{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T02:17:19Z",
    "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": "2026-10-06T07:08:19+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:52:01+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q577211",
    "registryId": "vr.tr.practical-number",
    "name": "practical number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q577211/",
  "model": {
    "registry_id": "vr.tr.practical-number",
    "name": "practical number",
    "purpose": "Let an agent define practical numbers, relay their characterisation, properties and known results from number theory references, and distinguish them from related classes such as perfect and abundant numbers.",
    "definition": "A positive integer n such that every smaller positive integer can be written as a sum of distinct divisors of n, the sequence beginning 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, named by Srinivasan in 1948, characterised by a condition on prime factorisation by Stewart and Sierpinski, and including all powers of two and all perfect numbers; practical numbers behave in some ways like primes, and Goldbach-type results and density estimates have been proved for them.",
    "what_it_is_for": "Not applicable; a class of integers.",
    "affordances": [
      "define and characterise",
      "relay properties and results",
      "relay analogies with primes",
      "distinguish related classes"
    ],
    "distinguishing_features": [
      "Subset sum property of divisors",
      "Stewart-Sierpinski characterisation",
      "Prime-like distribution",
      "All even except 1"
    ],
    "appearance": "The sequence 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, ...",
    "visual_identification": [
      "Integers whose divisors sum to all smaller integers",
      "Sequence OEIS A005153",
      "Perfect numbers equal their aliquot sum; abundant numbers exceed it"
    ],
    "physical_properties": [
      {
        "quantity": "smallest",
        "typical_range": "1",
        "unit": "number",
        "note": ""
      },
      {
        "quantity": "smallest odd practical number",
        "typical_range": "1",
        "unit": "number",
        "note": "the only odd one"
      }
    ],
    "families_and_kinds": [
      "practical numbers in general",
      "powers of two and perfect numbers as practical numbers",
      "primitive practical numbers",
      "practical numbers in Egyptian fraction problems"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "in registry terms"
      },
      {
        "relation": "is characterised by",
        "target": "prime factorization",
        "why": "via the Stewart-Sierpinski condition"
      },
      {
        "relation": "includes",
        "target": "perfect number",
        "why": "as a subclass"
      },
      {
        "relation": "is listed in",
        "target": "OEIS",
        "why": "as A005153"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A005153",
        "note": ""
      }
    ],
    "standards_and_regulation": [
      "No regulation; mathematical conventions"
    ],
    "failure_modes_and_hazards": [
      "Confusing with perfect or abundant numbers",
      "Misstating the characterisation",
      "Computation errors"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Test whether an integer is practical and list its divisors as a witness.",
        "Explain properties of practical numbers and their link to Egyptian fractions."
      ],
      "must_not": [
        "Present a conjectured property as proven.",
        "Confuse practical numbers with similar sequences such as highly composite numbers.",
        "Credit the naming to someone other than Srinivasan."
      ],
      "requires_human": [
        "Publishing new results on practical numbers as established."
      ]
    },
    "ethics": {
      "considerations": [
        "Accurate credit and proof standards keep mathematics trustworthy.",
        "Students rely on correct definitions."
      ],
      "affected_parties": [
        "Mathematicians",
        "Students"
      ]
    },
    "owners": {
      "steward": "The public: held in common as mathematical knowledge.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q577211: practical number",
      "url": "https://www.wikidata.org/wiki/Q577211",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Practical number",
      "url": "https://en.wikipedia.org/wiki/Practical_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "define",
        "name": "Define",
        "description": "Definition.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a practical number, and how is it checked?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about practical, perfect or abundant numbers?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "characterisation",
            "name": "Characterisation",
            "description": "Characterisation.",
            "findings": [
              {
                "id": "characterisation-finding",
                "name": "Characterisation",
                "description": "Characterisation.",
                "questions": [
                  {
                    "text": "What is the Stewart-Sierpinski characterisation in terms of prime factorisation?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits prime factorisation?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "results",
        "name": "Results",
        "description": "Results.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "properties",
            "name": "Properties",
            "description": "Properties.",
            "findings": [
              {
                "id": "properties-finding",
                "name": "Properties",
                "description": "Properties.",
                "questions": [
                  {
                    "text": "What properties do practical numbers have, including closure under products?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "primes",
            "name": "Primes",
            "description": "Analogies with primes.",
            "findings": [
              {
                "id": "primes-finding",
                "name": "Primes",
                "description": "Primes.",
                "questions": [
                  {
                    "text": "What Goldbach-type and density results hold for practical numbers, as proved by Melfi, Weingartner and others?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "apply",
        "name": "Apply",
        "description": "Applications.",
        "rationale": "Practice.",
        "layers": [
          {
            "id": "egyptian",
            "name": "Egyptian",
            "description": "Egyptian fractions.",
            "findings": [
              {
                "id": "egyptian-finding",
                "name": "Egyptian",
                "description": "Egyptian.",
                "questions": [
                  {
                    "text": "How are practical numbers used in Egyptian fraction representations?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits Egyptian fraction?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "computation",
            "name": "Computation",
            "description": "Computation.",
            "findings": [
              {
                "id": "computation-finding",
                "name": "Computation",
                "description": "Computation.",
                "questions": [
                  {
                    "text": "How are practical numbers computed and enumerated?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits computational number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did Srinivasan, Stewart and Sierpinski introduce and characterise practical numbers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "related",
            "name": "Related",
            "description": "Related classes.",
            "findings": [
              {
                "id": "related-finding",
                "name": "Related",
                "description": "Related.",
                "questions": [
                  {
                    "text": "How do practical numbers relate to perfect, abundant and highly composite numbers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits perfect number?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "How should the density and Goldbach-type results be linked?",
    "How should OEIS and literature be linked?",
    "How should primitive practical numbers be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}