{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T03:52:47Z",
    "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": "2026-10-06T07:09:00+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "revisions": [
      {
        "at": "2026-10-06T22:42:20+00:00",
        "engine": "codex",
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          "distinguishing_features"
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        "answering_review": "16d817adb989242f68d59441100877724f5a4cb9fa2798e32b374500a288f3d2"
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    "gate": {
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      "version": "1.0",
      "at": "2026-10-07T10:10:28+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q2063121",
    "registryId": "vr.tr.refactorable-number",
    "name": "refactorable number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q2063121/",
  "model": {
    "registry_id": "vr.tr.refactorable-number",
    "name": "refactorable number",
    "purpose": "Let an agent explain refactorable numbers, relay their definition, examples and known properties from number theory references, and distinguish them from related divisor-based number classes.",
    "definition": "A positive integer that is divisible by the number of its divisors, also called a tau number, such as 1, 2, 8, 9, 12, 18, 24, 36, 40 and 56, so that 12 with six divisors is refactorable because 6 divides 12; refactorable numbers have natural density zero, can occur in consecutive pairs such as 8 and 9, and were studied by Kennedy and Cooper and by Zelinsky.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the definition",
      "relay examples and properties",
      "describe results",
      "distinguish related classes"
    ],
    "distinguishing_features": [
      "The divisor count divides the number.",
      "Refactorable numbers have natural density zero.",
      "Consecutive refactorable numbers include 8 and 9.",
      "The definition uses the divisor-counting tau function."
    ],
    "appearance": "Not a visible object; integers with a divisibility property.",
    "visual_identification": [
      "Integers divisible by their number of divisors",
      "Tau numbers 1, 2, 8, 9, 12, 18, 24, 36, 40, 56",
      "Perfect numbers use divisor sums; highly composite numbers maximise divisor counts"
    ],
    "physical_properties": [
      {
        "quantity": "first terms",
        "typical_range": "1, 2, 8, 9, 12, 18, 24, 36, 40, 56",
        "unit": "sequence",
        "note": "OEIS A033950"
      },
      {
        "quantity": "natural density",
        "typical_range": "0",
        "unit": "value",
        "note": "proved by Kennedy and Cooper"
      },
      {
        "quantity": "consecutive refactorable numbers",
        "typical_range": "only 1 and 2",
        "unit": "result",
        "note": "Zelinsky"
      }
    ],
    "families_and_kinds": [
      "refactorable numbers",
      "refactorable numbers that are perfect squares",
      "refactorable primes powers such as 8 and 9",
      "related classes such as highly composite and tau-rich numbers"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "in registry terms"
      },
      {
        "relation": "is defined by",
        "target": "divisor function",
        "why": "tau(n)"
      },
      {
        "relation": "is listed in",
        "target": "OEIS",
        "why": "as A033950"
      },
      {
        "relation": "is contrasted with",
        "target": "perfect number",
        "why": "defined by divisor sums"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A033950",
        "note": "refactorable numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; standard mathematical definitions"
    ],
    "failure_modes_and_hazards": [
      "Confusing tau numbers with other divisor classes",
      "Miscounting divisors",
      "Overstating open results"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Explain refactorable numbers and test given integers.",
        "Count divisors correctly and show the working.",
        "Cite known results with their authors."
      ],
      "must_not": [
        "Miscount divisors or present wrong examples.",
        "Overstate open results as proved.",
        "Confuse tau numbers with other divisor classes.",
        "Misattribute results."
      ],
      "requires_human": [
        "Publishing a new claimed theorem about refactorable numbers."
      ]
    },
    "ethics": {
      "considerations": [
        "Correct credit for results matters to mathematicians.",
        "Errors in teaching material mislead students."
      ],
      "affected_parties": [
        "Students",
        "Researchers",
        "Readers"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; the mathematical community maintains it.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q2063121: refactorable number",
      "url": "https://www.wikidata.org/wiki/Q2063121",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Refactorable number",
      "url": "https://en.wikipedia.org/wiki/Refactorable_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 a refactorable number is.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a refactorable number, and why is 12 refactorable while 10 is not?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about refactorable numbers or another divisor-based class?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "examples",
            "name": "Examples",
            "description": "Examples.",
            "findings": [
              {
                "id": "examples-finding",
                "name": "Examples",
                "description": "Examples.",
                "questions": [
                  {
                    "text": "What are the first refactorable numbers, and which perfect squares and prime powers qualify?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Which entry fits the specific number?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theory",
        "name": "Theory",
        "description": "Results.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "results",
            "name": "Results",
            "description": "Known results.",
            "findings": [
              {
                "id": "results-finding",
                "name": "Results",
                "description": "Results.",
                "questions": [
                  {
                    "text": "What is proved about density and consecutive refactorable numbers, and by whom?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "open",
            "name": "Open",
            "description": "Open questions.",
            "findings": [
              {
                "id": "open-finding",
                "name": "Open",
                "description": "Open.",
                "questions": [
                  {
                    "text": "What questions about refactorable numbers remain open?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "compare",
        "name": "Compare",
        "description": "Related classes.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "divisor",
            "name": "Divisor",
            "description": "Divisor function.",
            "findings": [
              {
                "id": "divisor-finding",
                "name": "Divisor",
                "description": "Divisor.",
                "questions": [
                  {
                    "text": "How does the divisor counting function tau behave, and how does it determine refactorability?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits divisor function?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "classes",
            "name": "Classes",
            "description": "Other classes.",
            "findings": [
              {
                "id": "classes-finding",
                "name": "Classes",
                "description": "Classes.",
                "questions": [
                  {
                    "text": "How do refactorable numbers relate to highly composite, perfect and harmonic divisor numbers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits highly composite number?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and computation.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How were refactorable numbers introduced by Kennedy and Cooper and named tau numbers by Colton?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "computation",
            "name": "Computation",
            "description": "Computation.",
            "findings": [
              {
                "id": "computation-finding",
                "name": "Computation",
                "description": "Computation.",
                "questions": [
                  {
                    "text": "How can refactorable numbers be computed programmatically?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits integer sequence?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should the divisor function be the primary linked entry?",
    "How should number theory references be linked?",
    "How should open problems such as Colton conjectures be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}