{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-contraction-mapping-theorem",
  "semantic_family_id": "sf-contraction-mapping-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "壓縮映射定理"
  },
  "summary": {
    "zh-Hant": "非空完備度量空間上的self-map若以固定q<1壓縮所有距離，則有唯一fixed point且Picard orbit收斂。"
  },
  "statement": {
    "zh-Hant": "題族以完整finite metric與total self-map逐對驗d(Tx,Ty)≤q d(x,y)，固定0≤q<1。從任意x0完整產生Picard orbit，枚舉fixed points並核唯一性；exact rational metric下驗d(x_n,x*)≤q^n/(1-q)d(x_1,x_0)。finite metric自動complete但只證指定模型；q=1、非self-map、非complete空間與sampled-pairs不足均不得套定理。"
  },
  "formula": {
    "tex": "C(X,T,q)=x",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>C</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "C",
          "glyph": "C"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "X",
            "glyph": "X"
          },
          {
            "type": "symbol",
            "name": "T",
            "glyph": "T"
          },
          {
            "type": "symbol",
            "name": "q",
            "glyph": "q"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "x",
        "glyph": "x"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "555ec5e64e3ef3006607c275dd9f494202a0bf9529eb7f523d6530773c397295"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "finite metric matrix完整驗nonnegative、identity、symmetry與triangle inequality。"
    },
    {
      "id": "a2",
      "zh-Hant": "mapping為complete total self-map，q為exact rational。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "contraction_mapping_operator",
      "role_zh": "壓縮映射定理",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "complete_metric_space",
      "role_zh": "完備度量空間",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "contraction_self_map",
      "role_zh": "壓縮self-map",
      "scope": "representative_formula"
    },
    {
      "token": "q",
      "role": "contraction_constant",
      "role_zh": "壓縮常數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "unique_fixed_point",
      "role_zh": "唯一不動點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-orbit-fixed-periodic",
      "title_zh": "軌道、固定點與週期點",
      "reason_zh": "Picard orbit與fixed point。"
    },
    {
      "id": "mko-complete-metric-space",
      "title_zh": "完備度量空間",
      "reason_zh": "completeness。"
    },
    {
      "id": "mko-metric-convergence",
      "title_zh": "度量收斂",
      "reason_zh": "Cauchy/convergence。"
    },
    {
      "id": "mko-infinite-series",
      "title_zh": "無窮級數",
      "reason_zh": "geometric tail bound。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整metric/self-map/contraction資格先於定理。",
      "Picard orbit與唯一fixed point分開核對。",
      "geometric tail給exact error bound。"
    ],
    "common_misconception_zh": "只抽樣幾對點、允許q=1或忽略self-map/completeness仍宣稱唯一fixed point。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite model證書不取代一般complete-space theorem proof。",
    "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Picard–Lindelöf等應用。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-contraction-mapping-theorem.md 整理；2026-10-15 脈衝 R01（使用者手動接續 2026-11-16 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}