{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T01:09:16Z",
    "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": 2,
    "engine": "claude",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-05T23:55:00Z",
      "ingestedAt": "2026-10-05T20:50:48+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:51:53+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q1149466",
    "registryId": "vr.tr.sociable-number",
    "name": "sociable number",
    "version": "0.2.0-wave.2",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q1149466/",
  "model": {
    "registry_id": "vr.tr.sociable-number",
    "name": "sociable number",
    "purpose": "Let an agent explain sociable numbers and aliquot cycles, relate them to perfect and amicable numbers, describe known cycles and open problems, and support computation and learning.",
    "definition": "A positive integer belonging to a cycle of numbers in which each is the sum of the proper divisors of the previous one, returning to the start after a fixed number of steps, generalising perfect numbers with cycle length one and amicable pairs with cycle length two to cycles of length three or more, such as the cycle of five numbers beginning with 12496; sociable numbers are studied in number theory, with cycles found by computation and open questions about their existence and distribution.",
    "what_it_is_for": "Numbers in aliquot cycles.",
    "affordances": [
      "explain aliquot cycles",
      "relate to perfect and amicable numbers",
      "describe known cycles",
      "support computation"
    ],
    "distinguishing_features": [
      "Aliquot cycle",
      "Cycle length",
      "Computationally found",
      "Open problems"
    ],
    "appearance": "Not physical; sets of integers.",
    "visual_identification": [
      "Aliquot sum cycle of length three or more",
      "Generalises perfect and amicable numbers",
      "Perfect numbers have cycle length one; amicable pairs length two"
    ],
    "physical_properties": [],
    "families_and_kinds": [
      "sociable numbers of order four",
      "sociable numbers of order five",
      "longer cycles",
      "related perfect and amicable numbers",
      "aliquot sequences generally"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "category"
      },
      {
        "relation": "is related to",
        "target": "Catalan number",
        "why": "another integer sequence entry"
      },
      {
        "relation": "is related to",
        "target": "Fermat number",
        "why": "another number theory entry"
      },
      {
        "relation": "is related to",
        "target": "cyclic process",
        "why": "cycles in general"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "Mathematical conventions and notation",
      "Sequence database conventions"
    ],
    "failure_modes_and_hazards": [
      "Confusing sociable with amicable numbers",
      "Computational errors in divisor sums",
      "Overstating what is proven"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "State known sociable cycles and their lengths with a reference.",
        "Search for or verify aliquot cycles computationally, reporting the search range.",
        "Explain how sociable numbers relate to perfect and amicable numbers."
      ],
      "must_not": [
        "Claim a new sociable cycle without verification that others can repeat.",
        "Present open questions, such as whether cycles of length three exist, as settled.",
        "Attribute a discovery to a person without evidence.",
        "Confuse sociable numbers with amicable or perfect numbers."
      ],
      "requires_human": [
        "Announcing a claimed new discovery publicly."
      ]
    },
    "ethics": {
      "considerations": [
        "Mathematical claims depend on credit given to the right discoverers.",
        "False announcements waste others' effort and erode trust."
      ],
      "affected_parties": [
        "Mathematicians and amateur researchers",
        "Students"
      ]
    },
    "owners": {
      "steward": "Nobody: a mathematical concept held in common; integer sequence databases record known cycles.",
      "master_systems": [
        "Integer sequence databases"
      ]
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1149466: sociable number",
      "url": "https://www.wikidata.org/wiki/Q1149466",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Sociable number",
      "url": "https://en.wikipedia.org/wiki/Sociable_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "understand",
        "name": "Understand",
        "description": "What sociable numbers are.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition and aliquot sums.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is the aliquot sum, and how do sociable numbers form cycles that generalise perfect and amicable numbers?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits perfect or amicable numbers?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "examples",
            "name": "Examples",
            "description": "Known cycles.",
            "findings": [
              {
                "id": "examples-finding",
                "name": "Examples",
                "description": "Examples.",
                "questions": [
                  {
                    "text": "Which sociable cycles are known, including the cycle beginning with 12496 and the 28-cycle, and how were they found?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which databases list them?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "compute",
        "name": "Compute",
        "description": "Computing cycles.",
        "rationale": "Practice.",
        "layers": [
          {
            "id": "calculate",
            "name": "Calculate",
            "description": "Aliquot sequences.",
            "findings": [
              {
                "id": "calculate-finding",
                "name": "Calculate",
                "description": "Computing.",
                "questions": [
                  {
                    "text": "How is the aliquot sequence of this number computed, and does it cycle, terminate or grow?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits divisor functions?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "search",
            "name": "Search",
            "description": "Searching for cycles.",
            "findings": [
              {
                "id": "search-finding",
                "name": "Search",
                "description": "Searching.",
                "questions": [
                  {
                    "text": "How are sociable cycles searched for computationally, and what are the limits?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "open",
        "name": "Open",
        "description": "Open problems.",
        "rationale": "Research.",
        "layers": [
          {
            "id": "problems",
            "name": "Problems",
            "description": "Open questions.",
            "findings": [
              {
                "id": "problems-finding",
                "name": "Problems",
                "description": "Open questions.",
                "questions": [
                  {
                    "text": "What is unknown about sociable numbers, such as cycles of length three, odd cycles and the Catalan-Dickson conjecture?",
                    "kind": "boundary"
                  },
                  {
                    "text": "Which entry fits aliquot sequences?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "theory",
            "name": "Theory",
            "description": "Theoretical results.",
            "findings": [
              {
                "id": "theory-finding",
                "name": "Theory",
                "description": "Theory.",
                "questions": [
                  {
                    "text": "What theoretical results on the density of sociable numbers are known?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which findings are recent?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "learn",
        "name": "Learn",
        "description": "History and teaching.",
        "rationale": "Education.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did the study of sociable numbers develop from Poulet in 1918 to modern computation?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits history of number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "teach",
            "name": "Teach",
            "description": "Teaching.",
            "findings": [
              {
                "id": "teach-finding",
                "name": "Teach",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How can sociable numbers be used to teach divisors and computation?",
                    "kind": "action"
                  },
                  {
                    "text": "Which misconceptions arise?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should aliquot sequences be a separate entry?",
    "How should databases of cycles be linked?",
    "How should open problems be documented?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}