{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-11T11:32:52Z",
    "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": 1,
    "engine": "claude",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-05T17:00:00Z",
      "ingestedAt": "2026-10-05T13:25:37+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "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-Q1972803",
    "registryId": "vr.tr.perfect-power",
    "name": "perfect power",
    "version": "0.2.0-wave.1",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q1972803/",
  "model": {
    "registry_id": "vr.tr.perfect-power",
    "name": "perfect power",
    "purpose": "Let an agent handle perfect powers by definition, testing, examples and notable theorems.",
    "definition": "A positive integer that can be written as m to the power k with integers m greater than 1 and k at least 2, such as squares (4, 9), cubes (8, 27) and higher powers; Catalan conjecture, proved by Mihailescu, shows 8 and 9 are the only consecutive perfect powers.",
    "what_it_is_for": "Number theory and computation.",
    "affordances": [
      "test whether a number is a perfect power",
      "list squares, cubes and higher powers",
      "explain related theorems",
      "implement exact tests in code"
    ],
    "distinguishing_features": [
      "Integer power of an integer",
      "Includes squares and cubes",
      "Rare among large numbers",
      "Subject of deep theorems"
    ],
    "appearance": "Integers such as 16, 27, 32 and 64.",
    "visual_identification": [
      "Forms such as m squared or m cubed",
      "Squares and cubes are special cases",
      "Powers of a fixed base such as 2 form a different sequence"
    ],
    "physical_properties": [],
    "families_and_kinds": [
      "square numbers",
      "cube numbers",
      "fourth and higher powers",
      "powers of two, three and five"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "non-negative integer",
        "why": "category"
      },
      {
        "relation": "includes",
        "target": "square number",
        "why": "kind"
      },
      {
        "relation": "is related to",
        "target": "Catalan conjecture",
        "why": "theorem"
      },
      {
        "relation": "is studied in",
        "target": "number theory",
        "why": "field"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A001597",
        "note": "integer sequence"
      }
    ],
    "standards_and_regulation": [],
    "failure_modes_and_hazards": [
      "Floating point errors in root tests",
      "Confusing perfect powers with powers of a base"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Test whether an integer is a perfect power using exact integer arithmetic.",
        "List perfect powers and their bases and exponents for teaching or research."
      ],
      "must_not": [
        "Rely on floating point roots alone to decide that a large number is a perfect power.",
        "Confuse perfect powers with powers of a fixed base.",
        "Present an unproved conjecture about powers as a theorem."
      ],
      "requires_human": [
        "Publishing a claimed new result about perfect powers as established."
      ]
    },
    "ethics": {
      "considerations": [
        "Correct mathematics matters where perfect power tests feed into cryptographic and number theory software."
      ],
      "affected_parties": [
        "Students and teachers",
        "Users of number theory software"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; it belongs to mathematics.",
      "master_systems": []
    },
    "facets_not_applicable": {
      "interfaces": "No specific standard, law or official classification applies to perfect powers as an abstract mathematical concept."
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1972803: perfect power",
      "url": "https://www.wikidata.org/wiki/Q1972803",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Perfect power",
      "url": "https://en.wikipedia.org/wiki/Perfect_power",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "test",
        "name": "Test",
        "description": "Is it a perfect power.",
        "rationale": "Exact tests are needed.",
        "layers": [
          {
            "id": "check",
            "name": "Check",
            "description": "Roots.",
            "findings": [
              {
                "id": "check-finding",
                "name": "Check",
                "description": "Perfect power test.",
                "questions": [
                  {
                    "text": "Is this number a perfect power, and of which base and exponent?",
                    "kind": "boundary"
                  },
                  {
                    "text": "How is it tested exactly?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "code",
            "name": "Code",
            "description": "Implementation.",
            "findings": [
              {
                "id": "code-finding",
                "name": "Code",
                "description": "Implementation.",
                "questions": [
                  {
                    "text": "How can integer roots be computed without floating point errors?",
                    "kind": "action"
                  },
                  {
                    "text": "Which libraries help?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "examples",
        "name": "Examples",
        "description": "Sequences.",
        "rationale": "Examples illustrate.",
        "layers": [
          {
            "id": "list",
            "name": "List",
            "description": "Early terms.",
            "findings": [
              {
                "id": "list-finding",
                "name": "List",
                "description": "List.",
                "questions": [
                  {
                    "text": "What are the first perfect powers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "How dense are they among integers?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          },
          {
            "id": "kinds",
            "name": "Kinds",
            "description": "Squares and cubes.",
            "findings": [
              {
                "id": "kinds-finding",
                "name": "Kinds",
                "description": "Kinds.",
                "questions": [
                  {
                    "text": "Is it a square, cube or higher power?",
                    "kind": "definition"
                  },
                  {
                    "text": "Can it be several at once, such as 64?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theorems",
        "name": "Theorems",
        "description": "Results.",
        "rationale": "Deep results exist.",
        "layers": [
          {
            "id": "catalan",
            "name": "Catalan",
            "description": "Consecutive powers.",
            "findings": [
              {
                "id": "catalan-finding",
                "name": "Catalan",
                "description": "Catalan conjecture.",
                "questions": [
                  {
                    "text": "What does the Catalan conjecture, now a theorem, state?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Who proved it and when?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "open",
            "name": "Open",
            "description": "Open problems.",
            "findings": [
              {
                "id": "open-finding",
                "name": "Open",
                "description": "Open problems.",
                "questions": [
                  {
                    "text": "Which related problems remain open?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Is their status stated accurately?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "learning",
        "name": "Learning",
        "description": "Teaching.",
        "rationale": "Patterns help.",
        "layers": [
          {
            "id": "patterns",
            "name": "Patterns",
            "description": "Visual models.",
            "findings": [
              {
                "id": "patterns-finding",
                "name": "Patterns",
                "description": "Patterns.",
                "questions": [
                  {
                    "text": "How can squares and cubes be shown with arrays and blocks?",
                    "kind": "action"
                  },
                  {
                    "text": "Which exercises help?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "notation",
            "name": "Notation",
            "description": "Exponents.",
            "findings": [
              {
                "id": "notation-finding",
                "name": "Notation",
                "description": "Exponent notation.",
                "questions": [
                  {
                    "text": "How is exponent notation written and read?",
                    "kind": "definition"
                  },
                  {
                    "text": "What common mistakes occur?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should cubes be a separate entry?",
    "How should OEIS be linked?",
    "How should related conjectures be tracked?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}