{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T01:10:31Z",
    "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": "2026-10-06T07:04:07+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:51:54+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q1667469",
    "registryId": "vr.tr.prime-power",
    "name": "prime power",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q1667469/",
  "model": {
    "registry_id": "vr.tr.prime-power",
    "name": "prime power",
    "purpose": "Let an agent define prime powers, relay their properties and roles in number theory, algebra and finite fields, and distinguish them from primes, perfect powers and composite numbers generally.",
    "definition": "A positive integer that is a power of a single prime number, of the form p to the k for a prime p and a positive integer k, such as 2, 4, 8, 3, 9, 27, 5 and 25, including the primes themselves as the case k equals 1; prime powers are the building blocks of integers through the fundamental theorem of arithmetic, are exactly the orders of finite fields and appear in group theory, cyclotomic polynomials and many number-theoretic results.",
    "what_it_is_for": "Not applicable; a class of integers.",
    "affordances": [
      "define and characterise",
      "relay properties",
      "explain roles in algebra",
      "distinguish related classes"
    ],
    "distinguishing_features": [
      "Single prime factor",
      "Orders of finite fields",
      "Cyclic unit groups for odd primes",
      "Convention on including 1"
    ],
    "appearance": "Numbers such as 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32.",
    "visual_identification": [
      "Powers of a single prime",
      "Sequence OEIS A000961 including primes",
      "Perfect powers may have composite bases; semiprimes are products of two primes"
    ],
    "physical_properties": [
      {
        "quantity": "count below 100",
        "typical_range": "35",
        "unit": "numbers",
        "note": "including 1 by some conventions"
      }
    ],
    "families_and_kinds": [
      "primes as prime powers with exponent 1",
      "proper prime powers with exponent at least 2",
      "powers of 2",
      "prime powers as finite field orders",
      "prime power groups in group theory"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "in registry terms"
      },
      {
        "relation": "includes",
        "target": "prime number",
        "why": "as the exponent-one case"
      },
      {
        "relation": "is the order of",
        "target": "finite field",
        "why": "for every prime power"
      },
      {
        "relation": "is contrasted with",
        "target": "perfect power",
        "why": "which allows composite bases"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A000961",
        "note": "with 1"
      },
      {
        "scheme": "OEIS",
        "value_or_pattern": "A246655",
        "note": "without 1"
      }
    ],
    "standards_and_regulation": [
      "No regulation; mathematical conventions"
    ],
    "failure_modes_and_hazards": [
      "Confusing with perfect powers",
      "Convention disputes on including 1",
      "Errors in factorisation"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "State the definition of a prime power and give correct examples.",
        "Test primality and prime power form within stated limits and methods."
      ],
      "must_not": [
        "Treat a composite number with several prime factors as a prime power.",
        "Present a probabilistic primality test result as a proof.",
        "Rely on untested prime powers in cryptographic or field construction work.",
        "Omit the method and bound when reporting a computed result."
      ],
      "requires_human": [
        "Choosing parameters for a cryptographic system."
      ]
    },
    "ethics": {
      "considerations": [
        "Weak or wrongly tested parameters break cryptography that people depend on.",
        "Teaching material is widely copied, so errors propagate."
      ],
      "affected_parties": [
        "Users of cryptographic systems",
        "Students",
        "Software developers"
      ]
    },
    "owners": {
      "steward": "Nobody owns a mathematical fact.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1667469: prime power",
      "url": "https://www.wikidata.org/wiki/Q1667469",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Prime power",
      "url": "https://en.wikipedia.org/wiki/Prime_power",
      "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 prime power, and does the definition include primes and 1?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the number a prime power, a perfect power or a general composite?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "test",
            "name": "Test",
            "description": "Testing.",
            "findings": [
              {
                "id": "test-finding",
                "name": "Test",
                "description": "Test.",
                "questions": [
                  {
                    "text": "How is a number tested for being a prime power?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits integer factorisation?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "properties",
        "name": "Properties",
        "description": "Properties.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "arithmetic",
            "name": "Arithmetic",
            "description": "Arithmetic properties.",
            "findings": [
              {
                "id": "arithmetic-finding",
                "name": "Arithmetic",
                "description": "Arithmetic.",
                "questions": [
                  {
                    "text": "What properties do prime powers have, such as totient values and divisor counts?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "distribution",
            "name": "Distribution",
            "description": "Distribution.",
            "findings": [
              {
                "id": "distribution-finding",
                "name": "Distribution",
                "description": "Distribution.",
                "questions": [
                  {
                    "text": "How are prime powers distributed among the integers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "algebra",
        "name": "Algebra",
        "description": "Algebraic roles.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "fields",
            "name": "Fields",
            "description": "Finite fields.",
            "findings": [
              {
                "id": "fields-finding",
                "name": "Fields",
                "description": "Fields.",
                "questions": [
                  {
                    "text": "Why does a finite field exist exactly for each prime power order?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits finite field?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "groups",
            "name": "Groups",
            "description": "Groups.",
            "findings": [
              {
                "id": "groups-finding",
                "name": "Groups",
                "description": "Groups.",
                "questions": [
                  {
                    "text": "What are p-groups and Sylow theorems, and how do prime powers enter group theory?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits p-group?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "Applications and history.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "applications",
            "name": "Applications",
            "description": "Applications.",
            "findings": [
              {
                "id": "applications-finding",
                "name": "Applications",
                "description": "Applications.",
                "questions": [
                  {
                    "text": "Where do prime powers appear in coding theory, cryptography and combinatorics?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the specific application?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did the study of prime powers develop with the fundamental theorem of arithmetic and Galois fields?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of algebra?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should the convention on 1 be recorded explicitly?",
    "How should reference sequences be linked?",
    "Which algebraic applications should be primary links?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}