      "Confusing with natural density",
      "Forgetting that missing 1 gives density zero",
      "Overstating the Schnirelmann constant results"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Compute and explain Schnirelmann density for given sets of natural numbers.",
        "Cite the theorems that use it with their sources."
      ],
      "must_not": [
        "Confuse Schnirelmann density with asymptotic or natural density.",
        "Present conjectures as proved results.",
        "Misattribute results in additive number theory."
      ],
      "requires_human": [
        "Announcing a claimed proof of an open problem using the density."
      ]
    },
    "ethics": {
      "considerations": [
        "Credit for results belongs to those who proved them.",
        "False claims about open problems mislead students and waste reviewers' time."
      ],
      "affected_parties": [
        "Mathematicians",
        "Students"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; it is held in common.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q2482753: Schnirelmann density",
      "url": "https://www.wikidata.org/wiki/Q2482753",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Schnirelmann density",
      "url": "https://en.wikipedia.org/wiki/Schnirelmann_density",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
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  ],
  "structure": {
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        "id": "understand",
        "name": "Understand",
        "description": "What Schnirelmann density is.",
        "rationale": "Mathematics.",
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          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "How is Schnirelmann density defined, and why does missing 1 give density zero?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about Schnirelmann density, natural density or another density?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
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            "id": "examples",
            "name": "Examples",
            "description": "Examples.",
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                "id": "examples-finding",
                "name": "Examples",
                "description": "Examples.",
                "questions": [
                  {
                    "text": "What are the Schnirelmann densities of the odd numbers, the squares and the primes?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Which entry fits the specific set?",
                    "kind": "action"
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              }
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          }
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          {
            "id": "sumsets",
            "name": "Sumsets",
            "description": "Sumset inequalities.",
            "findings": [
              {
                "id": "sumsets-finding",
                "name": "Sumsets",
                "description": "Sumsets.",
                "questions": [
                  {
                    "text": "What do the Schnirelmann and Mann inequalities say about sumsets?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "primes",
            "name": "Primes",
            "description": "Sums of primes.",
            "findings": [
              {
                "id": "primes-finding",
                "name": "Primes",
                "description": "Primes.",
                "questions": [
                  {
                    "text": "How does Schnirelmann theorem show every integer is a sum of boundedly many primes, and what is known about the constant?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits Goldbach conjecture?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
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        "id": "compare",
        "name": "Compare",
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        "rationale": "Mathematics.",
        "layers": [
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            "id": "densities",
            "name": "Densities",
            "description": "Other densities.",
            "findings": [
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                "id": "densities-finding",
                "name": "Densities",
                "description": "Densities.",
                "questions": [
                  {
                    "text": "How does Schnirelmann density compare with natural, lower, upper and Banach densities?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
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            "id": "uses",
            "name": "Uses",
            "description": "Uses.",
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                "id": "uses-finding",
                "name": "Uses",
                "description": "Uses.",
                "questions": [
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                    "text": "Where else is Schnirelmann density used in additive combinatorics?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits additive combinatorics?",
                    "kind": "action"
                  }
                ]
              }
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        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History.",
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            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
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                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "Who was Lev Schnirelmann, and how did his work lead to later results by Mann, Vinogradov and Helfgott?",
                    "kind": "provenance"
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                  {
                    "text": "Which entry fits the history of number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
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            "id": "teaching",
            "name": "Teaching",
            "description": "Teaching.",
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              {
                "id": "teaching-finding",
                "name": "Teaching",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How is Schnirelmann density presented in additive number theory courses?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits mathematics education?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should Schnirelmann theorem be a separate entry?",
    "How should mathematical references be linked?",
    "How should the constant history be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}