{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-complex-modulus-conjugate",
  "semantic_family_id": "sf-complex-modulus-conjugate",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "複數模與共軛"
  },
  "summary": {
    "zh-Hant": "複數共軛保留實部並反轉虛部；模平方為實部與虛部平方和。"
  },
  "statement": {
    "zh-Hant": "對z=a+bi，共軛為a-bi，模平方為a²+b²，且z乘共軛等於模平方的實數嵌入。共軛保持加法與乘法。"
  },
  "formula": {
    "tex": "M(z)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>M</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "M",
        "glyph": "M"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "z",
          "glyph": "z"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "cc862bcabe3f4dfc24177603a3535d72d8842f64cccd9ab9809c150b9aa271b4"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫用整數有序對表示複數，只計算共軛與模平方，不把平方根近似冒充精確模。"
    },
    {
      "id": "a2",
      "zh-Hant": "本輪不處理argument、極式或複可微性。"
    }
  ],
  "symbols": [
    {
      "token": "M",
      "role": "complex_invariant_record",
      "role_zh": "複數模／共軛紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "z",
      "role": "complex_number",
      "role_zh": "複數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-complex-number",
      "title_zh": "複數",
      "reason_zh": "共軛與模建立在複數有序對運算上。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "複數模推廣實數絕對值的大小語義。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "冪與根式",
      "reason_zh": "模平方使用平方和。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "共軛(a,b)=(a,-b)。",
      "z乘共軛的虛部為零，實部為a²+b²。",
      "模平方保持精確整數，避免對非完全平方數引入小數近似。"
    ],
    "common_misconception_zh": "只改實部符號，或把模平方誤當模本身。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核整數有序對代數；不宣稱複分析結果。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入複數除法、極式與複函數。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-complex-modulus-conjugate.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-23 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}