{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T20:13:51Z",
    "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": "2026-10-06T07:00:12+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:52:03+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q948010",
    "registryId": "vr.tr.primitive-root-modulo-n",
    "name": "primitive root modulo n",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q948010/",
  "model": {
    "registry_id": "vr.tr.primitive-root-modulo-n",
    "name": "primitive root modulo n",
    "purpose": "Let an agent define primitive roots modulo n precisely, state when they exist and how many there are, test and find them for a given modulus, and relate them to orders, discrete logarithms and cryptography without giving operational key material.",
    "definition": "In number theory, an integer g such that every integer coprime to n is congruent to some power of g modulo n, that is, g generates the multiplicative group of integers modulo n; primitive roots exist exactly when n is 1, 2, 4, a power of an odd prime, or twice a power of an odd prime, and they underlie discrete logarithms and several cryptographic protocols.",
    "what_it_is_for": "Generating the multiplicative group modulo n; basis for discrete logarithms.",
    "affordances": [
      "test whether an integer is a primitive root modulo n",
      "find primitive roots for a modulus",
      "state the existence theorem and count",
      "explain the link to discrete logarithms"
    ],
    "distinguishing_features": [
      "Generator of the multiplicative group modulo n",
      "Existence depends on the form of n",
      "Order equals Euler phi of n",
      "Basis of the discrete logarithm"
    ],
    "appearance": "Not visible; a property of an integer relative to a modulus.",
    "visual_identification": [
      "Multiplicative order equal to Euler phi of n",
      "Exists only for n equal to 1, 2, 4, p to the k, or 2 p to the k with p an odd prime",
      "A primitive nth root of unity in the complex numbers is a related but different object"
    ],
    "physical_properties": [
      {
        "quantity": "number of primitive roots when they exist",
        "typical_range": "phi of phi of n",
        "unit": "integers",
        "note": "Euler totient applied twice"
      },
      {
        "quantity": "smallest primitive root modulo 7",
        "typical_range": "3",
        "unit": "integer",
        "note": "example"
      }
    ],
    "families_and_kinds": [
      "primitive roots modulo a prime",
      "primitive roots modulo prime powers and twice prime powers",
      "least primitive roots as a studied sequence",
      "generalisations to generators of cyclic subgroups"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "root of unity modulo n",
        "why": "the case of maximal order"
      },
      {
        "relation": "generates",
        "target": "multiplicative group of integers modulo n",
        "why": "the group it generates"
      },
      {
        "relation": "is used in",
        "target": "discrete logarithm",
        "why": "the inverse problem"
      },
      {
        "relation": "is related to",
        "target": "Euler totient function",
        "why": "counts and orders"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A001918",
        "note": "least primitive root of the nth prime"
      }
    ],
    "standards_and_regulation": [
      "No regulation; definitions follow standard number theory texts"
    ],
    "failure_modes_and_hazards": [
      "Assuming a primitive root exists for every modulus",
      "Confusing with complex roots of unity",
      "Using small or weak generators in cryptographic settings, which belongs to security guidance"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Compute and check primitive roots modulo n and explain when they exist.",
        "Use primitive roots in teaching and in authorized cryptographic work."
      ],
      "must_not": [
        "State that a primitive root exists for an n where it does not.",
        "Use small or weak parameters in a real security system.",
        "Present a conjecture, such as Artin's, as proven."
      ],
      "requires_human": [
        "Choosing parameters for a cryptographic system that protects people's data."
      ]
    },
    "ethics": {
      "considerations": [
        "Mistakes in number-theoretic parameters can weaken cryptography that protects people.",
        "Mathematical credit matters to researchers."
      ],
      "affected_parties": [
        "Users of cryptographic systems",
        "Students"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; it belongs to mathematics.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q948010: primitive root modulo n",
      "url": "https://www.wikidata.org/wiki/Q948010",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Primitive root modulo n",
      "url": "https://en.wikipedia.org/wiki/Primitive_root_modulo_n",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "define",
        "name": "Define",
        "description": "The definition.",
        "rationale": "Precision.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a primitive root modulo n, and how does it relate to multiplicative order?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about primitive roots modulo n or about complex roots of unity?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "existence",
            "name": "Existence",
            "description": "Existence theorem.",
            "findings": [
              {
                "id": "existence-finding",
                "name": "Existence",
                "description": "Existence.",
                "questions": [
                  {
                    "text": "For which n do primitive roots exist, and how many are there?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits the multiplicative group modulo n?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "compute",
        "name": "Compute",
        "description": "Finding and testing.",
        "rationale": "Computation.",
        "layers": [
          {
            "id": "test",
            "name": "Test",
            "description": "Testing.",
            "findings": [
              {
                "id": "test-finding",
                "name": "Test",
                "description": "Test.",
                "questions": [
                  {
                    "text": "How is an integer tested as a primitive root using the prime factors of phi of n?",
                    "kind": "action"
                  },
                  {
                    "text": "What is the result for the modulus in question?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          },
          {
            "id": "find",
            "name": "Find",
            "description": "Finding.",
            "findings": [
              {
                "id": "find-finding",
                "name": "Find",
                "description": "Find.",
                "questions": [
                  {
                    "text": "How are primitive roots found in practice, and what is known about the least primitive root?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "use",
        "name": "Use",
        "description": "Applications.",
        "rationale": "Applications.",
        "layers": [
          {
            "id": "logarithm",
            "name": "Logarithm",
            "description": "Discrete logarithms.",
            "findings": [
              {
                "id": "logarithm-finding",
                "name": "Logarithm",
                "description": "Logarithm.",
                "questions": [
                  {
                    "text": "How do primitive roots define discrete logarithms and index tables?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits the discrete logarithm problem?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "cryptography",
            "name": "Cryptography",
            "description": "Cryptographic use.",
            "findings": [
              {
                "id": "cryptography-finding",
                "name": "Cryptography",
                "description": "Cryptography.",
                "questions": [
                  {
                    "text": "How are generators used in protocols such as Diffie-Hellman, in general terms?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Is the user asking for parameter choices for a real system, which needs security guidance?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theory",
        "name": "Theory",
        "description": "Results and history.",
        "rationale": "Theory.",
        "layers": [
          {
            "id": "results",
            "name": "Results",
            "description": "Theorems and conjectures.",
            "findings": [
              {
                "id": "results-finding",
                "name": "Results",
                "description": "Results.",
                "questions": [
                  {
                    "text": "What is proved and conjectured about primitive roots, such as Artin conjecture?",
                    "kind": "provenance"
                  },
                  {
                    "text": "What is proven versus conjectured?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did Euler, Lagrange and Gauss develop the theory of primitive roots?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits Gauss?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should the discrete logarithm be a separate entry?",
    "How should OEIS sequences be linked?",
    "How should Artin conjecture and related open problems be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}