{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-11T07:52:46Z",
    "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": "2026-10-05T13:18:24+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-Q11567",
    "registryId": "vr.tr.complex-number",
    "name": "complex number",
    "version": "0.2.0-wave.1",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QLT"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q11567/",
  "model": {
    "registry_id": "vr.tr.complex-number",
    "name": "complex number",
    "purpose": "Let an agent handle complex numbers by form, operations, representation and applications.",
    "definition": "A number of the form a + bi, where a and b are real numbers and i is the imaginary unit with i squared equal to -1; complex numbers extend the reals, and phasors in electrical engineering use them.",
    "what_it_is_for": "Solving polynomial equations, analysing waves and circuits, and complex analysis.",
    "affordances": [
      "add, multiply and divide them",
      "convert between rectangular and polar forms",
      "use phasors in AC circuits",
      "plot on the complex plane"
    ],
    "distinguishing_features": [
      "Real and imaginary parts",
      "Algebraically closed field",
      "Polar form with modulus and argument",
      "No natural ordering"
    ],
    "appearance": "Expressions a + bi or r(cos t + i sin t); points on the complex plane.",
    "visual_identification": [
      "Imaginary unit i (or j in engineering)",
      "Argand diagrams",
      "Phasors for AC quantities"
    ],
    "physical_properties": [],
    "families_and_kinds": [
      "rectangular form",
      "polar and exponential form",
      "phasors",
      "Gaussian integers (subset)",
      "extensions such as quaternions"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "number",
        "why": "category"
      },
      {
        "relation": "extends",
        "target": "real numbers",
        "why": "field"
      },
      {
        "relation": "is generalised by",
        "target": "quaternion",
        "why": "extension"
      },
      {
        "relation": "is used in",
        "target": "electrical engineering and quantum mechanics",
        "why": "applications"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "ISO 80000-2 notation"
    ],
    "failure_modes_and_hazards": [
      "Branch cut errors with roots and logarithms",
      "Confusing i and j conventions"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Compute with complex numbers, stating the conventions used for branch cuts and the imaginary unit.",
        "Explain complex arithmetic and its use in engineering and physics."
      ],
      "must_not": [
        "Present a result that depends on a branch choice without saying which branch was used.",
        "Mix the i and j conventions in one calculation without noting it.",
        "Report a computed value with more precision than the method supports.",
        "Call complex numbers unreal or meaningless in a way that misleads learners."
      ],
      "requires_human": [
        "Relying on complex-valued calculations for safety-critical design, such as power systems."
      ]
    },
    "ethics": {
      "considerations": [
        "Calculation errors in engineering models can lead to unsafe systems.",
        "Mathematics teaching should be honest about where intuition fails."
      ],
      "affected_parties": [
        "Engineers and the users of what they build",
        "Students"
      ]
    },
    "owners": {
      "steward": "Nobody: a mathematical concept held in common.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q11567: complex number",
      "url": "https://www.wikidata.org/wiki/Q11567",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Complex number",
      "url": "https://en.wikipedia.org/wiki/Complex_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "form",
        "name": "Form",
        "description": "Representation.",
        "rationale": "Forms convert.",
        "layers": [
          {
            "id": "rectangular",
            "name": "Rectangular",
            "description": "a + bi.",
            "findings": [
              {
                "id": "rectangular-finding",
                "name": "Rectangular",
                "description": "Rectangular form.",
                "questions": [
                  {
                    "text": "What are its real and imaginary parts?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Is it written with i or j?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          },
          {
            "id": "polar",
            "name": "Polar",
            "description": "Modulus and argument.",
            "findings": [
              {
                "id": "polar-finding",
                "name": "Polar",
                "description": "Polar form.",
                "questions": [
                  {
                    "text": "What are its modulus and argument?",
                    "kind": "measurement"
                  },
                  {
                    "text": "In which range is the argument?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "operations",
        "name": "Operations",
        "description": "Arithmetic.",
        "rationale": "Operations follow rules.",
        "layers": [
          {
            "id": "arithmetic",
            "name": "Arithmetic",
            "description": "Basic operations.",
            "findings": [
              {
                "id": "arithmetic-finding",
                "name": "Arithmetic",
                "description": "Arithmetic.",
                "questions": [
                  {
                    "text": "What is the result of this operation?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Which form is easiest to use?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "functions",
            "name": "Functions",
            "description": "Roots and logs.",
            "findings": [
              {
                "id": "functions-finding",
                "name": "Functions",
                "description": "Complex functions.",
                "questions": [
                  {
                    "text": "Which branch is used for roots or logarithms?",
                    "kind": "boundary"
                  },
                  {
                    "text": "Is the result unique?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "applications",
        "name": "Applications",
        "description": "Uses.",
        "rationale": "Complex numbers are practical.",
        "layers": [
          {
            "id": "circuits",
            "name": "Circuits",
            "description": "Phasors.",
            "findings": [
              {
                "id": "circuits-finding",
                "name": "Circuits",
                "description": "AC circuits.",
                "questions": [
                  {
                    "text": "How are phasors used to analyse AC circuits?",
                    "kind": "definition"
                  },
                  {
                    "text": "What does impedance represent?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          },
          {
            "id": "signals",
            "name": "Signals",
            "description": "Fourier analysis.",
            "findings": [
              {
                "id": "signals-finding",
                "name": "Signals",
                "description": "Signal analysis.",
                "questions": [
                  {
                    "text": "How do complex exponentials describe signals?",
                    "kind": "definition"
                  },
                  {
                    "text": "Why are they convenient?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "learning",
        "name": "Learning",
        "description": "Teaching.",
        "rationale": "Intuition helps.",
        "layers": [
          {
            "id": "visual",
            "name": "Visual",
            "description": "Complex plane.",
            "findings": [
              {
                "id": "visual-finding",
                "name": "Visual",
                "description": "Visualisation.",
                "questions": [
                  {
                    "text": "How can multiplication be seen as rotation and scaling?",
                    "kind": "definition"
                  },
                  {
                    "text": "How is it plotted?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "misconceptions",
            "name": "Misconceptions",
            "description": "Errors.",
            "findings": [
              {
                "id": "misconceptions-finding",
                "name": "Misconceptions",
                "description": "Misconceptions.",
                "questions": [
                  {
                    "text": "What mistakes are common?",
                    "kind": "boundary"
                  },
                  {
                    "text": "How are they avoided?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should forms be separate entries?",
    "How should phasors be linked?",
    "Registry aliases include decimal fraction; should it be moved?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}
