{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-metric-convergence",
  "semantic_family_id": "sf-metric-convergence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "度量空間中的收斂"
  },
  "summary": {
    "zh-Hant": "度量空間序列收斂到x，表示每個正epsilon終有所有n≥N落入x的epsilon球。"
  },
  "statement": {
    "zh-Hant": "序列x_n在度量空間中收斂到x，若對每個epsilon>0存在N，使所有n≥N都有d(x_n,x)<epsilon。有限樣本尾段通過不足以證一般收斂；明示最終常值的無限序列則可給完整witness。"
  },
  "formula": {
    "tex": "L(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "L",
        "glyph": "L"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b6c8d98a5365da9a202f6b86d5a227b64bd4ec4fb4d38370c28e1430c70f7ecc"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用完整有限度量表。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限序列題明示為bounded audit；最終常值模型明示prefix後所有項恆等於tail。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "limit_record",
      "role_zh": "極限紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "candidate_limit",
      "role_zh": "候選極限",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-metric-space",
      "title_zh": "度量空間",
      "reason_zh": "epsilon球與距離來自度量。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "收斂對自然數索引序列量化。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "定義保留forall epsilon exists N forall n責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "開球使用嚴格d<epsilon。",
      "有限尾段只能核已列索引，不足以證所有未來項。",
      "最終常值序列對任意epsilon可取prefix長度作N；度量分離性給極限唯一。"
    ],
    "common_misconception_zh": "從長但有限的尾段看似接近就斷言收斂。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只有明示最終常值模型提供完整無限witness。",
    "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Cauchy序列、完備性與連續映射。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-metric-convergence.md 整理；2026-10-13 脈衝 R04（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"
  }
}