{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-chinese-remainder-theorem",
  "semantic_family_id": "sf-chinese-remainder-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "中國剩餘定理"
  },
  "summary": {
    "zh-Hant": "正兩兩互質模數的同餘系統在模乘積意義下存在唯一解。"
  },
  "statement": {
    "zh-Hant": "residues正規化，使用modular inverses構造x並核全部congruences。唯一性是mod product而非integer unique。modulus0無效；noncoprime不在CRT假設內但可相容，需pairwise gcd compatibility且一般mod lcm唯一。"
  },
  "formula": {
    "tex": "C(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "edb25cc2211e62a78b4da558ef318ebd833eb6f3b66a4334a6411cf8263d225b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "主CRT題使用正且pairwise-coprime moduli。"
    },
    {
      "id": "a2",
      "zh-Hant": "全部integer arithmetic exact。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "chinese_remainder_solution",
      "role_zh": "CRT解",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "residue_system",
      "role_zh": "同餘系統",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-integer",
      "title_zh": "整數",
      "reason_zh": "解與residue。"
    },
    {
      "id": "mko-modular-congruence",
      "title_zh": "模同餘",
      "reason_zh": "系統條件。"
    },
    {
      "id": "mko-gcd-coprime",
      "title_zh": "互質",
      "reason_zh": "pairwise hypothesis。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "逐方程與唯一性。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "構造正規化solution。",
      "逐congruence核對。",
      "noncoprime需compatibility，不能自動說inconsistent。"
    ],
    "common_misconception_zh": "把唯一mod product當唯一integer，或認為所有noncoprime系統矛盾。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "pairwise-coprime theorem與noncoprime boundary分開。",
    "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": "加入一般化CRT。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-chinese-remainder-theorem.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-11-04 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}