{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-bisection-method",
  "semantic_family_id": "sf-bisection-method",
  "type": "algorithm",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "二分法的有限 bracket 更新"
  },
  "summary": {
    "zh-Hant": "二分法在已提供的端點與中點符號資料上取中點並保留符號相反或零值的半區，有限步寬度減半。"
  },
  "statement": {
    "zh-Hant": "給定有序區間 [a,b] 及已提供的端點符號 bracket，每步取 m=(a+b)/2。若已觀察 f(m)=0 就停在 m；否則保留端點符號相反的半區。沒有另行使用中間值定理時，這只是 bracket 資料更新，不自行證明根存在。"
  },
  "formula": {
    "tex": "m=B(a,b)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>m</mi><mo>=</mo><mrow><mi>B</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "symbol",
        "name": "m",
        "glyph": "m"
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "B",
          "glyph": "B"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "b",
            "glyph": "b"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "03f471a5479d6cacb38904ff28663e0590b693447a7d39b7c93df3f6d28761b8"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題目直接提供端點與每步中點的符號；不由函數圖形或連續性暗推資料。"
    },
    {
      "id": "a2",
      "zh-Hant": "若初始寬度為 w，n 次有效二分後寬度為 w/2^n；此界不等於根存在證明。"
    }
  ],
  "symbols": [
    {
      "token": "m",
      "role": "midpoint",
      "role_zh": "區間中點",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "left_endpoint",
      "role_zh": "左端點",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "right_endpoint",
      "role_zh": "右端點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-numerical-approximation-error",
      "title_zh": "數值近似與誤差",
      "reason_zh": "有限步寬度可作位置誤差界。"
    },
    {
      "id": "mko-continuity",
      "title_zh": "實函數在一點的連續性與序列判準",
      "reason_zh": "一般根存在連結需要額外連續性與定理，本輪不自動使用。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "bracket端點與中點需保持次序。"
    },
    {
      "id": "mko-equation",
      "title_zh": "方程式",
      "reason_zh": "零值是方程解的直接觀察。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "符號bracket是演算法輸入資料，不等於本條目已證其中必有根。",
      "每步只依明示端點與中點符號選半區；符號資料若不能保留任何半區，輸入不合格。",
      "寬度界是純幾何減半結果；要把它解讀為根的誤差界，還需根確實位於bracket的獨立責任。"
    ],
    "common_misconception_zh": "看到異號就省略中間值定理而直接宣稱有根，或把區間寬度當函數值誤差。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核有限bracket更新與寬度；根存在及收斂到根未在本輪證明。",
    "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": "補上中間值定理MKO、根存在receipt與停止準則。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-bisection-method.md 整理；2026-10-13 脈衝 R05（使用者手動接續 2026-10-21 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"
  }
}